Citation
Counterfactuals

Material Information

Title:
Counterfactuals
Creator:
Mayer, John Clyde, 1945-
Publication Date:
Language:
English
Physical Description:
vii, 250 leaves : diagrs. ; 28 cm.

Subjects

Subjects / Keywords:
Contrafactuals ( jstor )
Inference ( jstor )
Logical antecedents ( jstor )
Logical consequents ( jstor )
Logical theorems ( jstor )
Modal logic ( jstor )
Modal realism ( jstor )
Semantics ( jstor )
Syntactical antecedents ( jstor )
Truth condition ( jstor )
Counterfactuals (Logic) ( lcsh )
Dissertations, Academic -- Philosophy -- UF
Philosophy thesis Ph. D
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Bibliography: leaves 243-249.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by John Clyde Mayer.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
023407697 ( ALEPH )
06976594 ( OCLC )

Downloads

This item has the following downloads:


Full Text











OUNERFACTUALS


BY

JOHN CLYDE MAYER





























A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFUlN OF THE REQUIREMENTS
FOR THE DEGREE OF IDCIOR OF PHILOSOPHY



UNIVERSITY OF FLORIDA


1980

























Copyright 1980

by

John Clyde Mayer














ACKNOWLEDII NrS


I am indebted to my advisor, J. Jay Zeman, for introducing me to

modal logic, conditional functions, possible wrlds, and a more empir-

ical view of logic than I might otherwise have had. For better or

worse, I am (only slightly) less a Platonist as a result.

To my typist, Joyce Pandelis, I owe a particular debt of thanks

for the time and effort she has spent in working with me through revi-

sions, corrections, and re-revisions. As a philosopher in her own right,

her comments and suggestions have been valuable, and only occasionally

mischievous.














TABLE OF (ONrENTS



ACKNOWLEDGU = S ...................................................... iii

ABSTRACT .............................................................. vi



ONE WHAT ARE QDUNTERFACrUALS? ..................................... 1

1.1 A Central Concept of Conditionality ...................... 1
1.2 The Metalinguistic Analysis .............................. 14
1.3 Notes .................................................... 34

TWO POSSIBLE WORLDS ANALYSIS OF COUN=AERFATUAS ................... 35

2.1 Possible Worlds: History ................................ 35
2.2 Possible Worlds: Modality and the Strict Conditional .... 41
2.3 Lewis' -Analysis of the Counterfactual Conditional ........ 50
2.4 Modal Logic and the System of Spheres .................... 64
2.5 Counterfactual Inferences and Fallacies .................. 67
2.6 The Limit Assumption ................... 90
2.7 Possible Worlds: Realism and Explanation.............. 106
2.8 Notes .................................................... 110

THREE ORDERINGS OF POSSIBLE WORLDS .................................. 112

3.1 Comparative Similarity ................................... 112
3.2 Varieties of Order ....................................... 122
3.3 Notes .................................................... 134

FOUR MODAL AND O0NDITIONAL LOGICS .................................. 135

4.1 A Modal/Conditional Lmguage and Modal Systems
E, M, R, K ......................................... 135
4.2 Neighborhood Semantics for Modal Logic ................... 144
4.3 Neighborhood and Relational Semantics .................... 159
4.4 Conditional Logic: The Systems Ce, Ck, CE, (1, CR,
CK ....................................................... 162
4.5 Neighborhood Semantics for Conditional Logic ............. 170
4.6 Alternative Semantics for Conditional Logic .............. 182
4.7 Extensions of CK ......................................... 189
4.8 Semantics for Extensions of CK ........................... 193
4.9 Notes .................................................... 217














FIVE COUNIERFACTUALS AND COMPARISON OF WORLDS ...................... 219

5.1 An Adequate Counterfactual Iogic ......................... 219
5.2 Comparative Order Analysis ............................... 230

REFERENCES ............................................................ 243

BIOGRAPHICAL SKETCH ................................................... 250














Abstract of Dissertation Presented to the Graduate Council
of the aLiversity of Florida in Partial Fulfillment of the
Requirents for the Degree of Doctor of Philosophy


QOUNIERFA LS

By

John Clyde Mayer

August 1980



Chairman: J. Jay Zeman

Major Department: Philosophy

Recently, possible world semantics has provided a basis for several

accounts of the counterfactual conditional that offer theories of counter-

factual deliberation superior to that of previous, so-called metalinguistic,

accounts. The present essay is a survey of a number of these possible

world accounts, with particular emphasis on that of David K. Lewis. It

is argued that possible world accounts more closely resemble scientific

theories than they do traditional conceptual analyses. The view that

what is at issue is a central concept of conditionality, rather than a

more narrow notion of counterfactuality or subjunctivity, is espoused.

In addition, a formal comparison of modal and conditional logics is

undertaken, using neighborhood semantics, as well as a formal comparison

of a variety of normal conditional logics, using selection function seman-

tics. Various families of normal conditional logics are thereby identified.

vi









These families are classified in terms of tw dimensions: one of in-

creasing materiality of the conditional connective, and the other of

increasing strength in the comparison of possible worlds implicit in

any semantics for the logics. Much of this work is a continuation of

that of Brian Chellas and Donald Nute.

Coparative order semantics, a generalization of Lewis' comparative

similarity semantics, is developed. In comparative order semantics, pos-

sible worlds are ordered relative to each world as a basis for comparison.

The smallest logic, CP, in which the order relative to each world is a

partial order, is identified. A number of logics of the counterfactual

conditional that have been suggested contain CP, though in some cases the

results presented in this essay are surprising, and at variance with what

has been claimed elsewhere. In particular, a family of logics is identified

which lies between previously identified logics whose comparative order

semantics are partially ordered (such as system SS of John Pollock), and

certain of the V-logics of Lewis (whose comparative order semantics are

a weak total order, relative to each base world). The smallest number

of this intervening family is characterized by a semiconnected partial

order.

The view that some version of comparative similarity ordering is

capable of supporting an analysis of the counterfactual conditional is

defended.


vii














CHAPTER ONE

WHAT ARE O FACrULS?



1.1 A Central Concept of Conditionality

Suppose that you and I are taking a road trip in your somewhat

beat-up 1965 roadster. Before starting I do you the favor of checking

the oil, and noticing that while you do not need any additional oil,

the oil you have is very dirty, I then assert "If you do not change

the oil, your engine will seize up." Based upon my assurances that I

know about this sort of thing, you go ahead and change the oil.

Later, while we are travelling, your thoughts turn to the ad-

ditional time and expense my interference has put you to, and you be-

gin to wonder aloud if I really knew what I was talking about. In the

process of the conversation I assert, "If you had not changed the oil,

your engine would have seized up." Of course, this present assertion

is nothing new; I am making the same assertion now as I did in the past.

My locution has changed to reflect our new perspective.

Building a little more on this example, suppose that your doubts

about my expertise arise even as you are draining the crankcase. By

way of reassuring you I say,"If the oil were not changed, the engine

would seize up." Again, I seem to have said the same thing. Upon ar-

riving at our destination, your doubts and my confidence unswerving, we

consult a third party, describing the state of the oil before we set

out. The third party says, "If that oil was not changed, then your

1








engine seized up." I feel vindicated, since the third party has made

the same claim as I did.

The four assertions above differ in tense and/or mood, yet all

seem to be making the same claim, though from differing perspectives

and background knowledge. This same point is made by Ellis [19] and

Stalnaker [96, P. 166]. The difference in perspective is a temporal

one; the difference in background knowledge, whether or not the oil

was changed on the occasion in question.

It is the conmmn conditional expressed by all four of the above

sentences with which I am principally concerned. From this example

as a beginning I hope to draw certain preliminary conclusions re-

garding the assertability conditions and the truth conditions of such

conditionals, whether or not the indicative or subjunctive mod is es-

sentially involved, in what sense the conditional is counterfactual,

whether the conditional is a material conditional, and what other

conditional-types may be sufficiently closely related to this paradig-

matic one so as to be subsumable to a conmon analysis.

Consider the conditional "If the oil were not change, the engine

would not seize up." In the circumstance described, since the oil is

changed, both of these conditionals have false antecedents. Hence, if

they are material conditionals, they are both true. If this is the

case, then we must look elsewhere than to their truth conditions for

why our behavior is so different depending upon which we base our ac-

tions on. This is highly implausible and flies in the face of the

fact that you change the oil just because you believe my assertion to

be true.









One might still charge that the conditional is a material con-

ditional because it is equivalent to a disjunction: i.e., the first

assertion could be re-expressed as 'ither the oil is changed or the

engine seizes up," while the fourth can be re-expressed in the past

tense as "Either the oil was changed or the engine seized up." These

are appropriately asserted just when one does not know which one of

the disjuncts is true. However, this equivalence does not hold for

the second formulation, "If you had not changed the oil, the engine

would have seized up," precisely because in that case the antecedent

is known to be false, so in the corresponding disjunction, one dis-

junct is known to be true, hence the disjunction is not assertable, yet

clearly the conditional is assertable. Furthermore, the disjunction is

true, while the conditional is still debatable. That is, you could

consistently deny the conditional, while accepting the disjunction,

because you accept the falsity of the antecedent. Hence the falsity of

the antecedent is not sufficient grounds to accept the truth of the

conditional, not merely improper grounds to assert it.

If I am correct in asserting that all four sentences express the

same conditional relationship between antecedent and consequent, then

this reveals something about how "counterfactual" counterfactuals must

be, and about whether subjunctivity is essential. Consider again the

third party who asserted,"If that oil was not changed, your engine

seized up." That party could just as well have said,"If that oil were

not changed, your engine would have seized up." In either case, the

third party would have been vindicated in his judgment by the subse-

quent discovery that the oil was not changed and the engine did seize

up. Since the first of the above assertions is indicative, and the









second subjunctive, the subjunctive mod is not essential to such con-

ditionals. Clearly the actual falsity of antecedent and consequent is

not essential either, though from what has been said so far one might

claim that the conditional is only assertable when the antecedent and

consequent are not known to be true. Thus the conditional would still

be "potentially" counterfactual. It is certainly not correct to say

that such a conditional is only assertable on the assumption that the

antecedent and consequent are false, since the third party above makes

no such assumption.

So a conditional of the sort we are interested in need not be

subjunctive, nor need it be either actually counterfactual or assmed

to be counterfactual in order to be asserted appropriately, as Goodman

[271 recognized. Nor apparently does the actual truth of the antece-

dent and consequent count against the truth of the conditional. (In-

deed, Lewis' account [511 incorporates the contrary view.)

The third party's choice of a past indicative or subjunctive

sentence to express himself does not, I believe, reflect a choice be-

tween tw different propositions which he could alternatively express,

but rather a choice between two ways of expressing the same proposition.

So he "says" nothing different by expressing himself one way rather than

the other. However, he may show something different about his attitude

toward the situation described to him by his choice of wrds. The use

of the subjunctive mod in the second case strongly suggests that he

regards the situation on which he has been asked to pass judgnent as

hypothetical. From this it is an easy step to counterfactual, and the

association of subjunctive conditionals with counterfactual ones.









Trying to identify a particular kind of conditional, not on the

basis of its grammatical form and variants but by appeal to examples,

has the drawback of needing to specify a set of fairly clearcut ex-

amples covering all serious possibilities. One example is not suffi-

cient since special features that it possesses may cause the kind of

conditional to be circtmscribed too narrowly, thus narrowing the scope

of any subsequent analysis. There are two features of the example we

have considered that bear mentioning in this connection: the example

given is essentially a conditional prediction (see Ellis [19]) or a

sequential conditional. ("Sequential counterfactual" is Jackson's

term in [36].) That is, it refers to two events, one, the antecedent,

temporally preceding the consequent. Furthermore, among the grounds

for accepting the conditional is surely the belief that there is a

connection between the antecedent and consequent. Either or both might

be taken to be an identifying characteristic of such conditionals.

Whether we do so will have a bearing on two matters: the scope of the

basic analysis, and the number of distinct kinds of conditionals we

will be forced to deal with.

Though recalling that expression in the subjunctive mood is not

essential, let us take as the first in our set of examples the simple

subjunctive form of the conditional involved in the oily engine:

E1.1.1: If the oil were not changed, then the engine would seize up.

This example is sequential, and there is a connection between antece-

dent and consequent. As a second example, let us retain connection,

but at least cloud sequentiality as in this example from Lewis [51]:

E1.1.2: If kangaroos had no tails, they would topple over. Next,

let us retain sequentiality, but omit connection as in this example









from Pollock [801:

E1.1.3: If the witch doctor were to do a rain dance, then it wuld

(still) not rain.

E1.1.4: Even if the witch doctor were to do a rain dance, it would

(still) not rain.

Finally, we can "reverse" sequentiality:

El.1.5: If the engine were to seize up, then the oil would not have

been changed.

I have included another version of El.1.3 because it might be

thought that the El.1.4 is a more natural way of expressing the lack

of a connection between antecedent and consequent which appears to

underlie such a conditional. Such conditionals were called by Goodman

[27, p. 5] "semifactuals" and by Pollock [80, p. 29] "even if'-condi-

tionals. The former appellation cores from the fact that in the typi-

cal cases of semifactuals the consequent is already true and the ante-

cedent's being true cannot alter that. A modification of our engine ex-

ample may raise some doubts about whether the truth of the consequent

is essential to such conditionals.

Suppose that upon checking your oil and finding it excessively

dirty I also notice the reprehensible shape you have allowed your

fifteen-year-old engine to get into. In fact, I am convinced that your

engine would seize up whether or not you changed the oil. You notice

the dirty oil, but are ignorant of the engine's sad state generally. I

tell you:

E1.1.6: Even if the oil were changed, the engine would (still) seize

up.









I am denying a connection between changing the oil and the seizing up

of the engine; in fact, as Goodman [27] points out, by negating the

consequent, I seem to be denying the counterfactual "If the oil wre

changed, then the engine would not seize up." Suppose you go on to

have your engine overhauled, and after we finally set out on our trip

your complaints about the cost force me to remind you:

E1.1.7: Even if you had changed the oil, the engine would have seized

UP.
And this seems to be fully counterfactual.

The dilema is this: if I maintain that E1.1.6 and E1.1.7 ex-

press the same proposition, one prospectively and the other retrospec-

tively, then I cannot maintain that in the first the consequent is true

(making it a semifactual) and in the second it is false (making it a

counterfactual). The difficulty, of course, is that the examples are

evaluated with respect to the same hypothetical situation, but the

judgment as to their factual status (semi- or counter-) is made with

respect to different actual situations.

However, this dilemma will not prevent me from being able to ac-

cept or reject the conditional(s) in question, since I do this upon

the basis of the actual situation at the time the engine was inspected

with the additional assumption that the oil is thereupon changed and

my knowledge of what generally happens to such messed up engines. And

it seems I can do this whether I am in the position of making a condi-

tional prediction before we set out, a contemporary lament as we sit

beside the road with a seized up engine, or a retrospective reminder

after we have safely arrived without a seized up engine because the

overhaul took place.








I think that an "even if'-conditional can best be viewed as

denying a connection between antecedent and consequent, but this need

not make it less of a conditional, nor do we necessarily need a sepa-

rate analysis for such. The "even if" is not invariably a signal that

the consequent is true and the antecedent cannot change that, but

rather a signal that this conditional is not grounded on a connection

between antecedent and consequent, but rather a lack thereof. We may

distinguish between those cases where an "even if'-conditional predic-

tion is asserted as opposed to a "standard" conditional prediction.

If we believe that a certain condition's being fulfilled will either

not change something that is already the case, or not in itself pre-

vent something that is going to coe about, then an "even if' condi-

tional prediction is appropriate. On the other hand, if a certain

condition's being fulfilled will bring something about, then a simple

conditional prediction is appropriate. However, these are conditions

of assertibility, not truth conditions. It remains to be seen whether

a single set of truth conditions can handle both conditionals.

Up to this point we have considered conditionals, whether sub-

junctive or indicative, counterfactual or senifactual, that are at

least closely associated with conditional predictions. Example El. 1.2

above and the following:

E1.1.8: If kangaroos had no tails, then they would (still) be vegetar-

ians,

fail to have an obvious sequential character, though El.1.2 is pre-

sumably based upon the presence of a connection between antecedent and

consequent, and E1.1.8 upon a lack thereof.









What suggests that these conditionals are accessible to the same

analysis as conditional predictions is (very roughly) the similarity

in the considerations that go into our judgment to accept or reject

conditionals of either type. The following is the conmun starting

point of many analyses of counterfactuals [13, 14, 19, 26, 36, 51, 60,

66, 67, 80, 96]. Recall that a counterfactual in our view is often a

conditional prediction viewed retrospectively against the knowledge

that the condition did not obtain at the time the prediction was ap-

propriate.

To our information about the actual situation at the time of the

conditional prediction we add the assumption that the antecedent con-

dition is fulfilled, changing whatever is required in our assumptions

about the actual situation to "fit" this added assumption. We then

consider what has occurred in similar situations, which knowledge may

be present for us in the form of various laws, causal and otherwise.

On this basis we determine whether or not the consequent wuld be re-

alized in such a situation. Something roughly like this method is what

I might apply in the engine example in making the prediction that I do.

Based upon my prior knowledge of similar situations, the actual state

of the engine now under inspection, and the assumption that the oil is

not changed I predict that in the near future under normal driving the

engine will seize up.

To evaluate E.1.2 on the other hand I may take into account my

knowledge of the physiognomy of kangroos, their skeletal structure, and

laws concerning balance and center of gravity. To this I add the as-

stumption that kangaroos have no tails, changing the known facts about

kangaroos no more than necessary to accommodate this assumption. I now









have a set of facts and laws, and if it is a consequence of this set

that kangaroos topple over, then the conditional in question is ac-

cepted. The only difference in the two cases is the specific temporal

order in the former not in the latter. Presumably this difference is

incorporated in large part in the laws applicable to the differing

situations. Otherwise it seems I can handle them quite similarly.

In applying a similar procedure to El.1.8 I find that the changes

I make in the known facts and laws in order to accommodate the assump-

tion that kangaroos have no tails leave unchanged that fact that kan-

garoos are vegetarians. Hence this fact appears in the set of facts

and laws, so as a consequence of it. Hence the "even if '-conditional

is accepted.

Note also that an asequential conditional (see Jackson [36]) is

still not that far removed from a sequential conditional. If El.1.2

is acceptable, and we were to somehow bring it about that kangaroos

became tailless, we wuld expect them to topple over. Hence we can

also make the prediction that "If kangaroos are de-tailed, then they

will topple over. "

The similarity in the informally sketched methods above suggests

that both sequential and asequential conditionals may be accessible to

the same analysis in terms of truth conditions. The reverse or back-

wards sequential (see Jackson [36]) of E1.1.5 can also be seen as

similar. Where with a forwards sequential we consider whether the

antecedent is sufficient for the consequent in terms of the laws, for

the backwards sequential we consider whether the consequent is neces-

sary for the antecedent to be subsequently realized in terms of the

laws involved.









While the thrust of the above remarks is to broaden the scope of

the conditionals with which we will be concerned, and to indicate that

"counterfactual" or "subjunctive" is not a necessary mark of such con-

ditionals, nevertheless most of the conditionals we are concerned with

can be expressed as subjunctive conditionals with propositional con-

stituents. For example, we can paraphrase "If kangaroos had no tails,

then they would topple over" as "If it were the case that kangaroos had

no tails, then it would be the case that kangaroos topple over." With

this in mind we offer the following symbolization of such a conditional:

Wpq = df "If it were the case that p, then it would be the
case that q."

Without implying the need for separate truth conditions, but for

ease of reference, we shall also adopt the following symbolizations:

Tpq = df "Even if it were the case that p, it would (still)
be the case that q."

Upq = df "If it were the case that p, then it could not be
false that q."

The former is intended to symbolize those conditionals where there is

an absence of a connection between antecedent and consequent ("even if'-

conditionals) and the latter those conditionals where there is a con-

nection between antecedent and consequent, i.e., where the antecedent

"brings about" the consequent (what Pollock [80, p. 271 calls "necessi-

tation"'-conditionals). We shall also adopt the standard Polish, or

prefix, notation for the usual logical operations of material condi-

tionality, material bioconditionality, negation, conjunction, and (in-

clusive) disjunction. These and the symbolization for strict condi-

tionality, necessity, and possibility in modal logics are listed below:








Cpq =df "Ifp, then q."

Epq = df "p if, and only if, q."

Np = df I'bt p."

Kpq =df "Bothp and q."

Apq = df 'Either p or q."

1p =df "p is necessary."

Mp =df "p is possible."

cpq = df "p strictly implies q."

Since in most systems of modal logic Cpq is true if, and only if, L(pq

is true, we shall usually use the latter in place of the former unless

it becomes necessary to distinguish between them. In like manner LEpq

will denote strict equivalence. We also introduce:

Fq = df KpqWqp

for what might be called counterfactual equivalence.

There is one further type of conditional for which we would expect

different truth conditions will be required.

Consider the pair of conditionals:

El.1.9: If Bizet and Verdi were compatriots, they would both be French.

El.1.10: If Bizet and Verdi were compatriots, they ould both be Italian.

We would be inclined to reject both of these conditionals but wuold ac-

cept both of the following:

E1.1.11: If Bizet and Verdi were compatriots, they might both be French.

El.1.12: If Bizet and Verdi were compatriots, they might both be Italian.

To symbolize the "might"-conditional we introduce:

Vpq = df "If it were the case that p, then it might be the
case that q."








At this point we have made reference to six conditionals: Wpq,

Tpq, and Upq, where we expect one set of truth conditions, the latter

two conditionals presumably being subclasses of the former; Vpq, dis-

tinct from the above three; and Cpq, and LCpq, which are the traditional

material and strict conditionals, respectively. By way of terminology

we will refer to the first four indiscriminately (and somewhat inac-

curately) as counterfactuals, the first three as "would"-counterfactuals,

the fourth as the 'ight"-counterfactual, the second as the "even if'-

counterfactual, the third as the "necessitation"-counterfactual. The

same prefixes with the suffix "conditional" will also be used. When

the term "counterfactual" alone is used, this will usually refer to the

'\%ould '-conditional.

It would be appropriate at this point to consider the various in-

ference patterns that are intuitively valid for the conditionals we have

mentioned. This would serve to illustrate some of their differences,

while providing criteria of adequacy for any purported analysis. In

keeping with a long tradition in analytical philosophy, I should like to

postpone these considerations until we have a preliminary analysis to

test them against. One inference pattern, however, is of such paramount

importance that it bears mentioning now. I refer to Strengthening the

Antecedent (also called Augmentation). It is well known that both of the

following inferences are valid in classical propositional and modal logics:

Cpq LCpq

CKprq . LCKprq

however, consideration of a single example will show that the correspond-

ing pattern for counterfactuals








Wpq

WKprq

is not valid. Speaking of a certain dry match in favorable conditions

(enough oxygen, etc.) I may say,"If this match were struck, it would

light." But it does not follow fran this that "If this match were

soaked in water and struck, it would light." The failure of Strength-

ening the Antecedent is one of the striking peculiarities of counter-

factuals, and the single strongest argment against the counterfactual

conditional being a strict conditional.

1.2 The Metalinguistic Analysis

I referred earlier to a certain procedure whereby a counterfac-

tual could be evaluated as the starting point of a number of analyses

of counterfactuals. This is what has been called the "linguistic" or

" netalinguistic" account (by Pollock [80] and Lewis [51], respectively).

Because most hold that the consequent of a counterfactual is a logical

consequence of the antecedent conjoined with other statements, they are

also called "consequence theories." According to such accounts the

truth of a counterfactual conditional is largely based upon the rela-

tions anng certain linguistic entities, such as sentences or, in some

cases, beliefs. (Such accounts have been offered by Goocnan [26],

Chisholm [13], Mackie [59], Rescher [83], Jackson [36], Ellis [19], and

others.? I shall sketch a general outline of such an account which

does not do full justice to any of those that have actually been of-

fered, but is sufficient to form a starting point for criticism.

Consider the well-worked-over example concerning a certain pres-

ently unlit match:








El.2.1: If that match were struck, then it would light.

We will symbolize El.2.1 as Wpq. The linguistic account attempts to

formalize our earlier procedure:

D1.2.1: Wpq is true just in case there is a set of true factual

statements F and a set of laws L such that the conjunction of F, L

and p logically implies q.

As Goodman [27] pointed out, determining just what should go into F and

L is no easy task. Certainly such facts as that the match is well-made,

there is enough oxygen present, the match is dry, etc., should belong to

F, while certain chemical and physical laws belong to L. In fact, if we

let L consist of the single physical law 'Matches satisfying conditions

C light when struck," where C incorporates the circumstances referred

to as facts above, then this law together with F and p logically imply

q, since the truths in F guarantee the satisfaction of conditions C.

We can just check each condition in C and see if it is satisfied by the

circumstances surrounding this particular match.

This approach wuld require that for each counterfactual we have

a highly specific covering law, the law itself specifying what mst go

into F. This shifts the problem of determining the truth of the counter-

factual to a problem of determining whether a certain highly specific law

is true, perhaps on the basis of other less specific, more general laws.

In either case, we somehow have to identify the relevant conditions F.

Shifting the problem to specifying a particular law of limited

generality does not solve it, since the problem of determining the spe-

cific facts F is now transformed into the problem of determining the spe-

cific conditions C under which the law holds. Furthermore, this approach

wuld not wrk for "even if '-conditionals where there is no covering law








connecting the antecedent (and relevant conditions) to the conse-

quent.

We can retreat to the original definition, let the laws be of

reasonable generality and concentrate on the problem of specifying

the facts F and laws L for a given counterfactual. But perhaps both

these issues can be sidestepped: presunably our laws are consistent

as a set, likewise the facts embodied in a description of all the cir-

cumstances surrounding the antecedent. Why not take all laws and all

true facts obtaining and conjoin them with the antecedent. The pro-

blem with this is that the falsity of the antecedent is one of the

facts, and from Np and p, q follows. Also, it is a logical law that if

p is false Cpq is true, and from p and Cpq, q logically follows. So

neither all facts, nor all laws can go into F and L.

Obviously we must eliminate Np from F. Clearly we must also

eliminate Nq because we do not want to validate both Wpq and WpNq, as

admission of Nq to F would do. In fact, if we were to admit to F any

statement r such that r would be false if p were true, we may validate

conditionals that under the circumstances ue would want to deny. For

instance, in the case of the match it also follows by law that "If the

match were struck, it would not be dry," since from the truths that

there is enough oxygen, the match does not light, it is well-made, ad-

ding that it is struck implies by a suitable law that it was not dry.

So we may include in F only such statements r which are not only

true, but would not be false if p were true, i.e., for which NWpNr is

true. Goodman [27, p. 15] calls such statements those "cotenable" with

p, and rightly observes, that now we are analyzing a counterfactual in

terms of other counterfactuals, so our account is irredeemably circular.









Pollock [80] observes that if all that is required for inclusion

in F, as Goodman appears to believe, is truth and cotenability, then

this implies an even stronger requirement on the truths in F. To show

this we require acceptance of two obvious principles regarding counter-

factuals:

(A) If Wpq is true and lCqr is true, then Wpr is true.

(B) If WpCpq is true, then Wpq is true.

We postulate the following in accordance with Goodman's proposals:

(a) Wpq is a counterfactual to be evaluated and p is false.

(b) Cotenability and truth are sufficient for inclusion of
r in set F; i.e., r is true and hpNr is true.

Noting that LCKpNrNr is true and using (A) contrapositively, it follows

that h KpNr is true. Noting that LEKpNrNCpr is true and using (A)

contrapositively again, it follows that NWpNCpr is true. Since p is

false, Cpr is true. Hence by (b) Cpr is included in F as it is true

and cotenable with p. Since F together with p logically implies any-

thing included in F, Goodman's proposal validates WpCpr. Hence by (B),

Wpr is true. So "r is cotenable with p" amounts to Wpr is true on

Goodman's assumptions [80, p. 11].

If F is to include everything cotenable with p, then F includes

the consequent of all true statements of the form Wpr. That is, F in-

cludes everything that wuxld be the case if p were true. Counterfac-

tuals are analyzed in terms of cotenability (and other elements), but

then cotenability is analyzed in terms of counterfactuals, and so our

analysis is circular. Because he did not see a way out of this vicious

circle in analyzing counterfactuals, Goodman shifted his concern to a

weaker notion, that of dispositions.









Since analyzing counterfactuals in terms of counterfactually

defined cotenability is so obviously circular, it is curious that a

recent treatment of counterfactuals seems to make a virtue of it.

Ellis provides "a unified account of three kinds of conditionals" in

terms of his notion of a "rational belief system" [19, P. 107]. (See

[18] also.) One of these conditionals is that which we have been calling

counterfactual. While I am in complete agreement with Ellis' conclusion

"that indicative and subjunctive conditionals are usually variant locu-

tions for the one kind of conditional which is variably strict" [19,

p. 115], and have so argued in the first section, I do not see how his

account can be construed as an analysis of conditionals, particularly of

the "variably strict" conditional, which we shall see later is an appro-

priate way to refer to the counterfactual conditional. My reason for

this reservation is that his account uses the counterfactual conditional

to give the truth conditions for the counterfactual conditional in much

the same way as Goodman's self-adittedly failed account.

Ellis' truth condition for the conditional may be paraphrased as

follows [19, p. 108]:

DI.2.2: Wpq is held true in belief system B just in case in all ccm-

pleted extensions of a certain modification of B, B', Nq nowhere occurs.

A (rational) belief system is essentially a partial evaluation on all

the sentences of a language; certain sentences are held true, others

false, and others withheld (i.e., no firm belief one way or the other).

There are a number of rationality requirements on a belief system,

among which is DI.2.2 above. A completed extension of a belief system

is the replacement of all withheld evaluations by true or false evalu-

ations without violating any of the rationality requirements. These









notions are all unproblematic, as are the rationality requirements not

presented here. What is of concern is the definition of the modified

belief system B' constructed from B which serves to characterize the
P
conditional Wpq.

According to Ellis, belief system B' "can be thought of as the
p
assumed basis of reasoning from the supposition that p" [19, p. 109].

For the counterfactual conditional, Ellis' definition of B' may be

paraphrased as follows [19, p. 112]:

D1.2.3:

(1) r is held true in B' if either Lr or Wpr is held true in B.
P
(2) p is held true in B'.
P
(3) Otherwise B' is agnostic.
p
In Ellis' words B' "includes not only what we should take to be neces-
p
sarily true, but also what we think would still be or have been the case

if 'p' were . ." [19, p. 113].

In view of condition D1.2.3(1), as an analysis of the counterfac-

tual condition this account is circular. D1.2.3(1) replaces Goodman's

notion of cotenability, and D1.2.2 replaces the requirement that the

conjunction of F, L, and p logically implies q. What then does Ellis'

account accomplish? And why, considering the family resemblance of

D1.2.3(1) to cotenability should this account be thought to advance the

theory of conditionals? To answer these questions requires a digression

on the subject of what constitutes an analysis.

"Analysis" can mean one of two things, not necessarily exclusive.

Both are routes for the clarification of a concept. One is to explicate

or articulate the concept in terms of other, presumably better under-

stood, concepts. In this context an analysis is much like a definition;









for a complete analysis the analysandum offered is a definitional equiv-

alent for the analysans. It is, of course, a serious shortcoming in a

definition for the term defined to appear in the definition itself on

the analysandun side. (We are not speaking here of a recursive defini-

tion.) There are other constraints on analysis. Where clear usage is

evident in the pre-analytic concept, this usage should be preserved

under the analysis. A concept with no puzzling cases is in need of no

clarification, so no analysis; hence an analysis should go some way to-

ward resolving the puzzling cases. Puzzling cases for counterfactuals

involve the Bizet and Verdi examples of Section 1.1, counterlegals,

counteridenticals, and others where there seems to be sowe question as

to how to interprete the antecedent. At times even a failure to cover

all pre-analytic cases of clear usage is forgiveable if the analysis of-

fers advantages in other respects. Of course, one is then rightly sub-

ject to the charge of advocating a change in the concept.
A second method of analysis is to codify the rules governing the

operation of a concept. This is often expressed as making explicit the

"logic" of the concept. In this context an analysis is much like the

notion of "syntactic meaning' where the meaning of, say, a logical con-

nective, is said to be implicitly given by the axioms and rules of in-

ference that formalize its operation. For terms that appear as primitives

in a theory such a notion is valuable.2

An example of a relatively pure case of the first type of anal-

ysis would be the analysis of knowledge as justified true belief (or

more accurately nondefectively justified true belief). More relevant

to our subject would be Lewis' analysis of a "law of nature":









. a contingent generalization is a law of nature if
and only if it appears as a theorem (or axiom) in each
of the true deductive systems that achieves a best com-
bination of simplicity and strength. [51, p. 73]

An example of a relatively pure case of the latter kind of anal-

ysis is found in Wasserman [1021 wherein he presents a so-called "log-

ical analysis" of the counterfactual conditional. What Wasserman does

is provide a language containing a binary connective intended to repre-

sent the counterfactual conditional with a model-theoretic structure as

semantics. This procedure is intended to make explicit the logical

structure of the conditional in question. Of this semantics Wasserman

says:

The "philosophical" motivation for the formal semantics
provided for a statement of the form "If 4 were the case,
then ip would be the case" is that such a statement is
about some "world," "state-of-affairs," or, more formally,
some structure S, and that the statement "means" that *
holds in every structure which differs from S "just enough"
to make 4 true. [102, p. 396]

Wasserman indicates that he is providing an analysis of a logical

kind that can be construed as in some way giving the caningng" of

counterfactuals. (One wonders if meaningg is the same as meaning.)

This reflects a practice which has become standard in analytical philos-

ophy: the meaning of a concept can be given in terms of its truth con-

ditions. Thus Lewis states that the task involved in giving an analysis

of counterfactuals is to "give a clear account of their truth condi-

tions" [51, p. 1]. For Stalnaker the task is "to find a set of truth

conditions for statements having conditional form which explains why we

use the method we do use to evaluate them' [96, p. 169]. While the

claim may not be that the truth conditions constitute the meaning of the









concept, the claim certainly is that once one has grasped the truth

conditions one has grasped the meaning of the concept.

But here we must be careful. Not just any set of truth con-

ditions will do (as Pollock points out in a different connection in

[78, p. 8]). Stalnaker indicates this when he says above that the

truth conditions must "explain" something. Judging by his analysis

Lewis has something similar in mind. The question to be answered is:

When should we be satisfied with a purported analysis? Goodman rightly

rejects his own analysis as circular, but Ellis offers an analysis con-

taining formal elements with precisely the same characteristics.

Wasserman's analysis provides a logic for the conditional, but would we

be justified in claiming on that basis to have grasped its meaning?

Stalnaker, who shares with Ellis the conviction that the mood or

factual status of a conditional is a secondary consideration, distin-

guishes two problems involved in analyzing counterfactuals. The first

he calls "the logical problem of conditionals" which is "the task of

describing the formal properties of the conditional function

[96, p. 165]. The second is the "pragmatic problem of counterfactuals"

which concerns the fact that

. the formal properties of the conditional function,
together with all of the facts, may not be sufficient
for determining the truth value of a counterfactual; that
is, different truth valuations of conditional statements
may be consistent with a single valuation of all non-
conditional statements. [96, pp. 165-166]

The development of a semantic theory for counterfactuals Stalnaker re-

gards as part of the logical problem. The semantic theory that he

does develop sheds light on the second problem as well, in his view,








because it shows where the semantic component of the concept leaves off

and the pragmatic component begins [96, p. 1661.

I should think, however, that the logical problem actually in-

volves two problems: the task of describing the formal logical pro-

perties of the conditional and the task of devising a satisfactory

semantics. These two tasks are different. One could describe the

formal properties of the conditional in terms of a proof-theoretic

system: a set of axioms and rules of inference in which a conditional

connective occurs, and in which those sentences and rules of inference

our pre-analytic intuitions hold valid occur while those we regard as

invalid do not. We would be remiss to accept such an analysis as com-

plete for it is possible to understand the logic of a concept without

understanding the concept itself. .

For example, in [15] Chisholm makes use of a relation "more

reasonable than" holding between propositions. That is, a certain

proposition p may be more reasonable for subject S at time t than

another proposition q. This appears in his formal definitions as an

undefined relation, but to explicate it he offers certain basic prin-

ciples as axioms of the concept intended to make explicit its logical

structure [15, p. 131. If left at this point (which Chisholm does not

do) we may have in our grasp the logic of "more reasonable than" with-

out understanding what it is for one proposition to be more reasonable

than another. We do not know how to apply the relation to propositions,

only how to manipulate its previous application.

Devising a semantics to validate this axiomatic system may not in

itself be sufficient to convey an understanding of the concept. The









concepts in terms of which the semantic theory is itself stated must be

ones we can apply apat from the system they are designed to validate.

If this condition is not met, then we are in precisely the same predic-

ament as before. We may know how to determine what sentences and rules

of inference are valid in which the concept occurs, but we do not know

how to apply the concept itself. It is in this connection that the

role of analysis as explanation arises. If we do apply the concept and

have some idea of the method employed, then the truth conditions, pos-

sibly presented in the form of a semantics for sentences employing the

concept, must explain why the method wrks as it does.

Thus there are two constraints on a logical analysis of a concept

even when construed as a search for truth conditions:

1. The truth conditions (or semantics) must be applicable and under-
standable apart from the concept analyzed by them.

2. The truth conditions must explain how and why our pre-analytic
employment of the concept wrks.

Recognizing these constraints reduces the apparent distinction between

a traditional meaning analysis, the first kind discussed above, and a

logical analysis. Furthermore, it gives us a means for evaluating a

purported analysis of the counterfactual conditional.

Without a clearer delineation of what constitutes the structure

S against which as background Wpq is evaluated, Wasserman's analysis

remains incomplete. In fact since S is a truth set maximal with

respect to joint satisfiability with p [102, p. 397], S will be under-

constrained in any case, as Goodman has pointed out, for all that is

required of S is that it be true and consistent with p. Hence

Wasserman's analysis will fail to explain how our pre-analytic








employment of the concept wrks if only because it will fail to dis-

tinguish between correct and incorrect applications of it.

Ellis' analysis fails for just the reason w supposed: the

concepts in terms of which the truth conditions for the conditional

are stated require in part that we already understand the concept of

the conditional. We cannot construct the belief system B' unless we
p
antecedently understand under what conditions statements of the form

Wpr are held true in system B.

This is not to say that Ellis' semantics is in a formal sense

ill-defined, but rather that there are non-formal criteria that truth

conditions must meet in order to qualify as an analysis. Thus an anal-

ysis of the conditional mist be in terms of non-conditional notions

just because we are regarding the conditional notions as problematic.

Ellis' definitions amount to a recasting of cotenability in terms of

belief systems. Until we have an analysis of cotenability independent

of the concept of the conditional our analysis will fall short.

A general shortcoming of both linguistic accounts and related

belief-based accounts is that they attempt to model our informal pro-

cedure for evaluating conditionals rather than explain it. Of course,

we do take as our assumed basis of reasoning on the assumption that p

is true what we believe would still be the case on that assumption,

but this is just to say what it is we do, not to explain how or why it

works.

Returning once again to Goodman's analysis, suppose we could

satisfactorily settle the problem of what to include in the set of

specific facts F. Let us turn our attention to the set of laws L.

Three problems immediately arise:








1. What are we to make of counterfactuals 'whose antecedents

deny accepted laws, so-called counterlegals?

2. How do we determine which laws are relevant, or alternately

which laws are irrelevant and would lead to incorrect evaluation of

the counterfactual?

3. Is not the concept of law itself problematic, to perhaps as

great an extent as the concept of the conditional it is being taken to

clarify?

In reference to the first problem we could refuse to countenance

counterlegals, but this would be blatantly ad hoc. But if we permit

counterlegals then we will be faced with the cotenability problem all

over again in terms of which laws we shall retain and which reject in

population L and F.

In reference to the second problem consider the following pairs

of laws:

LI: All matches, well-made, dry, in sufficient oxygen, and
struck, light.

12: All matches, well-made, in sufficient oxygen, struck, and
not lit, are not dry.

On the one hand, Li would appear to validate

E1.2.1: If that match had been struck, it would have lit

while L2 appears to validate:

E1.2.2: If that match had been struck, it would not have been dry.

The consequents are incompatible, so both counterfactuals cannot be true,

yet what licenses our relying on one law rather than the other? It can-

not be that one law is true and the other false, for both are true.

Rescher [83, p. 161] considers a similar example in explicating

counterfactuals in terms of his "belief-contravening suppositions." We








have a covering law (Li), the beliefs that the match is not struck and

not lit, and the beliefs that the "auxiliary" conditions are met.

Rescher's analysis is simply that for counterfactuals for which a

covering law exists, so-called "nomological counterfactuals," when we

assume, contrary-to-belief, that the match is struck, we seem to have a

choice about rejecting the law, some auxiliary hypothesis or the instance

of the consequent of the law. But in fact we regard laws as inviolable,

and if we extend this to the hypotheses which "assure its applicability"

then our only choice is to reject that it did not light, thus validating

E1.2.1 rather than the competing E1.2.2. However, where there are two

laws, as above, this technique runs into difficulty.

This analysis works only if antecedently we have some reason to

choose LI over 12 as the relevant law. In a footnote [83, p. 161n]

Rescher notes an objection of Goodman's to this analysis which parallels

ours. According to Rescher LI and L2 are represented by Goodman as log-

ically equivalent, as would several other partial contrapositives of Li

be. Each would validate a differing counterfactual. Rescher's reply

takes the form that the other equivalents to the covering law LI may be

deductively equivalent to Li, but are not equivalent in the context of

inductive logic. (This claim is related to a solution of Hempel's "raven

paradox.") Rescher claims the covering law Li has primacy in the evalu-

ation of counterfactuals over its "equivalents."

Rescher's response misses one point and raises another of rele-

vance to our third problem. Contra Rescher, Goodman need not claim

that Li and I2 are equivalent, with Li being the "favored" formulation

of the covering law; rather Goodman can maintain that Li and 12 are both

inductively confirmed laws. Then the question is indeed what relevant









law do we choose, not whether we reject the consequent or an auxiliary

hypothesis of the "favored" formulation of the law.

The point of relevance to the third problem is that we do favor

LI, not because it is directly inductively confirmed as 12 is not

(which is a false claim), but rather because it has the form of a

causal law with a direction. This amounts to its being conditional in
4
nature, and not materia, as we shall see.

In more direct reference to the third problem, one might feel

that, as laws and counterfactuals are both problematic, to analyze one

in terms of the other is not to solve the problem. The immediate re-

joinder would be, better one problematic concept than two. If counter-

factuals can be analyzed in terms of laws, then we simply have to go on

to analyze laws. Rescher apparently holds this view, and regards the

analysis of counterfactuals to be laid at rest while more study is

needed of laws and confirmation theory [83, p. 164]. (In this con-

nection see also the rest of Goodman [27].)

This is a problem of metaanalysis and its appearance is not new

to philosophy. One is reminded of Quine's attacks on the concepts of

analyticity, meaning, and synonomy. When we have a set of systemati-

cally interrelated concepts all of a problematic nature, the reduction

of all the others to one may only be an apparent, not an actual, advance.

It is my feeling that the lack of advance is most pointedly felt as a

failure to explain any of the concepts at issue. Repeated failure to

explain any one of the interrelated concepts leads to one of two out-

comes:

1. "Sour grapes" in which the whole complex is given up as a bad idea.








2. "Sweet lemon" in which it is blissfully agreed that the concepts

must be acquired as a set, all are basic, and none has priority over

the others.
Otherwise it remains open season on the set of interrelated concepts

with repeated efforts to explicate one of them, and then the others

will fall into line.

We have not shown that laws fall into this analytic circle, nor

considered other escapes, such as the mve to dispositions. Goodman,

having analyzed counterfactuals in terms of cotenability and laws, and

noting the circle into which cotenability and counterfactuals fall,

and the problems with laws themselves, shifts the problem, like Rescher,

to laws and confirmation theory. Dispositions, as a weaker, but re-

lated, notion to counterfactuals are picked up along the way. Ellis re-

duces counterfactuals to cotenability, but this is flatly circular.

With the exception of Ellis, the metalinguistic accounts including be-

lief accounts such as Rescher's must look to a further clarification

of laws in order to pull off an explanatory analysis.

There is, however, good reason to believe laws do fall into this

analytic circle, particularly if we are searching for truth conditions

rather than being satisfied with justification conditions. Any belief-

based account of counterfactuals, or laws for that matter, terminates

ipso facto in justification conditions. It is not clear to me that

justification conditions ever have explanatory force, and not all truth

conditions do. At best they can codify what we do, but not illuminate

how or why it works. (For a defense of the opposing view see Pollock

[781.)








It is comonplace that laws (or more properly, lawlike state-

ments, of which the true ones are laws) are generalizations. It is

apparently equally comonplace that they are not material generaliza-

tions, many of the latter being clearly accidental rather than law-

like in nature, e.g.,

E1.2.3: All the coins now in my pocket are silver

as opposed to

E1.2.4: All pulsars are neutron stars.

Thus one of the problems of laws is to distinguish in a noncircular way

between accidental and lawlike generalizations. Now it is clear that

laws support counterfactuals, but this cannot be the distinguishing

characteristic of laws, or if it is, then we have placed laws squarely

in the analytic circle with counterfactuals (if we continue to analyze

counterfactuals in terms of laws).

A material generalization is conclusively confirmed in virtue of

the vacuity of its antecedent or by exhaustive enumeration. Such is

not the case with laws. A law, for example Newton's second law of no-

tion, may indeed have a vacuous antecedent, but it is not true in

virtue of that. Other examples could be cited, but this may be beating

a dead horse. It is generally admitted that laws are not material gen-

eralizations.
However, it is equally obvious that laws are generally conditional

statements of some sort as well as generalizations of some sort, partic-

ularly the causal laws usually taken to be intimately related to coun-

terfactuals. In fact, considering the match example again, the "fa-

vored" law U is a generalized conditional prediction, while its aber-

rant relatives are conditional postdictions. Indeed Stalnaker suggests









that laws are just universally quantified counterfactual conditionals

[96, p. 177]. If this be admitted, then laws share with counterfac-
tuals the property of being conditional in nature, but not material in

nature.

On the other hand, laws extend to contrary-to-fact situations

where material statements do not. This is amply illustrated by E1.2.4

where not only is it held that each actual pulsar is a neutron star,

but that anything else which could be a pulsar (but is not) would be a

neutron star. Pollock notes this as the subjunctive nature of laws,

which he then calls subjunctive generalizations as opposed to material

generalizations [80, pp. 13, 48]. However, at this point whether we

have two concepts to analyze, conditionality and subjunctivity, or one,

conditionality, is beside the point. In either case laws will share

with counterfactuals a characteristic which our analysis of either (or

both) must explain. For convenience I will continue to refer to the

concept of the conditional as what is to be explicated.

This places laws squarely in the analytic circle with conditionals.

And to break out of the circle and avoid "sour grapes" or "sweet lemons"

some one of the problematic concepts must be given an explanatory anal-

ysis. There seem to be three approaches to the resulting problem of

breaking out of the circle, each with its attendant problems and virtues.

One approach is to accept that counterfactuals can be analyzed in

terms of laws and cotenability, and then to provide a more basic, explan-

atory, analysis of laws and a resolution of the circularity in coten-

ability. This is fundamentally an attack on the law problem. The anal-

ysis of laws takes the form in Pollock's approach of analyzing them in

terms, not of their truth conditions, but of their justification








conditions, and is thus an exercise in confirmation theory. The con-

firmation theory is found in [78], while [801 takes the claim that

laws can be analyzed in terms of their justification conditions for

granted and proceeds to analyze counterfactuals in terms of laws and

cotenability. Then cotenability is given an explanatory analysis in

terms of possible worlds. Thus the analysis Pollock offers is only

partly an analysis in terms of truth conditions, resting as it does

upon an analysis of laws in terms of justification conditions.

In partial contrast, Goodman [27] also provides an analysis of

laws in terms of confirmation theory and so in terms of justification

conditions. However, he seems to regard the solution to the coten-

ability problem as fall-out from the analysis of laws [27, p. 122].

An account such as Jackson's [36] or Barker's [2] in terms of causal

laws, while having virtues and defects of their own (most counterlegals

become irredeemably ambiguous), certainly are predicated upon an anal-

ysis of causal laws if they are to have any explanatory force. The

same can be said of Rescher's [83] analysis of nomological counterfac-

tuals (all others are irredeemably ambiguous) in terms of laws.

With the exception of Pollock's reliance on possible worlds to

analyze cotenability, all of these accounts share the assumption that

an analysis of law is prior to an analysis of counterfactuals and that

this analysis occurs in the context of confirmation theory and is an

analysis in terms of justification conditions. The breakout from the

circle thus comes in the analysis of law. Pollock included, these

variants are all direct inheritors of the metalinguistic approach.

The second approach is to attack counterfactuals directly by pro-

viding an explanatory truth condition account of them. It is here that









possible worlds semantics makes its appearance as an explanation of

why and how our informal procedure for evaluating counterfactuals works

as it does. If Stalnaker is taken as exemplifying this approach, then

laws are analyzed in terms of counterfactuals, specifically as quanti-

fied counterfactual conditionals [96, p. 177]. Lewis [511 and Nute [68]

share with Stalnaker the assumption that counterfactuals are prior to

laws as far as breaking out of the analytic circle is concerned. The

concept of the conditional is provided with an analysis in terms of truth

conditions. Analyses of the second sort then break out of the circle at

the point of analyzing the concept of the conditional, not in terms of

the laws, but in terms of truth conditions based on possible worlds in a

manner as yet to be illustrated. The claim that such an analysis is also

an explanation will be defended in CHAPER TWO.
Lewis' approach, however, while sharing with Stalnaker's the as-

sumption that the conditional is prior, differs in its treatment of law,

and thus represents the third approach. Both the second and first ap-

proaches affirm the analytic circle while breaking out of it. Once one

of the concepts involved has been analyzed independently of the others,

the others can then be analyzed in terms of it. Lewis removes laws from

the analytic circle in such a way that counterfactuals neither depend

directly upon them, nor do they depend upon counterfactuals. Why and

how he does this will be covered in CHAFTER THREE.

Whenever an analytic circle demonstrably exists, then an analysis

of any one of the problematic concepts has a certain prima facie vir-

tue, in that analyses of the other concepts immediately follow. Con-

peting analyses will then differ in a combination of respects: they








may be analyses of the same or different sorts, of the same or different

concepts within the circle. If the analytic circle is only suggested by

the failure of prior attempts to arrive at a satisfactory analysis,

nevertheless, much the same situation obtains. He who maintains that

the circle is not only broken, but resolves itself into two lines has,

of course, an additional task: to undermine the evidence for circular-

ity.

The metalinguistic analyses, including belief analyses, lead in-

exorably to laws, confirmation theory, and analysis in terms of justi-

fication conditions. It will be maintained that such analyses are sub-

ject to the charge of failing to explain the concepts they take as pro-

blenatic. On the other hand, possible wrld accounts (with the excep-

tion of Pollock's mixed account) have the prima facie virtue of pro-

viding an explanation of why the concept works the way it does.


1.3 Notes

'A survey of early accounts of this sort may be found in Schneider
[87].

'For other logical analyses and criticisms thereof see Bode [7],
Fumerton [24], Lehmann [44], and Nute [69]. We discuss Wasserman [102]
as an example.
3This suggestion was made to me in conversation by Gary Fuller.

4For more discussion than we shall have space for of the relation-
ship anong laws, necessity, conditionals, and causation, see Barker [2],
Chisholm [14], Fine [21], Goosens [28], lbnderick [35], Jackson [36],
Kim [37], Kneale [38], Lewis [52], Loeb [55], Lyon [58], Mackie [60, 61,
62], Nute [70], Shorter [92], Sosa [95], Swain [98], Temple [99], Vendler
[101], and Yagisawa [103].














CHAPTER TW

POSSIBLE NORMDS ANALYSIS OF COUNIERFACTUALS


2.1 Possible Worlds: History

In the period between 1968 and 1973 several analyses of the counter-

factual conditional appeared that diverged sharply from the metalinguistic

accounts that had been produced in the preceding two decades. The diver-

gence was in the uniform reliance of these new approaches upon the pos-

sible world semantics for modal logic introduced by Saul Kripke [40, 41].

Of these accounts the most thoroughly worked out was that of David K.

Lewis [50, 51]. It is this account which we shall discuss in some detail

in the following sections. Within the same period accounts differing in

detail from that of Lewis also made an appearance. Those of particular

note include Aqvist [1], Stalnaker [961, Stalnaker and Thomason [97],

and Nute [67]. We shall discuss and compare several of these accounts in

CHAPTER THREE.

Elements of Lewis' book, Counterfactuals, appeared as early as
0
1970, and Aqvist's article was earlier(1971) published by Upsalla.

Nute's article was delayed two years in publication, so the initial de-

velopments in this field were grouped into the years noted above. The

appearance of these efforts sparked a resurgence of competing accounts

of counterfactuals too numerous to mention, as well as spirited defenses

of the possible worlds account. A notable effort intended to cover the

entire range of subjunctive constructions is that of Pollock [80], which

35








makes use of both possible worlds and the insights of the metalinguistic

account.

However, this recent history owes n ch to earlier developments in

modal logic, particularly the algebraic semantics of Lemun [45, 46] and

other workers in this area. Nodal logics were studied systematically by

C. I. Lewis [48], actually significantly earlier than the publication

date of the cited work. A survey of these systems bringing together many

strands of the treatment of modal logics is to be found in Zeman [104].

It is doubtful that the counterfactual conditional would have yielded at

all to logical analysis were it not for the groundwork laid in the study

of the strict conditional.

The concept of possible worlds, which makes its first appearance

in a formal semantics for modal logic with Kripke, is usually credited

to Leibnitz. Its use in semantics for modal logic was prefigured in

significant ways by other authors. A. N. Prior [81, 82] had made use of

possible worlds as uments of time in his study of tense logics. A full

bibliography of his works in this connection may be found in Zeman [1041.

Earlier C. I. Lewis had identified possible worlds with the "comprehen-

sion" of a proposition [471 in an attempt to explicate meaning. The

same tactic was followed by Carnap [10], who specifically identified pro-

positions as sets of "Leibnitz possible worlds," or Wittgensteinian "pos-

sible states of affairs."

One striking anticipation of possible world semantics for modal

logic occurs earlier than any of the works cited above. I refer to the

Existential Graphs of C. S. Peirce. The incomplete development by

Peirce of his system of Existential Graphs is traced in Roberts [84].

Of interest to us is the review of Roberts' book by Zeman [105] which









brings out the significant parallels between Peirce's "ganma" system

and possible wrld semantics. Evidently, having developed graphical

systems for propositional and quantificational calculus (the "alpha"

and 'beta" systems), Peirce experimented with a third system (or frag-

ments of several systems) in which he endeavored to make possible the

representation of universes of discourse other than the actual:

. these would be "worlds of possibility." . .
He proposed that instead of considering just one SA
. we think of ourselves as working with a book
of such sheets, with each sheet in the book repre-
senting a possible world much as Kripkean semantics
correlates a semantic tableau with each possible
world. [105, p. 252]

In the above quotation from Zeman, SA refers to the Sheet of Assertion

upon which graphical signs are written as assertions about the universe

of discourse.

Peirce did not quite reach conceptual closure on this idea, due

to the fact that he did not have an adequate way to represent the ac-

cessibility relation. Though even here, Zeman notes [105, p. 253], he

came close. Peirce did hit upon a predicate which bears interpretation

as an accessibility relation, but did not develop it.

In addition to relational possible worlds semantics of the Kripkean

variety, Scott [88] and Montague [65] introduced a variant approach:

neighborhood semantics. A comprehensive treatment of modal logics in

terms of neighborhood semantics is found in Segerberg [91] which forms

an important basis for our presentation in Sections 4.1-4.3. Most of

the results therein are first brought together by Segerberg. The ap-

plication of neighborhood semantics to conditionals is developed by

Chellas [11], and a systematic comparison of the varieties of relational








and neighborhood semantics has been carried out by Nute [74]. Sections

4.4-4.7 are considerably indebted to these last two mentioned works.

The logician, qua logician, is interested in the adequacy of a

formal semantics quite apart from whether or not it affects an analysis

of the concepts the system is intended to formalize. It is possible to

regard relational possible world semantics as providing an analysis of

important concepts of modality (see Bradley and Swartz [8], Foulis and

Randall [22, 23], and Zeman's application and development of the latter

[106, 107]). This is of interest to the logician qua philosopher. The

semantics for conditional logic developed by Lewis, Stalnaker, Nute,

and others are intended as analyses, and must therefore meet constraints

we suggested in Section 1.2 and will explore further in Section 2.7 and

CHAPTER THREE. What is lacking in the application of neighborhood

semantics to conditional logic is the idea of an analysis as opposed to

a formalization. Though Nute [741 compares various semantics for con-

ditionals on a formal basis, his comparison of them for philosophical

adequacy is limited to those which have been explicitly developed with

analysis in mind: Stalnaker, Lewis, and himself. The philosophical

adequacy of neighborhood semantics for the analysis of conditionals is

largely unexplored. It is this gap in exposition for which we ultimately

hope to provide some filling.

The history of possible worlds, even such a sketchy account as

that offered here, would be incomplete without mention of two further

areas where possible worlds have had an impact: science fiction and

the interpretation of quantum mechanics. We will return to the former

in more detail, but of the latter we only note that the Everett-Wheeler

interpretation of quantum mechanics employs the notion of a "reality









composed of many worlds" [17, p. v]. For the difficulties involved

in drawing any significant positive philosophical conclusions from

this theory one should see Skyrms' [93] criticism of "realistic!' pos-

sible worlds views.

Making no pretense to realism are the speculative excursions by

many contemporary science fiction authors into the realm of possible

worlds. On the one hand are the many '"hat if' themes which concentrate

on alternate histories of the actual world. Of more interest are those

speculations which postulate the simultaneous existence of a variety

of "parallel" worlds, usually with some means of enabling access from

one to another. In this connection the Lord Kalvan of Otherwhen

stories of H. Bean Piper [771 are typical, being adventure stories with

little conceptual meat.

A more highly developed parallel worlds theory is found in Worlds

of the Imperium, by Keith Iaumer [43], which is suggestive of Lewis'

employment of comparative overall similarity as a way of ordering pos-

sible worlds for the purpose of determining the truth value of counter-

factuals. As we shall discover in Section 2.3, it is not enough to

analyze counterfactuals in terms of a single accessing relation on a

set of possible worlds. In addition to the concept of possible worlds

themselves, we must also have a concept of "distance" of them from our

actual world, which ones are in our immediate neighborhood, and which

father away? Lewis suggests that the concept needed here is the quite

ordinary one of comparative overall similarity [51, p. 1]. That is, we

can compare possible worlds, much as we compare other things, in respect

of their overall similarity to a given, possibly the actual, world. If

we imagine an instant of time in our world as a point, then in Laumer's








story our world is a line of such points, and it has neighbors, other

world-lines lying parallel to ours, defining a plane of points, with

two dimensions of similarity. Either in the normal temporal direc-

tions or at right angles to them, worlds farther from ours (actually

from an instant of ours) will be less similar to this instant than

world-instants closer by. If we travel along a world-line, the in-

stants gradually become less similar to our starting point; likewise

if we travel along a right angle line to our world-line, a gradual

sequence of alterations will obtain. Laumer describes several such

imaginary journeys, and others combining both directions. Assuning

that we could come up with a uniform metric for this plane of world-

instants, then it seems obvious that there are both degrees of sim-

ilarity to a given world-instant, and numbers of worlds that are

equally similar to our present world-instant, though differing from

it in different respects.

Of course, Lewis does not suggest that we can actually put a

metric on the space of possible worlds. Even in the science fiction

tale above it is hard to see how that could be done. But he does

put a certain organization on that space, a topology of sorts, if

not a metric.

In Saberhagen's Mask of the Sun [85], while parallel worlds are

not accessible in terms of physical transference of the protagonist

to them, the wearing of a certain mask enables the wearer to view

future possibles. The author develops this idea in a context of

branching time: the wearer is not seeing the future but one of many

possible futures branching out from the mask's temporal point of view.

The explanation of this capacity suggested in the novel is that the








mask computes the possibilities based upon a comprehensive access to

facts about the present. This is reminiscent of Stalnaker's claim

that "one can sometimes have evidence about non-actual situations"

[96, p. 166]. Such evidence is acquired from the actual situation

in non-mysterious ways [96, pp. 178-179].

Both the actual and the speculative history of possible worlds

have something to offer us. They give us a variety of analogies

with which to test our grasp of the concept of possible worlds.


2.2 Possible Worlds: Modality and the Strict Conditional

The naive concept of a possible world seems natural and obvious:

we all understand what is meant by saying "things could be otherwise."

If the actual world is the way things are, then a possible world is

another way things could have been. We can think of possible worlds

as variants on the actual world. A critic might suggest that intro-

ducing possible worlds, when we have enough difficulty determining

what the actual world is, is to compound our problems to no purpose.

Our problems, however, are already compounded: the difficulty in

determining what the actual world is lies in the fact that the extent

of our knowledge and belief (or true belief) seriously underdeter-

mines it.

The critic would express this underdetermination by saying we

do not or cannot know everything about the actual world, while I would
express it by saying we do not or cannot know which world is actual,

except within certain limits. While the critic can only say that our

knowledge underdetermines the world that is, I can make sense of a

positive assertion as to what it does determine: the set of worlds

that, for all we know, any one of which might be the actual world. If









we must operate within a context of indeterminacy, then plurality among

what is indeterminate allows room for greater future determinacy. As

the positive benefits of this way of looking at things accumulate the

critic will, of course, adopt a point of view closer to mine. 1_

Possible worlds make their appearance in recent efforts to pro-

vide a semantics for nodal logic. As a mathematical tool of formal

logic there is no serious question as to its utility. However, as a

device for the analysis of concepts there is neither a shortage of

users nor of critics. The only defense that I can think of for

utilizing the concept of possible worlds for analyzing other concepts

is that it, unlike some of its alternatives, provides an explanation

for how and why the concepts analyzed work the way they do (though for

a detailed defense see [64]). But this defense mist wait until Section

2.7. It will be mre appropriate anyway once we have a purported anal-

ysis in terms of possible worlds as a concrete example.

We may take mdal logic to be the logic of possibility and neces-

sity. Various systems of modal logic may have application, or be de-

signed to have application, outside the bounds of these notions, such

as tense logic, deontic logic, epistemic logic, etc. Pbwever, we seem

to have sufficient opportunity for variation within the bounds of pos-

sibility alone: there is the logically possible, the physically pos-

sible, the technologically possible, and the actually possible, to

name but a few. There are these kinds of necessity as well, in ad-

dition to necessity in terms of need, or in terms of keeping certain

things fixed. (What will kill the aphids without doing in the roses?)

We should begin with our widest sense of possibility and our

narrowest sense of necessity: that which is in some way possible and








that which is necessary no matter what. Focusing on the latter

characterization, the following definition seems appropriate:

D2.2.1: Lp is true iff p is true in every possible world.

Correspondingly, for possibility we have:

D2.2.2: Ip is true iff p is true in some possible world.

These definitions have the virtue of making what is not possibly false

equivalent to what is necessarily true.

To handle our notions of physically possible, technologically pos-

sible, etc., we could simply substitute these terms for "possible" in

the above definitions. There are several drawbacks to this, chief

anong which is that our various notions of possibility appear irreduc-

ible in our definitions, while in fact the various notions of possi-

bility may be systematically interrelated. Surely the physically pos-

sible worlds are a subset of the logically possible, and the techno-

logically possible a subset of the physically possible. For some kinds

of possibility not all the possible worlds in the broadest sense need

to be taken into consideration.

Also, there are circumstances under which the operative concept

of possibility does not determine a static set of possibilities. Con-

sider for the mmnent possible worlds as the possible futures of this

present world. The actual world is the present instant. Relative to

it certain futures are possible. However, from the point of view of

one of those possible futures, its possible futures may not contain

some of the futures possible with respect to the present. In getting

from here to there some possibilities may be forever lost. This view

introduces two new considerations: possibility can be possibility
relative to a given world; what is possible relative to one world my








not be possible relative to another. Further, as we saw above, not every

possible world need be relevant to what is possible in some restricted

sense.

The technique for handling these considerations in possible world

semantics, and what gives it its considerable flexibility for permit-

ting the representation of a variety of conceptions of possibility, is

the notion of a frame. A frame consists not only of a set of possible

worlds, but also of a relation among these possible worlds that repre-

sents, for each world, the worlds to which it has access, or which are

it alternatives, in term of possibility. We can alter our definitions

D2.2.1 and D2.2.2 so as to incorporate this structure:

D2.2.3: Let our frame be F = < U,R > where U is the set of possible

worlds and R is the accessibility relation on U.

We adopt "@" as an abbreviation for "is true at." The truth of modal

prepositions may then be defined by:

D2.2.4: Lp @ u iff for allw in U, if uRw, thenp @w.

D2.2.5: Mp @ uiff for somew in U, uRwandp @w.

Now by specifying different conditions on relation R we mdel different

concepts of possibility. If, for example, we specify that for all u in

U, u1W for all w in U, then D2.2.4 and D2.2.5 reduce to D2.2.1 and D2.2.2.

The same would be accomplished by specifying that R be reflexive, transi-

tive, and symmetric with every world accessing at least one other. The

possible futures situation could be modeled by an R which is just reflex-

ive and transitive. These and other alternatives form the basis of a

great variety of systems of modal logic. Details need not concern us

now. If we assume that the accessing relationis universal, then every

world has access to every other.








A pictorial representation much used by Lewis for the accessibility

relation is that of a circle, where the center represents the given world

u, and all the points bounded by the circle represent the worlds acces-

sible from u. The circle and its interior is the "sphere of acces-

sibility" around u.


Figure 2.2.1

This suggests that we my define our frame in a different, but equiva-

lent, fashion:

D2.2.6: F = < U,S > where U is again the set of possible worlds, S is

a function from the set of possible worlds U to the power set of U,

P(U), the set of all subsets of U.

To each world u in U, S assigns a subset of U, designated Su, which will

be called the sphere of accessibility around u.

Our definitions of truth may be altered accordingly:

D2.2.7: Lp @ u iff for allw e U, ifw e Su, thenp @ w.

D2.2.8: Mp @ u iff for somew e U, w e Suandp @ w.

The requirement that R be universal now translates into the requirement

that Su = U for all u in U. Given the usual interpretation of the

quantifiers, we may shorten the above to:

D2.2.9: Lp @ u iff for allw e Su, p @ w.

D2.2.10: Mp @ u iff for some w S u, p @ w.








What, you may ask, has this to do with conditionals? Recall that

a principal objection to the material conditional as an analysis of con-

ditionals in English is that it simply is not plausible that a condi-

tional be true just because its antecedent is false or its consequent

true. Something more is called for, and one of the first things to try

is to formalize the notion that sonm connection obtains between the an-

tecedent and consequent. Now we have seen that this is not enough, but

it is a place to start.

In an effort to provide an alternative to the material condi-

tional for the analysis of "if. . then; . a number of modalized

conditionals have been developed, either as primitives in a logical sys-

tem, or defined in terms of the modalities of possibility and necessity.

The insight which these systems are formalizing is that the antecedent

of a conditional somehow necessitates the consequent: if the antecedent

is true, then the consequent is true, of Necessity. Though the debate

over the modal nature of the conditional dates back to antiquity, the

genesis of both modern modal logic and modalized conditionals can be

traced to C. I. Lewis, as we indicated in Section 2.1. C. I. lewis

developed several systems of modal logic incorporating what he called
"strict implication." (For a survey of Lewis-type modal systems one

should read J. Jay Zeman's Modal logic [104].f2 These systems are of

interest for our purposes only in their failure to provide an analysis

for counterfactuals, for while the material conditional is too weak to

serve as a counterfactual conditional, the strict conditional is too

strong, as we shall see.

The idea behind the strict conditional, expressed in terms of

possible worlds, is that, while neither the antecedent nor consequent








need be necessary in themselves, relative to those worlds where the

antecedent is true, the consequent is necessary, i.e., true in all of

them. There are variations on this, of course, but many of them are

amenable to rephrasing the claim in terms of accessibility. Returning

to our future possibles example, a certain conditional presently false

(since in those futures where the antecedent is true, it is not in all

the case that the consequent is true), may be true from the point of

view of one of those futures (since by then certain possibilities may

no longer be accessible, perhaps including those in which the antece-

dent was true and the consequent false). What is not inevitable today

may become so by tomorrow, as we often find out to our regret. Hence

the definition, or truth conditions, for the strict conditional should

permit at least the flexibility of D2.2.4.

With this in mind, the following suggests itself:

D2.2.11: cpq @ u iff for allw F Su, if p @ w, then q @ w.

Given the usual interpretation of Cpq and in view of D2.2.9 the above

reduces to:

D2.2.12: Cpq @ u iff LCpq @ u.

Henceforth we will use LCpq to denote the strict conditional, unless we

have reason to materially alter our definitions.

We indicated earlier that we could model various senses of pos-

sibility and necessity by placing various conditions on the accessi-

bility relation, or in our present parlance, on the sphere of acces-

sibility. The sphere of accessibility for a given world could range

from the empty set to the entire set of possible worlds or anywhere

in between. Suppose that for a certain world u we have a choice of two

different spheres: 1 and S2, each determining a sonxvhat different
U U









sense of what is possible relative to u. Corresponding to these we

have two necessity operators, L' and L2. Now if S1 and S2 are disjoint
U U
or properly intersect, then LI and L2 are not in any obvious way compa-
rable; if, however, S2 is a subset of S1 then Lip will inply L2p for
U U
any proposition p. If the containment is proper, it will not generally
2
be the case that L p implies L1p. Hence our two necessity operators

will be ordered. In this context Lewis [51, p. 12] describes one

operator (L ) as stricter than the other, and hence a conditional de-

fined in terms of one as a stricter conditional than the other.

The difficulty of taking the counterfactual to be a strict con-

ditional lies in the variations on strictness of the conditional. For

for any fixed degree of strictness of the conditional, it is always pos-

sible to strengthen the antecedent:

E2.2.1:
L.Cpq


LCKprq

is valid for any operator of fixed strictness, L, as the following argu-

ment shows.

LCpq is true at u iff at everyw e Su Cpq is true. But if Cpq is

true at any wrld w, then CKprq is true, since strengthening the antece-

dent is valid for the material conditional. Hence the truth of IJpq

leads inexorably to the truth of LCKprq, with no particular conditions

of the function s, since with CKpqr true at every wrld in Su, LCKprq

will be true at u.

However, for any given counterfactual (or at least those with

contingent antecedents) it is possible to "undermine" the antecedent








by conjoining another proposition to it. For example, the following

inference is certainly invalid:

E2.2.2:

If this match were struck, then it would light.

If this match were soaked in water and struck,
then it would light.

Hence, in general, the inference from Wpq to WKprq is invalid. Conse-

quently, as Lewis concludes, Wpq cannot be a conditional of a fixed

degree of strictness. Variability must be built into the truth con-

ditions for Wpq.

There is another alternative, and that would be to take the actual

antecedent as elliptical for a more fully expressed antecedent which was

so constructed as to neutralize the problem of undermining. On this

view, "If the match were struck, then it would light" is elliptical for

"If the match were struck and not wet and well-made and in sufficient

oxygen and . then it would light." There are two arguments against

this view which I shall not elaborate: first, it is implausible that

anyone would mean the latter conditional when uttering the former [80,

p. 91; second, this really raises the issue of cotenability over again

in a slightly altered context.

Lewis raises still another argument against this view which is

decisive as far as I am concerned. If the antecedent of a counterfac-

tual is elliptical for something much more complex, then it strongly de-

pends upon the exact context of utterance for its interpretation.

Which means the counterfactual is pragmatically abiguous to a high

degree. On Lewis' view









It consigns to the wastebasket of contextually resolved
vagueness something much more amenable to systematic
analysis than most of the rest of the mess in that waste-
basket. [51, p. 13]

Hence in the subsequent section we will present Lewis' analysis of the

counterfactual conditional as a "variably strict" conditional with the

expectation that ambiguity will be kept within more acceptable bounds.


2.3 Lewis' Analysis of the Counterfactual Conditional

That the counterfactual conditional is a variably strict condi-

tional is generally admitted by advocates of both the metalinguistic

and possible worlds approaches. The difference in treatment involves

in part fixing the boundary between semantic and pragmatic ambiguity

to which effect we quoted Stalnaker earlier. The essential requirement

is to have a system for resolving as much of the apparent ambiguity in

conditionals as possible. Once ambiguity is seen as systematic, it is

no longer a barrier to analysis. (See Lewis [49].)

To motivate his construction of the truth conditions for counter-

factuals, Lewis [51, p. 1] invites us to consider what has become my

favorite example: !If kangaroos had no tails, they would topple over."

Previously we said that in evaluating this conditional we construct for

ourselves a situation, altering what we must in what we take to be

actually true, in which kangaroos have no tails. If it is the case in

such a situation that kangaroos topple over, then the conditional is

true. However, as we saw, a critical element of this procedure is the
"altering what we must," for it is surely possible to imagine situations

in which kangaroos have no tails, but evolved that way, so also evolved

a structure which otherwise permits them to be balanced. Or, a situation

in which kangaroos have no tails, but have learned to use crutches, so








do not topple over. The problem of excluding these situations on the

ground that they would not be the case is precisely the problem of co-

tenability noted by Goodman.

Instead of speaking of "situation" we can speak of imagining a

world where kangaroos have no tails. This world is not the actual

world, but rather a possible world, differing from ours just eough so

that kangaroos have no tails. But the cotenability problem arises

anew in that we can consider possible worlds where kangaroos have

evolved tailless or use crutches. In practice these considerations do

not deter us from evaluating the conditional as true. What Lewis pro-

vides is an analysis which explains why that is the case.

We are concerned only with worlds very much like ours: that are

similar to a certain degree to the actual world. The more imaginative

worlds above are less similar to the actual world than are worlds where

less has changed. Lewis' suggestion is that we can compare worlds in

terms of overall similarity to a given world [51, p. 14] in much the

same way that we can copare facial expressibns, or cities, or cultures.

Now while it seems obvious that we have a good grasp on various more

restricted notions of similarity-in-certain-respects it may not be clear

that we have a sufficient grasp of comparative overall similarity to

make it a useful notion for an analysis. Our first thought on it being

suggested that we can compare cities overall, is to break this comparison

down into similarity in various respects. I shall return to the concept

of comparative overall similarity later, so for now we shall assume that

it is relatively unproblematic so as to get on with the analysis.

We can think of our accessing relation on possible worlds as
modifiedd" by similarity considerations. Thus for the kangaroo example









we are concerned with worlds similar to ours to a certain fixed, though

somewhat vague, degree. There are a number of equivalent ways in which

Lewis' formal semantics can be set up so as to carry information about

accessibility and similarity. I choose that which is apparently com-

patible with the neighborhoods semantics to be discussed later.

For a strict conditional we need one sphere of accessibility for

each world given by the function S: U P(U), where % denotes the

subset of U which is the sphere about u. For a variably strict con-

ditional we will in general require more than one sphere about each

world u, or as we might say, u will have many neighborhoods. (Though

we shall see these are not quite the neighborhoods of neighborhood

semantics.)

D2.3.2: Let U represent the set of possible worlds, and let $: U

P(P(U)) be a function from U to the power set of the power set of U.

That is, $ assigns to each u in U, not a single subset of U (a single

sphere about u) but a set of subsets of U (a series of spheres about u).

We shall designate the image of u under $ u where each Su in $ is a

single sphere about u.

lewis places four conditions on $ in order that it plausibly carry in-

formation about similarity [51, p. 14]. To these we add a fifth which

is optional for Lewis, and determines, in part, the kind of modal logic

that is validated by this framework. The conditions on $ are:

C2.3.1: {u} is an element of $

C2.3.2: For all A,B in $u either A is a subset of B or B is a subset
of A.

C2.3.3: If X is a subset of $ then the union of X is an element of
$u









C2.3.4: If the nonempty set Y is a subset of $u then its intersection
is an elenmnt of $u"

C2.3.5: For each u,v in U the union of $u equals the union of $v.

Following Lewis' terminology, we shall call these conditions

respectively (strong) centering, nesting, closure under unions, closure

under (nonempty) intersections, and uniformity [51, pp. 14, 117]. Lewis

calls $ a system of spheres. We shall depart from Lewis slightly by

calling $ a sphere function, and by calling $ the system of spheres

about u. A picture suggestive of a system of spheres about u which we

shall have occasion to use repeatedly is that of Figure 2.3.1.





UU






Figure 2.3.1

Each circle represents the boundary of one of the spheres of acces-

sibility about u. In what follows for brevity we will use the following

symbols:

= is an element of"
C = "is a subset of'

U = "the union of"

A = df "the intersection of'

0 = df"t e ipty set"
In this notation the conditions listed above may be more briefly stated

as








C2.3.1: {u}s$u

C2.3.2: If A,B s $u then Ac B or B cA.

C2.3.3: If X c $ ,then UX $"

C2.3.4: If 0 # Yc $ then AY $u"

C2.3.5: For all u,v U, U$u= U$v.

In Lewis' view, these conditions, or rather the first four, are

necessary for the system of spheres to be plausibly considered to convey

information about comparative similarity. In what follows we continue

to adhere to Lewis' presentation, except where noted. (See [51, pp. 14-

16].)

It is reasonable that the actual world, or any given world, is

more similar to itself than any other possible world, hence the centering

requirement. The singleton set {ul is one of the spheres about u, since

a sphere represents a set of worlds similar to u to at least a certain

degree. It should be kept in mind that a sphere does not represent a

set of worlds equally similar to u, but rather a set of worlds more

similar to u than any worlds not in the set. And u is more similar to

u than any other world, so belongs in a set by itself.

Suppose there were a pair A,B E $ which did not satisfy the

nesting condition. Then there are worlds v and w such that v e A and

v j B and w J A and w E B, as suggested by Figure 2.3.2.


Figure 2.3.2








Since each set in $u is a set of worlds more similar to u than any worlds

outside the set, it follows that v is more similar to u than w (from

v e A and w { A) but also that w is more similar to u than v (from

w s B and v b). Hence nesting is required if $ is to carry informa-

tion about comparative overall similarity. Of course, if we are con-

cerned only with simiarity-in-certain-respects, then "similar" has dif-

ferent (to be specified) senses in the apparently inconsistent state-

ments above, so in that case they would be compatible. However, in

Lewis' analysis worlds are compared in terms of overall similarity to

the given world u for each system of spheres, so nesting is required.

The justification of closure under unions and intersections is

based on the following consideration: suppose there is a set of worlds

such that any world inside it is more similar to the given world u than

any world outside it. Then this set should be a sphere about u in vir-

tue of being similar to u to at least a certain degree. But UK where

X c $u is just such a set, since any world w E UK is an element of some
U u
Su c X, hence is more similar to u than any world v Su. Since any
v 4 UX is not an element of any Su in X, it follows that any w e UX is

more similar to u than any world v j UX. Dual considerations apply in

the case of intersections.

Closure under unions and intersections has other implications also.

First, it implies that there is both a largest and smallest sphere in

$u. The smallest sphere is A$u and the largest sphere is U$u, since

$u is a subset of itself, so falls under the hypotheses of conditions

3 and 4. Since closure under unions is not restricted to nonenpty sets

X, and the union of the empty set is empty, it follows that 0 e $u'
hence that A$u = 0. If we were to restrict condition 3 to nonenpty









sets, then {ul would be the smallest sphere about u. As we shall see

below, the largest sphere in $u may be identified with U in virtue of

condition 5, uniformity.

Lewis carefully points out a consequence that might be overlooked

[51, p. 15]. While closure under unions and intersections guarantees

an upper bound and a lower bound on each subset of $u, it is not neces-

sarily the case that these bounds must be in the subset of $ under con-

sideration. That is, for X c $ (assume nonenpty) while UK E $ and

AX e $u ,it does not follow that UX e X or that AX e x. This is pre-

cisely analogous to the set of rational numbers less than 1 and mre

than 0; there is neither a greatest nor a least element of that set of

rational numbers, but the set is bounded above and below. This is of

importance in connection with the limit assumption which we shall dis-

cuss in Section 2.6.

Conditions 1 and 5 together imply that the largest sphere in $u

is U. Consider any pair of wrlds u,v E U. By centering {u} $u and

{v} e $v' hence u e U$u and v E U$ v. But by uniformity, U$ = U$v, hence

u s U$v and v e U$ u. But u and v were arbitrarily chosen elements of U,

so for all u s U, for allv e U, v e U$ u. Hence for all u e U, U$u= U.

Thus $ is universal, in the sense that every world has access to every

other at the level of the largest spheres about each. Lewis does not im-

pose uniformity in general on the sphere function and allows for the pos-

sibility that U$u my not exhaust U for some or all u c U. We do so in

order to provide for a simpler characterization of the modal logic this

semantics validates. (As we shall see, it is S5.)









There are many sphere functions which would satisfy these con-

ditions. Any particular function will be determined by nonformal

considerations. We will consider some of these when we look at simi-

larity again in CHAPTER THREE.

Given the sphere function and the resulting system of spheres

for each world, we can now state the truth conditions for the counter-

factual conditional Wpq. First we will follow Lewis in adopting the

convention that a world at which proposition p is true will be called

a p-world, and the convention that any sphere containing a p-world

will be called a p-permitting sphere. We may then state the truth con-

ditions for the counterfactual conditional [51, p. 16]:

D2.3.2: Wpq @ u iff either

(1) there is no p-permitting sphere in $u, or

(2) for some p-permitting sphere in $u Cpq is true at
every world in that sphere.

Thus there are two ways for a counterfactual to be true: it may be

that the antecedent is not true in any world in U$u, in which case,

with Lewis, we call the antecedent not entertainable. For example

this would be the case with the counterfactual "If the circle were

squareable, mathematicians would be confused." There are some problems

with assigning a uniform truth value to all such counterfactuals but

we will not consider that issue.

On the other hand, if there is an antecedent-permitting sphere

such that every antecedent-world in that sphere is also a consequent

world, then the counterfactual is true. A situation under which the

conditional Wpq is true in virtue of D2.3.2(2) is diagramed in Figure

2.3.3.


























Figure 2.3.3

In the fourth nonempty sphere outwards there are p-worlds, so this is

a p-permitting sphere; at each of the p-wrlds in that sphere q is also

true, so Cpq is true at every world in that sphere, hence Wpq is true

at u.

What, you may ask, about the fifth sphere, where there are p-

worlds which are not q-worlds? Since these are all worlds less similar

to u than the worlds in the fourth sphere, our intuitions should not be

offended. Recalling the kangaroo example, our concern is with the

status of tailless kangaroos in the wrlds sufficiently similar to ours.

Perhaps in those worlds in the fifth sphere kangaroos haw learned to use

crutches.

These truth conditions are intended to apply to our '%ould"-

conditional of Section 1.1. This includes the "even if'- and
"necessitation'-conditionals. We will show that at least some reasonable

adequacy conditions are met when we review counterfactual inferences

and fallacies in Section 2.5. However, these truth conditions are not








intended to apply to the 'ight"-counterfactual, also discussed in

Section 1.1. For this conditional we may use the same basic semantics,

but will require a different set of truth conditions.


(a) (b)


(c) (d)


Figure 2.3.4








Figure 2.3.4 represents various distributions of the truth values

of propositions p and q over the system of spheres $u. We shall refer

to them in what immediate follows.

Figure 2.3.3 and figures 2.3.4(a) and 2.3.4(d) all represent cases

where Wpq is true. As we shall see in Section 4, 2.3.4(a) represents a

case where Lpq is true as well. In 2.3.4(d) q is true at u and p being

true will not change this, that is "even if p were true, q would be."

This is a case where an "even-if'-conditional is appropriate, but no

further definition of truth for the conditional in question, the '"ould"-

conditional, is required. The condition that every p-world be a q-world

in some p-permitting sphere suffices.

In both 2.3.4(b) and 2.3.4(c), the conditional Wpq is not true.

The situations differ in that in 2.3.4 (b) there is some p-permitting

sphere where some of the p-worlds are q-worlds, but not all, while in

2.3.4(c) there is a p-permitting sphere where none of the p-worlds are

q-worlds. In the latter case, 2.3.4(c), WpNq is true, but in the former

case, 2.3.4(b), neither Wpq nor WpNq is true. For entertainable antece-

dents, Wpq and WpNq behave as contraries: they may not both be true,

but they may both be false. In traditional quantificational logic, to

each contrary corresponds a subcontrary. Similarly, in Lewis' analysis,

to each of the contraries Wpq and WpNq corresponds a subcontrary Vpq and

VpNq. These are the aforementioned "might"-conditionals. In those cases

where p is entertainable, but neither Wpq nor WpNq are true, then both

Vpq and VpNq are true. Recall the pair of conditionals concerning Bizet

and Verdi as an example.









The definition of truth for the "might"-conditional is then given

as follows [51, p. 211]:

D2.3.3: Vpq @ u iff both

(1) there is some p-permitting sphere in $u, and

(2) every p-permitting sphere contains at least one Kpq-world.

Note that the traditional debate over whether or not univerally quanti-

fied propositions presume existence arises anew in the case of the counter-

factual conditional where it reappears as a debate over whether the con-

ditional presupposes that its antecedent is entertainable. As we have

defined the '%%uld"-conditional, it is "vacuously" true when the antece-

dent is not entertainable, i.e., when there is no p-permitting sphere.

In such a case the subalternate '!might"-conditional will be false, the

contrary '\%ould'-conditional true, and its subalternate "uight"-conditional

false. Hence these definitions do not support the conditional analog of

the traditional square of opposition:


Wpq WpNq






Vpq VpNq

Figure 2.3.5

Given the definitions D2.3.2 and D2.3.3, the only relation that

does obtain is the contradictory relation along the diagonals, i.e.,

both of the following are validated:

EWpqNVpNq

EWpNqNVpq









In view of these equivalence the 'ould"- and "might"-conditionals are

interdefinable on Lewis' analysis.

Returning to the Bizet and Verdi examples, it seems reasonable to

deny "If Bizet and Verdi had been compatriots, then they would have been

Italian" on the grounds that among those worlds most similar to the actual

world will be found some where they are both Italian, but also some where

they are not both Italian, though compatriots. In fact, I would agree

with Lewis in judging the following to be true:

E2.3.1: If Bizet and Verdi had been compatriots, then they would either

both have been Italian or both have been French.

E2.3.2: If Bizet and Verdi had been compatriots, then they both might

have been Italian.

E2.3.3: If Bizet and Verdi had been compatriots, then they both might

have been French.

In effect, we are adopting a similarity ordering of the possible worlds

where the closest worlds where Bizet and Verdi are compatriots contain

exclusively worlds where they are both French and other worlds where

they are both Italian, while the worlds where are both Chinese, say,

are more distant.

In view of this definition of the '"ight"-conditional, Lewis'

semantics necessarily fails to validate the principle of conditional

excluded middle (CEM). In classical propositional logic ACpqCpNq is

a theorem. The corresponding ALCpqLCpNq is not generally a theorem of

modal logic, i.e., CRA fails for the strict conditional. The counter-

factual conditional shares this property with the strict conditional.

(The principle of CEM is validated by Stalnaker's semantics, for reasons

which we shall discuss in CHAPTER THREE.)








If for some reason it is desirable to preserve all of the relations

of the traditional square of opposition, then we must exclude vacuous

truth for the conditional Wpq. That is, if the antecedent of the con-

ditional is impossible, we require it to be false. For this purpose the

following two definitions suffice:

D2.3.4: Wpq @ u iff there is some p-permitting sphere in $ and Cpq

@ every world in that sphere.

D2.3.5: Vpq @ u iff every p-permitting sphere in $ contains at least

one Kpq-world.

These have the virtue of still preserving the interdefinability of the

'"ould'- and "might"-conditionals.

Lewis' definition (either one) of the "might"-conditional is of

considerable importance, offering as it does a way of resolving problems

which had hertofore been labeled irredeemable. Rescher gives the fol-

lowing examples of "purely hypothetical counterfactuals," i.e., those

not thoroughly grounded in laws [83, p. 162]:

E2.3.4: If Bizet and Verdi were compatriots, Bizet would be an Italian.

E2.3.5: If Bizet and Verdi were compatriots, Verdi would be a Frenchman.

E2.3.6: If Georgia included New York City, this city would lie south of

the Mason-Dixon line.

E2.3.7: If Georgia included New York City, this state would extend

north of the Mason-Dixon line.

His view is that "these opposed results cannot be avoided" because

"The contextual ambiguity of the antecedent gives us no way of choosing

among the various mutually rebutting counterfactuals" [83, p. 162].

It is clear that these are rebutting only if one holds that con-

ditional excluded middle is valid for the counterfactual conditional.








However, Lewis' analysis allows us to avoid having to toss these into

the "irredeemably ambiguous" bin. Both E2.3.4 and E2.3.5 are false,

hence not rebutting, while the corresponding "might"-conditionals, as

already noted, are true. Similar considerations may be applicable to

E2.3.6 and E2.3.7, however; the so-called ambiguity may be even more

simply resolved. E2.3.6 is false and E2.3.7 is true because satisfying

the antecedent does not require we go so far as to change the physical

location of political subdivisions. If we had such a possibility in

mind then we would utter instead:

E2.3.8: If New York City were in Georgia, then this city would lie

south of the Mason-Dixon line.

As Goodman pointed out, the direction of the expressions in the ante-

cedents of E2.3.6 and E2.3.7 assume importance which allows us to view

them as unambiguous [27, p. 151. It is assigning too much to contextual

ambiguity not to take note of this fact.

With respect to E2.3.4 and E2.3.5 however, it seems the only

alternative to consigning these to the fog of ambiguity is to give

serious consideration to "night"-conditionals. That Lewis' analysis

provides for this is a point in its favor.


2.4 Modal logic and the System of Spheres

In the preceding sections we indicated that lewis' analysis of

the counterfactual conditional is an extension or application of possible

worlds semantics developed for modal logic. In the version of Lewis'

analysis that we have given, the logical modalities, including the strict

conditional, can be expressed.








With more than one sphere of accessibility assigned to each wrld

u in U, the concepts of possibility and necessity in their widest sense

need to be correlated with the "largest" sphere. The conditions ve have

placed on the neighborhood function $ require that the largest sphere be

U$u for each u in U, and furthermore that U$u = U for all u in U. What

Lewis calls the "outer modalities" [51, p. 22] are defined as follows:

D2.4.1: Lp @ u iff every world in U$u is a p-wrld.

D2.4.2: Np @ u iff some world in U$ is a p-world.

In view of the fact that U$ = U for all u in U, these outer modalities

correspond to the logical modalities of S5.

Hence the requirement that p be entertainable can be expressed as

the requirement that Mp be true. Given the definitions D2.3.2 and D2.3.3

of Wpq and Vpq, it then follows that Wpq does not while Vpq does entail

that Mp is true.

We may also, given the above definition of necessity, define a

strict conditional which will be the strict conditional of S5:

D2.4.3: LCpq @ u iff every world in U$u is a Cpq-wrld.

Referring back to Figure 2.3.4(a), that diagrams a situation in which

LCpq is true. It is then readily seen that LCpq entails Wpq on Lewis'

analysis, since if Cpq is true at all worlds in U$u, it nst then be true

at every world in some p-permitting sphere. The converse, however, does

not hold, as Figures 2.3.3 and 2.3.4(d) illustrate: Wpq may be true,

though there are NCpq-wrlds (that is, KpNq-worlds).

One may easily confirm from the definition of Wpq that it entails

Cpq, hence we have a hierarchy of conditionals: Lcpq entials Wpq entails

Cpq. However, in no case does the converse entailment hold.








If from the conditions on $ we were to drop C2.3.5, the uniformity

condition, then it would not generally be the case that U$u = U for all

u in U. In such a case the outer modalities as above defined would not

correspond to the mdalities of S5, but rather to the system known

generally as T. (The only condition on the accessibility relation being

the reflexivity condition implied by centering.) Adopting a condition

such as

C2.4.1: For all u,v,wE U, if u e U$v andv c U$w, then U U$w

would impose transitivity on the accessing relation, and hence would

validate the modal logic S4, which is properly contained in S5. For our

purposes it is convenient to adopt a formulation which makes the outer

modalities the best known modalities, and hence uniformity is the standard.

We have indicated that LCpq is in a sense the "outer limit" of Wpq.

It is of interest to consider what the corresponding outer limit of Vpq

might be. One plausible suggestion is that it could be MKpq, however

inspection of the following figure reveals that that is easily counter-

exampled:


Figure 2.4.1








In the situation diagrammed in Figure 2.4.1, MKpq is true because there

is a Kpq-wrld in U$U; however, Vpq is false because though there is a

p-permitting sphere, it is not the case that every p-permitting sphere

contains a Kpq-world; the second non-trivial sphere out does not. (0, {u}

are trivial spheres.)

Rather than introduce the necessity operator in terms of truth

conditions under a sphere function, $, we could define it in terms of

the counterfactual conditional. From the assumption that every world

in U$u is a p-wrld, it follows that no world is a Np-world, hence p is

not entertainable. Therefore, WNpp is true, and conversely. Hence we

may state:

D2.4.4: Lp @ u iffWNpp @ u.

Of course, D2.4.4 is suitable only if we use the first definition of the

'"ould" -counterfactual. Keeping the equivalence between NINp and NP, we

may define the latter in terms of the "might"-conditional:

D2.4.5: Mp @ u iffVpp @ u.

We introduced Lewis' analysis by considering modal logic in Section

2.2, and we will return to modal logic again when we consider neighborhood

semantics in CHAPTER FOUR. In the next section, where we discuss counter-

factual inferences and fallacies, we will have occasion to compare infer-

ences valid in classical propositional and modal logics with those in

counterfactual logic. Rather than set forth those proper to modal logic
here, we shall mention them as we come to them in the next section.


2.5 Counterfactual Inferences and Fallacies

One test of adequacy for an analysis of the counterfactual condi-
tional is to see if it validates inference patterns recognized as valid








and invalidates inference patterns recognized as invalid, that is,

generally preserves our pre-analytic intuitions as to the "logic" of

the concept involved. lewis' analysis withstands this test admirably

as we shall show in this section.

First I wish to define a notion of semantic entailment for the

analysis so far presented. To distinguish it from our syntactic forma-

lations we shall use infix notation: the symbol "1=" is intended to

denote semantic entailment which is defined as follows:

D2.5.1: p[=qifflIpl[~IqJJ

where JrJJ denotes the set of worlds where r is true. (This notion will

have to be relativized to a model when we shift to formal semantics in

CHAPTER FOUR.)

Consider again Figure 2.3.4(a). With the help of this figure and

definition D2.4.3 it is clear that pI=q iff LCpq is true. In what follows,

where LCpq is used as a premise, substitution of pl=q will not alter our

conclusions with regard to the inference pattern. Difficulty arises only

if we define semantic entailment for sets of propositions:

D2.5.2: GJ=q iff {w: if p c G, then p @ w} c liqlJ.

We cannot take LCGq as equivalent to GJ=q as the strict conditional holds

only between propositions. Nor will the conjunction of all propositions

in G work, since G could be infinite, and it is not our intention to repre-

sent infinite conjunctions in our object language. (Strictly speaking, we

have not really indicated what our object language is, except informally.

This will be done in CHAPTER FOUR. I believe it would be distracting at
this point. In Section 2.6 we will have to make use of D2.5.2 and so it

is stated at this time.)








We may divide our concerns into those inference patterns that ought

to be valid for the counterfactual conditional and those which ought not

to be valid. Among the first will be those patterns we wuld expect any

conditional to adhere to. Many authors (including Zeman [104], Hardegree

[32]) consider two requirements absolutely minimal for a conditional

function:

C2.5.1: If pl= q then Wpq is true.

C2.5.2: If Wpq and p are true, then q is true.

The first is a simplified semantic version of the deduction theorem, and

the second is modus ponens or detachment. Both are satisfied by the

material conditional as well as the strict conditional (as we have defined

it). But they also hold for the counterfactual conditional. From pl= q

we have observed that LCpq follows. But from LCpq it follows that Wpq,

as we noted in Section 2.4. Hence C2.5.1 holds for the counterfactual.

Since Cpq follows from Wpq, also noted in Section 2.4, and modus ponens

holds for the material conditional, C2.5.2 holds as well for the counter-

factual.

lewis draws particular attention to three inference patterns valid

for both the material and the strict conditional which are not valid for

the counterfactual conditionals on his semantics [51, pp. 31-35]; and

demonstrably ought not to be valid for conditionals in general. These

are strengthening the antecedent, hypothetical syllogism (transitivity

of implication), and contraposition. The first two are immediate corol-

laries to a stronger principle identified by J. Jay Zeman, that of semi-

substiutivity of implication (respectively, strict implication) which is
valid for the material (respectively, strict) conditional [104, pp. 11,









162]; the strongest version of SSI(SSS) holds only for the material

conditional and for strict conditionals at least as strong as that

of S30.

We have already considered the inference of strengthening the

antecedent; the counterexamnple and corresponding inference pattern

are repeated below:

E2.5.1:


Wpq


If this match were struck, it would light.


WKprq If this match were soaked in water and struck,
it would light.

The conjoining of r with p removes us to more remote worlds where the

consequent is no longer true, as Figure 2.5.1(a) illustrates.








qq

rr








Figure 2.5.1
The same figure also serves as a counternudel to transitivity, a

counterexample to which is given on the following page:






71


E2.5.2:

Wrp If J. Edgar Hoover had been born Russian, he would
have been a Coumunist.
Wpq
If he had been a Comnunist he would have been a
traitor.
Wrq

If he had been born Russian, he would have been
a traitor.

(This particular example is due to Stalnaker [96, p. 173].)

If one is inclined to try to retain transitivity in spite of such

counterexamples because it is felt it is essential to any conditional

function, the following observation should be persuasive of abandoning

the attempt. Recall that LCpq entails Wpq, LCKprp is valid, hence WKprp

is valid. To abandon this wuld be either to abandon the validity of

LCKprp, or the entailment, so WKprp is valid. If transitivity is ac-

cepted, then from Wpq and the valid WKprp, WKprq follows, thus again

validating strengthening the antecedent.

It will not be sufficient to strengthen the first premise of

transitivity to a strict conditional (or entailment) as the following

counterexample shows (due to Lewis [51, p. 32]):

E2.5.3:

LCpq Necessarily, if I started at 5 a.m., I started
before 6 a.m.
Wqr
If I had started before 6 a.m., I would have
arrived before noon.
Wpr
If I had started at 5 a.m., I would have arrived
before noon.

Figure 2.5.1(b) is a countermodel to this inference pattern. The infer-

ence fails in the following situation: suppose that I actually started

just a few minutes after 6 a.m. and actually arrived just after noon, so









the second premise is true. However, if I had started at 5 a.m. I

would have been very tired, and so wuld have forgotten to take the

shortcut that I actually did take, thus lengthening my trip by over

an hour. Then the conclusion fails.

There are several patterns related to transitivity that are

valid for the counterfactual conditional: (a) half of substitution

under strict equivalence (SSE), (b) substitution under counterfactual

equivalence (SCE), (c) the other half of SSE, the consequence princi-

ple (CP), and (d) a strengthened version of transitivity (RRT). The

patterns, and diagrams corresponding, which suggest how one might argue

for their validity on Lewis' semantics, are represented in E2.5.4 and

in Figure 2.5.2.

E2.5.4:


LEpq

Wqr


Wpr



Fpq

Wqr


. Wpr



Wpq

LCqr


. Wpr
(c) CP


or




(a) SSE



or




(b) SCE


LEpq

Wpr


* Wqr



Fpq

Wpr


* Wqr



Wpq

WKpqr


Wpr
(d) RRT


























(a) (b)


p


0 P







(c) (d)



Figure 2.5.2

E2.5.3 and E2.5.4(c) present an interesting contrast. Each can

be considered half of a principle of semisubstitutivity of the counter-

factual conditional with respect to the strict conditional. The in-

validity of the first is essential if we are to avoid the fallacy of

strengthening the antecedent, since from LCKprp and Wpq, WKprq follows









if we accept the pattern of E2.5.3, thus once again validating strength-

ening the antecedent. The fact that the following inference is valid

may provide pause:

E2.5.5:

p
L~pq

Wqr


r

However, consideration of the fact that at the world where p is true, Wqr

may not be true, as wuld be the case in the example considered, shows we

have nothing to fear on that account.

Rejection of E2.5.4(c), on the other hand, would be extremely im-

plausible, for then w would be in the position of holding that q would

be true if p were, but that something entailed by q would not be true.

An argument for the validity of SCE (E2.5.4(b)), and consequently

for RRT (E2.5.4(d)) which follows from it may be found in Lewis [51,

pp. 33-35]. We may note that SSE (E2.5.4(a)) also follows from SCE

(E2.5.4(b)), since LEpq entails Fq (i.e., KWpqWqp).

The consequence principle (CP = E2.5.4(c)) is of special note'

since a related principle which seems to have the same plausibility

as the consequence principle fails on lewis' semantics. This is inti-

mately tied up with the limit assumption, so w shall postpone consid-

eration of it until Section 2.6.

The third inference pattern valid for both the material and strict

conditional is that of contraposition. It ought not be valid for the

counterfactual conditional as the following example shows:








E2.5.6:

Wpq


WNqNp


If the U. S. were to threaten to cut off wheat sales
to OPEC, then OPEC wuld not embargo oil sales to
the U. S.


If OPEC were to embargo oil sales to the U. S.,
then the U.S. wuld not threaten to cut off
wheat sales to OPEC.

One could argue that the threat to cut off wheat sales becomes even more

likely if an oil embargo goes through, so the premise could be true,

while the conclusion is false. Figure 2.5.3 is a couttermndel to this

inference pattern. Once should note that though contraposition fails,

modus tollens does not, as Wpq entails Cpq and mdus tollens is valid

for the latter.


Figure 2.5.3

The question arises as to why hypothetical syllogism should ever

have been thought to be valid for ordinary language conditionals? The

reason is simple: it often is. And it often is under conditions which

are weaker than those of SCE, but stronger than those of Rr. The









following pair of examples was suggested by remarks of Donald Nute

[75]3 (the order of the premises has been reversed for a reason to

be explained):

E2.5.7:

(a) Wqr If Prof. X were to work less, she would be less


Wpq


Wpr



Wqr

Wsq


tense.

If Prof. X were to delegate her authority, she
would work less.


If Prof. X were to delegate her authority, she
would be less tense.

If Prof. X were to work less, she would be less
tense.

If Prof. X were to be canmed, she would work less


If Prof. X were to be canned, she would be
less tense.

If your intuition is to accept (a) and reject (b) then I would agree.

What sense or system can we make of this on Lewis' analysis? Figure

2.5.4 diagrams a situation where the premises of the two arguments

above are true, (a) succeeds, and (b) fails (i.e., Wpr is true and Wsr

is false). It is instructive to compare that figure with 2.5.2(b) and

(d). Fpq requires that there be some p-permitting sphere where p-worlds

exactly coincide with q-worlds, as diagrammed in 2.5.2(b). But 2.5.4 is

weaker since there the p-worlds need merely be a subset of the q-worlds
in some p-permitting sphere. On the other hand, in 2.5.2(d), while the

set of worlds where r is true and the set of worlds where q is true must

intersect in such a way as to contain all the p-worlds in some p-permitting

sphere, it is not necessarily the case that this intersection exhausts the

q-worlds in that sphere, as it must to make Wqr true. Those cases where a









hypothetical syllogism 'works," but for which SCE is too strong, may

be like that diagranmed in Figure 2.5.4, while those where SCE fails

must be. (Except that there could be some Krs-worlds in the s-permitting

sphere, but then also some KrNs-worlds, else Wsr is true contrary to the

assumption that the hypothetical syllogism failed.)


Figure 2.5.4

Figure 2.5.5 also represents a case in which hypothetical syllogism works,

but I shall argue that this is less usual.


Figure 2.5.5








E2.5.7(a) succeeds because the q-permitting sphere where all

q-worlds are r-worlds is the same as the p-permitting sphere where all

p-worlds are q-worlds. If we imagine these conditionals as uttered in

the order given in a conversation, then agreeing on the truth of Wqr is

to tacitly agree upon a particular q-permitting sphere wherein the

worlds are no more dissimilar to the actual world than they have to be

to make the conditional true. If the second conditional does not require

altering this basis of evaluation, that is, if the same sphere will do to

make Wpq true, then the conclusion Wpr must follow. But it follows rela-

tive to the selection of a single sphere for evaluating both conditionals.

E2.5.7(b) fails because the q-permitting sphere where all q-worlds

are r-worlds is not the same as the p-permitting sphere where all s-worlds

are p-worlds, and in the latter sphere r is not still true at those s-

worlds. If we imagine these conditionals as uttered in the order given,

then agreement on the truth of Wqr followed by the utterance of Wsr in-

vites the response: "But I was not thinking of working less that way,"

thus reserving the right not to accept the inference. The first condi-

tional established the boundaries of the strictness required to validate

it. The second conditional tacitly violates those boundaries.

It is these considerations that lead Nute [75] to regard hypo-

thetical syllogism as pragmatically valid but not semantically valid,

depending as it does upon the context of utterance. However, it is to

be noted, that the dependence is systematic rather than merely ambiguous.

The situation diagranmmd in Figure 2.5.5 is, I think, exemplified

by the following:









E2.5.8:

Wqr If Prof. X were to work less, she would be less tense.

Wpq If Prof. X were to die, she would work less.


Wpr If Prof. X were to die, she would be less tense.

If these conditionals were to be uttered in the order given, it would be

to invite the same surprised response, and for the same reason: the

basis for evaluation has been unexpectedly altered. In such a case I

think we would be correct in charging that the premises offer no support

for the conclusion, though the conclusion happens to be true anyway.

An inference is semantically valid if it is impossible for the

premises to be true and the conclusion false. Correspondingly, an in-

ference is pragmatically valid if it is impossible for the premises to

be true with respect to the same sphere, and the conclusion false.

E2.5.7(a), unlike either E2.5.7(b) or E2.5.8, is pragmatically valid.

Adopting such a view of hypothetical syllogism, and hence of transitivity

of counterfactual implication, allows us to make sense both of those

instances where it seems to hold, and those where it certainly fails.

A similar situation arises in the case of the inference pattern

simplification of disjunctive antecedents (SDA). A considerable litera-

ture has arisen in recent papers on the topic of the counterfactual con-

ditional with respect to this inference alone [20, 53, 56, 63, 68, 73,

74, 76]. The inference pattern, an example, and the appropriate diagram

follow (adapted from Nute [681):









E2.5.9:

WApqr


K.prWqr


If the sun were to grow cold or we were to
have a mild winter, we would have a
bumper crop.


If the sun were to grow cold, we ould
have a bumper crop, and if we were
to have a mild winter, we would have
a bumper crop.


Figure 2.5.6

While it is clear that Figure 2.5.6 is a countermodel to SDA, it

is not clear that E2.5.9 contains a counterexmple. One could argue that

Lewis' semantics is inadequate just because it permits our example of

WApqr to be true. Rather one must take both p-worlds and q-worlds into

account in evaluating conditionals with disjunctive antecedents: Figure

2.5.6 illustrates that if we must find both a p-permitting and a q-

permitting sphere, then WApqr is not true at u. One would argue this

way if one wanted to retain SDA as a valid inference pattern for counter-

factuals.









Nute [68, 74] argues for the retention of SDA based upon its

initial intuitive plausibility. However, this has the consequence that

SSE and the stronger SCE must then be rejected, since together they imply

that the counterfactual is a strict conditional. To prove this we need

the following obviously valid inference patterns and sentences:

E2.5.10:

(a) Wpq/.. WpAqr Weakening the consequent

(b) IEAKpqKpNqp

(c) LEKpNqANpq DeMorgan' s Laws

(d) LECpqANpq

(e) WNpp/. Lp D2.4.4

We already have that LCpq implies Wpq. The following suffices to prove

the converse, thus proving the equivalence desired:

Wpq Assumed

WAKpqKpNqq SSE & (b)

WKpNqq SDA

WKpNqANpq (a)

WNANpqANpq SSE & (c)

WNCpqCpq SSE & (d)

LCpq (e)

Nute rejects SCE which implies SSE in order to retain SDA. It has

been argued, I think successfully, by Loewer [56], and others [53, 63],

that this is too high a price to pay for SDA. But perhaps we can save

both our reluctance to abandon SSE and our intuitions about SDA. It is

my understanding that Nute has since come to this position through ap-

plying the category of pragmatic validity to SDA, while recognizing that

it is semantically invalid.






82


As Loewer point out [56, p. 535], the conditions under which a

counterfactual with a disjunctive antecedent is uttered are usually such

that we would be prepared to defend either SDA conjunct; our intention

is to make a more inclusive statement than either counterfactual with

single antecedent alone. The following examples and accompanying diagrams

will illustrate my point.

E2.5.11:

(a) WApqr If Prof. X were to work less or were under


. K4prwpq


WApsr


. prWsr


less pressure to publish, she would be
less tense.


If Prof. X were to work less, she would
be less tense, and, if she were under
less pressure to publish, she would
be less tense.

If Prof. X were to work less or to die, she
would be less tense.


* If Prof. X were to work less, she would
be less tense, and, if she were to
die, she would be less tense.


(a) (b)


Figure 2.5.7








If the premise of E2.5.11(a) is uttered in conversation we evalu-

ate it by tacitly considering a sphere of wrlds no more dissimilar

than necessary to make the counterfactual true. There is such a sphere

which is both p- and q-permitting. In such a case the inference must

'"rk." However, the second example jars our sensibilities, since to

make it true we must move farther out in the system of spheres for one

disjunct than for the other. Even though the premise may be true, w

are inclined to feel it is an odd way to support either conjunct in the

conclusion of SDA. Of the three examples, only E2.5.11(a) is prag-

matically valid, and none are semantically valid; no instance of SDA is.

Our characterization of pragmatic validity as requiring evaluation

of premises and conclusion with respect to the same sphere is, unfortu-

nately, too weak. Part of our intuitive notion is that the order of ut-

terance of the premises should not require us to change spheres if we

have already focused on one that suffices to make the first premise true.

However, the inferences diagranmmd in Figure 2.5.8 satisfy our definition

of pragmatic validity, but run counter to this intuition.


\r r
(a) Wqr, Wpq/. Wpr (b) WApqr/.. KWprWqr


Figure 2.5.8








One should note that these figures are only slight variations of

Figures 2.5.5 and 2.5.7(b), where we agreed the inference should not

be considered pragmatically valid. But both of the above meet the con-

dition that all statements be evaluated with respect to the same sphere

(the third non-trivial one). For the SDA example, both disjuncts occur

in the same larger sphere. However, in both cases there is a smaller

sphere that would make some premise true. It would therefore seem that

we should define pragmatic validity as follows:

D2.5.1: An inference is pragmatically valid iff it is impossible for

the premises to be true and the conclusion false under the conditions

that:

(1) all premises are evaluated as true with respect-to the

same sphere, and

(2) no premise is true with respect to any smaller sphere.

This has the effect of requiring that all variably strict conditionals

in the inference be of the same degree of strictness, and squares with

our intuition that we need consider no worlds more dissimilar to the

actual world than to some fixed degree. This would eliminate the situ-

ations of Figure 2.5.8, but it has two consequences which may be unwelcome.

First, it imposes on the system of spheres the limit assumption:

C2.5.1: If 0 7 Xc $u, then AXe X.

That is, there is some smallest sphere in the set of spheres making

the premises true. Lewis, for reasons we shall discuss in Section 2.6,

wants to reject the limit assumption.

Second, it makes the following inference appear pragmatically in-


valid:


























Wpq, LCqr/. Wpr

Figure 2.5.9

But this inference is semantically valid, as an instance of the conse-

quence principle, and examples of it all seem quite objectionless.

What our definition overlooks is that our concern in the examples

cited was with essentially counterfactual conditionals, those for which

the corresponding strict conditional need not be true. Also our con-

cern in fixing a sphere for evaluation was to ensure that no smaller

sphere made one premise true without making the others true as well.

Hence the following annynded definition should meet both objections:

D2.5.2: An inference is pragmatically valid iff it is impossible for

the premises to be true and the conclusion to be false under the con-

ditions that:

(1) all premises are evaluated as true with respect to the

same sphere, and

(2) no nonnecessary premise is true with respect to any smaller

sphere which does not make every premise true, and

(3) no nonnecessary statement is vacuously true.








There is widespread agreement on the semantic invalidity of

transitivity for the counterfactual conditional, and nearly equally

widespread agreement on the invalidity of SDA, however, there are

several other inference patterns upon which agreement is not as easy

to find. Fortunately, adopting different positions on these does not

greatly alter the nature of the conditional in question. Of the fol-

lowing inferences, the first and last are invalid on Lewis' preferred

semantics, as well as on that version which we have presented:

E2.5.12:

Wpq Kpq Kpq LNpLNp

..Vpq Wpq ..Vpq ..Wpq ..Vpq
(a) (b) (c) (d) (e)

The invalidity of (a) and (e) and the validity of (d) rest upon the

fact that the conditional Wpq can be vacuously true, while Vpq is never

vacuously true, given the preferred definitions. If a given proposition

p is not entertainable, then LNp is true, as is Wpq, but Vpq is then false,

since there is no p-permitting sphere at all. The validity of (b) and (c)

follows from the definitions directly, as {u} is then a p-permitting

sphere in which every p-world is also a q-world.

Depending upon one's tastes, the validity or invalidity of these

inferences can be altered by minor alterations in the conditions upon the

$ function, or the truth conditions for the '"ould'- and '"night"-

conditionals. By switching to the alternate definitions of Wpq and

Vpq (D2.3.4 and D2.3.5) we preserve their interdefinability, but make

(a) and (e) valid, and (d) invalid. If we are agreeable to abandoning

the interdefinability of Wpq and Vpw, then the combination of D2.3.4 and








D2.3.3 will render all vacuous counterfactuals false, thus invalidating

both (d) and (e) but rendering (a) valid.

We can invalidate both (b) and (c) by abandoning the centering

condition on $ (C2.3.1), since then the smallest sphere about u may not

even contain u, so it would be possible for Kpq to be true at u but

either Wpq or both Wpq and Vpq to be false. If (c) seems desirable, but

(b) not, then we can employ Lewis' condition of weak centering in place

of centering:

C2.5.2: For all0 0 A e $ u e A.

In such a case we have a smallest non-empty sphere about u, A$u' of

worlds indistinguishable from u in terms of our similarity ordering.

As Lewis suggests, we may want to vary the conditions anyway for

different applications of the analysis of conditionals. However, herein

we will continue with the analysis presented without remarking on the

otherwise desirable flexibility of Lewis' full analysis. I am satisfied

with the present assigrmient of validity and invalidity to all of the in-

ferences in E2.5.12 except (a) and (d). However, my dissatisfaction is

all but evaporated by the realization that while (a) is semantically in-

valid, it is always pragmatically valid on our definition (D2.5.2). And

(d), though semantically valid, is never pragmatically valid. Clause (3)

of our definition rules out vacuous premises or conclusions, so every

time Wpq is nonvacuously true, Vpq must be, and though when INp is true,

so must Wpq be, the latter's truth is vacuous.

One could argue that E2.5.12(d) should be retained because of the

following intuitively valid argument: Suppose it is not the case that

if p were true, then q would be true. Then it seems to follow that there

are circumstances under which if p held, then Np might hold. Otherwise








we could hardly defend the supposition. In fact, our best defense is

that both p and Nq are possible simultaneously. So, of course p is pos-

sible, and hence not impossible. The following chain of inferences sun-

marizes the above argument:
E2.5.13: N~pq


VpNq

N Kpq






E2.5.13 is semantically valid on the analysis we have given, as I believe

it should be, but it is also pragmatically valid, intuitively, and in

terms of our definition. But we cannot hold it semantically valid with-

out holding E2.5.12(d) valid as well, since it is the contrapositive of

the above argument.

There are two apparent oddities about pragmatic validity as we have

defined it: first, semantic validity does not guarantee pragmatic valid-

ity, as witness E2.5.12(d); second, the contrapositive argument to a

pragmatically valid one may not itself be pragmatically valid (same ex-

ample). These situations depend upon the presence of clause D2.5.2(3),

so may be avoided if that clause is dropped. I an reluctant to call any

argument "pragmatically valid" when it contains vacuously true counter-

factuals, so an willing to put up with these oddities. I an influenced

perhaps by my feeling that though the inference from "All unicorns are

furry" to "It is not the case that some unicorns are bald" is semantically

valid, it is decidedly odd, in view of the nonexistence of unicorns.








As for the validity of E2.5.12(b) and (c), we remarked in Section

1.1 that the antecedent and consequent being true was no bar to the truth

of the conditional, though assertion of a conditional usually presupposes

the utterer does not know the antecedent to be true. Nute [67, 68, 74]

argues that (b) is counterintuitive on the grounds that in many situa-

tions where both p and q happen to be true we wuld deny that if p were

true, then q wuld be because we deny that the connection between p and

q, if any, guarantees the truth of q just because p is true. That is,

we hold Kpq and VpNq as compatible. But they are not on Lewis' seman-

tics as we have presented it because {u} is the smallest non-empty sphere

about u. So not every p-permitting sphere contains an Nq-wrld. This

argument has merit, and so one might want to retreat to weak centering,

though I will argue that perhaps the intuition Nute calls upon is actually

something else. To E2.5.12(c) I know of no objection; it wuld seem that

the fact that p and q are both true is sufficient prima facie evidence

for "if p were true, then q might be." We may observe that if NVpq is

asserted, then Kpq is an entirely sufficient rebuttal.

I believe Nute's argument really rests upon the following: we hold

Kpq and MKpNq as compatible, that is, though p and q are both true, it is

possible, we think, that p could be true and q false. But MKpNq is, as

we observed in Section 2.4, weaker than VpNq. It is really the former

we hold compatible with Kpq rather than the latter.

Lewis' analysis generally preserves our pre-analytic notions con-

cerning the validity and invalidity of various counterfactual inferences.

Coupled with the concept of pragmatic validity (due in part to Lewis

himself [54], but more to Nute [75]), it also explains why we accept many
counterfactual inferences to which there seem to be counterexamples. We








do so when the strictness of the variably strict conditionals involved

coincide. That the analysis here presented does help to explain this,

is, I think, a strong point in its favor.

In the next tw sections and in CHAE THREE we shall consider

aspects of Lewis' analysis that are more problematic: his rejection of

the limit assumption, his possible worlds realism and the notion of

comparative overall similarity itself.


2.6 The Limit Assumption

In Section 2.3 we observed that closure under unions and inter-

sections imposes a certain kind of bounding condition on $ and subsets

thereof. That is, there is a smallest and largest sphere in $u' A$u

and U$u, respectively. Also any nonempty subset X of $ is bounded both

above and below by spheres UX and AX respectively. However, the closure

conditions are not as strong as they might be; it is not the case that

for all X c $u, either UX X or that AXE X. We restate here what Lewis

calls the Limit Assumption, LA (actually it is a "lower-limit" assumption):

C2.5.1: If 0 Xc $u, then AX s X.

If we replaced the closure under (nonenpty) intersections condition

(C2.3.4) on $ by this, we would have a sphere function which satisfied

the limit assumption.

A sphere function which satisfied LA would allow us to speak of a
"closest" sphere satisfying any given condition, since it would be the

intersection of all spheres satisfying that condition. For instance, for

proposition p we would be assured of a closest p-permitting sphere. We

could then define truth for the '"uld"-conditional more simply:









D2.6.1: Wpq @ u iff every p-wrld in the closest p-permitting sphere

in $u is a q-wrld,

which would be nonvacuously true in case there was a p-permitting sphere,

and vacuously true otherwise. The spheres (though not the worlds they

contain) would be well-ordered with respect to subset containment. That

is, every subset of $u would have a least element.

But Lewis questions whether this would in general be a suitable

restriction on $. He argues:

Suppose we entertain the counterfactual supposition that
at this point


there appears a line more than an inch long. (Actually
it is just under an inch.) There are worlds with a line
2" long; worlds presumably closer to ours with a line
1 long; worlds presumably still closer to ours with a
line 1 long; worlds presumably still closer......
But how long is the line in the closest worlds with a
line more than an inch long? If it is 1+x" for any x
however small, why are there not other worlds still
closer to ours in which it is l+ x", a length still
closer to its actual length? . Just as there is no
shortest possible length above 1", so there is no
closest world to ours among the worlds with lines more
than an inch long.. ... [51, p. 21]

On the basis of this example, Lewis rejects the limit assumption. Before

considering the consequences of this rejection, we note that Lewis recog-

nizes an even stronger version of LA: the uniqueness assumption, UA.

(Called Stalnaker's assumption by Lewis because it is associated with

Stalnaker's semantics for conditionals. We use Nute's [74, p. 100] more

descriptive term.).

We will discuss Stalnaker's semantics in CHATTER THREE as part of

a general discussion of similarity. For now it suffices to indicate that

the uniqueness assumption requires that the closest p-permitting sphere

for any proposition p contains but one p-world. This means the sphere








function under this condition can be viewed as placing a well-order

on the worlds themselves. Since at such a closest p-world, either q

or Nq is true for any proposition q, UA validates conditional excluded

middle (CM4) for counterfactuals, and renders Lewis' definition of the

eightt -conditional superfluous. (Lewis does discuss other ways to de-

fine this conditional on the uniqueness assumption, all of which he

deems unsatisfactory [51, pp. 80-81].) As both of these consequences are

unwelcome, we shall agree with Lewis in rejecting the uniqueness assunp-

tion.

Adopting IA, though not UA, does not have the consequence of

validating CEM, since the closest p-permitting sphere may well contain

more than one p-world. Under Lewis' analysis it is entirely possible

for worlds to tie in comparative overall similarity to a given world.

This is the rationale for defining the "might"-conditional. Accepting LA

does have the consequence of going against the intuition Lewis draws on

in his example; however, the game of intuitions has two sides, and

Pollock [80] and Herzberger [34] bring out the other side with respect

to LA.

Starting with an adaptation of Lewis' example quoted above sup-

porting the rejection of IA, Pollock shows that this rejection also re-

quires the rejection of a generalization of the consequence principle

(CP = E2.4.4(c)) that seems to have as much claim for validity as the

original consequence principle. In a related fashion Herzberger argues

that rejection of LA introduces "counterfactual inconsistencies" on

Lewis' analysis. We will consider each argument in turn.

In the passage quoted above Lewis claims that it is reasonable

to suppose that for each world where the line is 1+x" long, there is








a closer world where it is between I" and 1+x" long, for each positive

value of x. Granted this assumption, Pollock then claims that the fol-

lowing sentence is true on Lewis' analysis for each positive value of

x [80, p. 19]:

E2.6.1: If the line were more than an inch long, it would not be

lix" long.

For this to be true it must be the case that in some antecedent-

permitting sphere every antecedent-world is a consequent-world. And

this for each x. That is, for each x, there is a sphere where the line is

more than an inch long at some wrld, and at every world where it is

mre than an inch long in that sphere, it is not l+x" long. let us as-

sune this condition is met, though it is not clear that Lewis meant it,

and certainly does not need it to make his point.

With x going to zero, it follows that the line would not be 1+x'

long for all positive values of x, hence the line would not be more than

one inch long; for if it is, it is by some positive aunt. So, Pollock

concludes, if the line were mre than one inch long, it would not be

more than one inch long, a flat contradiction [80, p. 19].

All that saves Lewis' semantics from evident inconsistency is

that the key principle used above is not valid on that semantics [80,

p. 20]:

E2.6.2: The Generalized Consequence Principle (GCP):

If G is a set of sentences and for each q c G Wpq is true, and G j=r,

then Wpr is true.

GCP is the version of CP generalized to all sets of sentences, including,
as in Pollock's example, infinite sets. While CP is, as we have noted,

valid, and its finite generalization is valid since we can then take G




Full Text
200
L4.8.20: In any selection function frame, (id), (md), (co), (ca),
and (cem) imply (cv).
L4.8.21: In any selection function frame, (id), (11), (ca), and (cb)
imply (md) and (co).
The proofs of the completeness of the various extensions of CK
will again require the notions of canonical frame and model. As is
the case in our treatment of neighborhood semantics for classical con
ditional logics, the frames defined in D4.8.3 are not unique, so, fol
lowing Chellas, we will designate a certain subclass of canonical frames
as proper, and make our selection of specific canonical frames from among
those.
D4.8.3: Let L be a normal conditional logic and U the set of maximally
L-consistent extensions of L. Let f: U x p(U) -> P(U) be any function
such that for all sentences a,b and all u e U:
Wab £ u iff f(u, |a|^) c |bL.
Furthermore, if V:P -> P(U) is a valuation such that V(P) = |p|^, then
F = < U,f > and M = < U,f ,V > are canonical selection function frame and
model for L, respectively.
The following theorems are analogs to those for canonical neighbor
hood frames and models for conditional logics so no proofs are presented.
(See Chellas [11, pp. 139-140] and Section 4.5.)
T4.8.1: Let M be a canonical (selection function) model for L, a normal
conditional logic. Then for all sentences a and all worlds u in U,
|= a iff a e u.
Thus, ||a||W = |a|L.
T4JL2.: With M,L as in T4.8.1, for every sentence a:
1= a iff |j- a,


83
If the premise of E2.5.11(a) is uttered in conversation we evalu
ate it by tacitly considering a sphere of worlds no more dissimilar
than necessary to make the counterfactual true. There is such a sphere
which is both p- and q-permitting. In such a case the inference must
"work." However, the second example jars our sensibilities, since to
make it true we must move farther out in the system of spheres for one
disjunct than for the other. Even though the premise may be true, we
are inclined to feel it is an odd way to support either conjunct in the
conclusion of SDA. Of the three examples, only E2.5.11(a) is prag
matically valid, and none are semantically valid; no instance of SDA is.
Our characterization of pragmatic validity as requiring evaluation
of premises and conclusion with respect to the same sphere is, unfortu
nately, too weak. Part of our intuitive notion is that the order of ut
terance of the premises should not require us to change spheres if we
have already focused on one that suffices to make the first premise true.
However, the inferences diagrammed in Figure 2.5.8 satisfy our definition
of pragmatic validity, but run counter to this intuition.
Figure 2.5.8


101
of all q^ defined above. Each is a subset of the set of worlds iden
tified with p, which set violates the limit assumption. Suppose AQ f 0.
Then there would be a set of p-worlds, namely AQ, which was contained in
every set of p-worlds, hence the set of p-worlds wuld not violate the
limit assumption, contrary to hypothesis. Thus AQ = 0. Since Q c 0p,
it follows that A0p c AQ, so A0p = 0. Hence 0p is not satisfiable, so
inconsistent.
On the other hand, if p satisfies the limit assumption, then there
is a set of worlds, namely the set of closest p-worlds, each of which
satisfies every member of 0p, and A0p f 0.
Hence the counterfactual inconsistency on the level of propositions
and the rejection of LA imply each other. Thus also, counterfactual con
sistency and LA imply each other.
The limit assumption can now be shown to be equivalent to GCP being
valid. Below we restate the definition for entailment of a proposition
by a set of propositions in a slightly different, though equivalent,
form to D2.5.2:
D2.6.5: G | = p iff AG c ||p||.
Here we are identifying p with a set of worlds, and G with a set of sets
of worlds. The set of worlds at which every proposition in G is true
is AG.
First we show that the limit assumption implies the validity of
GCP: suppose LA holds, Wpq is true for each q in Q, and Q | = r. We must
show Wpr is true. By definition, Q c 0p, hence A0p c AQ. Since AQ is
identified with a proposition, let it be s. We claim Wps is true, for,
if not, then s is not in 0p, so by LA A0 0 s. But then A0p 0 AQ, a
contradiction. So Wps is true. But Q|=r, so AQ c ||r||, hence ||s||c ||r||,


236
The instances of E5.2.4 are true on the second reading also, though
trivially, since all the counterfactuals in E5.2.4 are false. However,
consider the following propositions: a = {v,z}, b = {w,y}, and c = {v,y}.
Figure 5.2.1(b) is a countermodel to the instance of CB:
E5.2.5: CWAabcAWacWbc
That is, there is some disjunctive condition (Aab = {v,w,y,z}) which
results in exactly one horse getting around the pile-up, but neither dis
junct is counterfactually sufficient for this.
As I stated earlier, I cannot think of a counterexample to CB, and
so do not think this example is one. The situation of E5.2.5 seems highly
implausible to me: if some disjunctive condition has the result indicated,
then some one of the disjuncts does. If one is looking for a way to ex
press propositions a and b perhaps "the Flyer/Galloper just barely
finishes" will do, as this leaves open whether the other horse gets
around the pile-up more easily, or not at all. To accomnodate Pollock's
account to this view requires either the change that results in w or z to
contain the changes that result in v and y, thus restoring the missing
links of Figure 5.2.1(b), or to deny that any of the changes contain any
of the others, as in Figure 5.2.2(a).
(a) (b)
Figure 5.2.2
It was argued in Section 3.2, against a similar example of Nute's
(E3.2.3), that once we have reached the situation depicted in Figure


149
(b)
||Aab||
= II a
II U ||b||
(c)
||Na||
= U -
l|a||
(d)
1= Eab
iff,
|= a iff
|= b
u
u
'u
(e)
1= Kab
iff both 1= a
and 1= b
u
u
u
(f)
1= Aab
iff either 1=
a or 1= b
u
'U
u
(g)
1= Na
'u
iff ]j
1 a
(h)
1= Ma
'u
iff Ij
LNa
L4.2.3: E + S is consistent wrt (with respect to) C.
O
Proof: In view of L4.2.2 it is sufficient to show S is valid in C. So
b
assume M is any model on any frame in C and u is any world. By condition
(s), = 0, hence ||Np|| l N So by the truth definition LNp, hence
|=MP- QED
L4.2.4: M = E + M is consistent wrt C .
m
Proof: It is sufficient to show M is valid in C Let M be any model
on any frame in and suppose |=LKpq. Then by truth definition
||Kpq|[ e N^, hence ||p|| A ||q|| e N Hence by condition (m), both ||p||,
||q|| e N^. So by truth definition, |= Lp and |= Lq, and so |=KLpLq. QED
L4.2.5: E + R is consistent wrt C .
Proof: It is sufficient to show R is valid in C Let M be any model on
any frame in C^. Assume |= KLpLq. Then by the truth definition, |= Lp
and '|= Lq. So by the truth definition, ||p[| e and ||q|| e N^. Hence
by condition (r), ||p|| A ||q|| e N^. So ||Kpq|| e N^, hence by truth defini
tion |= LKpq. QED
L4.2.6: E + Q is consistent wrt C .
- q
Proof Letting M be any model on any frame in and u any world, by
condition (q), = P(U). Let p be any proposition. Then ||p|| c U,
hence by (q) ]|p|j E Nu> Therefore, |= Lp. QED


150
L4.2.7: E + N is consistent wrt C .
Proof: Letting M be any model on any frame in C and u any world, it
follows from condition (n) that U e N^. PC considerations show ||1|| = U,
hence by truth definition, |= Ll. QED
Combinations of the above lemmas establish the following results for the
logics we have primarily identified:
LA.2.8: R = E+ M + Ris consistent wrt C .
mr
Proof: Follows from lenrnas L4.2.4 and L4.2.5. QED
L4.2.9: K = E + M + R + N is consistent wrt C
mm
Proof: Follows from lenrnas L4.2.7 and L4.2.8. QED
We might establish similar results for other extensions of E. One
extension in particular of interest is K + T, the modal logic usually
denoted by T. (K is also denoted T, Zeman [104].)
L4.2.10: T is consistent wrt C (indeed wrt Cj).
mmt t
Proof: It is sufficient to show T is valid in C (in view of L4.2.9).
mmt
Assume |= Lp. Then ||p|| e N^, so N f 0. So by condition (t), u e AN^,
hence u e ||p||. Therefore, |= p. QED
System T, along with systems S4 (= T + U) and S5 (= T + U + E) are three
of the modal logics that have drawn the most attention as reasonable logics
for our ordinary notions of possibility and necessity. (Recall we Imposed
on the sphere function conditions sufficient to guarantee the modal logic
validated thereby was S5.)
We now have consistency results for E, M, R, K, and T. The proof
of completeness of these logics with respect to the frames already noted
for them is more complicated. The notion of maximally L-consistent sets
plays a major role. It is in terms of these that special frames of the
requisite classes are constructed. In model theoretic semantics such


180
(b) The supplementation of that frame still satisfies (c),
including (cm), by L4.5.14, below.
(c) Hence, the supplemented frame is canonical for L, by
LA.5.13.
The following lemma is useful in this connection:
L4.5.14: If F = < U,N > is a proper canonical frame satisfying some per
missible combination of conditions (cr), (cn), (cq), and (cs), then
JL
F = < U,N > satisfies that combination of conditions and (cm).
Proof: That F* satisfies (cm) is obvious. The proofs for each separate
condition and the combinations are precisely similar to those of LA.2.14.
QED
In view of EA. 5. A, as supplemented above, we will only have to show
EA.5.4(2) holds in our completeness proofs. Several immediately follow:
LA.5.15: CE is complete wrt C.
Proof: Clearly the frame of is in the class of all frames, C, so the
lemma follows by E4.5.4. QED
LA.5.16: CE + CS is complete wrt C .
Proof: Let be a proper canonical model for CE + CS satisfying (cs)
whenever X / |a| for every formula a, that is, the smallest proper canonical
model. That satisfies (cs) when X = |a| for some formula a follows:
Vab e u for any world u and any formulas a and b since CE + CS is the
deductive closure of the system containing CS. Consequently, NWcd e u
for all u and all formulas c and d, so Wed l u. So by definition of Nq ,
Ng(u,X) = 0 for all u and X. Consequently, (cs) is satisfied. By E4.5.4
the lemma follows. QED


175
E4.5.2:
CQ: Wpq
CS: Vpq
In the following series of lemmas we state the consistency of
various conditional logics with respect to certain classes of proposi
tional frames. Hereafter, unless otherwise noted, frames will be pro-
positional neighborhood frames. We prove only a sample of the following
lemmas since the proofs follow those of comparable lemmas in Section 4.2
so closely. In view of L4.5.4 we need only check the validity of the
conditional axioms in the designated class of frames.
L4.5.5: CE + CS is consistent wrt C .
s
Proof: Assume M is any model on any frame in C and u any world. By
s
condition (cs), N(u, ||p||) = 0, hence ||Nq|| l N(u,p). So by the truth
definition we have WpNq, and so by D4.5.9, we have |= Vpq. QED
(Compare L4.1.3.)
L4.5.6: CE + OQ is consistent wrt C^. (Compare L4.1.6.)
L4.5.7: CM = CE + CM is consistent wrt C (Compare L4.2.4.)
Proof: It is sufficient to show that CM is valid in C Let M be any
m J
model on any frame in C^, and suppose |= WpKqr. Then by the truth defini
tion, ||Kqr11 e N(u,p). But then ||q|| A ||r|| e N(u,p), and so by (cm),
II q 11II r 11 e N(u,p). So by the truth definition, both |= Wpq and |=Wpr,
and hence |= KWpqWpr. QED
L4.5.8: + CR is consistent wrt C' (Compare L4.2.5.)
L4.5.9: + CN is consistent wrt C (Compare L4.2.7.)
L4.5.10: . = ffi + CM + CRis consistent wrt C (Compare L4.2.8.)


165
While our choice between the two languages is one of inscriptional
preference, our intuitions about conditionality and indexed modality may
differ. For example, if the truth of a is irrelevant to the truth of b,
should it follow that b is both a-necessary and Na-necessary? The con
ditional parallel is the typical "even if'-conditional, where there is
intuitively no necessitation connecting the antecedent to the consequent.
Comparing the indexed modal rules and axioms above to the modal
rules and axioms in E4.1.1 and E4.1.2 reveals the clear parallel between
the constructions. Excepting RCEA for the moment, RCEC corresponds to
RE, RCM to KM, etc., where an arbitrary sententially indexed modality
has been substituted for the single modality expressed by "L" in the lat
ter series. Though in what follows we will be considering our basic lan
guage to be CW, in view of the close correspondence between indexed
modal languages and conditional languages, the terminology and many of
the techniques developed in Sections 4.1 and 4.2 will apply to our pre
sent endeavors.
We have already observed informally that the presence of RCEA.
renders sententially indexed modalities equivalent to propositionally
indexed ones (in the sense that an equivalence class of sententially in
dexed modalities corresponds to a propositionally indexed modality).
The formal semantic distinction will be developed in Section 4.5. One
might wonder why, if RE "divides" into two conditional rules, the same
practice is not followed for KM and the others. For example, what about:
RCMA: From Cab infer CWacWbc,
or, expressed in terms of indexed modalities:
From Cab infer CL.
The answer is that they could have been included and would have been but


62
In view of these equivalence the "would"- and "might"-conditionals are
interdefinable on Lewis' analysis.
Returning to the Bizet and Verdi examples, it seems reasonable to
deny "If Bizet and Verdi had been compatriots, then they would have been
Italian" on the grounds that among those worlds most similar to the actual
world will be found some where they are both Italian, but also some where
they are not both Italian, though compatriots. In fact, I would agree
with Lewis in judging the following to be true:
E2.3.1: If Bizet and Verdi had been compatriots, then they would either
both have been Italian or both have been French.
E2.3.2: If Bizet and Verdi had been compatriots, then they both might
have been Italian.
E2.3.3: If Bizet and Verdi had been compatriots, then they both might
have been French.
In effect, we are adopting a similarity ordering of the possible worlds
where the closest worlds where Bizet and Verdi are conpatriots contain
exclusively worlds where they are both French and other worlds where
they are both Italian, while the worlds where are both Chinese, say,
are more distant.
In view of this definition of the "might"-conditional, Lewis'
semantics necessarily fails to validate the principle of conditional
excluded middle (CEM). In classical propositional logic ACpqCpNq is
a theorem. The corresponding ALCpqLCpNq is not generally a theorem of
modal logic, i.e., GEM fails for the strict conditional. The counter-
factual conditional shares this property with the strict conditional.
(The principle of CEM is validated by Stalnaker's semantics, for reasons
which we shall discuss in CHAPTER THREE.)


[85]
[86]
248
Saberhagen, Fred, Mask of the Sun, Ace, New York, 1979.
Schlossberger, Eugene, "Similarity and Counterfactuals," Analysis
38 (1978), pp. 80-82.
[87] Schneider, Ema, "Recent Discussion of Subjunctive Conditionals,"
Review of Metaphysics 6 (1953), pp. 623-647.
[88] Scott, Dana, "Advice on Modal Logic," in K. Lambert, Philosophical
Problems in Logic, D. Reidel, Dordrecht, 1970.
[89] Segerberg, Krister, "Decidability of S4.1," Theoria 34 (1968),
pp. 7-20.
[90] "Decidability of Four Modal Logics," Theoria 34 (1968),
pp. 21-25.
[91] An Essay in Classical Modal Logic, University of Uppsala,
Uppsala, Sweden, 1971.
[92] Shorter, J. M. "Causality and a Method of Analaysis," in R. J.
Butler, editor, Analytical Philosophy, 2nd series, Basil Blackwell,
Oxford, 1965.
[93] Skyrms, Brian, "Possible Worlds, Physics, and Metaphysics,"
Philosophical Studies 30 (1976), pp. 323-332.
[94] Slote, Michael A. "Time in Counterfactuals," Philosophical Review
87 (1978), pp. 3-27.
[95] Sosa, Ernest, ed., Causation and Conditionals, University Press,
Oxford, 1975.
[96] Stalnaker, Robert C., "A Theory of Conditionals," from Studies in
Logical Theory, American Philosophical Quarterly, Monograph Series
2, editor N. Rescher, Blackwell, Oxford, 1968, pp. 98-112. Reprinted
in Causation and Conditionals [95].
[97] Stalnaker, Robert C. and Thomason, Richmond H., "A Semantic Analysis
of Conditional Logic," Theoria 36 (1970), pp. 23-42.
[98] Swain, Michael, "A Counterfactual Analysis of Event Causation,"
Philosophical Studies 34 (1978), pp. 1-19.
[99] Temple, Dennis, "Nomic Necessity and Counterfactual Force," American
Philosophical Quarterly 15 (1978), pp. 221-227.
[100]Thomason, R. and Gupta, A., "A Theory of Conditionals in the Context
of Branching Time," Philosophical Review 89 (1980), pp. 27-53.
[101]Vendler, Zeno, "Causal Relations," Journal of Philosophy 64 (1967),
pp. 704-713.


I certify that I
conforms to acceptable
adequate, in scope and
Doctor of Philosophy.
I certify that I
conforms to acceptable
adequate, in scope and
Doctor of Philosophy.
I certify that I
conforms to acceptable
adequate, in scope and
Doctor of Philosophy.
have read this study and that in my opinion it
standards of scholarly presentatiori and is fully
quality, as a dissert^tionN^or the degree of
Philosophy
have read this study and that in my opinion it
standards of scholarly presentation and is fully
quality, as a dissertation for the degree of
TT-
Thomas Auxter
Associate Professor of
Philosophy
have read this study and that in my opinion it
standards of scholarly presentation and is fully
quality, as a dissertation for the degree of
R. G.j Selfrid&
Professor of Comptter and
Information Sciences, and
Mathematics
This dissertation was submitted to the Graduate Faculty of the Department
of Philosophy in the College of Liberal Arts and Sciences and to the
Graduate Council, and was accepted as partial fulfillment of the require
ments for the degree of Doctor of Philosophy.
August 1980
Dean, Graduate School


164
then what we really have are propositionally indexed modalities, and
this requirement could be expressed by the following rule:
RE": From Eab infer EL^cL^c.
We would then not distinguish among equivalent sentences in either
operator or operand.
Let ios designate a language of sententially indexed modalities as
CLI. Below are parallel sets of rules of inference and axioms as they
would appear in a conditional language, CW, and in CLI:
E4.4.1: Rules of inference:
Conditional:
Indexed modal:
RCEA: From Eab infer EWacWbc
From Eab infer EL^cL^c
RCEC: From Eab infer EWcaWcb
From Eab infer EL aL b
c c
RCM: From Cab infer CWcaWcb
From Cab infer CL aL b
c c
RCR: Frcm CKabc infer CKWdaWdbWdc
From CKabc infer CKL^aL^bL^c
RCN: From b infer Wab
From b infer L b
a
RCK: From CKK . Ka^^ . a^b
Frcm CKK . Ka^ . anb
infer CKK . KWca^Wca2 . .
infer CKK . KL a-,L a0 . .
c 1 c 2
Wca Web, for n > 0
L a L b, for n > 0
c n c
E4.4.1: Axioms:
Conditional:
Indexed modal:
CM: CWpKqrKWpqWpr
CL KqrKL qL r
P P P
CR: CKWpqWprWpKqr
CKL qL rL Kqr
P P P
CN: Wpl
L 1
P
CK: CWpCqrCWpqWpr
CL CqrCL qL r
p pM p
It is clear that there is a one-to-one correspondence between formulas
of CW and CLI defined by identifying formulas of the form Wab with those
of the form L b for all formulas a and b.
cl


85
Wpq, LCqr/. Wpr
Figure 2.5.9
But this inference is semantically valid, as an instance of the conse
quence principle, and examples of it all seem quite objectionless.
What our definition overlooks is that our concern in the examples
cited was with essentially counterfactual conditionals, those for which
the corresponding strict conditional need not be true. Also our con
cern in fixing a sphere for evaluation was to ensure that no smaller
sphere made one premise true without making the others true as well.
Hence the following ammended definition should meet both objections:
D2.5.2: An inference is pragmatically valid iff it is impossible for
the premises to be true and the conclusion to be false under the con
ditions that:
(1) all premises are evaluated as true with respect to the
same sphere, and
(2) no nonnecessary premise is true with respect to any smaller
sphere which does not make every premise true, and
(3) no nonnecessary statement is vacuously true.


23
because it shows where the semantic component of the concept leaves off
and the pragmatic component begins [96, p. 166].
I should think, however, that the logical problem actually in
volves two problems: the task of describing the formal logical pro
perties of the conditional and the task of devising a satisfactory
semantics. These two tasks are different. One could describe the
formal properties of the conditional in terms of a proof-theoretic
system: a set of axioms and rules of inference in which a conditional
connective occurs, and in which those sentences and rules of inference
our pre-analytic intuitions hold valid occur while those we regard as
invalid do not. We would be remiss to accept such an analysis as com
plete for it is possible to understand the logic of a concept without
understanding the concept itself.3
For example, in [15] Chisholm makes use of a relation "more
reasonable than" holding between propositions. That is, a certain
proposition p may be more reasonable for subject S at time t than
another proposition q. This appears in his formal definitions as an
undefined relation, but to explicate it he offers certain basic prin
ciples as axioms of the concept intended to make explicit its logical
structure [15, p. 13]. If left at this point (which Chisholm does not
do) we may have in our grasp the logic of "more reasonable than" with
out understanding what it is for one proposition to be more reasonable
than another. We do not know how to apply the relation to propositions,
only how to manipulate its previous application.
Devising a semantics to validate this axiomatic system may not in
itself be sufficient to convey an understanding of the concept. The


167
E4.4.3: Conditional systems:
(a)
Ce = PC + RCEC
half-classical
(b)
Ck = PC + RCK
half-normal
(c)
CE = PC + RCEA + RCEC
classical
(d)
CM = PC + RCEA + RCM
monotonic
(e)
CR = PC + RCEA + RCR
regular
(f)
CK = PC + RCEA + RCK
normal
Parallel to the convention of Section 4.
1, a logic containing CE, for
example, will be called classical, and a logic containing CM, monotonic,
etc., as indicated to the right in E4.4.3. The presence of Ce and Ck
should especially be noted, as there is no obvious modal parallel to
these conditional logics outside of sententially indexed modalities.
They are only half-classical in the terminology of Chellas because the
classical requirement that sentences expressing the same proposition not
affect the truth value of compounds containing them differently extends
only to the consequent position in the conditionals.
From the following lemnas the containment relations of the logics
follow. The proofs are easy adaptations of the parallel proofs of the
modal lemnas in Section 4.1. We prove only L4.4.3 and otherwise refer
to the parallel modal lemnas.
L4.4.1: RCEC is derivable in PC + RCM. (Compare 14.1.1.)
L4.4.2: RCM is derivable in PC + RCR. (Compare L4.1.4.)
L4.4.3: RCK is derivable in PC + RCR + RCN, and conversely.
Proof: RCN is RCK for n = 0 and RCR is RCK for n = 2. Conversely, RCN
gives us RCK for n = 0 and RCR gives us RCK for n = 2. The associativity
of conjunction and an induction give us RCK for n > 2. QED


230
the reasonablesness depends upon how likely p, q, and r are. If they
are all about equally likely, then the antecedent of E5.1.6 is true.
But then, by the same reasoning, so is the consequent: if either p or
q were true, then r might also be true. Given three equally likely un
related propositions, I am inclined to think that if either of any two
were true the other might still be false, or might also be true. Thus
Pollock's alleged counterexample is of dubious import. If the unrelated
propositions are about equally likely, as they are in Pollock's example,
then the consequent is true along with the antecedent of E5.1.6. On the
other hand, if the unrelated propositions are not equally likely, then
one or the other conjunct of the antecedent is not true. In either case
we have no counterexample.
Even if Pollock's argument is thus weakened, I can think of no
decisive argument for CV, and so am willing to regard it as problematic
along with CC, and perhaps CB. Nevertheless, we are left with a minimal
counterfactual logic which requires a partial order of possible worlds.
It is not clear that even so, a partial order must form a part of an
analysis of counterfactuals. This is the point made by Loewer:
. . although our reasoning with counterfactuals does in
volve a similarity ordering of worlds, the concept of sim
ilarity is primitive and does not support an analysis of
counterfactuals. [57, pp. 113-114]
We shall argue in the next section that, on the contrary, the partial
ordering required by any adequate counterfactual logic can be regarded
as emerging from an analysis of counterfactuals.
5.2 Comparative Order Analysis
We have returned repeatedly in this essay to the distinction between
the logic of a concept and an analysis of a concept. The issue is raised


206
In the previous section we said that CP, CA, SS could be considered
a family of partially-ordering logics, and that V, VC, VW, a family of
weak-totally ordering logics. In this section we shall show that a logic
containing CP can be partially ordered and a logic containing V can be
weak-totally ordered and conversely. We shall also show that there is a
family of partially-ordering logics between the twa families noted above,
which has not previously been noted, to our knowledge.
Initially we show that CP is well-chosen on the grounds that 0 does
not contain CA, and a partial ordering of possible worlds implies that
(ca) is satisfied. We shall go on to establish comparative ordering
semantics, a generalization of Lewis' comparative similarity semantics,
for those extensions of CK which contain CP.
To show that 0 does not contain CA we require a set of possible
worlds U and a selection function f that satisfies (id), (md), and (co),
but does not satisfy (ca). It will be instructive if we so construct
< U,f > that with a relation vR^w defined by f(u,{v,w}) = {v}, R will
not be transitive. Let U = {u,v,w,z} and define f so that f satisfies
(id), (md), (co), and even (ca), for worlds v,w,z. For world u and any
X c U, we define f(u,X) by f(u,X) = {u}, if u e X, and otherwise by the
following table:
X
f (u,X)
0
0
{v}, {v,w}
{V}
{w}, {w, z}
{w}
{z}, {v,z}
{z}
{V,w,z}
{v,W,z}


110
"actual" is analogous to "present, is to convey understanding of what
a possible world is through analogy to something we already under stand,
rather than reduction to something we already understand. To go on to
explain why possible worlds are not moments of time is to deepen our
understanding by exhibiting the limits of the analogy.
If the concept of a possible world can be grasped by analogy as
I think it can, then certain applications, say to an analysis of pos
sibility and necessity, or to consistency, can serve as paradigmatic
applications which extend our understanding. That is, we come to see
possible worlds as the kind of thing which can serve in an explanation.
The parallel in scientific explanation should not go unremarked.
Hence I think the view that possible worlds analysis conveys no
genuine understanding is due to a mistaken view of what constitutes an
explanation: reduction to the familiar.
Lewis' possible worlds realism is a dispensable part of his anal
ysis, but it is not dispensable in favor of some identification of pos
sible worlds with other more familiar entities. These identifications
result in too narrow a view, points which Lewis makes successfully. But
to dispense with Lewis' realism is to take a check now to avoid accepting
the cash of an imnediate metaphysical commitment. Its bearing on what
one does in the meantime may well be minimal.
2.8 Notes
1This objection was raised by Tan Simon, one of my thesis advisors,
in discussing an earlier draft of some of the material in this essay.
Goodman [27] raises similar objections to unactualized possibles. I have
no a priori objection to a ncminalistic reduction of possible worlds to
something else. The utility of possible worlds as theoretical constructs
will remain.


194
selection functions, since for a proposition and a world they select those
worlds which are of concern in evaluating a counterfactual. We could use
propositional neighborhood functions, and would have to if were were going
to consider nomormal conditional logics. Since we shall limit our at
tention to extensions of CK, such generality must give way to the relative
simplicity of selection functions.
Fran the point of view of an analysis of counterfactuals, these
choices all have one significant drawback: the selection of worlds
formally depends upon both a world and a proposition. One notable ad
vantage of Lewis' account is that the arrangement of possible worlds for
the purpose of counterfactual deliberation is stable among different
antecedents. This is a consequence of similarity ordering. The antece
dent should not condition our judgment as to which worlds are more simi
lar to the actual world.
For sufficiently strong extensions of CK we will be able to regain
part of the advantage of Lewis' antecedent-independent account of the ar
rangement of possible worlds. We will be able to define an ordering re
lation R relative to each world u, which locates possible worlds in a
corparative ordering. That this is the case, and that certain logics are
determined by a comparative ordering semantics, are among the principle
topics of this section.
In order to discuss all of the extensions of CK mentioned in the
previous section in the same context, we will first discuss them in the
context of selection function semantics. We repeat the basic definitions
of selection functions, frames, models, and the truth of conditional
formulas below. The definitions of consistency, completeness, truth of


154
Proof: For M, assume | = LKpq. Then ||Kpq|| e N*, so ||p|| A |[q|| e
Hence by (m) both ||p||,||q|| e Nj\ Thus both |=Ip and |=Lq, and so
| = KLpLq.
For R assume |=KLpLq. Then both ||p||,||q|| eNJ, so by (r) ||p|| A ||q|[
e N*\ Ifence ||Kpq [ | e N^, and so | = KLpq. QED
So augmentation of would, contrary to the (expected) distinctness of
M and R, force the satisfaction of axiom R. Indeed, using an example of
Chellas and McKinney [12, p. 382n], we can present a countermodel for R
that satisfies (m).
Figure 4.2.2
In the above diagram U = {u,v}, while is designated by the circles con
nected to u by straight lines. In brackets the atomic formulas true at
u (respectively v) are indicated. While Lp and Lq are true at u, ||Kpq||
is not in N^, so CKLpLqLKpq fails. Assuming is the same as N the
model satisfies (m). Hence the above belongs to the class of frames


178
We noted above that canonical frames are generally not uniquely
specified by condition D4.5.10(c). Suppose X c P(U^) such that X f |a|^
for all formulas a. What element of P(P(U^)) do we assign to the pair
(u,X)? Depending upon our choice we will get different canonical frames.
A particular family of canonical frames, called proper by Chellas [11,
p. 145], is identified by the following specification of N^:
E4.5.3: N(u, |a|L) = {jb|L : |b|L c and Wab e u}.
That this determines a set of canonical frames is evident by canparing
C4.5.1 to D4.5.10(c). For subsets of P(U^) such as X above, a smallest
proper canonical frame is determined by setting N^(u,X) = 0 for all such
X and all worlds u; a largest proper canonical frame is determined by
setting N^(u,X) = P(U^). (See [11, p. 140], but note the adaptation to
neighborhood models is used above.)
Completeness proofs for conditional logics are similar to those for
modal logics. In all cases, for a classical conditional logic L we may
proceed according to the following plan:
E4.5.4:
(1) Assume some formula a is valid in class of frames C .
c
(2) Show that the frame of some (proper) canonical model M.^
satisfies (c).
(3) Then a is true in M~.
(4) Hence a is a theorem of L by T4.5.1 and the corollary to
Lindenbaum's Lenina.
(5) Therefore, L is complete with respect to C .
In the simplest cases we will be able to find a proper canonical model
which satisfies the stated conditions on the frames. However, analogous


98
consistent demands that everything that would be the case were such-and-
such true must likewise be consistent.
It is precisely this that Herzberger wishes to claim for the counter-
factual conditional: "... all things that would be true under any pro
perly entertainable hypothesis are things that at least could be jointly
true" [34, p. 83]. And it is this that he shows the rejection of IA by
Lewis violates. More precisely, he shows that the collection of all
counterfactual consequents of a given entertainable antecedent is not
always consistent on Lewis' semantics without LA.
The definitions and argument that follow are adapted from
Herzberger. For simplicity assume u is fixed so we are only considering
one $u under $ [34, p. 83]:
D2.6.1: The set 0p, called the counterfactual theory for sentence p is
the set of all sentences q such that Wpq is true.
We have stated this definition for "sentences," though it could have
been stated for propositions interpreted as sets of worlds, those worlds
at which the proposition is true:
D2.6.2: The set 0p, called the counterfactual theory for proposition
p, is the set of all propositions q such that Wpq holds.
0p is then the set of all counterfactual consequents of p, the set of
sentences (propositions) that would be true if p were.
It is clear that if LNp is true, then by Lewis' truth conditions
(for vacuous truth) Wpq is true for any sentence q whatsoever. So ep
is certainly inconsistent for nonentertainable sentence p. What about
entertainable sentence p?
Using Pollock's schema, let p be the entertainable sentence "The
line is more than an inch long." Then 0p includes, on Pollock's


28
law do we choose, not whether we reject the consequent or an auxiliary
hypothesis of the "favored1' formulation of the law.
The point of relevance to the third problem is that we do favor
Ll, not because it is directly inductively confirmed as L2 is not
(which is a false claim), but rather because it has the form of a
causal law with a direction. This amounts to its being conditional in
nature, and not material as we shall see.4
In more direct reference to the third problem, one might feel
that, as laws and counterfactuals are both problematic, to analyze one
in terms of the other is not to solve the problem. The imnediate re
joinder would be, better one problematic concept than two. If counter
factuals can be analyzed in terms of laws, then we simply have to go on
to analyze laws. Rescher apparently holds this view, and regards the
analysis of counterfactuals to be laid at rest while more study is
needed of laws and confirmation theory [83, p. 164]. (In this con
nection see also the rest of Goodman [27].)
This is a problem of metaanalysis and its appearance is not new
to philosophy. One is reminded of Quine's attacks on the concepts of
analyticity, meaning, and synonony. When we have a set of systemati
cally interrelated concepts all of a problematic nature, the reduction
of all the others to one may only be an apparent, not an actual, advance.
It is my feeling that the lack of advance is most pointedly felt as a
failure to explain any of the concepts at issue. Repeated failure to
explain any one of the interrelated concepts leads to one of two out
comes:
1. "Sour grapes" in which the whole complex is given up as a bad idea.


249
[102] Wasserman, Howard C., "An Analysis of the Cbunterfactual Condi
tional," Notre Dame Journal of Formal Logic 17 (1976), pp. 395-400.
[103] Yagisawa, T., "Counterfactual Analysis of Causation and Kim's
Examples," Analysis 39 (1979), pp. 100-105.
[104] Zeman, J. Jay, Modal Logic, Clarendon, Oxford, 1973.
[105] "Pierce's Logical Graphs," Semitica 12 (1974), pp.
239-256.
[106] Orthomodular Logic, chapters 1 and 2, preprint, 1974.
[107] "Generalized Normal Logic," Journal of Philosophical
Logic 7 (1978), pp. 225-243.
[108] "Normal Implications, Bounded Posets, and the Existence
of Meets," Notre Dame Journal of Formal Logic 20 (1979), pp. 685-
688.
[109] "Two Basic Pure-Implicational Systems," Notre Dame
Journal of Formal Logic 20 (1979), pp. 674-684.
[110] "Normal, Sasaki, and Classical Implications," Journal of
Philosophical Logic 8 (1979), pp. 243-245.


92
function under this condition can be viewed as placing a well-order
on the worlds themselves. Since at such a closest p-world, either q
or Nq is true for any proposition q, UA validates conditional excluded
middle (CEM) for counterfactuals, and renders Lewis' definition of the
"might"-conditional superfluous. (Lewis does discuss other ways to de
fine this conditional on the uniqueness as simp t ion, all of which he
deems unsatisfactory [51, pp. 80-81].) As both of these consequences are
unwelcome, we shall agree with Lewis in rejecting the uniqueness assump
tion.
Adopting LA, though not UA, does not have the consequence of
validating CEM, since the closest p-permitting sphere may well contain
more than one p-world. Under Lewis' analysis it is entirely possible
for worlds to tie in comparative overall similarity to a given world.
This is the rationale for defining the "might"-conditional. Accepting LA
does have the consequence of going against the intuition Lewis draws on
in his example; however, the game of intuitions has two sides, and
Pollock [80] and Herzberger [34] bring out the other side with respect
to LA.
Starting with an adaptation of Lewis' example quoted above sup
porting the rejection of LA, Pollock shows that this rejection also re
quires the rejection of a generalization of the consequence principle
(CP = E2.4.4(c)) that seems to have as much claim for validity as the
original consequence principle. In a related fashion Herzberger argues
that rejection of LA introduces "counterfactual inconsistencies" on
Lewis' analysis. We will consider each argument in turn.
In the passage quoted above Lewis claims that it is reasonable
to suppose that for each world where the line is 1+x" long, there is


145
world semantics in our discussion of modal logic in Section 2.2; we shall
have occasion to refer to it again below in discussing the connection
between neighborhood and relational semantics.
The central notions of neighborhood semantics, indeed, of model-
theoretic semantics, are those of a frame, a model on that frame, and
truth of a formula with reference to that model. The definitions below
are adopted from Segerberg [91, pp. 13-14].
D4.2.1: A neighborhood frame F = < U,N > is an ordered pair such that
(a) U is a set (of possible worlds), and
(b) N:U -* P(P(U)) is a function.
for each u e U, is a set of subsets of U, called the set of neighbor
hoods of u.
D4.2.2: A model M = < U,N,V > on frame F = < U,N > is an ordered triple
such that V:P -> U is a function, where P is the set of propositional let
ters. (If P has been ordered by the set of natural numbers, and so is
countable, then V:N -* U will do as well.)
D4.2.3: The truth in M of a formula a at world u in U, symbolized as
|= a, is defined as follows:
(a) For all p e P, p iff u e V(p)
(b) Not |=o
(c) | = Cab iff if | = athen |=b
(d) | = La iff for some A e N^, A = { v : v e U and | = a }
We are presently restricting our attention to language CL, so no
clause is required for Wab at this tine. The set A above is referred
to as the set of worlds where formula a is true and is conventionally
symbolized ||a{[^. A formula not true by D4.2.3 will be said to be false,
symbolized j y a. We will drop the superscripts when safe to do so.


129
argument here is linked to SDA, which in my opinion greatly weakens it,
a similar point can be made without bringing in SDA or disjunctive ante
cedents .
In [74] Nute argues against the total similarity ordering of Lewis'
analysis in a way that is not linked to the acceptance of SDA, but
rather to the kinds of discriminations we are capable of making in judg
ments of comparative similarity. Therein he suggests that counterfactual
deliberation "proceeds through a search for counterexamples" [74, p. 5].
That is, we construct "reasonable" situations in which the antecedent is
true, and if the consequent is true in all of them, then the counter factual
is true. Of course, the familiar problem is determining what are the
reasonable potential counterexamples.
Nute considers the following example [74, p. 108]:
E3.2.3: If Carter had never served as Governor of Georgia, he would never
have been President of the United States.
There are many ways in which Carter might never have been Governor, of
which Nute chooses four for particular attention: he loses two bids for
Governor; he wins a Senate bid and establishes a good record; he is de
feated for the Senate; he wins a Sentate bid but establishes a poor record.
Only in the second case does Carter still become President. Nute then
claims that it is difficult enough to decide which situations are suf
ficiently like the actual situation for comparison at all without having
to rank them in similarity order as well.
The weakness of this argument is that we could easily consider all
the situations described to be equally similar to the actual world on a
sufficiently loose sense of comparative similarity. The collection would
then constitute an equivalence class, so this would not differ from Lewis'


148
Franks may then be classified based on the worlds in them:
D4.2.5:
(a) singular frames consist of singular worlds exclusively.
(b) monotonic frames consist of monotonic worlds exclusively.
(c) regular frames consist of regular worlds exclusively.
(d) normal frames consist of normal worlds exclusively.
Note that both monotonic and regular frames may contain sigular worlds,
though not so normal frames. We will denote by C the class of all frames
and by Cc the class of frames, each frame of which satisifes (possibly
multiple) condition (c). Thus, for example, C is the class of normal
frames. The figure below represents the containment relations of some
of these classes, reading the arrow as "contains."
Figure 4.2.1
In the next series of lemmas we prove the consistency of various
of the logics we have mentioned with respect to an appropriate class of
frames. (Appropriate because it will turn out to be the largest class
of frames with respect to which each is consistent.) First however we
make some observations about defined connectives, in effect expanding
upon the truth definition through our definitions of the connectives and
set theoretic considerations.
E4.2.2: For any model:
(a) ||Kab|| ||a|| A ||b||


192
pp. 75-76]. From our point of view this is a misnomer, as 0 itself does
not impose an order relation on the set of possible worlds, as will be
shown in the next section. We shall show that an order relation
R c U x U x U may be defined for logics containing CP in Section 4.8.
Relative to a given world u, the relation will be shown to be a partial
order. If MP is present, u will be R^-minimal, and if CC and MP are pre
sent u will be R -least.
u
An ordered logic containing both MP and CA is additive, thus CA is
the smallest additive logic. CK+ID+MD + CA + CB can also be partially
ordered, and we shall investigate the relationship among that logic, 0,
CD, CP, and CA in the next section.
An ordered logic containing CV will be called variably strict.
Thus V is the smallest variably strict logic. We will show that the order
relation R relative to world u for a variably strict logic is a weak
total order. If MP is present, u is R^-minimal and if CC and MP are pre
sent, u is R least. All of Lewis' V-logics are variably strict and VW
and VC are among them, VW being weakly material, and VC being material.
SS, the smallest additive material logic, is studied under that
name by Pollock [80]. By adding CEM to SS or VC we get the logic C2,
the conditional logic of Stalnaker's analysis. C2 is the smallest
singular logic, in Nute1 s terminology, which should not be confused with
the term "singular" applied to worlds of nonnormal logics in Section 4.5
and which satisfy CS: Vpq.
Figure 4.7.1 diagrams the containment relations of the logics we
have discussed above. We shall show in the next section that there is a
logic between CP and V, and consequently logics between CA and VW, and
between SS and VC, which have not previously been noted.


216
C2
SS SS + CB VC
CA CA + CB VW
CP CP + CB V
Figure 4.8.4
We close this section with sane observations about conditions (up),
(cc), and (md). From our examples, it has become evident that (mp) is
equivalent to the condition that if u e X, then u e R^/X, while (cc) is
equivalent to the condition that if u e X, then Ru/X = {u}. Thus u is
R^-minimal in a comparative order frame satisfying (mp), and u is R^-least
in a comparative order frame satisfying (cc). This has certain implica
tions for an analysis of counterfactuals which will be developed in
CHAPTER FIVE. In view of these observations we may state:
T4.8.32: CA, SS are determined by the class of partially ordered compara
tive order frames with u R -minimal for each u e U, and u R -least for
u u
each u e U, respectively.
T4.8.33: VW, VC are determined by the class of weak-totally ordered com
parative order frames with u R -minimal for each u e U and u R -least for
u u
each u e U, respectively.
We may also state, though without full proof, the following:
T4.8.34: CP + CB, CA + CB, SS + CB are determined by the class of
partially ordered comparative order frames satisfying (cb), (cb) + (mp),
and (cb) + (mp) + (cc), respectively.


66
If from the conditions on $ we were to drop C2.3.5, the uniformity
condition, then it would not generally be the case that U$u = U for all
u in U. In such a case the outer modalities as above defined would not
correspond to the modalities of S5, but rather to the system known
generally as T. (The only condition on the accessibility relation being
the reflexivity condition implied by centering.) Adopting a condition
such as
C2.4.1: For all u,v,w e U, if u e U$v and v £ U$w, then u e U$w
would impose transitivity on the accessing relation, and hence would
validate the modal logic S4, which is properly contained in S5. For our
purposes it is convenient to adopt a formulation which makes the outer
modalities the best known modalities, and hence uniformity is the standard.
We have indicated that LCpq is in a sense the "outer limit" of Wpq.
It is of interest to consider what the corresponding outer limit of Vpq
might be. One plausible suggestion is that it could be MKpq, however
inspection of the following figure reveals that that is easily counter
examp led:
Figure 2.4.1


8
I think that an "even if'-conditional can best be viewed as
denying a connection between antecedent and consequent, but this need
not make it less of a conditional, nor do we necessarily need a sepa
rate analysis for such. The "even if" is not invariably a signal that
the consequent is true and the antecedent cannot change that, but
rather a signal that this conditional is not grounded on a connection
between antecedent and consequent, but rather a lack thereof. We may
distinguish between those cases where an "even if'-conditional predic
tion is asserted as opposed to a "standard" conditional prediction.
If we believe that a certain condition's being fulfilled will either
not change something that is already the case, or not in itself pre
vent something that is going to come about, then an "even if' condi
tional prediction is appropriate. On the other hand, if a certain
condition's being fulfilled will bring something about, then a sduple
conditional prediction is appropriate. However, these are conditions
of assertibility, not truth conditions. It remains to be seen whether
a single set of truth conditions can handle both conditionals.
Up to this point we have considered conditionals, whether sub
junctive or indicative, counterfactual or semifactual, that are at
least closely associated with conditional predictions. Example El.1.2
above and the following:
El. 1.8: If kangaroos had no tails, then they would (still) be vegetar
ians,
fail to have an obvious sequential character, though El. 1.2 is pre
sumably based upon the presence of a connection between antecedent and
consequent, and El. 1.8 upon a lack thereof.


210
For (md), we must show R^/X = 0 implies (R^/Y) A X = 0. Suppose
v e (R /Y) A X. Then v e X. Since v e Ru/Y it follows that v e Dorn R^.
So, by L.8.22, Ru/X f 0. Contrapose. QED
For (co), we must show Ru/X c Y and Ru/Y c X inplies R^/X = Ru/Y.
Suppose that Ru/X c Y and R^/Y c X. By way of contradiction and without
loss of generality, suppose v e R /X and v i R^/Y. Since v e R^/X and
R /X c Y, v e Y. As v l R /Y, there is sane w e R /Y such that w f v
u u u
and wR v. Also w e Y, by (id), and w e X since R /Y c X. Since wR v
u J u u
and v e R /X, w l R /X. But as w e X, there is some z e R /X such that
u u u
zR^w and z e Y and z e X. We then have zR^w and wR v, so by transitivity
of R we have zR v. But zR v and z e X implies v t R /X, a contradiction.
For (ca), we must show R / (X U Y ) c (R^/X) U (R /Y). Suppose
v e Ru/(X U Y). Then for all w e X U Y such that w f v, wjl v. By (id)
v e X U Y. By way of contradiction assume v i R^/X and v t R /Y. Without
loss of generality assume v e X. Since v l R^/X, there is some z e R^/X
such that z ^v and zR^v. But z e X U Y and z ^ v inplies zR^v, a contra
diction. QED
Loewer [57, pp. 111-112] defines a partial order for each world u
of a frame satisfying (id), (cc), (11), (ca), and (cb). In light of this,
the following is an interesting result:
T4.8.21: There is a partially ordered comparative ordering frame that
does not satisfy (cb).
Proof: Let U = {u,v,w,y,z}. For each x e U let R be a partial ordering
of U, and specifically for world u, let R^ be given by Figure 4.8.1.


202
frame, so f(u,X) = U. We show F satisfies (mp) when X = |a| for some
sentence a. Suppose u e |a| and show u e f(u, |a|). So we must show
{b : Wab e u} c u. Suppose b is such that Wab e u. By MP, Cab e u,
and as a e u, we have b e u by deductive closure. QED
Chellas points out that neither the largest nor the smallest proper
canonical frame satisfies both (id) and (mp). This dictates a different
choice of f in the following [11, p. 143]:
T4.8.6: B = CK + ID+MPis determined by the class of dependable weakly
material frames. (LA.8.3.)
Proof: Let F = be a proper canonical frame for B that satisfies
(id) and (mp) whenever X f |a| for every sentence a. For instance, let
f(u,X) = X. The proofs of T4.8.5 and T4.8.6 suffice to show (id) and
(mp) are satisfied whenever X = |a| for some sentence a. QED
T4.8.7: G = B + CC is determined by the class of material frames.
(L4.8.3.)
Proof: Let F < U,f > be a proper canonical frame for G that satisfies
(id), (mp), and (cc) whenever X f |a| for every sentence a. Indeed, for
X f |a| for every sentence a define f by:
{u}, if u e X
f (u,X) =
X, otherwise.
That f satisfies (id) and (mp) whenever X = |a| for some sentence a is
obvious from T4.8.6. We show f satisfies (cc) whenever X = |a| for some
sentence a. Let u e |a| Show f(u, |a|) = {u}. That is, we must show
(v : {b : Wab e u} c^ v} = {u}. Suppose v is such that b e v and Wab e u.
We claim v = u. Suppose c e u. As a e u, Kac e u, so by CC, Wac e u.


159
Let u e Up and assume Np(u) / 0. World u contains all substitution
instances of T: CLpp, so for each formula a, if La e u, then a e u, by
closure under MP. So for each such formula a, |a| e Np(u) and u e |a[.
Hence u e ANp (u). QED
In view of the above lemmas and the earlier lemmas on consistency
we can state the following results concerning the determination of logics
by classes of frames:
T4.2.2
T4.2.3
T4.2.4
T4.2.5
T4.2.6
E is determined by C.
M is determined by C .
R is determined by C
mr
K is determined by C
mm
T is determined by C
'mmf
(L4.2.2 + L4.2.15)
(L4.2.4 + L4.2.16)
(L4.2.8 + L4.2.17)
(L4.2.9 + L4.2.18)
(L4.2.10 + L4.2.19)
Completeness results for modal logics containing R or K are more
easily established using the notion of relational frames, which we
briefly discussed in a previous section. It is this tactic which
Segerberg [89, 90, 91] follows; however, we shall not go so far. Above
we have explored rather basic systems expressible in CL. Before turning
to the systems expressible in CW, we shall investigate the realtionship be
tween relational and neighborhood semantics and determine whether or not
the sphere function is a neighborhood function on Lewis' interpretation.
4.3 Neighborhood and Relational Semantics
Following Segerberg [91, p. 23], we may define an alternative
relation on any regular neighborhood frame.
D4.3.1: Let F = < U,N > be a regular frame. We define an alternative
relation R c U U by:
uRv iff u is normal and v e AN .
u


7
I am denying a connection between changing the oil and the seizing up
of the engine; in fact, as Goodman [27] points out, by negating the
consequent, I seem to be denying the counterfactual "If the oil were
changed, then the engine would not seize up." Suppose you go on to
have your engine overhauled, and after we finally set out on our trip
your complaints about the cost force me to remind you:
El.1.7: Even if you had changed the oil, the engine would have seized
up.
And this seems to be fully counterfactual.
The dilemma is this: if I maintain that El. 1.6 and El. 1.7 ex
press the same proposition, one prospectively and the other retrospec
tively, then I cannot maintain that in the first the consequent is true
(making it a semifactual) and in the second it is false (making it a
counterfactual). The difficulty, of course, is that the examples are
evaluated with respect to the same hypothetical situation, but the
judgment as to their factual status (semi- or counter-) is made with
respect to different actual situations.
However, this dilemna will not prevent me from being able to ac
cept or reject the conditional(s) in question, since I do this upon
the basis of the actual situation at the time the engine was inspected
with the additional assumption that the oil is thereupon changed and
my knowledge of what generally happens to such messed up engines. And
it seems I can do this whether I am in the position of making a condi
tional prediction before we set out, a contemporary lament as we sit
beside the road with a seized up engine, or a retrospective reminder
after we have safely arrived without a seized up engine because the
overhaul took place.


89
As for the validity of E2.5.12(b) and (c), we remarked in Section
1.1 that the antecedent and consequent being true was no bar to the truth
of the conditional, though assertion of a conditional usually presupposes
the utterer does not know the antecedent to be true. Nute [67, 68, 74]
argues that (b) is counterintuitive on the grounds that in many situa
tions where both p and q happen to be true we would deny that if p were
true, then q would be because we deny that the connection between p and
q, if any, guarantees the truth of q just because p is true. That is,
we hold Kpq and VpNq as compatible. But they are not on Lewis' seman
tics as we have presented it because {u} is the smallest non-empty sphere
about u. So not every p-permitting sphere contains an Nq-world. This
argument has merit, and so one might want to retreat to weak centering,
though I will argue that perhaps the intuition Nute calls upon is actually
something else. To E2.5.12(c) I know of no objection; it would seem that
the fact that p and q are both true is sufficient prima facie evidence
for "if p were true, then q might be." We may observe that if NVpq is
asserted, then Kpq is an entirely sufficient rebuttal.
I believe Nute's argument really rests upon the following: we hold
Kpq and MKpNq as compatible, that is, though p and q are both true, it is
possible, we think, that p could be true and q false. But MKpNq is, as
we observed in Section 2.4, weaker than VpNq. It is really the former
we hold compatible with Kpq rather than the latter.
Lewis' analysis generally preserves our pre-analytic notions con
cerning the validity and invalidity of various counter factual inferences.
Coupled with the concept of pragmatic validity (due in part to Lewis
himself [54], but more to Nute [75]), it also explains why we accept many
counterfactual inferences to which there seem to be counterexamples. We


44
not be possible relative to another. Further, as we saw above, not every
possible world need be relevant to what is possible in some restricted
sense.
The technique for handling these considerations in possible world
semantics, and what gives it its considerable flexibility for permit
ting the representation of a variety of conceptions of possibility, is
the notion of a frame. A frame consists not only of a set of possible
worlds, but also of a relation among these possible worlds that repre
sents, for each world, the worlds to which it has access, or which are
it alternatives, in terms of possibility. We can alter our definitions
D2.2.1 and D2.2.2 so as to incorporate this structure:
D2.2.3: Let our: frame be F = < U,R > where U is the set of possible
worlds and R is the accessibility relation on U.
We adopt as an abbreviation for "is true at." The truth of modal
prepositions may then be defined by:
D2.2.4: Lp @ u iff for all w in U, if uRw, then p @ w.
D2.2.5: Mp @ u iff for some w in U, uRw and p @ w.
Now by specifying different conditions on relation R we model different
concepts of possibility. If, for example, we specify that for all u in
U, uRw for all w in U, then D2.2.4 and D2.2.5 reduce to D2.2.1 and D2.2.2.
The same would be acconplished by specifying that R be reflexive, transi
tive, and symmetric with every world accessing at least one other. The
possible futures situation could be modeled by an R which is just reflex
ive and transitive. These and other alternatives form the basis of a
great variety of systems of modal logic. Details need not concern us
now. If we assume that the accessing relation is universal, then every
world has access to every other.


246
[48] Lewis, C. I. and Langford, C. H., Symbolic Logic, 2nd edition,
Dover, New York, 1959,
[49] Lewis, David K., "General Semantics," Synthese 22 (1970), pp. 18-67.
[50] "Counterfactuals and Comparative Possibility," Journal
of Philosophical Logic 2 (1973), pp. 418-446.
[51] Counterfactuals, Harvard, Cambridge, 1973.
[52] "Causation," Journal of Philosophy 70 (1973), pp. 556-
567. Reprinted in Causation and Conditionals [95].
[53] "Possible-Worlds Semantics for Counterfactual Logics:
A Rejoinder," Journal of Philosophical Logic 6 (1977), pp. 359-363.
[54] "Conversational Score," Journal of Philosophical Logic,
to appear.
[55] Loeb, Louis E., "Causal Overdeterminism and Counterfactuals
Revisited," Philosophical Studies 31 (1977), pp. 211-214.
[56] Loewer, Barry, "Counterfactuals with Disjunctive Antecedents,"
Journal of Philosophy 73 (1976), pp. 531-537.
[57] "Cotenability and Counterfactual Logic," Journal of
Philosophical Logic 8 (1979), pp. 99-115.
[58] Lyon, Ardoon, "The Inmutable Laws of Nature," Proceedings of the
Aristotlean Society 77 (1976-77), pp. 107-126.
[59] Mackenzie, Nollaig, "Analyzing with Subjunctives," Dialogue (Canada)
17 (1978), pp. 131-134.
[60] Mackie, John L., "Counterfactuals and Causal Laws," in R. J. Butler,
Analytical Philosophy, Blackwell, Oxford, 1962, pp. 66-80.
[61] "Caiases and Conditions," American Philosophical Quarterly
2 (1965), pp. 245-264. Reprinted in Causation and Conditionals [95].
[62] The Cement of the Universe, Oxford University Press,
Oxford, 1974.
[63] McKay, Thomas and Van Inwagen, Peter, "Counterfactuals with Dis
junctive Antecedents," Philosophical Studies 31 (1977), pp. 353-356.
[64] Merrill, G. H., "Formalization, Possible-Worlds, and the Foundations
of Modal logic," Erkenntnis 12 (1978), pp. 305-327.
[65] Montague, Richard, "Pragmatism," in R. Klibarsky, editor, Contemporary
Philosophy I. Logic and Foundations of Mathematics, La Nuova Italie,
Florence, 1970.


168
In view of these lemmas we may assert that CK contains CR contains CM
contains CE. It is also clear that CE contains Ce and that CK contains
Ck, though it is not the case that CM or CR are extensions of Ck.
Furthermore, an alternative basis for CK, in view of L4.4.3 is:
CK = PC + RCEA + RCR + RCN.
The following lemmas enable us to give alternative bases for the
classical conditional logics. We prove only the first two, and both
there and for the remainder refer to the parallel modal lemmas. The
proofs of the first two show how easily the modal proofs that correspond
may be adapted.
L4.4.4: RCM is derivable in Ce + CM.
Proof: Assume Cab. By PC CaKab follows, but CKaba is a PC theorem.
Hence we have EaKab. Then by RCEC we have EWcaWcKab, and so we have
CWcaWcKab. Now CWcKabKWcaWcb is an instance of 01, so by PC we have
CWcaKWcaWcb. But CKWcaWcbWcb is a PC theorem, so we have by PC,
CWcaWcb. QED (Compare L4.1.5.)
L4.4.5: CM is derivable in PC + RCM.
Proof: Both CKqrq and CKqrr are PC theorems. So by RCM we have both
CWpKqrWpq and CWpKqrWpr, and hence by PC, CWpKqrKWpqWpr, which is CM.
QED (Conpare L4.1.8.)
L4.4.6: RCR is derivable in Ce + CM + CR. (Compare L4.1.2.)
L4.4.7: CR is derivable in PC + RCR. (Conpare L4.1.3.)
L4.4.8: RCN is derivable in Ge + CN. (Compare L4.1.6.)
L4.4.9: CN is derivable in Ce + RCN. (Conpare L4.1.7.)
In view of L4.4.4-9 the following are alternative bases for the
classical conditional logics noted above:


224
We conclude that, as (cp) and (co) stand in validity-preserving
equivalence in the presence of (id), (md), and (ca), CP and 00 are
deductively equivalent in CK + ID + MD + CA.
T5.1.5: CP = ID+MD + 00 + CA=ID+MD + CP + CA.
In what follows we shall argue that certain theses are requisite
for any adequate logic of counterfactuals. These minimal theses are
listed below:
E5.1.2: Minimal counterfactual theses:
LID: CLCpqWpq
MP: CWpqCpq
IMD: CLpWqp
CP: CKWpqWprWKprq
CA: CKWprWqrWApqr
All but LID and IMD are recognizable from our previous lists. We observe
that LID and IMD are the modal equivalents of ID and MD, respectively, pro
vided the logic permits the definition of a modal operator. To accept LID
and IMD is to accept ID and MD.
In addition to the above minimal theses, the following also deserve
seme consideration:
E5.1.3: Optional counterfactual theses:
CC: CKpqWpr
CB: CWApqrAWprWqr
CV: CKWpqVprWKprq
We have discussed these previously in a formal context, as well.
We assume without further argument that any adequate counterfactual
logic must be normal, that is, contain CK. We argued in Section 2.5 that
the strict conditional was stronger than the counterfactual conditional,


31
that laws are just universally quantified counterfactual conditionals
[96, p. 177]. If this be admitted, then laws share with counterfac
tual s the property of being conditional in nature, but not material in
nature.
On the other hand, laws extend to contrary-to-fact situations
where material statements do not. This is amply illustrated by El.2.4
where not only is it held that each actual pulsar is a neutron star,
but that anything else which could be a pulsar (but is not) would be a
neutron star. Pollock notes this as the subjunctive nature of laws,
which he then calls subjunctive generalizations as opposed to material
generalizations [80, pp. 13, 48]. However, at this point whether we
have two concepts to analyze, conditionality and subjunctivity, or one,
conditionality, is beside the point. In either case laws will share
with counterfactuals a characteristic which our analysis of either (or
both) must explain. For convenience I will continue to refer to the
concept of the conditional as what is to be explicated.
This places laws squarely in the analytic circle with conditionals.
And to break out of the circle and avoid "sour grapes" or "sweet lemons"
some one of the problematic concepts must be given an explanatory anal
ysis. There seem to be three approaches to the resulting problem of
breaking out of the circle, each with its attendant problems and virtues.
One approach is to accept that counterfactuals can be analyzed in
terms of laws and cotenability, and then to provide a more basic, explan
atory, analysis of laws and a resolution of the circularity in coten
ability. This is fundamentally an attack on the law problem. The anal
ysis of laws takes the form in Pollock's approach of analyzing them in
terms, not of their truth conditions, but of their justification


147
we have either both | = Laand | = Lbor neither, for each u in U. Hence,
we have ELaLb. QED
The other logics we have mentioned are consistent, and, as we shall
see, complete, with respect to subclasses of the class of all frames.
To distinguish these we make note of the following conditions which may
be placed on a frame:
E4.2.1:
(m) A A B s N implies A e N and B e N
u v u u
(r) A,B e implies A A B e N
(n) U e Nu
(q) Nu = P(U)
(s) N = 0
u
(t) N^ ^ 0 implies u e AN
Referring to these conditions we may classify worlds as follows:
D4.2.4:
(a) singular worlds satisfy (s)
(b) monotonic worlds satisfy (m)
(c) regular worlds satisfy both (m) and (r)
(d) normal worlds satisfy (m), (r), and (n) jointly
Note that monotonic and regular worlds may be singular, in which case
(m) or (m) and (r) are vacuously satisfied. Also the satisfaction of
(n) precludes (s) and conversely, so normal worlds are those for which
Nu i- 0 and A A B e iff A,B e N^, or in other worlds N is a filter in
the subset algebra of U. (This is Segerberg's definition of normal worlds
[91, p. 18]. Note further that (m) and not (s) together imply (n), thus
showing the equivalence of these two definitions of normal worlds.)


146
The following items of terminology are standard, and will be used
throughout what follows. A formula true at every u in U is true in M.
A formula true in every M for F is valid in F. If every formula in set
S of formulas is true in M, then M is a model for S. If some formula in
set S is false in M3 then M is a countermodel for S. If every formula
in set S is valid in F, the F is a frame for S. If a set S is valid
in every F in a class C of frames, then S is valid in C. Two models
are S-equivalent if they both make exactly the same formulas in S true
(and exactly the same false).
If L is a logic and C a class of frames, then L is determined by
C provided L is both consistent and complete with respect to C3 where
L is consistent with respect to C if every F in C is a frame for L,
and L is complete with respect to C if every formula valid in C is a
theorem of L. If C contains only F, then L is determined by F. If each
member of C satisfies fixed condition c, then L is determined by c.
We may make the immediate observation that every non-modal axiom,
being truth-functionally valid, is valid in the class of all neighbor
hood frames, while US and MP preserve validity, so no contradiction
can be derived. Hence,
L4.2.1: PC is consistent with respect to the class of all frames.
Thus is will be necessary only to check the modal axioms and rules of
inference to show any modal logic consistent with respect to some class
of frames. The consistency of E is thus nearly as iimediate:
L4.2.2: E is consistent with respect to the class of all frames.
Proof: We show RE preserves validity. Suppose Eab is valid (in the class
of all frames). Then for any model M on any frame ||a|| = ||b||. For if
not, then there is some world where Eab does not hold. Hence by D4.2.3(d),


Ill
2Much of the material on modal logic in this section and elsewhere
in this essay is drawn from the cited work and class notes from Jay Zeman's
classes in Modal and Quantum Logic which I attended at the University of
Florida in 1978 and 1979. I am also indebted to Bradley and Swartz [8],
Hacking [30, 31], and Lewis and Langford [48].
3The remarks in this section regarding pragmatic validity are largely
based upon a paper delivered by Nute at Valdosta State College in May, 1980.
Further suggestions emerged from subsequent conversation between myself and
Nute. Hence, any divergence, particularly into error, of my account from
his, is my responsibility.
4A fuller exposition of mechanical explanation and the mechanical
philosophy of the seventeenth and early eighteenth centuries can be found
in Kuhn [42] and most particularly in Boas [6].


Figure 4.8.1
In view of T4.8.20, we know that f defined by f(u,X) = R^/X satisfies (id),
(md), (co), and (ca) for all worlds in U. Let X = {y,v} and Y = {w, z}.
Then X U Y = {v,w,z,y}. Note that R^/X = {y,v} and Ru/Y = {w,z}, and
RU/(X U Y) = {y,z}. Then Ru/X i Ru/(X U Y) and Ru/Y i RJ (X U Y), so (cb)
is not satisfied. QED
Observe that the frame of T4.8.21 provides a countermodel to
CB: CWApqrAWprWqr. Furthermore, we could have assuned that R satisfied
(mp) or (mp) and (cc), as indeed R does. Thus the following theorem is
immediate:
T4.8.22: CP, CA, SS do not contain CB.
Loewer [57, p. 115nl6] claims that his sytem G* determined by a
selection function semantics satisfying the conditions noted above
(including (cb)) is equivalent to Pollock's system SS. Evidently, this
is not the case. The fact that the frame of T4.8.21 satisfies the axioms
and rules of inference of Pollock's axiomatization of SS [80, pp. 42-43]
can be directly verified.
The following theorems show that further containments are proper,
and that (cb) is satisfied when the ordering is a weak total order.
T4.8.23: Every weak-totally ordered comparative ordering frame satisfies
(cb).
Proof: Let F = < U,R > be a weak-totally ordered comparative order frame
and define f by f(u,X) = R^/X. As before, the selection function frame


87
D2.3.3 will render all vacuous counterfactuals false, thus invalidating
both (d) and (e) but rendering (a) valid.
We can invalidate both (b) and (c) by abandoning the centering
condition on $ (C2.3.1), since then the smallest sphere about u may not
even contain u, so it would be possible for Kpq to be true at u but
either Wpq or both Wpq and Vpq to be false. If (c) seems desirable, but
(b) not, then we can employ Lewis' condition of weak centering in place
of centering:
C2.5.2: For all 0^Ae$,ueA.
u
In such a case we have a smallest non-empty sphere about u, A$u, of
worlds indistinguishable from u in terms of our similarity ordering.
As Lewis suggests, we may want to vary the conditions anyway for
different applications of the analysis of conditionals. However, herein
we will continue with the analysis presented without remarking on the
otherwise desirable flexibility of Lewis' full analysis. I am satisfied
with the present assignment of validity and invalidity to all of the in
ferences in E2.5.12 except (a) and (d). However, my dissatisfaction is
all but evaporated by the realization that while (a) is semantically in
valid, it is always pragmatically valid on our definition (D2.5.2). And
(d), though semantically valid, is never pragmatically valid. Clause (3)
of our definition rules out vacuous premises or conclusions, so every
time Wpq is nonvacuously true, Vpq must be, and though when LNp is true,
so must Wpq be, the latter's truth is vacuous.
One could argue that E2.5.12(d) should be retained because of the
following intuitively valid argument: Suppose it is not the case that
if p were true, then q would be true. Then it seems to follow that there
are circumstances under which if p held, then Np might hold. Otherwise


187
nothing about Q and assume it is empty, thus F = < U,S,0 > is normal.
Suppose |= Wab. Then either ||a|| A U$u = 0 or not. If so, then
0 = S(u, ]ja11) c ||b||. If not, then by the truth definition D4.6.7, for
some X e $ 0 f (||a|| AX) c ||b||. But by definition, S (u, || a| |) _c
(|| a 11 AX), so S(u, ||a||) c ||b||. Suppose S (u, || a 11) c ||b||. If
S(u, ||a11) = 0, then by definition, ||aj| A U$u = 0, so |= Wab. So assume
otherwise. Let X be the smallest sphere such that ||a|| A X c ||b[| so
S (u, || a 11) = || a || AX. We may do this by definition of S and because F
is limited. Since S(u, ||a||) c ||b||, it follows that 0 f (||a|| A X) £
||b[| for some X e $ hence |= Wab. QED
The requirement that F be limited cannot be removed from T4.6.3,
else S will not produce "correct" results for truth of nonvalid conditional
formulas. If we had defined S(u,X) as the set of p-worlds in the smallest
p-permitting sphere, then for nonlimited frames, S would not even be well-
defined. The requirement that F be limited need not appear in the fol
lowing theorem, adapted from Nute [74, p. 66]. It should be noted that
Nute assumes limited sphere function frames are all that one need consider.
While this is true for questions of validity of the theorems of a logic,
it is insufficient for truth in general, so we prefer the greater gener
ality obtained by assuming the sphere function frame is not necessarily
limited.
T4.6.4: Each sphere function frame is equivalent to some normal neighbor
hood frame.
Proof: Let F = < U,$ > be a sphere function frame. Let M be any model
on F. We define a corresponding neighborhood function N: U x p(U) P(P(U))
by:


131
With reference to Lewis' much disputed line example, Pollock claims
that changing the line to 1%" long or to 2" long are distinct minimal
changes, neither containing a smaller change, nor the other, but which
likely result in worlds of differing similarity to the actual world
[80, p. 21].
Pollock's full analysis need not concern us now. He devotes much
effort to analyzing the notion of minimal change itself, which Lewis
does not and perhaps cannot do for comparative similarity. We will take
a simpler route and construct a semantics in terms of an unexplicated
notion of minimal change appropriate to Pollock's logic of the "simple
subjunctive." Nute [67, 68] and Chellas [11] use a semantics for con
ditionals based upon selection functions. We shall discuss these in more
detail in CHAPTER POUR. Lewis' account can also be reformulated in terms
of selection functions, though only in truth-preserving equivalence to
sphere functions satisfying the limit assumption [51, pp. 57-60].
Basically, a selection function is intended to pick out those worlds
that must be considered in evaluating a particular counterfactual at a
particular world. We have already seen an example of such a function in
Stalnaker's world-selection function. In more general terms we may define
a selection function as follows:
D3.2.9: Let U be a set of possible worlds. A selection function is any
function f: U x p(U) -> P(U).
We assume that all WDrIds have access to all others in D3.2.9. Otherwise
we would need to add an accessibility condition to the definition of f.
We can get Lewis' semantics for situations satisfying the limit as
sumption by adding the first four of the following conditions to our defi
nition of f:


27
have a covering law (LI), the beliefs that the match is not struck and
not lit, and the beliefs that the "auxiliary" conditions are met.
Rescher's analysis is simply that for counterfactuals for which a
covering law exists, so-called "nomo logical counterf actuals," when we
assume, contrary-to-belief, that the match is struck, we seem to have a
choice about rejecting the law, some auxiliary hypothesis or the instance
of the consequent of the law. But in fact we regard laws as inviolable,
and if we extend this to the hypotheses which "assure its applicability"
then our only choice is to reject that it did not light, thus validating
El.2.1 rather than the competing El.2.2. However, where there are two
laws, as above, this technique runs into difficulty.
This analysis works only if antecedently we have some reason to
choose LI over L2 as the relevant law. In a footnote [83, p. 161n]
Rescher notes an objection of Goodman's to this analysis which parallels
ours. According to Rescher LI and L2 are represented by Goodman as log
ically equivalent, as would several other partial contrapositives of LI
be. Each would validate a differing counterfactual. Rescher's reply
takes the form that the other equivalents to the covering law Ll may be
deductively equivalent to Ll, but are not equivalent in the context of
inductive logic. (This claim is related to a solution of Hempel's "raven
paradox.") Rescher claims the covering law Ll has primacy in the evalu
ation of counterfactuals over its "equivalents."
Rescher's response misses one point and raises another of rele
vance to our third problem. Contra Rescher, Goodman need not claim
that Ll and L2 are equivalent, with Ll being the "favored" formulation
of the covering law; rather Goodman can maintain that Ll and L2 are both
inductively confirmed laws. Then the question is indeed what relevant


232
While generally we expect an analysis to clarify seme of the puzzling
cases and otherwise to conform to our more firmly held intuitions about
the concept undergoing analysis, a well-constructed analysis can some-
tirres persuade us to revise our intuitions for the sake of greater clarity
elsewhere. The abandonment of existential import for universal generali
zations is a case in point.
We saw in the previous section that a minimally adequate logic for
the counter factual conditional must include CA, and thus conforms to a
semantics with a partial ordering of possible worlds relative to each
base world. However, comparative order frames constitute a formal seman
tics. Without something external to this formal structure upon which to
base the comparative order, that order remains merely primitive to the
semantics, and incapable of supporting an analysis. Restricting our at
tention to normal logics, Lewis, Stalnaker, and Pollock all propose
logics containing CA for the counterfactual conditional. Nute suggests
the closest normal logic to his favored nonclassical logic is GA [74,
i
p. 98]. Thus there is some agreement on a formal requirement of order.
There is also agreement on an informal requirement of order, though
based upon different ordering principles. For Lewis, it is the notion of
comparative overall similarity, which we have discussed in some detail.
For Stalnaker, it is the notion of a minimum difference, which amounts to
maximun similarity. For Pollock it is the notion of a change in the base
world, particularly those minimal changes which render the antecedent of
the conditional true. For Nute, it is the selection of worlds similar
enough to the base world to provide reasonable potential counterexamples
to the conditional. Thus we find general agreement on the presence of
some kind of ordering principle in the analyses offered, as well.


78
E2.5.7(a) succeeds became the q-permitting sphere where all
q-worlds are r-worlds is the same as the p-permitting sphere where all
p-worlds are q-worlds. If we imagine these conditionals as uttered in
the order given in a conversation, then agreeing on the truth of Wqr is
to tacitly agree upon a particular q-permitting sphere wherein the
worlds are no more dissimilar to the actual world than they have to be
to make the conditional true. If the second conditional does not require
altering this basis of evaluation, that is, if the same sphere will do to
make Wpq true, then the conclusion Wpr must follow. But it follows rela
tive to the selection of a single sphere for evaluating both conditionals.
E2.5.7(b) fails because the q-permitting sphere where all q-worlds
are r-worlds is not the same as the p-permitting sphere where all s-worlds
are p-worlds, and in the latter sphere r is not still true at those s-
worlds. If we imagine these conditionals as uttered in the order given,
then agreement on the truth of Wqr followed by the utterance of Wsr in
vites the response: "But I was not thinking of working less that way,"
thus reserving the right not to accept the inference. The first condi
tional established the boundaries of the strictness required to validate
it. The second conditional tacitly violates those boundaries.
It is these considerations that lead Nute [75] to regard hypo
thetical syllogism as pragmatically valid but not semantically valid,
depending as it does upon the context of utterance. However, it is to
be noted, that the dependence is systematic rather than merely ambiguous.
The situation diagrammed in Figure 2.5.5 is, I think, exemplified
by the following:


18
Since analyzing counterfactuals in terms of count erf actually
defined cotenability is so obviously circular, it is curious that a
recent treatment of counterfactuals seems to make a virtue of it.
Ellis provides "a unified account of three kinds of conditionals" in
terms of his notion of a "rational belief system" [19, p. 107]. (See
[18] also.) One of these conditionals is that which we have been calling
counter factual. While I am in complete agreement with Ellis' conclusion
"that indicative and subjunctive conditionals are usually variant locu
tions for the one kind of conditional which is variably strict" [19,
p. 115], and have so argued in the first section, I do not see how his
account can be construed as an analysis of conditionals, particularly of
the "variably strict" conditional, which we shall see later is an appro
priate way to refer to the counterfactual conditional. My reason for
this reservation is that his account uses the counterfactual conditional
to give the truth conditions for the counterfactual conditional in much
the same way as Goodman's self-admittedly failed account.
Ellis' truth condition for the conditional may be paraphrased as
follows [19, p. 108]:
Dl.2.2: Wpq is held true in belief system B just in case in all com
pleted extensions of a certain modification of B, B^, Nq nowhere occurs.
A (rational) belief system is essentially a partial evaluation on all
the sentences of a language; certain sentences are held true, others
false, and others withheld (i.e., no firm belief one way or the other).
There are a number of rationality requirements on a belief system,
among which is Dl.2.2 above. A completed extension of a belief system
is the replacement of all withheld evaluations by true or false evalu
ations without violating any of the rationality requirements. These


59
intended to apply to the "might"-counterfactual, also discussed in
Section 1.1. For this conditional we may use the same basic semantics,
but will require a different set of truth conditions.
Figure 2.3.4


115
[51, p. 75]. Thus Lewis' brief characterization of lawhood serves to
avoid the circle, while providing some account of the subjunctive nature
laws exhibit.
That D3.1.1 is an adequate definition of being a law is by no
means self-evident. It entirely bypasses the traditional quest for
distinguishing characteristics of lawlike, as opposed to accidental,
generalizations. Indeed, a generalization regarded as accidental in
one possible world may well be a law in another, though both are true
in their respective worlds. On the other hand, Lewis' analysis is
sympathetic to the Cartesian ideal of developing all science as a de
ductive system. D3.1.1 cannot serve as a working criterion of lawhood,
since our lamentable failure to be omniscient bars us from achieving
(or knowing we have achieved) even one complete deductive world-system.
The delineation of lav/s from nonlaws in the actual world is, however, a
matter of justification conditions rather than truth conditions, and
Lewis is offering his analysis as in terms of truth conditions.
Would Lewis' analysis of counterfactuals be greatly imparied if we
abandoned his definition of law? I think not, for one alternative open
to us is that pursued by Pollock: to provide an analysis of laws in terms
of their justification conditions. I have other objections to such an
alternative, foremost being that I am doubtful that justification con
ditions ever constitute an analysis. An analysis must surely be truth
preserving, but what is justified is not invariably true. Furthermore,
even if we were to leave the concept of law unanalyzed, we would be bet
ter off -with Lewis' analysis of counterf actuals, or one like it, than an
analysis directly in terms of laws, for the latter can provide no ex
planation of our intuitions regarding nonnomo logical counterf actuals.


162
Then the neighborhood frame corresponding to F, F', is the ordered pair
< U,N > such that
N = f 0 if u e Q
u I
[_{A : Ac U and {v : v e U and uRv} £ A}, if u i Q
That this results in a regular frame is obvious. That exactly the same
frmalas are valid in any model on the frames is also clear. Of course
our assertion here is limited to CL; we have not yet defined models for
CW in terms of neighborhood frames. It is to this conditional language
and the neighborhood semantics for it that we turn in the next section.
Then we can answer the question as to whether each system of spheres
frame (or model) corresponds to some neighborhood frame (or model).
4.4 Conditional Logic: the Systems Ce, Ck, CE, CM, CR, CK
In Section 4.1 we introduced a language CLW permitting the expres
sion of both modal and conditional sentences. In the remainder of that
section and in Section 4.2 we restricted our attention to the model frag
ment of that language, CL, and several basic logics expressible therein.
The neighborhood semantics of Segerberg [91] was restructed in a way
which conduces to the comparison of our development of CL-logics to the
Cl*J-logics to be discussed in the following sections. We shall approach
CW-logics in a somewhat round-about fashion, returning first to some of
our earlier observations about conditionals.
One reason that the strict conditional was thought tobe an improve
ment over the material conditional for the formalization of "if . ,
then ..." locutions was that it captured the sense of the antecedent
necessitating the consequent. However, we saw that Strengthening the
Antecedent, valid for the strict conditional, was not valid for the


221
(md) If Ru/X = 0, then (R/Y) A X = 0.
(co) If Ru/X c Y and Ru/Y c X, then Ru/X = RJY.
(cp) If Ru/X c Y and Ru/X c Z, then Rj (X A Z) c Y.
(ca) Ru/(X U Y) c (Ru/X) U (Ru/Y).
(cb) R /X c R /(X U Y) or R /Y c R /(X U Y).
u u u u
(cv) If Ru/X c Y, then either Ru/X c U Z or R^/ (X A Z) c Y.
Substituting "f(u,X)M for "R^/X1 in the above conditions results in the
equivalent conditions on selection function f.
We also restate the conditional theses to which we shall have
occasion to refer in this section:
E5.1.1: Conditional theses:
ID: Wpp
MP: CWpqCpq
CC: CKpqWpq
MD: CWNppWqp
CO: CFpqEWprWqr
CP: CKWpqWprWKprq
CA.: CKWprWqrWApqr
CB: CWApqrAWprWqr
CV: CKWpqVpfWKprq '
This list is distinguished from E4.7.1 by the absence of CEM, which is
characteristic of a strong total order, and the presence of CP, which
is axiom A4 in Pollock's axiomatization of SS. We have decidedly rejected
CEM and shall not discuss it further.
A definition relevant to the comparative order frames characterizing
the family of logics between the partially ordered and totally ordered
families is the following:


197
of the large number of conditions, we will generally refer to classes
of frames by the terms given in D4.8.2.
First we show the consistency of various extensions of CK with
respect to an appropriate class of frames, pause to show that not all of
the conditions are independent, then show the completeness of various
extensions of CK with respect to the appropriate class of frames. Con
sideration of time and space require that complete proofs not be pro
vided in some cases. For consistency we show the validity of the char
acteristic axiom(s) in the class of frames indicated:
L4.8.1: CK + ID is consistent wrt the class of dependable frames.
Proof: For any dependable model and any world u e U, (id) requires that
f (u, ||p||) c ||p||. Hence, by the truth definition, | = Wpp. QED
L4.8.2: CK + MP is consistent wrt the class of weakly material frames.
Proof: For any weakly material model and any world u assume |= Wpq.
If u i l|p 11, then |= Cpq trivially. If u e ||p||, then by (mp)
u e f (u, ||p 11). As f (u, ||p ||) c ||q || by assumption, u e ||q||. Hence
|= Cpq. QED
L4.8.3: B = CK + ID+MPis consistent wrt the class of dependable
weakly material frames. (L4.8.1 and L4.8.2.)
L4.8.4: G = B + CC is consistent wrt the class of material frames.
Proof: In view of L4.8.3 we show the validity of CC. For any material
model and any world u assume |= Kpq. Then both |= p and |= q, so u e ||p||.
Hence by (cc), f(u, ||p||) = {u}. As u e ||q||, f(u, ||p||) c ||q||, so |= Wpq.
QED
L4.8.5: 0 = CK + ID+MD + O0is consistent wrt the class of ordered
frames.


29
2. "Sweet lemon" in which it is blissfully agreed that the concepts
must be acquired as a set, all are basic, and none has priority over
the others.
Otherwise it remains open season on the set of interrelated concepts
with repeated efforts to explicate one of them, and then the others
will fall into line.
We have not shown that laws fall into this analytic circle, nor
considered other escapes, such as the move to dispositions. Goodman,
having analyzed counterfactuals in terms of cotenability and laws, and
noting the circle into which cotenability and counterfactuals fall,
and the problems with laws themselves, shifts the problem, like Rescher,
to laws and confirmation theory. Dispositions, as a weaker, but re
lated, notion to counterfactuals are picked -up along the way. Ellis re
duces counterfactuals to cotenability, but this is flatly circular.
With the exception of Ellis, the metalinguistic accounts including be
lief accounts such as Rescher's must look to a further clarification
of laws in order to pull off an explanatory analysis.
There is, however, good reason to believe laws do fall into this
analytic circle, particularly if we are searching for truth conditions
rather than being satisfied with justification conditions. Any belief-
based account of counterfactuals, or laws for that matter, terminates
ipso facto in justification conditions. It is not clear to me that
justification conditions ever have explanatory force, and not all truth
conditions do. At best they can codify what we do, but not illuminate
how or why it works. (For a defense of the opposing view see Pollock
[78].)


9
What suggests that these conditionals are accessible to the same
analysis as conditional predictions is (very roughly) the similarity
in the considerations that go into our judgment to accept or reject
conditionals of either type. The following is the common starting
point of many analyses of counterfactuals [13, 14, 19, 26, 36, 51, 60,
66, 67, 80, 96]. Recall that a counterf actual in our view is often a
conditional prediction viewed retrospectively against the knowledge
that the condition did not obtain at the time the prediction was ap
propriate .
To our information about the actual situation at the time of the
conditional prediction we add the assumption that the antecedent con
dition is fulfilled, changing whatever is required in our assumptions
about the actual situation to "fit" this added assumption. We then
consider what has occurred in similar situations, which knowledge may
be present for us in the form of various laws, causal and otherwise.
On this basis we determine whether or not the consequent would be re
alized in such a situation. Something roughly like this method is what
I might apply in the engine example in making the prediction that I do.
Based upon my prior knowledge of similar situations, the actual state
of the engine now under inspection, and the assumption that the oil is
not changed I predict that in the near future under normal driving the
engine will seize up.
To evaluate El. 1.2 on the other hand I may take into account my
knowledge of the physiognomy of kangroos, their skeletal structure, and
laws concerning balance and center of gravity. To this I add the as
sumption that kangaroos have no tails, changing the known facts about
kangaroos no more than necessary to accommodate this assumption. I now


144
E + R, and that neither M nor R is derivable in E + K [91, pp. 43-45].
The proof of these results requires the semantics to be developed in
the next section. The following lemmas may be proved at this time, thus
giving us another basis for K.
L4.1.9: K is derivable in E + M + R.
Proof: Assume LCpq and Lp. By PC we have KLCpqLp. An instance of R
is CKLCpqLpLKCpqp, so by MP, LKCpqp follows. Now CKCpqpq is a PC theorem,
and as RM is derivable in E + M, CKLCpqpLq follows. Hence by MP we have
Lq. QED
L4.1.10: RM is derivable in PC + K + RN.
Proof: Assume Cab. By RN LCab follows. An instance of K is CLCabCLaLb,
so by MP, CLaLb follows. QED
L4.1.11: R is derivable in PC + K + RN.
Proof: Assume KLpLq. CpCqKpq is a PC theorem. By L4.1.10, RM holds,
so we have CLpLCqKpq. By simplification of our assumption and MP we
have LGqKpq. Now CLCqKpqCLqLKpq is an instance of K, so by MP we have
CLqLKpq. Hence by simplicat ion and MP again we have IKpq. QED
In view of L4.1.9-11, an alternative basis for K is given by:
K = PC + K + RN
In some respects this is a most economical basis, requiring as it does,
only one modal rule of inference.
To show the containments noted after E4.1.5 are proper requires
use of the semantical techniques to be presented. These techniques will
be developed in the next section.
4.2 Neighborhood Semantics for Modal Logic
The most comprehensive work utilizing neighborhood semantics is
our primary reference, Segerberg [91]. We utilized relational possible


56
sets, then {u} would be the smallest sphere about u. As we shall see
below, the largest sphere in may be identified with U in virtue of
condition 5, uniformity.
Lewis carefully points out a consequence that might be overlooked
[51, p. 15]. While closure under unions and intersections guarantees
an upper bound and a lower bound on each subset of $^, it is not neces
sarily the case that these bounds must be in the subset of $u under con
sideration. That is, for X c $u (assume nonempty) while UX e $u and
AX £ $u, it does not follow that UX e X or that AX e x. This is pre
cisely analogous to the set of rational numbers less than 1 and more
than 0; there is neither a greatest nor a least element of that set of
rational numbers, but the set is bounded above and below. This is of
importance in connection with the limit assumption which we shall dis
cuss in Section 2.6.
Conditions 1 and 5 together imply that the largest sphere in $u
is U. Consider any pair of worlds u,v e U. By centering {u} e $u and
{v} e $v, hence u e U$u and v e U$v- But by uniformity, U$u = U$v, hence
u e U$v and v e U$^. But u and v were arbitrarily chosen elements of U,
so for all u e U, for all v e U, v e U$ Ifence for all u e U, U$ = U.
u u
Thus $ is universal, in the sense that every world has access to every
other at the level of the largest spheres about each. Lewis does not im
pose uniformity in general on the sphere function and allows for the pos
sibility that U$u may not exhaust U for some or all u e U. We do so in
order to provide for a simpler characterization of the modal logic this
semantics validates. (As we shall see, it is S5.)


139
The definition and results below are standard1; in all of what
follows let L be a system containing PC:
D4.1.6:
(a) L is consistent provided the negation of a tautology is
not among its theorem, otherwise inconsistent.
(b) A set S of formulas (of CLW) is L-cons is tent provided the
negation of no tautology is derivable from S in L, otherwise
L-inconsistent.
(c) A set of formulas S is maximally L-consistent provided a
formula a is in S iff Na is not in S. (Alternately, provided
for all formulas a not in S, S U (a) is L-inconsistent.)
T4.1.1: If a set S of formulas is L-consistent, then there is a
maximally L-consistent extension of S, S* (Lindenbaum's Lenina). (See
Lewis [51, p. 125] for simple proof.)
T4.1.2: Cab is derivable in L from set of formulas S iff b is derivable
in L from S U {a} (Deduction Theorem).
T4.1.3: Every maximally L-consistent set S contains every theorem of L.
T4.1.4; S |a iff every maximally L-consistent extension of S contains a.
T4.1.5: A maximally L-consistent set S" has each of the following pro
perties :
(a) Na e S* iff a i S*
(b) Kab e S" iff both a e S and b e S*
(c) Aab e S iff either a e S or b e S*
(d) Cab e S' iff if a e S~ then b e S*
(e) Eab e S iff either a,b e S or a,b i S".
The axioms and rules stated in D4.1.5 may be called nonnndal
axioms and rules of inference. Following is a list of modal axioms and


179
to modal logics, no proper canonical frame for a monotonic conditional
logic satisfies (cm). A canonical frame that does satisfy (cm) can be
found using the notion of supp lement at ion defined by Chellas [11, p. 145].
In the case of monotonic modal logics, the supplemented frame was not
itself canonical. The case is otherwise with conditional logics because
the canonical frames are not unique.
The definition and lenma below, adapted from Chellas [11, pp. 145-
146], establish the requisite techniques:
D4.5.11: Let F = < U,N > be a canonical propositional neighborhood frame
for monotonic conditional logic L. Then the supplementation of F is
F* = < U,N* > such that
N*(u,X) = {Y : Y c U and Z c Y for some Z e N(u,X)}.
A frame identical with its own supplementation is said to be supplemented.
L4.5.13: If F = < U,N > is a proper canonical frame for mono tonic condi
tional logic L, then F* = < U,N >, the supplementation of F, is a canonical
frame for L.
Proof: Assume F is as stated and show Wab e u iff |b| e N*(u, |a|).
Suppose Wab e U. Then |b| e N(u, |a|), so by definition, |b| e N*(u, |a|).
Suppose |b| e N*(u, |a|). Then for some Z e N(u, |a|), Zc |b|. Since
Z e N(u, |a|), by definition of N, for some c, Wac e u and Z = |c|. Hence
lcl £_ ib| so ( Ccb. Hence by RCM, | CWacWab. Then, as maximally L-
consistent sets are closed under MP, Wab e u. QED
For mono tonic conditional logics we will have to show:
E4.5.4(2):
(a) The frame of some proper canonical model M1 satisfies condi-
J-j
tions (c) less (cm).


Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
COUNTERFACTUALS
By
John Clyde Mayer
August 1980
Chairman: J. Jay Zeman
Major Department: Philosophy
Recently, possible world semantics has provided a basis for several
accounts of the counterfactual conditional that offer theories of counter-
factual deliberation superior to that of previous, so-called metalinguistic,
accounts. The present essay is a survey of a number of these possible
world accounts, with particular emphasis on that of David K. Lewis. It
is argued that possible world accounts more closely resemble scientific
theories than they do traditional conceptual analyses. The view that
what is at issue is a central concept of conditionality, rather than a
more narrow notion of counterfactuality or subjunctivity, is espoused.
In addition, a formal comparison of modal and conditional logics is
undertaken, using neighborhood semantics, as well as a formal comparison
of a variety of normal conditional logics, using selection function seman
tics. Various families of normal conditional logics are thereby identified.
vi


CHAPTER THREE
ORDERINGS OF POSSIBLE WORLDS
3.1 Comparative Similarity
The two notions central to Lewis' analysis of counterfactuals are
those of possible worlds and comparative overall similarity. In CHAPTER
TWO we postponed any critical examination of comparative overall similar
ity, but at sane point the questions raised by the use of such a concept
must be answered. As we indicated at the beginning of Section 2.7, objec
tions to comparative similarity may take the form of questioning whether
similarity is the appropriate principle for analyzing what goes on in
counterfactual deliberation, or alternatively, may grant the appropriate
ness, but question Lewis' particular analysis.
As his defense of his foundations indicates ([51], pp. 91-95),
Lewis believes the apparent weakness of comparative overall similarity
as a tool of analysis is its evident vagueness. In its defense he shows,
and I think correctly, that the ill-understood vagueness of counterfac
tuals themselves is appropriately matched by the well-understood vagueness
of comparative similarity. A vague, though familiar, concept is justifi
ably employed in explicating a vague, but unfamiliar, concept.
However, Lewis' defense is misaimed. The most telling objections
to comparative overall similarity arise not from its vagueness, and there
fore the possibility of our being misled in an unsystematic and random
112


CHAPTER TWO
POSSIBLE WORLDS ANALYSIS OF OOUNTERFACTUALS
2.1 Possible Worlds: History
In the period between 1968 and 1973 several analyses of the counter-
factual conditional appeared that diverged sharply from the metalinguistic
accounts that had been produced in the preceding two decades. The diver
gence was in the uniform reliance of these new approaches upon the pos
sible world semantics for modal logic introduced by Saul Kripke [40, 41].
Of these accounts the most thoroughly worked out was that of David K.
Lewis [50, 51]. It is this account which we shall discuss in some detail
in the following sections. Within the same period accounts differing in
detail from that of Lewis also made an appearance. Those of particular
O
note include Aqvist [1], Stalnaker [96], Stalnaker and Thomason [97],
and Nute [67]. We shall discuss and compare several of these accounts in
CHAPTER THREE.
Elements of Lewis' book, Counterfactuals, appeared as early as
O
1970, and Aqvist's article was earlier(1971) published by Upsalla.
Nute's article was delayed tv years in publication, so the initial de
velopments in this field were grouped into the years noted above. The
appearance of these efforts sparked a resurgence of competing accounts
of counterfactuals too numerous to mention, as well as spirited defenses
of the possible worlds account. A notable effort intended to cover the
entire range of subjunctive constructions is that of Pollock [80], which
35


42
we must operate within a context of indeterminacy, then plurality among
what is indeterminate allows room for greater future determinacy. As
the positive benefits of this way of looking at things accumulate the
critic will, of course, adopt a point of view closer to mine.1
Possible worlds make their appearance in recent efforts to pro
vide a semantics for modal logic. As a mathematical tool of formal
logic there is no serious question as to its utility. However, as a
device for the analysis of concepts there is neither a shortage of
lasers nor of critics. The only defense that I can think of for
utilizing the concept of possible worlds for analyzing other concepts
is that it, unlike some of its alternatives, provides an explanation
for how and why the concepts analyzed work the way they do (though for
a detailed defense see [64]). But this defense must wait until Section
2.7. It will be more appropriate anyway once we have a purported anal
ysis in terms of possible worlds as a concrete example.
We may take modal logic to be the logic of possibility and neces
sity. Various systems of modal logic may have application, or be de
signed to have application, outside the bounds of these notions, such
as tense logic, deontic logic, epistemic logic, etc. However, we seem
to have sufficient opportunity for variation within the bounds of pos
sibility alone: there is the logically possible, the physically pos
sible, the technologically possible, and the actually possible, to
name but a few. There are these kinds of necessity as well, in ad
dition to necessity in terms of need, or in terms of keeping certain
things fixed. (What will kill the aphids without doing in the roses?)
We should begin with our widest sense of possibility and our
narrowsst sense of necessity: that which is in seme way possible and


141
for the basis for a system will also be an expression for the logic
of that system.
The following systems are central to Segerberg's classification
of modal logics:
E4.1.3:
(a) E = PC + RE
(b) R = PC + R + EM
(c) K = R + RN
In Segerberg's terminology, a logic which contains E is called classical,
that which contains R, regular, and that which contains K, normal. The
rule RE is sometimes called Interchange or Substitution of Equivalents,
EM, Regularity (in Segerberg [91], a misnomer), and RN, Necessitation.
That R is an extension of E follows from the fact that RE is
derivable in R. For if Eab be assumed, then by PC both Cab and Cba fol
low, whence by RM, both CLaLb and CLbLa follow, whence by PC, ETnTb fol
lows. (Where we say "by PC" we mean by some rule of inference, primitive
or derived, allowed in PC.) That K is an extension of R is obvious from
the definition.
The following lemmas will aid us in providing a more uniform char
acterization of classical modal logics:
L4.1.1: RE is derivable in PC + RM.
Proof: This is proved above, in showing R to be an extension of E.
L4.1.2: RR is derivable in R.
Proof: Assume CKabc. By RM CLKabLc follows, while CKLaLbLKab is an
instance of axiom R under US. Hence by PC (transitivity of C) CKLalbLc
follows. QED


33
possible worlds semantics makes its appearance as an explanation of
why and how our informal procedure for evaluating counterfactuals works
as it does. If Stalnaker is taken as exemplifying this approach, then
laws are analyzed in terms of counterfactuals, specifically as quanti
fied counterfactual conditionals [96, p. 177]. Lewis [51] and Nute [68]
share with Stalnaker the assumption that counterfactuals are prior to
laws as far as breaking out of the analytic circle is concerned. The
concept of the conditional is provided with an analysis in terms of truth
conditions. Analyses of the second sort then break out of the circle at
the point of analyzing the concept of the conditional, not in terms of
the laws, but in terms of truth conditions based on possible worlds in a
manner as yet to be illustrated. The claim that such an analysis is also
an explanation will be defended in CHAPTER TWO.
Lewis' approach, however, while sharing with Stalnaker's the as
sumption that the conditional is prior, differs in its treatment of law,
and thus represents the third approach. Both the second and first ap
proaches affirm the analytic circle while breaking out of it. Once one
of the concepts involved has been analyzed independently of the others,
the others can then be analyzed in terms of it. Lewis removes laws from
the analytic circle in such a way that counterfactuals neither depend
directly upon them, nor do they depend upon counterfactuals. Why and
how he does this will be covered in CHAPTER THREE.
Whenever an analytic circle demonstrably exists, then an analysis
of any one of the problematic concepts has a certain prima facie vir
tue, in that analyses of the other concepts immediately follow. Com
peting analyses will then differ in a combination of respects: they


189
4.7 Extensions of CK
The analyses of counterfactual conditionals by Lewis, Stalnaker,
and Pollock all result in logics for the conditional that are ex
tensions of CK. Loewer [57] shows that one plausible reading of
Goodman's failed analysis also yields a logic for the counterfactual
which is an extension of CK. In [74] Nute discusses in some detail a
number of extensions of CK, and our discussion in the next two sections
is largely based upon his. However, Nute's preferred analysis accepts
SDA and rejects substitution of equivalents, so is an extension of Ck,
the smallest half-normal conditional logic. We shall confine our at
tention to normal conditional logics.
In our view, the ordering relations, if any, imposed upon the set
of possible worlds by a given analysis of counterfactuals constitute the
single most important distinguishing characteristic of various analyses.
The theses stated in this section and the semantics of Section 4.8 are
chosen so as to make this order more transparent. The following condi
tional theses may be added to CK to yield various extensions (we continue
to assume our language is CW):
E4.7.1; Conditional Theses:
ID: Wpp
MP: (WpqCpq
CC: CKpqWpq
MD: CWNppWqp
00: CKWpqWqpEWprWqr
CA: CKWpfWqrWApqr
CB: CWApqrAWprWqr


54
C2.3.1: {u}£$u
C2.3.2: If A,B e $ then A c B or B c A.
C2.3.3: If X c $ then UX e $ .
C2.3.4: If 0 Y c $ then AY e $u-
C2.3.5: For all u,v e U, U$u = U$v-
In Lewis' view, these conditions, or rather the first four, are
necessary for the system of spheres to be plausibly considered to convey
information about comparative similarity. In what follows we continue
to adhere to Lewis' presentation, except where noted. (See [51, pp. 14-
16].)
It is reasonable that the actual world, or any given world, is
more similar to itself than any other possible world, hence the centering
requirement. The singleton set {u} is one of the spheres about u, since
a sphere represents a set of worlds similar to u to at least a certain
degree. It should be kept in mind that a sphere does not represent a
set of worlds equally similar to u, but rather a set of worlds more
similar to u than any worlds not in the set. And u is more similar to
u than any other world, so belongs in a set by itself.
Suppose there were a pair A,B e $u which did not satisfy the
nesting condition. Then there are worlds v and w such that v e A and
v i B and w l A and w e B, as suggested by Figure 2.3.2.
Figure 2.3.2


207
Note that f(u,{v,w}) = {v} and f(u,{w,z}) = {w}, while f(u,{v,z}) = {z}.
Hence, vR w and wR z, but v#. z, so R is not transitive.
Since f(u,X) c X, (id) is satisfied. Since f(u,X) f 0 for any X
except 0, (rad) is satisfied. Checking the satisfaction of (co) is more
tedious. In most cases where f (u,X) c Y is satisfied, f(u,Y) c X is not.
We present below the three cases where f(u,X) c Y and f(u,Y) c X for some
X,Y c U (other than those where X and Y are equal).
Let X = {v} and Y = {v,w}. Then {v} = f(u,X) c_Y= {v,w} and {v} =
f(u,Y) c X = {v}. But f(u,X) = {v} = f(u,Y), so (co) is satisfied.
Let X = {w} and Y = {w,z}. Then {w} = f(u,X) c Y = {w,z} and {w} =
f(u,Y) c X = {w}. But f(u,X) = {w} = f(u,Y), so (co) is satisfied.
Let X = (z) and Y = {v,z}. Then {z} = f(u,X) c Y = {v,z} and {z} =
f(u,Y) c X = {z}. But f(u,X) = {z} = f(u,Y), so (co) is satisfied.
That (ca) is not satisfied can be observed by setting X = {v,w} and
Y = {w,z}. Then f(u,X U Y) = f(u,{v,w,z}) = {v,w,z} and f(u,X) U f(u,Y) =
f(u,{v,w}) U f(u,{w,z}) = {v,w}. Hence (ca) is not satisfied. Note that
we could have chosen f to satisfy (mp) or (mp) and (cc), as it does for
world u. The following theorems are then imnediate:
T4.8.18: 0, 00, CG do not contain CA.
T4.8.19: 0, 00, CG are properly contained in CP, CA, SS respectively.
An algebraic proof of part of these theorems is found in Nute [74, pp. 93-
94].
In order to show that a partial ordering of possible worlds implies
that (ca) is satisfied, we shall require the definition of comparative
order relations, frames, models, and truth for conditionals Wab and
(derived from D4.7.1) Vab.


20
for a complete analysis the analysandum offered is a definitional equiv
alent for the analysans. It is, of course, a serious shortcoming in a
definition for the term defined to appear in the definition itself on
the analysandum side. (We are not speaking here of a recursive defini
tion.) There are other constraints on analysis. Where clear usage is
evident in the pre-analytic concept, this usage should be preserved
under the analysis. A concept with no puzzling cases is in need of no
clarification, so no analysis; hence an analysis should go some way to
ward resolving the puzzling cases. Puzzling cases for counterfactuals
involve the Bizet and Verdi examples of Section 1.1, counterlegis,
counteridenticals, and others where there seems to be seme question as
to how to interprete the antecedent. At times even a failure to cover
all pre-analytic cases of clear usage is forgiveable if the analysis of
fers advantages in other respects. Of course, one is then rightly sub
ject to the charge of advocating a change in the concept.
A second method of analysis is to codify the rules governing the
operation of a concept. This is often expressed as making explicit the
"logic" of the concept. In this context an analysis is much like the
notion of "syntactic meaning" where the meaning of, say, a logical con
nective, is said to be implicitly given by the axioms and rules of in
ference that formalize its operation. For terms that appear as primitives
in a theory such a notion is valuable.2
An example of a relatively pure case of the first type of anal
ysis would be the analysis of knowledge as justified true belief (or
more accurately nondefectively justified true belief). More relevant
to our subject would be Lewis' analysis of a "law of nature":


229
Pollock then argues that the antecedent of E5.1.5 is true, but the
consequent is false.
Pollock suggests that if p and q are unrelated, then their disjunc
tion cannot bring about the truth of one of the disjuncts, so WApqp is
false and NWApqp is thus true. Similarly, for the second conjunct of
the antecedent of E5.1.5, if Kpr and q are unrelated, then their disjunc
tion again cannot bring about one disjunct, so NWAKprqq is true. Thus
the antecedent of E5.1.5 is true. Now consider the consequent VApqr.
Since p and q are unrelated and irrelevant to r, it follows that even if
either p or q were ture, Nr would still be true. Hence WApqNr is true,
so VApqr is false. Thus goes Pollock's argument.
Loewer [57, p. Ill] writes E5.1.5 entirely in terms of the equiva
lent cotenability statements, or in our symbology, as "might"-conditionals:
E5.1.6: CKVApqNpVAKprqNqVApqr.
This is revealing on several counts. One can argue for the truth of both
conjuncts of the antecedent of E5.1.6 without employing the principle that
a disjunction of unrelated propositions cannot bring about one of the dis
juncts. This principle is questioned by Loewer on the grounds that if
one disjunct is much more likely than the other, it would be true if the
disjunction were [57, p. Ill]:
E5.1.7: If either Nixon had been impeached or Atilla the Hun were still
alive, then Nixon would have been impeached.
We can argue for the truth of the antecedent of E5.1.6 without this
principle. It seems reasonable that if p, q, and r are unrelated, then
if either p or q were true, p might be false, and also that if either
both p and r were true or q were, then q might be false. Of course, it
also seems reasonable, that if p or q were true, p might be true. However,


40
story our world is a line of such points, and it has neighbors, other
world-lines lying parallel to ours, defining a plane of points, with
two dimensions of similarity. Either in the normal temporal direc
tions or at right angles to them, worlds farther from ours (actually
from an instant of ours) will be less similar to this instant than
world-instants closer by. If we travel along a world-line, the in
stants gradually become less similar to our starting point; likewise
if we travel along a right angle line to our world-line, a gradual
sequence of alterations will obtain. Laumer describes several such
imaginary journeys, and others combining both directions. Assuming
that we could come up with a uniform metric for this plane of world-
instants, then it seems obvious that there are both degrees of sim
ilarity to a given world-instant, and numbers of worlds that are
equally similar to our present worId-instant, though differing from
it in different respects.
Of course, Lewis does not suggest that we can actually put a
metric on the space of possible worlds. Even in the science fiction
tale above it is hard to see how that could be done. But he does
put a certain organization on that space, a topology of sorts, if
not a metric.
In Saberhagen's Mask of the Sun [85], while parallel worlds are
not accessible in terms of physical transference of the protagonist
to them, the wearing of a certain mask enables the wearer to view
future possibles. The author develops this idea in a context of
branching time: the wearer is not seeing the future but one of many
possible futures branching out from the mask's temporal point of view.
The explanation of this capacity suggested in the novel is that the


247
[66] Nagel, Ernst, The Structure of Science, Harcourt, Brace, and World,
New York, 1961.
[67] Nute, Donald, "Counterfactuals," Notre Dame Journal of Formal Logic
16 (1975), pp. 476-482.
[68] "Counterfactuals and the Similarity of Worlds," Journal
of Philosophy 72 (1975), pp. 773-778.
[69] "Conditional Logic Potpourri," Relevance Logic News 1
(1976), pp. 72-78.
[70] "The Logic of Causal Conditionals of Universal Strength,"
Relevance Logic News 1 (1976), pp. 79-91.
[71] "David Lewis and the Analysis of Count erf actuals," Nous
10 (1976), pp. 355-361.
[72] "An Incompleteness Theorem for Conditional Logic," Notre
Dame Journal of Formal Logic 19 (1978), pp. 634-636.
[73] "Simplification and Substitution of Counterfactual Ante
cedents," Philosoghia (Israel) 7 (1978), pp. 317-326.
[74] Scientific Law and Nomological Conditionals, NSF Technical
Report SOC76-08970 (1978).
[75] "Conversational Score and Conditionals," Journal of
Philosophical Logic, to appear.
[76] Topics in Conditionals, D. Reidel, Boston, 1980.
[77] Piper, H. Beam, Lord Kalvan of Otherwhen, Ace, New York, 1965.
[78] Pollock, John L., Knowledge and Justification, Princeton University
Press, Princeton, 1974.
[79] "The 'Possible-Worlds' Analysis of Counterfactuals,"
Philosophical Studies 29 (1976), pp. 469-476.
[80] Subjunctive Reasoning, D. Reidel, Boston, 1976.
[81] Prior, A. N., Time and Modality, Clarendon, Oxford, 1957.
[82] Past, Present, and Future, Clarendon, Oxford, 1967.
[83] Rescher, Nicholas, "Belief-Contravening Suppositions and the Problem
of Contrary-to-fact Conditionals," Philosophical Review 60 (1961),
pp. 176-196. Reprinted in Causation and Conditionals [95].
[84] Roberts, Don, The Existential Graphs of C. S. Peirce, Approaches to
Semiotics 27, Mouton, The Hague, 1974.


26
1. What are we to make of counterfactuals whose antecedents
deny accepted laws, so-called counterlegis?
2. How do we determine which laws are relevant, or alternately
which laws are irrelevant and would lead to incorrect evaluation of
the counterfactual?
3. Is not the concept of law itself problematic, to perhaps as
great an extent as the concept of the conditional it is being taken to
clarify?
In reference to the first problem we could refuse to countenance
counter legis, but this would be blatantly ad hoc. But if we permit
counter legis then we will be faced with the cotenability problem all
over again in terms of which laws we shall retain and which reject in
population L and F.
In reference to the second problem consider the following pairs
of laws:
LI: All matches, well-made, dry, in sufficient oxygen, and
struck, light.
L2: All matches, well-made, in sufficient oxygen, struck, and
not lit, are not dry.
On the one hand, LI would appear to validate
El.2.1: If that match had been struck, it would have lit
while L2 appears to validate:
El.2.2: If that match had been struck, it would not have been dry.
The consequents are incompatible, so both counterfactuals cannot be true,
yet what licenses our relying on one law rather than the other? It can
not be that one law is true and the other false, for both are true.
Rescher [83, p. 161] considers a similar example in explicating
counterfactuals in terms of his "belief-contravening suppositions." We


133
This definition is used by Loewer [57, p. 110] and is equivalent to
that given earlier for limited sphere functions. It should be noted
that Loewer misstates condition 03.2.2(d), using p c f(u,q) as the
second condition of the antecedent of (d).
It is condition (d) that makes for the towering of the system of
spheres about u in the sphere function equivalent to f. Without condition
(d) the remaining conditions would not generally even yield a partial
order. Of some interest then are the conditions required for definition
of a partial order such that no extension to a weak total order is gener
ally possible.
If we take the notion of minimal change for granted, then Pollock's
account requires that we select for consideration all those p-worlds that
result from some minimal change in the actual world that makes the antece
dent true. Loewer claims [57, pp. 111-112] that the semantics determining
Pollock's logic of the "simple subjunctive" is a selection function seman
tics satisfying conditions 03.2.2(a) through (c) plus (e) and (f). (We
shall question this claim in CHAPTER POUR in our broader discussion of
conditional logics there.)2 In terms of this selection function an order
relation can be defined, as by Loewer [57, p. 112], by:
D3.2.13: Let U, f be as in D3.2.9 and 03.2.2(a)-(c), (e) and (f). We
define T c U x U x U by:
vT^w iff f(u,{v,w}) = {v}.
The relation T as defined does not permit ties, since if vT w and WT v,
we have v = w, so T^ is antisymmetric. While T^ is transitive, extending
Tu by adding the condition "or f(u,{v,w}) = {v,w}" results in a relation
which is not generally transitive [57, p. 115nl5]. As f(u,{u,w}) = {u},
u is the T -least element. While f does not pick out members of an


96
E2.6.5: If the line were more than an inch long, it might be less
than 1+x" long, and it might not be less than 1+x" long,
and this precludes the truth of E2.6.4. I think that Pollock's informal
argument is unrepairable, based as it is upon the indefensible E2.6.1 or
E2.6.4.
E2.3.3. and E2.3.5 raise the possibility of another argument con
cerning a sort of consequence principle. The following "might"-
consequence principle (VCP) is valid on Lewis' semantics:
E2.6.6: Vpq
LCqr
. Vpr
as Figure 2.6.1 suggests.
Figure 2.6.1


166
for the fact that no conditional or indexed modal logic with any pre
tensions to relevance to any of our naive notions would contain RCMA.
The following considerations make this evident. Suppose Cab is
a theorem and further that sane proposition c is a-necessary, i.e.,
L c holds. Since Cab is a theorem, a reasonable sonantics might well
cl
have the set of a-worlds a subset of the set of b-worlds, i.e.,
Ilall £ ||b||. It is evident that if we take "a-necessary" to mean,
quite reasonably, "true at every a-world" that it need not follow that
c be true at every b-world as well. Now the above would be a rather
strong truth condition for a-necessity, so the inference would certainly
fail for any weaker condition. On the other hand, similar reasoning
might well persuade one that the order-reversing
RCMA'': Fran Cab infer CL^cL^c
ought to hold, since if c is b-necessary, and every a-world is a
b-world, then c is a-necessary as well.
However, even here we should pause and recall that by the deduction
theorem (T4.1.2) RCMA" can be expressed as:
From Cab and L^c infer Lac
or, in conditional notation:
From Cab and Wbc infer Wac.
But we rejected this inference pattern for counterfactuals in Section 2.5
(see E2.5.3).
Following Chellas [11, pp. 136, 180] and Nute [74, p. 56] condi
tional logics may be classified on the basis of the following systems
(conpare E4.1.5):


80
E2.5.9:
WApqr If the sun were to grow cold or we were to
have a mild winter, we would have a
. bumper crop.
. KWprWqr
. If the sun were to grow cold, we would
have a bumper crop, and if we were
to have a mild winter, we would have
a bumper crop.
Figure 2.5.6
While it is clear that Figure 2.5.6 is a countermodel to SDA, it
is not clear that E2.5.9 contains a counterexample. One could argue that
Lewis' semantics is inadequate just because it permits our example of
WApqr to be true. Rather one must take both p-worlds and q-worlds into
account in evaluating conditionals with disjunctive antecedents: Figure
2.5.6 illustrates that if we must find both a p-permitting and a q-
permitting sphere, then WApqr is not true at u. One would argue this
way if one wanted to retain SDA as a valid inference pattern for counter-
factuals.


242
In conditional predictions that form part of a scientific experi
ment, a high degree of precision is desirable. In suppositions about
tailless kangaroos or future Presidents (or past ones who might not have
been), a precise order is neither desirable nor possible. Comparative
similarity can be relaxed sufficiently to leave room for both kinds of
situations. Anything more is likely a Procrustean bed for conditionals.
What we have is less, but not fairly construed as merely primitive.


157
Suppose F satisfies (t). To show F* satisfies (t) assume if / 0.
Then N t4 0, so u e AN But AN c AN", and so u e AN",
u u u u u
Clearly, if F satisfies some permissible combination of conditions,
then F* does as well. QED
Suppose we wish to prove that monotonic logic L is complete with
respect to class C where c denotes some combination of conditions.
r cm
The general format is as follows:
E4.2.3:
(1) Assume some formula a is valid in C .
cm
(2) Show the frame of satisfies c.
(3) Then the frame of Ai£ satisfies cm, by L4.2.14.
(4) So a is true in M*.
(5) Hence a is true in by L4.2.13.
(6) Hence a is a theorem of L by T4.2.1 and the corollary to
Lindenbaun's Lemma.
(7) Therefore, L is complete with respect to
Where the logic concerned is not monotonic, but classical nevertheless,
the simpler plan suggested earlier suffices in most cases (E + Q is one
exception [91, p. 42]):
E4.2.4:
(1) Assume some formula a is valid in C .
c
(2) Show the frame of M^ satisfies c.
(3) Then a is true in Af^.
(4) Hence a is a theorem of L by T4.2.1 and the corollary to
Lindenbaun's Lemma.
(5) Hence, L is complete with respect to CQ.


241
from the point of view of tg what will happen should a particular branch
be realized. Something of this sort is apparently part of the background
to Thomason and Gupta's study of counterfactuals in the context of
branching time [100].
Considerations of similarity, the probability of divergent branches,
a shared history up to some point with the actual world, would all appear
to be of significance in selecting worlds that serve as those closest to
actuality. That the resulting order would be partial rather than total
seems likely. What is of importance may differ from situation to situation.
So our ordering of worlds may be coarser at some times than at others.
In cases such as that of our favorite example "If kangaroos had
no tails, they would topple over," minimum divergence from the laws of
nature might require that we consider worlds where kangaroos have evolved
tailless, and so have evolved balanced without them. Since I believe the
counterfactual to be true, I do not believe such worlds are the closest
to actuality. To justify my ordering I would appeal to similarity in
matters of.fact over long periods of time at the cost of a small tail-
removing miracle as Lewis suggests in another context [51, p. 75]. An
account, such as Pollock's, which assigns a high priority to preserving
laws of nature, would not do justice to tailless kangaroos.
The pattern for counterfactual deliberation that I suggest generally
conforms to Lewis' account. In considering a given counterfactual, our
ordering of possible worlds at the coarsest level is a weak total order.
Often this will suffice. As we have a more precisely determined antecedent,
this order can be refined to a weak partial order, or even a partial order.
The degree to which we refine the order we impose will depend upon the de
gree to which we are capable of making the conditional precise.


119
possible worlds. This is not an ad hoc assumption, but a necessity of
extending the concept of comparative similarity to a new domain: pos
sible worlds.
I stated previously that an account must be in terms of previously
understood concepts in order to qualify as an analysis as opposed to a
formalization of the logic of the concept. However, "previously under
stood" should not be taken to mean ordinary or coirmon, but rather inde
pendent. Because we can achieve an understanding of possible worlds in
dependently of counterfactuals, possible worlds are acceptable in an
analysis of counterfactuals, but the concept of possible worlds is surely
not an ordinary or common one.
On the other hand, comparative similarity is an ordinary concept,
but is applied to an extraordinary domain in Lewis' analysis. Our previous
understanding here cannot be by way of seme independent grasp of compara
tive similarity as applied to possible worlds, since the application
would not have arisen but for Lewis' account. And the fact that compara
tive similarity of possible worlds involves different standards than com
parative similarity of more prosaic things suggests that independent under
standing is impossible.
As in our discussion of possible worlds realism, a scientific analogy
may be of some help in seeing Lewis' account as an analysis. To under
stand the kinetic molecular theory of gases requires that we understand
f
the application of the concept of motion outside the domain of the concept
of motion in our ordinary physical-object language. This does not make
for a new concept of motion, primitive to the kinetic molecular theory,
but rather an extension, by way of analogy, of our concept of motion to
a new domain. We grasp the microscopic theory through a macroscopic


233
The dilema for all of these analyses is that the ordering principle
is not sufficiently well understood to avoid disagreement about how to ap
ply it in particular cases. And the less said about the principle, the
more room there is for disagreement. So, as Lewis says little about
comparative similarity, there is considerable latitude for alleged counter
examples. As Pollock says much about the notion of minimal change, pro
viding an analysis of it as well, the latitude is greatly reduced, but
upon the penalty of having to bring in a number of subsidiary notions of
questionable pedigree, such as the concept of a simple proposition [80,
pp. 73, 91-93]. The fact that there is informed disagreement not only
about how to apply any given ordering principle, but also about what is
the operative ordering principle, seems only to strengthen Loewer's dilem
ma with reference to all of these analyses.
Counterfactuals, it has often been remarked, are notoriously vague.
I am in agreement with Lewis, that it is neither to be expected, nor is
it desirable, that an analysis should attempt to replace our vague concept
of conditionality with a sharply delimited one. For very likely, if we
were to do so, we could not consistently accorrmodate all of our intuitions
about conditionals. Hence Lewis argues that a correspondingly vague no
tion, that of comparative similarity, is just what is required.
One difficulty with comparative overall similarity, is that we are
bound to accept a weak total order, as we argued in Section 2.3. Whatever
the merits of Pollock's counterexample to CV, it seems to me that in
counterfactual deliberation we routinely consider as reasonable, situa
tions which do not lie at a single level of comparative similarity to the
actual world, that is, are not in a single equivalence class of possible
worlds, given a fairly narrow sense of comparative similarity.


124
Let us suppose $ satisfies the limit assumption. Then for any
proposition p, either there is no p-permitting sphere, or there is a
smallest p-permitting sphere. Let us call an equivalence class p-
permitting provided that it contains a p-world. Then for a limited $
function and a comparative similarity relation R defined in terms of
it, for any proposition p and R^, there is either no p-permitting class,
or a p-permitting class which is R^-least, that is, closest to u. We
may then state truth conditions for the "would"- and "might"-conditionals
in terms of the equivalence classes as:
D3.2.4: Wpq @ u iff, either
(a) there is no p-permitting class, or
(b) for some class [w], [w] is p-permitting and every world
in [w] is a Cpq-world.
D3.2.5: Vpq @ u iff, both
(a) there is a p-permitting class, and
(b) for the R^-least p-permitting class [w], some world
in [w] is a Kpq-world.
Note that R is then a well-order of equivalence classes, that is, every
set of equivalence classes has a least element. However, R^ itself is
not a well-order, since several worlds in a set may tie for minimality,
or there may be no R -minimal worlds, as is the case if a set of worlds
violates the limit assumption.
In summary, we may conclude that Lewis' account requires a weak
total order R^ with u strictly R -minimal for each world u. This
amounts to a strong total order on equivalence classes of worlds equally
similar to u. With the limit assumption, the latter becomes a well-order
of equivalence classes. Some of the accounts of counterfactuals in terms


245
[31] "What is Logic?" Journal of Philosophy 76 (1979),
pp. 285-319.
[32] Hardegree, Gary M. "Stalnaker Conditionals and Quantum Logic,"
Journal of Philosophical Logic 4 (1975), pp. 399-421.
[33] Hazen, Allen and Slote, Michael, "Even if," Analysis 39 (1979),
pp. 35-38.
[34] Herzberger, Hans, "Counterfactuals and Consistency," Journal of
Philosophy 74 (1979), pp. 83-88.
[35] Honderick, Ted, "Causes and Causal Circumstances as Necessitating,"
Proceedings of the Aristotlean Society 78 (1977-78), pp. 63-86.
[36] Jackson, Frank, "A Causal Theory of Counterf actuals," Australasian
Journal of Philosophy 55 (1977), pp. 3-21.
[37] Kim, Jaegwon, "Causes and Counterfactuals," Journal of Philosophy
70 (1973), pp. 570-572.
[38] Kneale, William, "Natural Laws and Contrary-to-fact Conditionals,"
Analysis 10 (1950), pp. 121-125.
[39] Krabbe, Erik, "Note on a Completeness Theorem in the Theory of
Counterfactuals," Journal of Philosophical Logic 7 (1978), pp.
91-93.
[40] Kripke, Saul, "Semantical Analysis of Modal Logic I. Normal
Propositional Calculi," Zeitschrift fur Mathematische Logic und
Grundlagen der Mathematic 9 (1963), pp. 67-96.
[41] "Semantical Analysis of Modal Logic II. Non-normal
Propositional Calculi," in J. W. Addison, L. Henkin, and A. Tarski,
editors, The Theory of Models, North-Holland, Amsterdam. 1965. pp.
206-220.
[42] Kuhn, Thomas S., The Structure of Scientific Revolutions, 2nd
edition, University of Chicago Press, Chicago, 1970.
[43] Laumer, Keith, Worlds of the Imperium, Berkeley, New York, 1977.
[44] letmann, Scott K. "A General Propositional Logic of Conditionals,"
Notre Dame Journal of Formal Logic 20 (1979), pp. 77-83.
[45] Lennon, E. J. "Algebraic Semantics for Modal Logics I," Journal
of Symbolic Logic 31 (1966), pp. 46-65.
[46] "Algebraic Semantics for Modal Logics II," Journal of
Symbolic Logic 31 (1966), pp. 196-218.
[47] Lewis, C. I., "The Modes of Meaning," Philosophical and Phenomeno-
logical Research 4 (1944), pp. 236-249.


191
SS = CA + CC
VC = VW + CC
C2 = SS + CEM
Following the terminology of Nute [74, pp. 70-72] (as we do throughout
this section) we call any normal logic containing ID, dependable, and
a normal logic containing MP, weakly material. A dependable, weakly
material logic containing CC is called material. Thus G is the smallest
material logic. G is discussed under that name by Loewer as the logic
corresponding to Goodman's account of the counterfactual conditional [57,
p. 109]. B, CK + ID, and CK + MP are discussed by Chellas [11]. B can
plausibly be considered the logic of the version of Goodman's account ad
vocated by Bennett [3]. (See Loewer [57, p. 107].) MP provides an analog
of the rule of inference of modus ponens or detachment for the conditional,
while ID guards against the possibility that if p were true, p might not
be true. Without these it is difficult to see how a logic could be con
sidered to represent the counterfactual conditional, so in some sense B
is the absolutely minimal normal logic for the counterfactual conditional.
A logic containing MD will be called modal. That this appellation
is appropriate will be shown in the next section. We note at this point
that if MD is a thesis of a logic, we may define a modal operator as fol
lows:
D4.7.2: La WNaa
That "L" corresponds to the modal operator of some normal modal logic will
be shown in Section 4.8.
A dependable logic containing both ID and 00 will be called ordered.
Thus 0 is the smallest ordered logic. Nute calls logics containing 0
ordered because of the properties of the algebras which model them [74,


52
we are concerned with worlds similar to ours to a certain fixed, though
somewhat vague, degree. There are a number of equivalent ways in which
Lewis' formal semantics can be set up so as to carry information about
accessibility and similarity. I choose that which is apparently com
patible with the neighborhoods semantics to be discussed later.
For a strict conditional we need one sphere of accessibility for
each world given by the function S: U -* P(U), where denotes the
subset of U which is the sphere about u. For a variably strict con
ditional we will in general require more than one sphere about each
world u, or as we might say, u will have many neighborhoods. (Though
we shall see these are not quite the neighborhoods of neighborhood
semantics.)
D2.3.2: Let U represent the set of possible worlds, and let $: U
P(P(U)) be a function from U to the power set of the power set of U.
That is, $ assigns to each u in U, not a single subset of U (a single
sphere about u) but a set of subsets of U (a series of spheres about u).
We shall designate the image of u under $, $u where each in $u is a
single sphere about u.
Lewis places four conditions on $ in order that it plausibly carry in
formation about similarity [51, p. 14]. To these we add a fifth which
is optional for Lewis, and determines, in part, the kind of modal logic
that is validated by this framework. The conditions on $ are:
C2.3.1: (u) is an element of $u
C2.3.2: For all A,B in $ either A is a subset of B or B is a subset
of A. U
C2.3.3: If X is a subset of $ then the union of X is an element of
$ U
u


99
assumption^ all consequents of the form "The line is not 1-hc" long"
for each positive x. But as Wpp is true for all sentences p, 0p also
includes p. We have already observed that all instances of qx and p are
not simultaneously satisfiable, so 0p is inconsistent. Of course, this
conclusion rests on Pollock's dubious assumptions about the similarity
ordering of possible worlds. But even setting this objection aside,
another difficulty presents itself in that all instances of qx are
not expressible in any language whose sentences are of finite length;
that is, 0p is not denumerable. It is here that Herzberger suggests
the less demanding schema for q of "the line is less than 1+x" long" as
a case where a denumerable set may serve for the subset of 0p to which
we need to call attention vis a vis satisfiability. Presumably he has
in mind something like x = - for each positive integer n as a
2n
denumerable set of such sentences q^. For each p-world, even should
there be nondenumerably many of them, some instance of q^ will make
Wpq^ true.
Since this still rests on Pollock's assumptions, we have not shown
Lewis' semantics to permit inconsistent expressible counterfactual
theories, though the possibility is there. However, Herzberger shows
that on the level of propositions, counterfactual inconsistency is un
avoidable. In considering propositions we are not bound by considerations
of expressibility.
The following definitions introduce the terminology needed to
make Herzberger's point [34, p. 85]:
D2.6.3: For proposition p and world w, q^ is a critical consequent
for (p,w) iff both


55
Since each set in $u is a set of worlds more similar to u than any worlds
outside the set, it follows that v is more similar to u than w (from
v e A and w l A) but also that w is more similar to u than v (from
w e B and v i b). Hence nesting is required if $ is to carry informa
tion about comparative overall similarity. Of course, if we are con
cerned only with simiarity-in-certain-respects, then "similar" has dif
ferent (to be specified) senses in the apparently inconsistent state
ments above, so in that case they would be compatible. However, in
Lewis' analysis worlds are compared in terms of overall similarity to
the given world u for each system of spheres, so nesting is required.
The justification of closure under unions and intersections is
based on the following consideration: suppose there is a set of worlds
such that any world inside it is more similar to the given world u than
any world outside it. Then this set should be a sphere about u in vir
tue of being similar to u to at least a certain degree. But UX where
X c $u is just such a set, since any world w e llX^is an element of some
£ X, hence is more similar to u than any world v i S^. Since any
v i UX is not an element of any in X, it follows that any w e UX is
more similar to u than any world v i UX. Dual considerations apply in
the case of intersections.
Closure under unions and intersections has other implications also.
First, it implies that there is both a largest and smallest sphere in
$u- The smallest sphere is A$u and the largest sphere is U$u> since
$u is a subset of itself, so falls under the hypotheses of conditions
3 and 4. Since closure under unions is not restricted to nonempty sets
X, and the union of the empty set is empty, it follows that 0 e $u,
hence that A$u = 0. If we were to restrict condition 3 to nonempty


190
CV: CKWpqM-JpNrWKprq
CEM: AWpqWpNq
The Theses of E4.7.1 may all be found in other sources, principally
Nute [74], though our naming conventions differ somewhat.3 In order to
simplify the statement of two of these theses we introduce (as in CHAPTER
ONE) the following defined symbols:
D4.7.1: For any formulas a,b:
(a) Vab =df NWaNb
(b) Fab KWabWba
Thus we may restate two of our theses as:
E4.7.2:
CO: CFpqEWprWqr
CV: CKWpqVprWKprq
Extensions of CK which we shall discuss in this and the succeeding
section include:
E4.7.3: Extensions of CK:
CK + ID
CK + MP
B = CK + ID + MP
G = B + CC
0 = CK + ID + MD + CO
00 = 0 + MP
CG = 00 + CC
CP = 0 + CA
V = 0 + CV
CA = CO + CA
VW = V + MP


223
(T4.8.11 and T4.8.28). This was based upon the fact that any partially
ordered comparative order frame satisfies conditions (id), (md), (co),
and (ca) (T4.8.20) and any frame satisfying these conditions can be
partially ordered (T4.8.26). The following theorems relate (cp) to
these observations:
T5.1.3: Every partially ordered comparative order frame satisfies (cp).
Proof: We know (id), (md), and (ca) are satisfied by T4.8.20. Suppose
that R-u/X c Y and Ru/Xc Z. By way of contradiction, suppose v e R^/ (X A Z)
and v i Y. Hence v l R^/X. Because v £ X A Z by (id) and since v l El /X,
it follows that there is some w e R /X such that w v and v#. w. Then by
u u J
(id) and our assumptions, weXAYAZ. If wR v and v e X, then v e R^/X.
Hence v e Y, a contradiction. So wR v. But wR v and w e X A Z implies
u u r
v i Ru/(X A Z), a contradiction. QED
T5.1.4: Any selection function frame satisfying (id), (md), (ca), and
(cp) can be partially ordered.
Proof: Let F = < U,f > be such a selection function frame. Define a
relation R c U x U x U by
vR^w iff f(u,{v,w}) = {v}.
In view of T4.8.26, we shall show that R is transitive. Recall that anti
symmetry follows by definition, and (id) plus (md) yield reflexivity.
Suppose vR^w and wR^z and show vR^z. We may then assume that
f(u,{v,w}) = {v} and that f(u,{w,z}) = {w}. Frcm the proof of T4.8.26, we
know that (ca) implies f(u,{v,w,z}) c {v}. Again, (md) implies
f(u,{v,w,z}) = {v}. Then in accordance with (cp), we have f(u,{v,w,z}) c {v}
and f(u,{v,w,z}) c (v,z). So we have f(u,{v,w,z} A {v,z}) c {v}. Hence
f(u,{v,z}) c {v}. From (md) we infer f(u,{v,z}) f 0. Therefore,
f(u,{v,z}) = {v}. QED


49
by conjoining another proposition to it. For example, the following
inference is certainly invalid:
E2.2.2:
If this match were struck, then it would ligjht.
. If this match were soaked in water and struck,
then it would light.
Hence, in general, the inference from Wpq to WKprq is invalid. Conse
quently, as Lewis concludes, Wpq cannot be a conditional of a fixed
degree of strictness. Variability must be built into the truth con
ditions for Wpq.
There is another alternative, and that would be to take the actual
antecedent as elliptical for a more fully expressed antecedent which was
so constructed as to neutralize the problem of undermining. On this
view, "If the match were struck, then it would light" is elliptical for
"If the match were struck and not wet and well-made and in sufficient
oxygen and . then it would light." There are two arguments against
this view which I shall not elaborate: first, it is implausible that
anyone would mean the latter conditional when uttering the former [80,
p. 9]; second, this really raises the issue of cotenability over again
in a slightly altered context.
Lewis raises still another argument against this view which is
decisive as far as I am concerned. If the antecedent of a counterfac
tual is elliptical for something much more complex, then it strongly de
pends upon the exact context of utterance for its interpretation.
Which means the counterfactual is pragmatically ambiguous to a high
degree. On Lewis' view


225
and, in some sense, was the outer limit of the counter factual. If pro
position p strictly implies proposition q, then surely q would be true
if p were. Similarly, if proposition p is necessary, then it would be
true under any condition. Consequently, we should accept both LID and
LMD, and so ID and MD.
Suppose we agree that if p were true, q would be true. If we sub
sequently discover that p is true, but q is false, this effectively re
buts our original counterfactual, that is, CKpNqNWpq. The contrapositive
of this is MP. Regarding Wpq as a conditional prediction leads to the
same conclusion. (Recall the oily engine of Section 1.1.) A conditional
prediction would hardly be informative if a rule of detachment were not
operative for it. So we are bound to accept MP.
We have previously observed that counterfactuals do not generally
permit strengthening the antecedent. However, in certain cases the ante
cedent can be strengthened. One case in which it seems evident that we
can conjoin a proposition to the antecedent of a counterfactual is when
the conjunct itself is a counterfactual consequent of the original antece
dent. Suppose kangaroos would topple over if they had no tails, and kanga
roos would lose their balance if they had no tails. It seems to follow
that if kangaroos had no tails and lost their balance, then they would top
ple over. As Pollock assert^ "if r would be true if p were, then in some
sense Kpr being true is not a different circumstance from p being true"
[80, p. 39]. So we are committed to accepting CP.
Suppose that r would be true if p were and also that r would be true
if q were. We then agree that whichever of p 'and q is true, r will be
true. Thus it would sean we agree to the validity of CA. The cases
where one might doubt the validity of CA are those where Apq somehow


114
In the making of judgments of comparative overall similarity of
possible worlds to a given possible world what kinds of things would
we take into account? Certainly we would not be concerned only with
matters of particular fact, but also with what laws held at the worlds
concerned. In general, a world whose laws are identical to ours is more
similar to the actual world than a world like ours in matters of partic
ular fact, but with radically different laws. Lewis suggests that simi
larities of fact and law are balanced one against the other in deter
mining comparative overall similarity, with similarities in laws being
generally of more importance [51, p. 75]. Suppose we were to analyze
laws in terms of counterfactuals. Our analysis would clearly be circular,
since we would have to employ laws to analyze counterfactuals in the first
place.
If comparative similarity involves in part comparison of possible
worlds on the basis of laws, then Lewis is under some obligation to de
velop an analysis of laws themselves that does not reduce his larger anal
ysis to just another circle. Lewis adopts the following (slightly reworded)
definition of a law of nature (see [51, p. 73]):
D3.1.1: A contingent generalization is a law of nature at world u iff it
appears as a theorem (or axiom) in every deductive system true at u that
achieves a best combination of simplicity and strength.
By this definition, a law is just a material generalization, rather than
some other sort of generalization, say, a subjunctive one. That laws
tend to have subjunctive force is a consequence of the similarity ordering:
since laws are weighed heavily in similarity, the closer spheres about a
world u will tend to be occupied by worlds with the same laws as at u


57
There are many sphere functions which would satisfy these con
ditions. Any particular function will be determined by nonformal
considerations. We will consider some of these when we look at simi
larity again in CHAPTER THREE.
Given the sphere function and the resulting system of spheres
for each world, we can now state the truth conditions for the counter-
factual conditional Wpq. First we will follow Lewis in adopting the
convention that a world at which proposition p is true will be called
a p-world, and the convention that any sphere containing a p-world
will be called a p-permitting sphere. We may then state the truth con
ditions for the counterfactual conditional [51, p. 16]:
D2.3.2: Wpq @ u iff either
(1) there is no p-permitting sphere in $^, or
(2) for some p-permitting sphere in $ Cpq is true at
every world in that sphere. u
Thus there are two ways for a counterfactual to be true: it may be
that the antecedent is not true in any world in U$u, in which case,
with Lewis, we call the antecedent not entertainable. For example
this would be the case with the counterfactual "If the circle were
squareable, mathematicians would be confused." There are some problems
with assigning a uniform truth value to all such counterfactuals but
we will not consider that issue.
On the other hand, if there is an antecedent-permitting sphere
such that every antecedent-world in that sphere is also a consequent
world, then the counterfactual is true. A situation under which the
conditional Wpq is true in virtue of D2.3.2(2) is diagranmed in Figure
2.3.3.


24
concepts in terms of which the semantic theory is itself stated must be
ones we can apply apart from the system they are designed to validate.
If this condition is not met, then we are in precisely the same predic-
amsnt as before. We may know how to determine what sentences and rules
of inference are valid in which the concept occurs, but we do not know
how to apply the concept itself. It is in this connection that the
role of analysis as explanation arises. If we do apply the concept and
have some idea of the method employed, then the truth conditions, pos
sibly presented in the form of a semantics for sentences employing the
concept, must explain why the method works as it does.
Thus there are two constraints on a logical analysis of a concept
even when construed as a search for truth conditions:
1. The truth conditions (or semantics) must be applicable and under
standable apart from the concept analyzed by them.
2. The truth conditions must explain how and why our pre-analytic
employment of the concept works.
Recognizing these constraints reduces the apparent distinction between
a traditional meaning analysis, the first kind discussed above, and a
logical analysis. Furthermore, it gives us a means for evaluating a
purported analysis of the counterfactual conditional.
Without a clearer delineation of what constitutes the structure
S against which as background Wpq is evaluated, Wasserman' s analysis
remains incomplete. In fact since S is a truth set maximal with
respect to joint satisfiability with p [102, p. 397], S will be under
constrained in any case, as Goodman has pointed out, for all that is
required of S is that it be true and consistent with p. Hence
Wasserman's analysis will fail to explain how our pre-analytic


218
2Much of the terminology and results of Sections 4.1 and 4.2 are
to be found in Segerberg [91] and Chellas and McKinney [12]. We have
altered some terminology in the interests of greater uniformity and to
bring our modal terminology into line with the terminology to be used
with conditional logics in subsequent sections. We provide the fol-
flowing list of correspondences where we differ with those of Segerberg
or Chellas and McKinney:
THIS ESSAY SEGERBERG
RM RR
RR
RK
M ER
R C
M R: CLKabLa
R K
K K'
U 4
(m) (r) A A B e N implies (r) A c B and A e N^ implies
A e N B e N
u u
(r) (k) (k)
3We discuss a number of logics not specifically discussed in Nute
[74], though where we have discussed the same logic, we have retained his,
or historically earlier, terminology. However, we have altered some names
of specific theses. MP, CC, MD, and CO are Called MP', CS, 00.1, and GO.2
in ute. CB is (17) of Loewer [57]. Logics B and G are discussed by
Loewer, as isCK + ID+MD + CC + CA + CB which he calls G* and, or so we
believe, mistakenly identifies with SS. 0, CG, and CP are implicit in
Nute, though first explicitly discussed here. The extensions of CP, CA,
and SS formed by adding CB as an axiom are, to my knowledge, first men
tioned in the present essay.
4For sets of sentences X,Y,Z, let X*,Y*,Z* denote the set of maxi
mally L-consistent extensions of X,Y,Z, respectively. It is easily veri
fiable that for all sets of sentences X,Y,Z, the following two principles
hold: (1) ifXcY, then Y* c X*; (2) (X A Y)* c X* U Y*. The applicable
principle for the proof of T4.8.11 is an immediate consequence of (1) and
(2); (3) if X A Y c Z, then Z* c X* U Y*.
CHEUAS
RR
RC
RK
R
C
R
K
5V is complete wrt a towered sphere function frame. See Lewis [51]
for proof, and Krabbe [39] for slight correction to proof.
6 For lack of maximal depth of both selection function and neighbor
hood semantics see Gerson [25] and Nute [72, 74], There is an extension
of S4 not complete wrt any class of neighborhood frames.


142
L4.1.3: R is derivable in PC + RR.
Proof: CKpqKpq is a theorem of PC. Hence by RR, CKLpLqLKpq follows,
but that is R. QED
In view of L4.1.1-3 we may designate four systems as follows:
E4.1.4:
(a) E = PC + RE
(b) M = PC + RE + RM = E + RM
(c) R = PC + RE + RM + RR = M + RR
(d) K = PC + RE + RM + RR + RN = R + RN
A further lemma allows us to simplify these bases:
L4.1.4: EM is derivable in PC + RR.
Proof: Assume Cab. By PC CKaab follows, hence by RR we have CKLaLaLb.
But CLaKLaLa is a theorem of PC, hence by PC we have CLalb. QED
In view of L4.1.1 and L4.1.4 we may restate El.1.4 as follows:
E4.1.5:
(a) E = PC + RE
(b) M = PC + EM
(c) R = PC + RR
(d) K = PC + RR + RN
Unfortunately, RR is not derivable in PC + RN. However, RR is derivable
from RK, indeed, is the case where n = 2 while RN is the case where n = 0,
hence we may replace (d) above by
(d) K = PC + RK
thus achieving a certain neatness. A logic which contains M is called
monotonic. (This terminology is introduced by Chellas and McKinney [12].
They designate the smallest monotonic logic by R, and the smallest regular
logic by C.) It is clear that K contains R contains M contains E.


214
{v}. Suppose f(u,{v,w,z}) = 0. Then by (md), f(u,{v,w}) A {v,w,z} = 0,
a contradiction. So f(u,{v,w,z}) = {v}.
We then have f(u,{v,w,z}) c {v,z} and f(u,{v,z}) c {v,w,z}, the
latter by (id). So by (co), f(u,{v,w,z}) = f(u,{v,z}). Hence, f(u,{v,z})
{v}. QED
T4.8.27: A selection function frame satisfying (id), (md), (co), and (cv)
can be weak-totally ordered.
Proof: A proof similar to that of T4.8.26 can be devised. Condition (cv)
implies both transitivity and connectedness in the presence of the other
conditions. QED
Loewer [57, p. 115nl5] contains a proof of the transitivity of a
relation defined as in T4.8.25 based upon a set of conditions including
(ca) and (cb), as previously mentioned. There are a number of typograph
ical errors in the proof, and it uses condition (cb), which the above
proof does not. It was Loewer's otherwise well-executed article which
impelled me to study ordering relations more closely.
In view of T4.8.26 and T4.8.27 we may immediately conclude:
T4.8.28: CP is determined by the class of partially ordered comparative
order frames.
T4.8.29: V is determined by the class of weak-totally ordered comparative
order frames.
Second, consider the frame F = < U,R > with U = {u,v,w,y,z}. Define
Rx as a total order for all x e U such that x + u. Let R be given by
Figure 4.8.3.


176
LA.5.11: CK=CE + CM+CR + CNis consistent wrt C (Conpare LA. 2.9.)
mm
Both + CS and CE + OQ are of some pathological interest. In the
former Vpq is an axiom, and in the latter Wpq is. Worlds which satisfy
either (cs) or (cq) are "unpredictable" worlds, though in differing
senses: in cs-worlds anything might happen, and in cq-worlds everything
will happen. Note that Vpq also has a propositionally indexed modality
corresponding to it. If we define M q as NL^Nq, then as NL^Nq corresponds
to NWpNq, by D4.5.9, Vpq corresponds to M^q.
We also remark upon the following without proof:
L4.5.12: Ck is consistent with respect to the class of sentential models
satisfying the sentential analogs of (cm), (cr), and (cn).
The proofs of completeness for the logics remarked above are again
complicated by the fact that the most natural canonical models are insuf
ficient in the case of a monotonic conditional logic. Furthermore, the
fact that N has as a second argument a subset of U means that canonical
frames are in general not unique for given logic L. Chellas [11, pp. 144-
145] provides the requisite techniques and we rely upon his methods and
terminology in what follows.
D4.5.10: Let L be a classical conditional logic, and
(a) denote the set of maximally L-consistent formulas.
(b) |a|^ denote the maximally L-consistent sets of which a
is a member (so = |1| ).
(c) N: x p(U^) * P(P(U^)) is any function such that
Wab e u iff |b|^ e N(u, |a|^).
(d) For each p e P, V^(p) = |p¡L-


204
T4.8.9: 00 = O + MP is determined by the class of ordered weakly
material frames. (L4.8.6.)
Proof: A proof can be constructed by inspecting the proofs of T4.8.5
and T4.8.8.
T4.8.10: CG = CO + CC is determined by the class of ordered material
frames. (L4.8.7.)
Proof: See T4.8.7 and T4.8.8.
T4.8.11: CP = 0 + CA is determined by the class of ordered frames
satisfying (ca). (L4.8.8.)
Proof: Let F = < U,f > be a proper canonical frame for CP that satisfies
(id), (md), (co), and (ca) whenever X f |aj for every sentence a. In
view of T4.8.8, we will show (ca) is satisfied whenever X = |a| and
Y = |b| for some sentences a and b. We must show f(u, |a| U |b|) £
f (u, j a |) U f(u,|bj).
By CA and deductive closure, if Wac e u and Wbc e u, then WAabc e u.
Hence {c : Wac e u} A {c : Wbc e u} c {c : WAabc e u}. Then by a theorem
of maximally L-consistent extensions,4 {v : {c : WAabc e u} £ v} c
(v : (c : Wac e u} £ v} U {v : {c : Wbc e u} c v}. Therefore, f(u, |Aab|) £
f (u, | a |) U f (u, | b j). QED
T4.8.12: V = 0 + CV is determined by the class of variably strict frames.
(L4.8.9.)
Proof: Let F = < U,f > be a proper canonical frame for V that satisfies
(id), (md), (co), and (cv) whenever X ^ |a| for every sentence a. In view
of T4.8.8 we show that (cv) is satisfied whenever X = |a|, Y = |b|, and
Z = |c| for some sentences a, b, and c. Suppose f(u,|a|) c |b| and
f(u,ja|) a [c| f 0. Show f(u,|Kac|) £ |b|. (This is because |Kac| =
| a | A | c |.) From our assumptions we have Wab e u and WaNc i u. So by


81
ute [68, 74] argues for the retention of SDA based upon its
initial intuitive plausibility. However, this has the consequence that
SSE and the stronger SCE must then be rejected, since together they imply
that the counter factual is a strict conditional. To prove this we need
the following obviously valid inference patterns and sentences:
E2.5.10:
(a)
Wpq/. WpAqr
Weakening the consequent
(b)
LEAKpqKpNqp
(c)
LEKpNqANpq
DeMbrgan's Laws
(d)
LECpqANpq
(e)
WNpp/. Lp
D2.4.4
We already have that LCpq implies Wpq. The following suffices to prove
the converse, thus proving the equivalence desired:
Wpq
WAKpqKpNqq
WKpNqq
WKpNqANpq
WNANpqANpq
WNCpqCpq
LCpq
Assumed
SSE & (b)
SDA
(a)
SSE & (c)
SSE Sc (d)
(e)
Nute rejects SCE which implies SSE in order to retain SDA. It has
been argued, I think success fully, by Loewer [56], and others [53, 63],
that this is too high a price to pay for SDA. But perhaps we can save
both our reluctance to abandon SSE and our intuitions about SDA. It is
my understanding that Nute has since come to this position through ap
plying the category of pragmatic validity to SDA, while recognizing that
it is semantically invalid.


193
C2
G CG SS VC
CK + MP
00 CA VW
CK CK + ID 0 CP V
Figure 4.7.1
A similar diagram is found in Nute [74, p. 72]. We note that the fourth
and fifth columns correspond to what we may call families of partially-
ordering logics and totally-ordering logics, respectively. In fact one
may regard the horizontal dimension as one of increasing strength of the
comparison of possible worlds required by the logic, while the vertical
dimension represents increasing materiality of the counterfactual condi
tional.
Of these logics, the weakest that can plausibly be considered a
logic of the counter factual conditional is CA. We shall support this
claim by arguing in CHAPTER FIVE that a logic for the counterfactual con
ditional must reflect at least a partial ordering of possible worlds, and
have world u R^-minimal in any such partial order R relative to world u.
4.8 Semantics for Extensions of CK
We have several choices open to us for an appropriate semantics
for extensions of CK. We saw in Section 4.6 that CK is determined by
the class of all normal propositional accessibility function frames. A
variety of many extensions of CK are determined by various conditions
placed upon such functions. In this section we will call such functions


74
if we accept the pattern of E2.5.3, thus once again validating strength
ening the antecedent. The fact that the following inference is valid
may provide pause:
E2.5.5:
P
LQjq
Wqr
. r
However, consideration of the fact that at the world where p is true, Wqr
may not be true, as would be the case in the example considered, shows we
have nothing to fear on that account.
Rejection of E2.5.4(c), on the other hand, would be extremely im
plausible, for then we would be in the position of holding that q would
be true if p were, but that something entailed by q would not be true.
An argument for the validity of SCE (E2.5.4(b)), and consequently
for RET (E2.5.4(d)) which follows from it may be found in Lewis [51,
pp. 33-35]. We may note that SSE (E2.5.4(a)) also follows from SCE
(E2.5.4(b)), since LEpq entails Fpq (i.e., KWpqWqp).
The consequence principle (CP = E2.5.4(c)) is of special note
since a related principle which seems to have the same plausibility
as the consequence principle fails on Lewis' semantics. This is inti
mately tied up with the limit assumption, so we shall postpone consid
eration of it until Section 2.6.
The third inference pattern valid for both the material and strict
conditional is that of contraposition. It ought not be valid for the
counter factual conditional as the following example shows:


2
engine seized up." I feel vindicated, since the third party has made
the same claim as I did.
The four assertions above differ in tense and/or mood, yet all
seem to be making the sane claim, though from differing perspectives
and background knowledge. This sane point is made by Ellis [19] and
Stalnaker [96, p. 166]. The difference in perspective is a temporal
one; the difference in background knowledge, whether or not the oil
was changed on the occasion in question.
It is the common conditional expressed by all four of the above
sentences with which I am principally concerned. Frctn this example
as a beginning I hope to draw certain preliminary conclusions re
garding the assertability conditions and the truth conditions of such
conditionals, whether or not the indicative or subjunctive mood is es
sentially involved, in what sense the conditional is counterfactual,
whether the conditional is a material conditional, and what other
conditional-types may be sufficiently closely related to this paradig
matic one so as to be subsumable to a common analysis.
Consider the conditional "If the oil were not change, the engine
would not seize up." In the circumstance described, since the oil is
changed, both of these conditionals have false antecedents. Hence, if
they are material conditionals, they are both true. If this is the
case, then we must look elsewhere than to their truth conditions for
why our behavior is so different depending upon which we base our ac
tions on. This is highly implausible and flies in the face of the
fact that you change the oil just because you believe my assertion to
be true.


51
do not topple over. The problem of excluding these situations on the
ground that they would not be the case is precisely the problem of co-
tenability noted by Goodman.
Instead of speaking of "situation" we can speak of imagining a
world where kangaroos have no tails. This world is not the actual
world, but rather a possible world, differing from ours just enough so
that kangaroos have no tails. But the cotenability problem arises
anew in that we can consider possible worlds where kangaroos have
evolved tailless or use crutches. In practice these considerations do
not deter us from evaluating the conditional as true. What Lewis pro
vides is an analysis which explains vhy that is the case.
We are concerned only wnLth worlds very much like ours: that are
similar to a certain degree to the actual world. The more imaginative
worlds above are less similar to the actual world than are worlds where
less has changed. Lewis' suggestion is that we can conpare worlds in
terms of overall similarity to a given world [51, p. 14] in much the
same way that we can conpare facial express ions, or cities, or cultures.
Now while it seems obvious that we have a good grasp on various more
restricted notions of similarity-in-certain-respect$ it may not be clear
that we have a sufficient grasp of comparative overall similarity to
make it a useful notion for an analysis. Our first thought on it being
suggested that we can conpare cities overall, is to break this comparison
down into similarity in various respects. I shall return to the concept
of comparative overall similarity later, so for now we shall assune that
it is relatively unproblematic so as to get on with the analysis.
We can think of our accessing relation on possible worlds as
"modified" by similarity considerations. Thus for the kangaroo example


121
However, in a new domain certain factors involved in comparison are
more important than others, including factors that might not have ap
peared in the analogous usages: application to cities or faces. The
new factors can be introduced through paradigmatic examples in the
context of the theory Lewis offers.
That the resulting analysis bears more resemblance to a scien
tific theory than a definition is not a point against it but a point
in its favor. All analyses are not of the form: a sister is a female
sibling. It would be remarkable indeed if every analysis was in terms
of antecedently understood concepts, where "antecedently understood"
meant that we already understood how to apply the concepts in every
domain to which the concepts could be appropriately applied.
In view of the liberties we have taken with the notion of analysis,
it might be more appropriate to call Lewis' account a theory of counter-
factuals rather than an analysis. But then, is there really much dif
ference between the two? Traditional analyses such as Russell's theory
of definite descriptions, are philosophical theories on analogy with
scientific theories. We test such theories against our intuitions rather
than with physical experiments. It is in this tradition that Lewis is
writing.
Even though we accept possible worlds and some kind of similarity
ordering of them as appropriate for an analysis of counterfactuals and
an explanation of counterfactual deliberation, there is room for dis
agreement on the type of ordering involved. The controversy over the
Limit Assumption provides one illustration of this. In Section 3.2 we
discuss some of the varieties of order that have appeared in possible
worlds analyses of counterfactuals alternative to Lewis'.


137
so, D4.1.2(e) and D4.1.3(g)). A modal operator, L, can be introduced
into CW by definition, though the reverse is not generally the case.
For the former purpose, the following datase may be added to D4.1.3:
(h) La = WNaa
The following definition fixes the notions of derivation, proof,
and theoremhood:
D4.1.4: A formula a of CLW is derivable (deducible) (in a designated
system) from set of formulas S provided there is a finite sequence of
formulas a-,..... a such that:
1 n
(a) a is a
n
(b) for each a^ (i = 1, . n) one of the following holds:
(1) a. is an axiom
i
(2) a^ belongs to S
(3) a^ follows from one or more previous members of the
sequence by a rule of inference
The formula a is provable, and so a theorem, if the above holds
where S = 0.
We will use S |^-a to symbolize a is derivable from S in L and |^-a
to symbolize a is a theorem of L, dropping the subscript when our at
tention is focused on one logic. The logic for a system is its set of
theorems. As is well-known, the same logic, i.e., same theorems in the
same language, may belong to different systems in virtue of their dif
fering axioms or rules of inference. We have been, and will continue
to be, more concerned with logics than with systems with which they may
be axiomitized, so for our purposes we may use the terms interchangeably.
(This would be a mistake if we were interested in comparing different
axiomatizations for the same logic.)


CHAPTER ONE
WHAT ARE COUNTERFACTUALS?
1.1 A Central Concept of Conditionality
Suppose that you and I are taking a road trip in your somewhat
beat-up 1965 roadster. Before starting I do you the favor of checking
the oil, and noticing that while you do not need any additional oil,
the oil you have is very dirty, I then assert "If you do not change
the oil, your engine will seize up." Based upon my assurances that I
know about this sort of thing, you go ahead and change the oil.
Later, while we are travelling, your thoughts turn to the ad
ditional time and expense my interference has put you to, and you be
gin to wonder aloud if I really knew what I was talking about. In the
process of the conversation I assert, "If you had not changed the oil,
your engine would have seized up." Of course, this present assertion
is nothing new; I am making the same assertion now as I did in the past.
My locution has changed to reflect our new perspective.
Building a little more on this example, suppose that your doubts
about my expertise arise even as you are draining the crankcase. By
way of reassuring you I say, "If the oil were not changed, the engine
would seize up." Again, I seem to have said the same thing. Upon ar
riving at our destination, your doubts and my confidence unswerving, we
consult a third party, describing the state of the oil before we set
out. The third party says, "If that oil was not changed, then your
1


103
LA. Herzberger notes that Lewis, in a deontic application of his anal
ysis, linked LA with counterfactual consistency. Therein Lewis remarks:
Semantically, a limited value structure is one that
guarantees (except in the case of vacuity) that the
full story of how things ought to be, given some cir
cumstance, is a possible story. That is not always
so. ("Semantic Analysis for Dyadic Deontic Logic,"
p. 6, quoted in [34, p. 87n])
A defense, then, of rejecting the limit assumption, to which presumably
Lewis would subscribe, is that in some applications the full generality
of his analysis without the limit assumption is preferable, e.g., deontic
applications.
However, our primary concern is not with cases of what ought to be,
but rather what would be. Therefore, it is on this level that LA is to
be accepted or rejected. If one is inclined on the basis of Lewis' sug
gestive argument to reject LA, then one must be prepared to defend coun
terfactual inconsistency for the ordinary "would"-conditional which
thereby follows. If we grant Lewis' point above, that the full story
of what ought to be the case need not always be consistent, does this
extend to the full story of what would be the case?
Let us consider again Levs' example of the counterfactual sup
position about the line. If we agree with Lewis that there is no
closest sphere containing worlds with the line more than one inch long,
then the counterfactual theory for this supposition is propositionally
inconsistent. I shall argue that this is precisely what we should ex
pect and accept for such a supposition considered in isolation from any
particular consequent.
Consider the foHewing example of a counterfactual incorporating
Lewis' supposition:


43
that which is necessary no matter what. Focusing on the latter
characterization, the following definition seems appropriate:
D2.2.1: Lp is true iff p is true in every possible world.
Correspondingly, for possibility we have:
D2.2.2: Mp is true iff p is true in some possible world.
These definitions have the virtue of making what is not possibly false
equivalent to what is necessarily true.
To handle our notions of physically possible, technologically pos
sible, etc., we could simply substitute these terms for "possible" in
the above definitions. There are several drawbacks to this, chief
among which is that our various notions of possibility appear irreduc
ible in our definitions, while in fact the various notions of possi
bility may be systematically interrelated. Surely the physically pos
sible worlds are a subset of the logically possible, and the techno
logically possible a subset of the physically possible. For some kinds
of possibility not all the possible worlds in the broadest sense need
to be taken into consideration.
Also, there are circumstances under which the operative concept
of possibility does not determine a static set of possibilities. Con
sider for the moment possible worlds as the possible futures of this
present world. The actual world is the present instant. Relative to
it certain futures are possible. However, from the point of view of
one of those possible futures, its possible futures may not contain
some of the futures possible with respect to the present. In getting
from here to there some possibilities may be forever lost. This view
introduces two new considerations: possibility can be possibility
relative to a given world; what is possible relative to one world may


227
either simplified counterfactual, thus we observed in Section 2.5 that
SDA was pragmatically valid. That is, both disjuncts are usually enter-
tainable with respect to the same sorts of possible worlds. If so, then
SDA works, if not, then it does not.
The situation is not the same with CB. Even if the two disjuncts
are true in quite dissimilar possible worlds, then at least one of the
two simplified counterf actuals would seem to hold. However, we are al
ready discussing an analysis, presuming some sort of defensible comparison
of possible worlds, so perhaps our further discussion of CB should be left
to the next section. I can think of no counterexample to CB, and so am
inclined to accept it.
Counterexamples to CC, such as those suggested by Bennett [3] and
Bigelow [5], are drawn from the vast philosophical storehouse of unintui
tive indicative conditionals where the antecedent and consequent are ir
relevant to each other. All of these counterexamples draw on the
irrelevancy principle: if p is irrelevant to q and conversely, then p
and q cannot be conditionally related. This principle is simply false.
It is clear that an "even if"-conditional is true because the antecedent
and consequent are irrelevant to each other. The only difficulty is
that we usually do not assert any conditional when both the antecedent
and consequent are known to be true. So the operative principle is: if
p and q are irrelevant to each other, and both are true, then they are
not conditionally related.
But the revised irrelevancy principle is also false. The fact that
p and q are true is certainly sufficient evidence for the truth of the
conditional Vpq, so the only difficulty is whether Wpq is also true in
such a case. The irrelevancy principle must be restricted to just those


127
Conditions D3.2.6(c) and (d), in particular, are required for showing
this, and the definition of f reveals that f selects the S^-least
p-world, for any proposition p. That is, a counterfactual Wpq is true
at u provided the S^-least p-world is a q-world. Stalnaker's account
imposes the limit assumption plus the even stronger assumption that
for entertainable antecedent p, there is a most similar p-world.
Though Stalnaker approaches the problem from the point of view of
locating the least different world that makes the antecedent true, this
amounts to locating the most similar world that makes the antecedent true.
Thus Stalnaker's account is simply a more restricted version of Lewis'.
One can arrive at Lewis' position by asking two questions about Stalnaker's
assumptions: for a given antecedent p, why should there be just one
minimally different world? Furthermore, why should there be any minimally
different world? Lewis' argument for there being more and more similar
worlds without end is an argument for there being less and less different
worlds without end. There may, for certain antecedents, be no minimally
different world. But even assuming that there are minimally different
worlds, there could be more than one. The Bizet and Verdi example, or
Jackson's example discussed in the previous section, illustrate this pos
sibility.
However, Lewis' and Stalnaker's accounts share a significant as-
sunption: that the order of the possible worlds is connected. Both
Pollock and Nute argue for analyses that incorporate a partial order
rather than a total order. We shall first discuss their informal argu
ments, and then, for Pollock only, define an order relation in terms of
a semantics adequate to his logic of the "simple subjunctive," as he
calls our "would"-counterfactual [80, p. 42].1


113
fashion, but rather from the possibility that we may be systematically
misled. Objections of this latter sort are raised, for example, by
Barker [2], Jackson [36], and Pollock [80]. Their objections pave the
way for analyses by each of counterfactuals in terms of laws, particu
larly causal laws in the former two cases, though Pollock also employs
possible worlds.
To discuss these alternative causal theories of counterfactuals in
any detail would go beyond the scope of our present concerns. Generally
such theories adopt the view that counterfactuals are by and large nomo-
logical, that is, essentially hypothetical instantiations of laws of
nature or causal laws. Pollock's is perhaps the most thoroughly worked
out of such accounts, but as his theory is, at least in part, also a pos
sible worlds theory, we shall consider it in Section 3.2. Those theories
which view counterfactuals as nomological cannot hope to do more than as
sign counterlegals and those counterfactuals not clearly based on laws to
the realm of the irredeemably ambiguous. However, it is evident that
laws do play some role in the evaluation of counterfactuals. Hence any
account which purports to be an analysis of counterfactuals must speak
to the role of laws.
We discussed in CHAPTER ONE the analytic circle that involved both
counterfactuals and laws. One way to break out of it, we indicated, was
to analyze counterfactuals independently, and then laws in terms of coun
terfactuals. That laws are generalized counterf actual conditionals was
suggested in Stalnaker's account. Utilizing a similarity comparison rules
out this simple solution for the following reason: it would reintroduce
the very circle we are trying to avoid.


68
and invalidates inference patterns recognized as invalid, that is,
generally preserves our pre-analytic intuitions as to the "logic" of
the concept involved. Lewis' analysis withstands this test admirably
as we shall show in this section.
First I wish to define a notion of semantic entailment for the
analysis so far presented. To distinguish it from our syntactic formu
lations we shall use infix notation: the symbol "|=" is intended to
denote semantic entailment which is defined as follows:
D2.5.1: p| = qiff ||p[[c|Iq||
where ||r|| denotes the set of worlds where r is true. (This notion will
have to be relativized to a model when we shift to formal semantics in
CHAPTER POUR.)
Consider again Figure 2.3.4(a). With the help of this figure and
definition D2.4.3 it is clear that p|=qiff LCpq is true. In what follows,
where LCpq is used as a premise, substitution of p|=q will not alter our
conclusions with regard to the inference pattern. Difficulty arises only
if we define semantic entailment for sets of propositions:
D2.5.2: G| = q iff {w: if p e G, then p @ w} c ||q|¡.
We cannot take LCGq as equivalent to G| = qas the strict conditional holds
only between propositions. Nor will the conjunction of all propositions
in G work, since G could be infinite, and it is not our intention to repre
sent infinite conjunctions in our object language. (Strictly speaking, we
have not really indicated what our object language is, except informally.
This wn.ll be done in CHAPTER FOUR. I believe it wrould be distracting at
this point. In Section 2.6 we will have to make use of D2.5.2 and so it
is stated at this time.)


61
The definition of truth for the "might"-conditional is then given
as follows [51, p. 21]:
D2.3.3: Vpq @ u iff both
(1) there is some p-permitting sphere in $ and
(2) every p-permitting sphere contains at least one Kpq-world.
Note that the traditional debate over whether or not univerally quanti
fied propositions presune existence arises anew in the case of the counter-
factual conditional where it reappears as a debate over whether the con
ditional presupposes that its antecedent is entertainable. As we have
defined the "would"-conditional, it is "vacuously" true when the antece
dent is not entertainable, i.e., when there is no p-permitting sphere.
In such a case the subaltrnate "might"-conditional will be false, the
contrary "would"-conditional true, and its subaltrnate "might"-conditional
false. Hence these definitions do not support the conditional analog of
the traditional square of opposition:
Given the definitions D2.3.2 and D2.3.3, the only relation that
does obtain is the contradictory relation along the diagonals, i.e. ,
both of the following are validated:
EWpqNVpNq
EWpNqNVpq


COUNTERFACTUALS
BY
JOHN CLYDE MAYER
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1980


244
[15] Theory of Knowledge, 2nd edition, Prentice-Hall,
Englewood Cliffs, NJ, 1977.
[16] Davis, W. A., "Indicative and Subjunctive Conditionals,"
Philosophical Review 88 (1979), pp. 544-564.
[17] DeWitt, Bryce and Graham, Neill, eds., The Many-Worlds Interpreta-
tion of Quantum Mechanics, Princeton University Press, Princeton,
1973.
[18] Ellis, Brian D., "Epistemic Foundations of Logic," Journal of
Philosophical Logic 5 (1976), pp. 187-204.
[19] "A Unified Theory of Conditions," Journal of
Philosophical Logic 7 (1978), pp. 107-124.
[20] Ellis, Brian; Jackson, Frank; and Pargetter, Robert, "An Objection
to Possible-Worlds Semantics for Counterfactual Logics," Journal
of Philosophical Logic 6 (1977), pp. 355-357.
[21] Fine, Kit, "Model Theory for Modal Logic: Part I: the De Re/De
Dicto Distinction," Journal of Philosophical Logic 7 (1978), pp.
125-156.
[22] Foulis, D. J. and Randall, C. H. "Operational Statistics I," Journal
of Mathematical Physics 13 (1972), pp. 1667-1675.
[23] "Operational Statistics II," Journal of Mathematical
Physics 14 (1973), pp. 1472-1480.
[24] Fumerton, R. A., "Subjunctive Conditionals," Philosophy of Science
43 (1976), pp. 523-538.
[25] Gerson, Martin, "The Inadequacy of Neighborhood Semantics for Modal
Logic," Journal of Symbolic Logic 40 (1975), pp. 141-148.
[26] Goodman, Nelson, "The Problem of Counterfactual Conditionals,"
Journal of Philosophy 46 (1947), pp. 113-128. Reprinted in Fact,
Fiction, and Forecast [27].
[27] Fact, Fiction, and Forecast, 3rd edition, Hackett,
Indianapolis, 1979.
[28] Goosens, W. K. "Causal Chains and Counterfactuals," Journal of
Philosophy 76 (1979), pp. 489-495, erratum 579.
[29] Haack, Susan, "Recent Publications in Logic," Philosophy 51 (1976),
pp. 62-79. ^
[30] Hacking, Ian, "What is Strict Implication?" Journal of Symbolic
Logic 28 (1963), pp. 51-72.


95
a par as far as intuitive attractiveness are concerned. I am willing
to let the issue be decided by other consequences.
The second problem is that Lewis only suggests that for each world
where the line is 1+x" long there is some closer world where the line
is between 1" and 1+x" long, not that for each world where the line
is more than an inch long, every world no farther away than some world
with a line between 1" and 1+x" is a world where the line is not 1+x"
long. In other words, take a world where the line is 2" long. There
are closer worlds where it is 1 %" long, but in the same sphere with a
world where it is 1%" long there may well be worlds where it is all
sorts of lengths, including 2", more, or less. The length of lines is
not the only element of comparative similarity. So what does follow
from Lewis' assumption is:
E2.6.3: If the line were more than an inch long, it might not be 1+x"
long, and it might be 1+x" long,
for each positive value of x. It is easy to imagine that in each sphere
there are worlds equally similar to the actual vrorld with widely varying
lengths of line. But, of course, E2.6.3 does not lead to the counter
example to GCP, because the truth of both parts of E2.6.3 precludes the
truth of the corresponding "wauld"-conditionals.
Herzberger, in an aside [34, p. 85], suggests a weakening of
Pollock's schema:
E2.6.4: If the line were more than an inch long, it would be less
than 1+x" long.
It is fortunate that his purpose in doing so is not to strengthen
Pollock's argument, but to make another point which we shall discuss
shortly, for the following is certainly true:


108
I consider the issue of possible worlds realism to be on a par
with the issue of the "reality" of gravitational force as it would have
appeared to a Newtonian physicist. A divergence of views on the reduc-
ibility of possible worlds to other entities is not in itself a bar to
their serving as a basis for explanation of other concepts. Just as
Newton's laws served to unify various areas of mechanics and permit the
articulation of specific applications in terms of a single theory, so
possible worlds semantics serves to unify various areas of logic, indeed
of philosophy more generally, and to permit the articulation of various
applications in terms of a single theory. The heuristic value of pos
sible worlds alone is sufficient justification for basing an explanation
on a theory of them.
The real issue then is not whether possible worlds are "real" or
not, but whether they are dispensable without loss of explanatory power.
It makes no difference if in one particular application or another they
are dispensable, unless they are dispensable in all without losing the
unity that possible world accounts provide of the various applications.
The explication of conditionals is just one of the applications, though
one of considerable importance. The list of topics about which something
revealing can be said in terms of possible worlds is steadily increasing;
it would be pointless to try to list them all.
To grant that possible world semantics has heuristic value is to
adopt a de facto realism with regard to possible worlds. This is suf
ficient for their use in explanations of other concepts. A more subtle
attack would then be to challenge the heuristic value that has been as
sumed above.


158
Since the rest is routine, our completeness proofs will only have to
establish E4.2.3(2) ( = E4.3.4(2)) above, in order to be sufficient.
We may therefore state the following completeness lernnas:
L4.2.15: E is complete wrt C.
Proof: Clearly, the frame of Afg is in the class of all frames, so by
E4.2.4 the lemma follows.
14.2.16: M = E + M is complete wrt C .
*
Proof: Though the frame of does not satisfy (m), the frame of
does, so by E4.2.3 the lenma follows.
14.2.17: R = E+ M + Ris complete wrt C .
Proof: We need only show the frame of satisfies (r). Then the lemma
follows according to E4.2.3.
Assume A,B e Ng(u). Then by definition there are formulas a and
b such that La,Lb e u and A = |a| and B = |b|. Since La, Lb e u, by the
properties of maximally R-consistent sets, KLaLb e u. But u e Ug, so is
closed under applying MP to instances of axiom R such as CKLaLbLKab e u
with KLaLb e u, hence LKab e u. Consequently, by definition, |Kab| e
Ng(u). So by the properties of maximally R-consistent sets, Ja| A |b| e
Ng(u), but that is A A B e Ng(u) QED
L4.2.18: K = E+ M + R + Nis complete wrt C
mm
Proof: In view of L4.2.17, it is sufficient to show the frame of Mv
K
satisfies (n).
Assume u e Ug. Then LI e u, as axiom N. So by definition |1|
e NgCu), but 111 = U. QED
14.2.19: T = K + T is complete wrt C .
mmt
Proof: In view of 14.2.18 it is sufficient to show that the frame of
Mrj, satisfies (t).


12
cpq = df
"If p, then q."
Epq = df
"p if, and only if, q.
NP =df
"Not p."
Kpq df
"Both p and q."
Apq df
"Either p or q."
*df
"p is necessary."
II
£
"p is possible."
Cpq = df
"p strictly implies q.
Since in most systems of modal logic Cpq is true if, and only if, LCpq
is true, we shall usually use the latter in place of the former unless
it becomes necessary to distinguish between them. In like manner LEpq
will denote strict equivalence. We also introduce:
Fpq = ^ KWpqWqp
for what might be called counterfactual equivalence.
There is one further type of conditional for which we would expect
different truth conditions will be required.
Consider the pair of conditionals:
El.1.9: If Bizet and Verdi were compatriots, they would both be French.
El. 1.10: If Bizet and Verdi were compatriots, they would both be Italian.
We would be inclined to reject both of these conditionals but would ac
cept both of the following:
El. 1.11: If Bizet and Verdi were compatriots, they might both be French.
El. 1.12: If Bizet and Verdi were compatriots, they might both be Italian.
To symbolize the "might"-conditional we introduce:
Vpq = "If it were the case that p, then it might be the
case that q."


86
There is widespread agreement on the semantic invalidity of
transitivity for the counterfactual conditional, and nearly equally
widespread agreement on the invalidity of SDA, however, there are
several other inference patterns upon which agreement is not as easy
to find. Fortunately, adopting different positions on these does not
greatly alter the nature of the conditional in question. Of the fol
lowing inferences, the first and last are invalid on Lewis' preferred
semantics, as well as on that version which we have presented:
E2.5.12:
Wpq
Kpq
Kpq
LNp
LNp
. Vpq
. Vpq
. Vpq
. Vpq
. Vpq
(a)
(b)
(c)
(d)
(e)
The invalidity of (a) and (e) and the validity of (d) rest upon the
fact that the conditional Wpq can be vacuously true, while Vpq is never
vacuously true, given the preferred definitions. If a given proposition
p is not entertainable, then Hip is true, as is Wpq, but Vpq is then false,
since there is no p-permitting sphere at all. The validity of (b) and (c)
follows from the definitions directly, as {u} is then a p-permitting
sphere in which every p-world is also a q-world.
Depending upon one' s tastes, the validity or invalidity of these
inferences can be altered by minor alterations in the conditions upon the
$ function, or the truth conditions for the "would"- and "might"-
conditionals. By switching to the alternate definitions of Wpq and
Vpq (D2.3.4 and D2.3.5) we preserve their interdefinability, but make
(a) and (e) valid, and (d) invalid. If we are agreeable to abandoning
the interdefinability of Wpq and Vpw, then the combination of D2.3.4 and


50
It consigns to the wastebasket of contextually resolved
vagueness something much more amenable to systematic
analysis than most of the rest of the mess in that waste
basket. [51, p. 13]
Hence in the subsequent section we will present Lewis' analysis of the
counter factual conditional as a "variably strict" conditional with the
expectation that ambiguity will be kept within more acceptable bounds.
2.3 Lewis' Analysis of the Counterfactual Conditional
That the counterfactual conditional is a variably strict condi
tional is generally admitted by advocates of both the metalinguistic
and possible worlds approaches. The difference in treatment involves
in part fixing the boundary between semantic and pragpnatic ambiguity
to which effect we quoted Stalnaker earlier. The essential requirement
is to have a system for resolving as much of the apparent ambiguity in
conditionals as possible. Once ambiguity is seen as systematic, it is
no longer a barrier to analysis. (See Lewis [49].)
To motivate his construction of the truth conditions for counter-
factuals, Lewis [51, p. 1] invites us to consider what has become my
favorite example: "If kangaroos had no tails, they would topple over."
Previously we said that in evaluating this conditional we construct for
ourselves a situation, altering what we must in what we take to be
actually true, in which kangaroos have no tails. If it is the case in
such a situation that kangaroos topple over, then the conditional is
true. However, as we saw, a critical element of this procedure is the
"altering what we must," for it is surely possible to imagine situations
in which kangaroos have no tails, but evolved that way, so also evolved
a structure which otherwise permits them to be balanced. Or, a situation
in which kangaroos have no tails, but have learned to use crutches, so


38
and neighborhood semantics has been carried out by Nute [74]. Sections
4.4-4.7 are considerably indebted to these last two mentioned works.
The logician, qua logician, is interested in the adequacy of a
formal semantics quite apart from whether or not it affects an analysis
of the concepts the system is intended to formalize. It is possible to
regard relational possible world semantics as providing an analysis of
important concepts of modality (see Bradley and Swartz [8], Foulis and
Randall [22, 23], and Zeman's application and development of the latter
[106, 107]). This is of interest to the logician qua philosopher. The
semantics for conditional logic developed by Lewis, Stalnaker, Nute,
and others are intended as analyses, and must therefore meet constraints
we suggested in Section 1.2 and will explore further in Section 2.7 and
CHAPTER THREE. What is lacking in the application of neighborhood
semantics to conditional logic is the idea of an analysis as opposed to
a formalization. Though Nute [74] conpares various semantics for con
ditionals on a formal basis, his comparison of them for philosophical
adequacy is limited to those which have been explicitly developed with
analysis in mind: Stalnaker, Lewis, and himself. The philosophical
adequacy of neighborhood semantics for the analysis of conditionals is
largely unexplored. It is this gap in exposition for which we ultimately
hope to provide seme filling.
The history of possible worlds, even such a sketchy account as
that offered here, would be incomplete without mention of two further
areas where possible worlds have had an impact: science fiction and
the interpretation of quantum mechanics. We will return to the former
in more detail, but of the latter we only note that the Everett-Wheeler
interpretation of quantum mechanics employs the notion of a "reality


109
The flexibility of possible world semantics in characterizing and
distinguishing among various modal logics illustrates what can be done
with such a semantics that could not have been done without it. That
one can thereby also provide characterizations of tense logic, deontic
logic, and epistemic logic, to name a few, is illustrative of the breadth
of applicability of the possible worlds approach. The volume of fruitful
applications creates a prima facie case for its heuristic value.
Arguments to the contrary must then take the form of attempts to
show that a possible worlds analysis is systematically misleading: in
each case the yielding of a concept to possible worlds analysis is a
purely formal exercise which conveys either no genuine understanding,
or else systematic misunderstanding. In effect, one is supposedly
dazzled by the formal scheme without closely considering its merits as
an analysis. That is, it does not tell us what our, say, modal concepts
"really" are.
But the claim that possible worlds convey real understanding
only if they can themselves be reduced to something we already under
stand is the claim that no concept can be explained unless in terms of
concepts we already understand. If that were true, then the concept
of gravitational force never explained anything. It was a mistake to
suppose that it did.
On the contrary, we grasp the concept of possible worlds not on
the basis of reducing them to something else, say, maximally consistent
sets of sentences, to which they are deemed equivalent, but rather in
much the same way we grasp any unfamiliar, yet primitive, concept: by
analogy and through recognizing paradigmatic applications of the concept.
Hence to say that possible worlds are analogous to moments of time, that


60
Figure 2.3.4 represents various distributions of the truth values
of propositions p and q over the system of spheres $ We shall refer
to them in what immediate follows.
Figure 2.3.3 and figures 2.3.4(a) and 2.3.4(d) all represent cases
where Wpq is true. As we shall see in Section 4, 2.3.4(a) represents a
case where LCpq is true as well. In 2.3.4(d) q is true at u and p being
true will not change this, that is "even if p were true, q would be."
This is a case where an "even-if"-conditional is appropriate, but no
further definition of truth for the conditional in question, the "would"-
conditional, is required. The condition that every p-world be a q-warld
in some p-permitting sphere suffices.
In both 2.3.4(b) and 2.3.4(c), the conditional Wpq is not true.
The situations differ in that in 2.3.4(b) there is seme p-permitting
sphere where some of the p-worlds are q-worlds, but not all, while in
2.3.4(c) there is a p-permitting sphere where none of the p-worlds are
q-worlds. In the latter case, 2.3.4(c), WpNq is true, but in the former
case, 2.3.4(b), neither Wpq nor WpNq is true. For entertainable antece
dents, Wpq and WpNq behave as contraries: they may not both be true,
but they may both be false. In traditional quant if icational logic, to
each contrary corresponds a subcontrary. Similarly, in Lewis' analysis,
to each of the contraries Wpq and WpNq corresponds a subcontrary Vpq and
VpNq. These are the aforementioned "might"-conditionals. In those cases
where p is entertainable, but neither Wpq nor WpNq are true, then both
Vpq and VpNq are true. Recall the pair of conditionals concerning Bizet
and Verdi as an example.


ACKNOWLEDGEMENTS
I am indebted to my advisor, J. Jay Zeman, for introducing me to
modal logic, conditional functions, possible worlds, and a more empir
ical view of logic than I might otherwise have had. For better or
worse, I am (only slightly) less a Platonist as a result.
To my typist, Joyce Pande lis, I owe a particular debt of thanks
for the time and effort she has spent in working with me through revi
sions, corrections, and re-revisions. As a philosopher in her own right,
her conments and suggestions have been valuable, and only occasionally
mischievous.
in


71
E2.5.2:
Wrp
Wpq
. Wrq
If J. Edgar Hoover had been bom Russian, he would
have been a Conmmist.
If he had been a Comnunist he would have been a
traitor.
. If he had been bom Russian, he would have been
a traitor.
(This particular example is due to Stalnaker [96, p. 173].)
If one is inclined to try to retain transitivity in spite of such
counterexamples because it is felt it is essential to any conditional
function, the following observation should be persuasive of abandoning
the attempt. Recall that LCpq entails Wpq, LCKprp is valid, hence WKprp
is valid. To abandon this would be either to abandon the validity of
LCKprp, or the entailment, so WKprp is valid. If transitivity is ac
cepted, then from Wpq and the valid WKprp, WKprq follows, thus again
validating strengthening the antecedent.
It will not be sufficient to strengthen the first premise of
transitivity to a strict conditional (or entailment) as the following
counterexample shows (due to Lewis [51, p. 32]):
E2.5.3:
LCpq
Wqr
Necessarily, if I started at 5 a.m., I started
before 6 a.m.
If I had started before 6 a.m., I would have
arrived before noon.
If I had started at 5 a.m., I would have arrived
before noon.
Figure 2.5.1(b) is a countermodel to this inference pattern. The infer
ence fails in the following situation: suppose that I actually started
just a few minutes after 6 a.m. and actually arrived just after noon, so


239
contexts. Thus, in giving a counterfactual analysis of causation, Lewis
introduces comparative similarity and warns:
I have not said just how to balance the respects of
comparison against each other, so I have not said just what
our relation of comparative similarity is to be. Not for
nothing did I call it primitive. But I have said what sort
of relation it is, and we are familiar with relations of
that sort. [52, p. 183]
Comparative similarity is not merely primitive. But also, what is of im
portance in comparing worlds for the purpose of counterfactual deliberation
is not self-evidently part of our ordinary notion of overall similarity.
Some factors are manifest only in applying the concept to possible worlds.
Thus it is proper to call Lewis' account a theory of counterfactuals.
Comparative overall similarity is not the full answer to the compara
tive order of possible worlds that occurs in counterfactual reasoning.
Lewis restricts his account to determinism, but it seems clear that in
determinism would not produce a total order. Consider the following morbid
example. A new method of execution has been devised by a modem Koko. The
condemned prisoner is placed in an execution chamber where either soma one
of two drugs will be administered to cause death, or the prisoner will be
released, depending upon the outcome of a certain quantum event. Say that
there are equal probabilities that in an hour some atom in a certain
sample of radioactive material will decay by one of two modes whose pro
ability sun is nearly unity, and a slight probability that no decay will
occur. Label these E^, E^, and E^. If either E^ or occurs, the
prisoner will die. The death will be prolonged and painful if E-^ occurs,
swift and painless if E2 occurs. If E^, no decay, occurs, the prisoner
will be released.


199
L4.8.10: CA = CP + MP is consistent wrt the class of additive frames.
(L4.8.9 and L4.8.2.)
L4.8.11: VW = V + MP is consistent wrt the class of variably strict
material frames. (L4.8.9 and L4.8.2.)
L4.8.12: SS = CA + CC is consistent wrt the class of additive material
frames. (L4.8.10 and L4.8.4.)
L4.8.13: VC = VW + CC is consistent wrt the class of variably strict
material frames. (L4.8.11 and L4.8.4.)
LA.8.14: C2 = SS + CEM is consistent wrt the class of singular frames.
Proof: In view of L4.8.12 we show the validity of CEM. Let any singular
model be given and u any world in U. By (cem), f(u, ||p||) is either a
singleton or 0. If the latter, then we have both f(u, ||p||) c ||q|| and
f (u, ||p 11) c ||Nq 11, so have |= Wpq and |= WpNq, and so certainly |= AWpqWpNq.
If the former, suppose f(u, ||p||) = {w}. As w e ||q|| or w e ||Nq11 = U j]q|| ,
we have either f(u, ||p¡|) c ||q|| or f(u, ||p||) c ||Nq11. Hence, either |= Wpq
or |= WpNq, and so |= AWpqWpNq. QED
The conditions of C4.8.1 are not completely independent, so before
proceeding to completeness theorems we state several of the relations among
the conditions:
L4.8.15: In any selection function frame, (cc) implies (mp).
L4.8.16: In any selection function frame, (id) and (md) imply (11).
L4.8.17: In any selection function frame, (id), (md), (co), and (cv)
imply (11) and (12).
L4.8.18: In any selection function frame, (id), (md), (co), and (cv)
imply (ca) and (cb).
L4-8.19: In any selection function frame, (id), (11), and (12) imply
(md), (co), and (cv).


5
Trying to identify a particular kind of conditional, not on the
basis of its grammatical form and variants but by appeal to examples,
has the drawback of needing to specify a set of fairly clearcut ex
amples covering all serious possibilities. One example is not suffi
cient since special features that it possesses may cause the kind of
conditional to be circumscribed too narrowly, thus narrowing the scope
of any subsequent analysis. There are two features of the example we
have considered that bear mentioning in this connection: the example
given is essentially a conditional prediction (see Ellis [19]) or a
sequential conditional. ("Sequential counterfactual" is Jackson's
term in [36].) That is, it refers to two events, one, the antecedent,
temporally preceding the consequent. Furthermore, among the grounds
for accepting the conditional is surely the belief that there is a
connection between the antecedent and consequent. Either or both might
be taken to be an identifying characteristic of such conditionals.
Whether we do so will have a bearing on two matters: the scope of the
basic analysis, and the number of distinct kinds of conditionals we
will be forced to deal with.
Though recalling that expression in the subjunctive mood is not
essential, let us take as the first in our set of examples the simple
subjunctive form of the conditional involved in the oily engine:
El.1.1: If the oil were not changed, then the engine would seize up.
This example is sequential, and there is a connection between antece
dent and consequent. As a second example, let us retain connection,
but at least cloud sequentiality as in this example from Lewis [51]:
El.1.2: If kangaroos had no tails, they would topple over. Next,
let us retain sequentiality, but omit connection as in this example


196
Most of the above conditions are found in Nute [74, p. 84]. He develops
algebraic semantics for extensions of OK and uses the selection function
conditions through the equivalence of the two types of semantics. The
completeness of CD, CA, SS, V, VW, VC, and C2 with respect to the appro
priate conditions is claimed by him. Conditions (11) and (12) are used
by Lewis [51, p. 58] in defining his set-selection functions. They are
included so that we may show the relation of them to the other conditions
of C4.8.1.
Following Nute, we extend the terminology applied to the extensions
of CK mentioned in Section 4.7 to the corresponding selection functions
and frames:
D4.8.2: A frame F = < U,f > and selection function f is:
(a) dependable, if (id) is satisfied.
(b) weakly material, if (mp) is satisfied.
(c) material, if (id), (mp), and (cc) are satisfied.
(d) modal, if (md) is satisfied.
(e) ordered, if (id), (md), and (co) are satisfied.
(f) variably strict, if (id), (md), (co), and (cv) are
satisfied.
(g) additive, if (id), (mp), (md), (co), and (ca) are
satisfied.
(h) singular, if (id), (mp), (cc), (md), (ca), and (cem)
are satisfied.
Proofs of the consistency and completeness of extensions of CK with
respect to the appropriate classes of frames will follow the methods of
Chellas [11]. Chellas calls selection function frames "standard frames"
and treats OK + ID, CK + MP, and IS as examples. In what follows, because


16
connecting the antecedent (and relevant conditions) to the conse
quent.
We can retreat to the original definition, let the laws be of
reasonable generality and concentrate on the problem of specifying
the facts F and laws L for a given counterfactual. But perhaps both
these issues can be sidestepped: presumably our laws are consistent
as a set, likewise the facts embodied in a description of all the cir
cumstances surrounding the antecedent. Why not take all laws and all
true facts obtaining and conjoin them with the antecedent. The pro
blem with this is that the falsity of the antecedent is one of the
facts, and from Np and p, q follows. Also, it is a logical law that if
p is false Cpq is true, and from p and Cpq, q logically follows. So
neither all facts, nor all laws can go into F and L.
Obviously we must eliminate Np from F. Clearly we must also
eliminate Nq because we do not want to validate both Wpq and WpNq, as
admission of Nq to F would do. In fact, if we were to adnit to F any
statement r such that r would be false if p were true, we may validate
conditionals that under the circumstances we would want to deny. For
instance, in the case of the match it also follows by law that "If the
match were struck, it would not be dry," since from the truths that
there is enough oxygen, the match does not light, it is well-made, ad
ding that it is struck implies by a suitable law that it was not dry.
So we may include in F only such statements r which are not only
true, but would not be false if p were true, i.e., for which NWpNr is
true. Goodman [27, p. 15] calls such statements those "cotenable" with
p, and rightly observes, that now we are analyzing a counterf actual in
terms of other counterfactuals, so our account is irredeemably circular.


152
That is, ||a||^L = |a|^.
Proof: The definition of M and closure of maximally L-consistent sets
under MP establishes the theorem for formula a a propositional letter,
o, or of the form Cab. So assume (as an inductive hypothesis) that a
is of the form Lb and the theorem holds for all worlds and formula b.
Then |= Lb iff ||b|| e by the truth definition. As ||b|| = |b| by
hypothesis, |= Lb iff |b| e N Since is unambiguous by P4.2.1,
|= Lb iff Lb e u, by definition of N. QED
Completeness with respect to a class of frames is established by
showing that every formula valid in the class of frames singled out is
a theorem of the logic. Classes of frames are generally specified by
stating a condition every member of the class must satisfy. If the
canonical frame is a member of that class, then the proof is imnediate
from T4.2.1 and the corollary to Lindenbaum's Lenma which states that
only theorems beong to every maximally L-consistent set. Segerberg
calls such logics "natural" [91, p. 40]. On the other hand, where the
canonical frame is not a member of the appropriate class of frames, the
techniques required become more complex and even cumbersome. Two alter
natives are available: for logics which do not succunb to the "easy"
method, one can redefine the canonical model for that logic so that it
does; alternately one can define canonical model once and for all, and
then establish a relationship between the canonical model and certain
other models in the desired class such that what is valid in one is in
the other also. The latter is the tactic pursued by Segerberg, and has
the virtue of providing, as Chellas and McKinney [12, p. 383n] suggest,
a more uniform approach.


67
In the situation diagrammed in Figure 2.4.1, MKpq is true because there
is a Kpq-world in U$u; however, Vpq is false because though there is a
p-permitting sphere, it is not the case that every p-permitting sphere
contains a Kpq-world; the second non-trivial sphere out does not. (0, iu)
are trivial spheres.)
Rather than introduce the necessity operator in terms of truth
conditions under a sphere function, $, we could define it in terms of
the counterfactual conditional. From the assunption that every world
in U$u is a p-world, it follows that no world is a Np-world, hence p is
not entertainable. Therefore, WNpp is true, and conversely. Hence we
may state:
D2.4.4: Ip> @ u iff WNpp @ u.
Of course, D2.4.4 is suitable only if we use the first definition of the
"would"-counterfactual. Keeping the equivalence between NLNp and MP, we
may define the latter in terms of the "might"-conditional:
D2.4.5: Mp @ u iff Vpp @ u.
We introduced Lewis' analysis by considering modal logic in Section
2.2, and we will return to modal logic again when ws consider neighborhood
semantics in CHAPTER FOUR. In the next section, where we discuss counter-
factual inferences and fallacies, we will have occasion to compare infer
ences valid in classical propositional and modal logics with those in
counterfactual logic. Rather than set forth those proper to modal logic
here, we shall mention them as we come to them in the next section.
2.5 Counterfactual Inferences and Fallacies
One test of adequacy for an analysis of the counterfactual condi
tional is to see if it validates inference patterns recognized as valid


3
One might still charge that the conditional is a material con
ditional because it is equivalent to a disjunction: i.e., the first
assertion could be re-expressed as "Either the oil is changed or the
engine seizes up," while the fourth can be re-expressed in the past
tense as "Either the oil was changed or the engine seized up." These
are appropriately asserted just when one does not know which one of
tie disjuncts is true. However, this equivalence does not hold for
the second formulation, "If you had not changed the oil, the engine
would have seized up," precisely because in that case the antecedent
is known to be false, so in the corresponding disjunction, one dis
junct is known to be true, hence the disjunction is not assertable, yet
clearly the conditional is assertable. Furthermore, the disjunction is
true, while the conditional is still debatable. That is, you could
consistently deny the conditional, while accepting the disjunction,
because you accept the falsity of the antecedent. Hence the falsity of
the antecedent is not sufficient grounds to accept the truth of the
conditional, not merely improper grounds to assert it.
If I am correct in asserting that all four sentences express the
same conditional relationship between antecedent and consequent, then
this reveals something about how "counterfactual" counterfactuals must
be, and about whether subjunctivity is essential. Consider again the
third party who asserted,"If that oil was not changed, your engine
seized up." That party could just as well have said,"If that oil were
not changed, your engine would have seized up." In either case, the
third party would have been vindicated in his judgment by the subse
quent discovery that the oil was not changed and the engine did seize
up. Since the first of the above assertions is indicative, and the


118
similar to each other as possible. Which is more similar to the actual
world?
As in the Bizet and Verdi example, the question in this form is
unanswerable. Neither world is more similar to the actual world, but
both are equally similar. This is precisely what validates the might
counterfactuals. Now if we assumed that runs tend to be continued,
this would be tantamount to an additional law (of probability) and would
displace such worlds further from the actual world, since there is pre
sumably no such law in the actual world. Similarly, we cannot assume
that runs tend to break. To argue as Jackson does regarding the A-C
frequency is to contradict the randomness of the very law he has assumed
is constant in the closest worlds.
Other counterexamples to Lewis' account reveal not a basic flaw in
similarity ordering, but rather the vague nature of our standards of com
parative overall similarity. We may differ on what counts for greater
similarity, and thus adopt different sphere functions. As in the case
above, a particular choice may not merely be normal variation, but rather
a misjudgment, and therefore correctable. Comparative similarity applied
to cities, faces, or wines is not applied univocally. Applied to cities
it is certainly based upon different factors than applied to faces. It
is only to be expected that further dissimilarities should appear when
applying it to possible worlds. If we are to reconcile comparative simi
larity to our causal intuitions, then this may dictate certain standards
to be observed in comparing possible worlds. For example, as is widely
recognized (see Jackson [36], Lewis [52], Ellis [19]), similarity before
the event hypothesized in the antecedent of a sequential counterfactual
is generally more significant than similarity after the event in comparing


Copyright 1980
by
John Clyde Mayer


203
Hence c e v. For the converse, suppose c i u, so Nc e u by maximallity.
Then KaNc e u and by CC, WaNc e u. Hence Nc e v. So by consistency,
civ. QED
T4.8.8: 0 = CK + ID+MD + CDis determined by the class of ordered
frames. (L4.8.5.)
Proof: Let F = < U,f > be a proper canonical frame for 0 that satisfies
(id), (md), and (co) whenever X f |a| for every sentence a. This can be
accomplished by setting f (u,X) = X whenever X f |a| for every sentence a.
(id) is clearly satisfied, and as f(u,X) = 0 only if X = 0, (md) is
satisfied. It is clear that (id) is satisfied for all Y c U from T4.8.4,
so for any Y, f(u,Y) c Y. So if f(u,X) = Xc Y and f(u,Y) c X, then
f(u,Y) c X A Y = X, Hence f(u,X) = f(u,Y). So we must show (md) and
(co) are satisfied whenever X = |a| for some sentence a.
For (md), suppose f(u, |a|) = 0. As |NNa| = |a|, we have
f(u, |NNa|) c |Na|, so WNNaNa e u. Hence by MD, WbNa e u. Thus
f(u, |b|) c |Na|, so f(u,|b|) A |a| =0.
For (co), suppose f(u,|a|) c |b| and f(u, |b|) c |a|. Then both
Wab e u and Wba e u, and so KWabWba e u. So by CD, EWacWbc e u, and hence
Wac e u iff Wbc e u. We must show f(u, |a|) = f(u, |b|), so by way of con
tradiction and without loss of generality assume v e f(u, |a|) and
v i f(u, Jb|). Then for seme c, c e v and Wac e u. Since v i f(u, |bj),
and as Wbc e u, civ, which is a contradiction. QED
We keep the remaining proofs brief by assuming that some proper
canonical frame can be found between the smallest and the largest which
satisfies the requisite conditions whenever X / |a| for every sentence a.
Though this is not the case for every extension of CK [11, p. 143], it is
for those we consider here.


58
Figure 2.3.3
In the fourth nonempty sphere outwards there are p-worlds, so this is
a p-permitting sphere; at each of the p-worlds in that sphere q is also
true, so Cpq is true at every world in that sphere, hence Wpq is true
at u.
What, you may ask, about the fifth sphere, where there are p-
worlds which are not q-worlds? Since these are all worlds less similar
to u than the worlds in the fourth sphere, our intuitions should not be
offended. Recalling the kangaroo example, our concern is with the
status of tailless kangaroos in the worlds sufficiently similar to ours.
Perhaps in those worlds in the fifth sphere kangaroos have learned to use
crutches.
These truth conditions are intended to apply to our "would" -
conditional of Section 1.1. This includes the "even if"- and
"necessitation"-conditionals. We will show that at least some reasonable
adequacy conditions are met when vre review counterfactual inferences
and fallacies in Section 2.5. However, these truth conditions are not


217
In the previous section we noted that a logic containing MD, and
determined by a semantics satisfying (md), could appropriately be con
sidered a modal logic.6 The following theorem establishes this:
T4.8.35: In a conditional logic containing CK + MD, La defined by
La WNaa
is a normal modal operator.
Proof: Let 2? = < U,f > be a selection function frame satisfying (md).
Define F:U -> P(U) by
F(u) = Q f (u,X).
We claim F is an accessibility function and so determines the logic of
the modal fragment of the logic containing CK + MD. The standard truth
condition for such frames is:
|= La iff F(u) c Ha|[,
provided that there are no singular worlds. That there are no singular
worlds follows from the normality of CK + MD. Hence it suffices to show
that |= WNaa iff F(u) c ||a||.
Suppose |= WNaa. Then by MD, |= Wba for all sentences b. Hence
f(u, ||b11) c ||a11 for all sets ||b||. If X f ||b|| for every formula b,
adding these X's to the intersection cannot enlarge it. Therefore,
F(u) c || a 11.
For the converse, we may equivalently show that F(u) A ||a|| f 0
inplies |= Vaa. So suppose F(u) A ||a[| ^ 0. Then, in particular, for
X= ||a11 f (u, ||a11) A ||a11 ^ 0. So we have |= Vaa. QED
4.9 Notes
1 Proof of these theorems may be found in Beth [4] or most any
work in the foundations of logic.


171
We take as our language CW, and, pending the definition of model
and truth below, define validity, consistency, and completeness in the
standard way of Section 4.2.
D4.5.1: A sentential neighborhood frame F = < U,N > is an ordered pair
such that
(a) U is a set of possible worlds, and
(b) N: U x CW -* P(P(U)) is a function.
D4.5.2: A propositional neighborhood frame F = < U,N > is an ordered
pair such that
(a) U is a set of possible worlds, and
(b) N: U x p(U) -* P(P(U)) is a function.
The same definition of model will do for both types of frames:
D4.5.3: A model M = < U,N,V > is an ordered triple such that F = < U,N >
is a frame and V:P -* P(U) is a function, where P is the set of propositional
letters. M is called a model on P.
The definition of truth for non-conditional sentences is as in D4.2.3
so only the clause for conditional formulas is stated below.
D4.5.4: The truth in sentential (neighborhood) model M of formula Wab
is defined as:
|= Wab iff ||b||M e N(u,a).
D4.5.5: The truth in propositional (neighborhood) model M of formula
Wab is defined as:
|= Wab iff ||b||^ e N(u, ||a||V
We will drop superscripts in |= a and [|a||W whenever possible to do so
without confusion. Note that in D4.5.1 we tacitly use CW to stand for
the well-formed formulas of CW.


143
Using the following lenmas we may provide alternative systems for
M, R, and K using only PC + RE + certain axioms as bases:
L4.1.5: RM is derivable in E + M.
Proof: Assume Cab. By PC CaKab follows and CKaba is a PC theorem. Hence
by PC EaKab follows. Then by RE we have ELaLKab, and so by PC CLaLKab.
Now CLKabKLaLb is an instance of M, and CKLalblb is a PC theorem, hence
by PC we have CLaLb (two applications of transitivity). QED
L4.1.6: RN is derivable in E + N.
Proof: Assune a is a theorem. By PC we have Ela, hence by RE ELlLa.
But LI is N, so by MP La follows. QED
These lenmas are proved in Segerberg [91, pp. 45-46]. Together with
L4.1.2 we may then conclude:
E4.1.6:
(a) M = E + M
(b) R = E + M + R
(c) K=E+M+R+N
Strictly speaking, we have not proved containment both ways for E4.1.6(a)
or (c). The following lenmas provide the necessary proofs.
L4.1.7: N is derivable in E + RN.
Proof: Cpp is a PC theorem, so by RN we have LCpp. But ECppl is a PC
theorem, hence by RE we have ELCppLl. So by MP we have Ll. QED
L4.1.8: M is derivable in PC + RM.
Proof: Both CKpqp and CKpqq are PC theorems. Hence by RM we have both
CLKpqLp and CLKpqLq. Therefore, by PC we have CLB^qKLpLq. QED
Axiom K is often found in bases for normal modal logics, as
Segerberg indicates [91, p. 46]. For example, see Zeman [104, p. 282].
However, Segerberg shows that K is not derivable in either E + M or


100
(a) is a counterfactual consequent of p, and
(b) w is a KpNq-world.
D2.6.4: Proposition p has a complete set of critical consequents iff
there is a critical consequent for each p-world.
Consider now 0p; if p has a complete set of critical consequents, then
no world will satisfy all of them and p which latter is in 0p, hence 0p
is unsatisfiable, i.e., inconsistent.
Herzberger than goes on to show that every proposition p which
violates LA has a complete set of critical consequents [34, p. 86].
Recall that a proposition is identified with a set of possible worlds
(or so we are assuming), whether that proposition is expressible or not.
Using definition D2.6.2, for each entertainable proposition p (identified
with a nonempty set of worlds) there is a propositional counterfactual
theory 0p consisting of the counterfactual consequents of p, i.e., those
propositions q for which Wpq holds. Suppose that p violates the limit
assumption. Then there is no maximally close p-world (or closest set of
p-worlds). For each p-world there is at least one closer p-world.
Let w be any p-world and let q^ be that proposition identified
with the set of p-worlds closer to u than w. By the violation of the
limit assumption we are guaranteed that for each w this set is nonempty.
Since qw is a set of p-worlds all closer than w, D2.6.3(a) is satisfied,
and since w is not in q^, D2.6.3(b) is satisfied as well. So each p-
world has a critical consequent, and hence p has a complete set of
critical consequents which is a subset of 0p. As p is also in 0p, 0p is
unsatisfiable.
To provide a slightly different proof, 0p is satisfiable (where
each element is a set of worlds), provided AQp f 0. Let Q be the set


BIOGRAPHICAL SKETCH
John Clyde Mayer, bom May 22, 1945, in Kingston, Pennsylvania,
received his B.A. degree in 1967 from Randolph-Macon College, Ashland,
Virginia. After one year of graduate work in philosophy at Yale
University, New Haven, Connecticut, he taught at a private secondary
school in Atlanta, Georgia.
Returning to graduate school in 1977 in philosophy, he received
an M.A. in philosophy in 1978 from the University of Florida, Gainesville,
Florida, and the Ph.D. from the University of Florida in the same field in
1980.
Mayer considers Florida his home, having lived in Sarasota, Florida,
most of his life, and expects to continue graduate work, though in topology,
for the next two years, prior to pursuing a career in university teaching
and/or research.
250


13
At this point we have made reference to six conditionals: Wpq,
Tpq, and Upq, where we expect one set of truth conditions, the latter
two conditionals presumably being subclasses of the former; Vpq, dis
tinct from the above three; and Cpq, and LCpq, which are the traditional
material and strict conditionals, respectively. By way of terminology
we will refer to the first four indiscriminately (and somewhat inac
curately) as counterfactuals, the first three as "would"-cornterfactuals,
the fourth as the "might"-counterfactual, the second as the "even if-
counterfactual, the third as the "necessitation"-counterfactual. The
same prefixes with the suffix "conditional" will also be used. When
the term "counterfactual" alone is used, this will usually refer to the
' would1' -conditional.
It would be appropriate at this point to consider the various in
ference patterns that are intuitively valid for the conditionals we have
mentioned. This would serve to illustrate some of their differences,
while providing criteria of adequacy for any purported analysis. In
keeping with a long tradition in analytical philosophy, I should like to
postpone these considerations until we have a preliminary analysis to
test them against. One inference pattern, however, is of such paramount
importance that it bears mentioning now. I refer to Strengthening the
Antecedent (also called Augmentation). It is well known that both of the
following inferences are valid in classical propositional and modal logics
Cpq LCpq

. CKprq . LCKprq
However, consideration of a single example will show that the correspond
ing pattern for counterfactuals


240
Suppose in the actual world a certain prisoner is released because
occurred. Entertain the counterfactual supposition that the prisoner
was executed. Certainly the worlds in which the prisoner dies a long
and painful death are significantly dissimilar (particularly for the
prisoner) to those in which the death is painless and swift. Yet in
entertaining the counterfactual supposition that the prisoner was executed,
we should obviously consider both inodes of death, in spite of the dis
similarity of the worlds concerned. Lewis' analysis, if extended to
this indeterministic example, would have it that both sets of worlds are
equally similar to the actual world, though significantly dissimilar to
each other. It is not clear just what items of comparison with respect to
the actual world could be balanced against each other to get this odd re
sult. I suspect there are none, and that a painless death is more similar
to continued life than a long and painful death, if death can be similar
to life at all.
Conditional predictions in an indeterministic context lead to pre
cisely the same result. We argued in CHAPTER ONE that counterfactuals
and conditional predictions were both aspects of a single pervasive con
cept of conditionality. We may analyze the prisoner's situation from this
point of view. Prior to the potential execution there is a time tQ when
the future can be seen as dividing into three main branches (more properly,
sets of branches). On two of the branches, the prisoner is executed at
though in different ways, and on one branch he is released at t^.
The outcome depends upon the occurrence of a particular event, among the
three possible, at t2 (which we may imagine as an interval rather than an
instant). Having taken one branch, we can always counterfactually sup
pose what would have happened had another been taken, or we can predict


21
... a contingent generalization is a law of nature if
and only if it appears as a theorem (or axiom) in each
of the true deductive systems that achieves a best com
bination of simplicity and strength. [51, p. 73]
An example of a relatively pure case of the latter kind of anal
ysis is found in Wasserman [102] wherein he presents a so-called "log
ical analysis" of the counterfactual conditional. What Wasserman does
is provide a language containing a binary connective intended to repre
sent the counterfactual conditional with a model-theoretic structure as
semantics. This procedure is intended to make explicit the logical
structure of the conditional in question. Of this semantics Wasserman
says:
The "philosophical" motivation for the formal semantics
provided for a statement of the form "If ({> were the case,
then ip would be the case" is that such a statement is
about some "world," "state-of-affairs," or, more formally,
some structure S, and that the statement "means" that i|>
holds in every structure which differs from S "just enough"
to make true. [102, p. 396]
Wasserman indicates that he is providing an analysis of a logical
kind that can be construed as in some way giving the "meaning" of
counterfactuals. (One wonders if "meaning" is the same as meaning.)
This reflects a practice which has become standard in analytical philos
ophy: the meaning of a concept can be given in terms of its truth con
ditions Thus Lewis states that the task involved in giving an analysis
of counterfactuals is to "give a clear account of their truth condi
tions" [51, p. 1]. For Stalnaker the task is "to find a set of truth
conditions for statements having conditional form which explains why we
use the method we do use to evaluate them" [96, p. 169]. While the
claim may not be that the truth conditions constitute the meaning of the


235
r: both place = {w,z}
s: the Galloper wins = {y,z}
t: the Flyer wins = {v,w}
I do not know, and doubt if anyone does, how to determine the simple
propositions which enable one to rank the worlds in terms of change fran
the actual war Id. However, I believe the ranking of Figure 5.2.1(a) is
preferable to that of 5.2.1(b).
(a) (b)
Figure 5.2.1
It requires less of a change from the actual world for both trailing
horses to get around the pile-up, than for just one of them to do so, hence
worlds w and z are the result of a lesser change than either v or y. On
the other hand, world v makes just propositions q and t true, and world
y makes just p and s true, while w makes p, q, r, and t true and z makes
p, q, r, and s true. So one might argue that the lesser change is the one
that makes fewer of the designated propositions true. Furthermore, as s
is not true at w and t is not true at z, there is no link between y and
w or between v and z on the latter reading.
Note that on the first reading, Figure 5.2.1(a), both of the following
instances of CB are true, nontrivially:
E5.2.4: CWApqrAWprWqr
CWAstrAWsrWtr


45
A pictorial representation much used by Lewis for the accessibility
relation is that of a circle, where the center represents the given world
u, and all the points bounded by the circle represent the worlds acces
sible from u. The circle and its interior is the "sphere of acces
sibility" around u.
Figure 2.2.1
This suggests that we may define our frame in a different, but equiva
lent, fashion:
D2.2,6: F = < U,S > where U is again the set of possible worlds, S is
a function from the set of possible worlds U to the power set of U,
P(U), the set of all subsets of U.
To each world u in U, S assigns a subset of U, designated S^, which will
be called the sphere of accessibility around u.
Our definitions of truth may be altered accordingly:
D2.2.7: Ip @ u iff for all w e U, if w e Su, then p @ w.
D2.2.8: Mp @ u iff for some w e U, w e and p @ w.
The requirement that R be universal now translates into the requirement
that = U for all u in U. Given the usual interpretation of the
quantifiers, we may shorten the above to:
D2.2.9: Lp @ u iff for all w e Su, p @ w.
D2.2.10: Mp @ u iff for sane w e Su, p @ w.


182
follows that for all u, Wao e u. Consequently, |o| e N(u, |a|). But
[o| =0, so by superset closure, N(u,|a|) = P(U). QED
The above lemmas and the earlier ones on consistency of classical
conditional logics immediately give us the following results concerning
the determination of logics by classes of frames.
T4.5.2: CE is determined by C, the class of all propositional neighbor
hood frames.
T4.5.3: CM is determined by C .
T4.5.4: CR is determined by C .
T4.5.5: CK is determined by g the class of all propositional neigh
borhood frames where N(u,X) is a filter.
T4.5.6: CE + 00 is determined by C .
T4.5.7: CE + CS is determined by C .
Completeness proofs for the half-classical conditional logic Ce
and the half-normal conditional logic Ck can be produced by slight modi
fication of the above techniques. Completeness and consistency proofs
for extensions of CK will be considered in Section 4.8 after several ex
tensions are presented in Section 4.7. Before turning to extensions of
CK, we will discuss alternative semantics for CK and the relationship
between neighborhood frames and sphere function frames, partially answer
ing the question of Section 4.3.
4.6 Alternative Semantics for Conditional Logics
In Section 4.3 we showed that relational semantics for modal logics
could be defined as a special case of neighborhood semantics for regular
modal logics. The analogous situation obtains for conditional logics:


105
way in which the printer made a mistake is irrelevant, so are the con
siderations that led us to take p to be a LA-violating supposition.
But this argument depends upon our taking both antecedent and consequent
into account in our ordering of possible worlds. This would require con
siderably more of a departure from Lewis' truth conditions than either
Pollock or Herzberger suggest. (Nute reviews such an account in [74,
pp. 110-117], but ultimately rejects it. Butcher [9] constitutes such
an account.)
Pollock concludes that Lewis is right about the failure of the
limit assumption in the similarity ordering of possible worlds, but
wrong in taking overall similarity ordering to be the appropriate basis
for judging the truth value of counterfactuals [80, p. 21]. We shall
discuss Pollock's view more fully in CHAPTER THREE, when we discuss in
general the topics of similarity and ordering of possible worlds for the
purpose of counterfactual deliberation. The order that Pollock adopts in
his analysis is still based upon the antecedent alone.
Rejecting LA guards against our making the mistake of accepting
Pollock's schema E2.6.1, because we can point to Lewis' example to
justify the "might"-conditional E2.6.3 which contradicts the former.
Pollock, oddly enough, believes the relevant "might"-conditionals are
true [80, p. 21], but does not see that this undermines his schema
E2.6.1. Neither the failure of GCP nor the appearance of counterf actual
inconsistency is a decisive objection to Lewis' rejection of LA. The
full story of what would be the case given any antecedent is not always
a consistent story, nor should we expect it to be.


186
D4.6.8: Let F be a sphere function frame. Then F is
(a) limited iff Xc $u implies AX e X.
(b) normal" iff U$u f 0.
In D4.6.6 it will be noted that the closure and centering conditions
are not present. The former are not required in order to determine the
set of valid formulas (see Lewis [51, p. 119n]) and the latter is not re
quired to determine the weakest logic Lewis recognizes, V. As we shall
see later, we shall add conditions to get the class of frames that deter
mines VC and VCU. In view of D4.6.8(a), a frame is limited just in case
it satisfies the limit assumption, LA. This will not affect validity
either. Lewis' definition of normal [51, p. 120] has been given here as
normal'', since his definition is at variance with the definitions we
have been using. Every limited sphere function frame is normal in our
sense since the accessibility function frame corresponding to it has
Q = 0. Lewis essentially proves this himself [51, p. 58], and we prove
a version below. (Lewis calls such functions "set selection functions.")
T4.6.3: Each limited sphere function frame is equivalent to some normal
accessibility function frame. (Alternately, class selection function
frame.)
Proof: Let F be a limited sphere function frame and M any model on that
frame. We define a corresponding accessibility function S: U x p(U) -> P(U)
by:
M
It is sufficient to show |= Wab iff S(u, ||a||) c ||b||. Note we have said


160
The following lemma is then immediate:
L4.3.1: For any regular frame and model, |=Laiff u is normal and
ANu c || a||, for all formulas a.
Proof: Suppose |=La. Then u cannot be singular, so u is normal as the
frame is regular and consists only of normal and singular worlds. Since
INI £ Nu> c IIa 11.
Suppose u is normal and AN^ c ||a||. Since u is normal, is a
filter, so closed under supersets, hence ||a|| e N^. Therefore, |=La. QED
For any regular neighborhood frame we may define a corresponding relational
frame:
D4.3.2: Let F = < U,N > be a regular neighborhood frame. Then the
relational frame corresponding to F, F#, is given by:
F^ = < U,R,Q > such that
(a) U is the same set as in F.
(b) R is the alternative relation defined on F.
(c) Q is the subset of U consisting of singular worlds.
By changing clause (d) of D4.2.3, the definition of truth in a model to:
(d') | = La iff u i Q and for all v, if uRv then | = a,
it is clear that we may carry out our semantics for a logic containing R
by considering the corresponding relational frames (see Kripke [40, 41]).
Of importance in the above is the fact that | = La iff u is normal
and AN _c ||a||. Consider the sphere function $; it is clear that it
satisfies the formal criteria for being a neighborhood function, since
its range is P(P(U)); however, in Lewis' semantics it does not operate as
a neighborhood function as in neighborhood semantics. For on Lewis'
semantics it is U$^ which corresponds to the set of worlds to which u
has access; i.e., we must define the alternative or accessing relation


34
may be analyses of the same or different sorts, of the same or different
concepts within the circle. If the analytic circle is only suggested by
the failure of prior attempts to arrive at a satisfactory analysis,
nevertheless, much the same situation obtains. He who maintains that
the circle is not only broken, but resolves itself into two lines has,
of course, an additional task: to undermine the evidence for circular
ity.
The matalinguistic analyses, including belief analyses, lead in
exorably to laws, confirmation theory, and analysis in terms of justi
fication conditions. It will be maintained that such analyses are sub
ject to the charge of failing to explain the concepts they take as pro
blematic. On the other hand, possible world accounts (with the excep
tion of Pollock's mixed account) have the prima facie virtue of pro
viding an explanation of why the concept works the way it does.
1.3 Notes
1A survey of early accounts of this sort may be found in Schneider
[87].
2For other logical analyses and criticisms thereof see Bode [7],
Fumerton [24], Lehmann [44], and Nute [69]. We discuss Wasserman [102]
as an example.
3This suggestion was made to me in conversation by Gary Fuller.
4For more discussion than we shall have space for of the relation
ship among laws, necessity, conditionals, and causation, see Barker [2],
Chisholm [14], Fine [21], Goosens [28], Honderick [35], Jackson [36],
Kim [37], Kneale [38], Lewis [52], Loeb [55], Lyon [58], Mackie [60, 61,
62], Nute [70], Shorter [92], Sosa [95], Swain [98], Temple [99], Vendler
[101], and Yagisawa [103].


106
2.7 Possible Worlds: Realism and Explanation
One may object to Lewis' analysis on three levels: the notion of
possible worlds is itself suspect and so cannot serve to clarify some
thing else (see [2, 3, 19, 27, 29, 33, 36, 94]); the notion of compara
tive similarity is either too vague or inappropriate for analyzing
counterfactuals (see [79, 86]); possible worlds and similarity are ac
ceptable, but the particular analysis is flawed (see [5, 16, 34, 68, 71,
74)]. In this section I will comment only on the first level of objec
tion.
Realizing the suspect nature of his foundations, Lewis undertakes
a defense of "possible worlds realism" which is essentially the view
that possible worlds are entities sui generis, not reducible to some
other sorts of things, and furthermore, are more entities of the same
kind as the actual world [51, pp. 84-91]. He specifically rejects
taking possible worlds to be a dispensable locution for maximally con
sistent sets of sentences, sets of beliefs, maximal states of affairs,
or mathematical entities of some sort. He would presumably also reject
the view that they are the "many worlds" of the many worlds interpretation
of quantum mechanics (see [17] and [93]).
Rather than repeat or modify Lewis' arguments, I would like to
pose an analogy between scientific explantion and the kind of explana
tory analysis I see as essential to clarifying a concept. The situation
I wish to consider is the status of physics and what counted as a physical
explanation in the decade following the general acceptance of Newton's
laws of motion and law of gravity.4
Even while Newton's achievement was generally accepted it was
recognized, notably by Newton himself, that the law of gravity conflicted


153
Segerberg [91, pp. 19-20] defines the notion of the augmentation
of a frame specifically to guarantee the satisfaction of conditions (m)
and (r), and then proves the lemma stated below:
D4.2.7: Let F = < U,N > be a frame. The augmentation of F is the frame
= < U,N+ > such that
u
0 if \ = 0
{A : A c U and AN c A}, if N ^ 0
u u
L4.2.11: Let L be a regular logic, M4 the canonical model for L, and
t is augmentation. Then for all formulas a and all u e U,
M-r Ml
|= a iff |=a.
'u *u
We should remark that the augmentation of a model is the augmentation of
its frame, and that a model identical with its augmentation is said to
be augmented.
It is clear that the definition of A in the definition of If1"
u
guarantees superset closure, thus satisfies (m), and closure under inter
sections, thus satisfies (r). In virtue of this, the definition of N+
u
preserves both normal worlds and singular worlds. In effect, then, for
a regular logic, L4.2.11 forms a bridge from the canonical model to the
application of T4.2.1 in completeness proofs.
Though augmentation is appropriate for regular logics, it is not
for monotonic logics that are not regular. Segerberg's error [91, p. 43]
in this regard motivated Chellas and McKinney [12]. We may express the
difficulty in the following lemma:
L4.2.12: If is an augmented model for classical logic L, then both
axioms M and R are true.


TABLE OF CONTENTS
ACKNOWLEDGEMENTS iii
ABSTRACT vi
ONE WHAT ARE OOUNTERFACTUALS? 1
1.1 A Central Concept of Conditionality 1
1.2 The Metalinguistic Analysis 14
1.3 Notes 34
TWO POSSIBLE WORLDS ANALYSIS OF COUNTERFACTUALS 35
2.1 Possible Worlds: History 35
2.2 Possible Worlds: Modality and the Strict Conditional.... 41
2.3 Lewis' Analysis of the Counter factual Conditional 50
2.4 Modal Logic and the System of Spheres 64
2.5 Counterfactual Inferences and Fallacies 67
2.6 The Limit Assumption 90
2.7 Possible Worlds: Realism and Explanation 106
2.8 Notes 110
THREE ORDERINGS OF POSSIBLE WORLDS 112
3.1 Comparative Similarity 112
3.2 Varieties of Order 122
3.3 Notes 134
POUR MODAL AND CONDITIONAL LOGICS 135
4.1 A Modal/Conditional Language and Modal Systems
E, M, R, K 135
4.2 Neighborhood Semantics for Modal Logic 144
4.3 Neighborhood and Relational Semantics 159
4.4 Conditional Logic: The Systems Ce, Ck, CE, CM, CR,
CK 162
4.5 Neighborhood Semantics for Conditional Logic 170
4.6 Alternative Semantics for Conditional Logic 182
4.7 Extensions of CK 189
4.8 Semantics for Extensions of CK 193
4.9 Notes 217
iv


46
What, you may ask, has this to do with conditionals? Recall that
a principal objection to the material conditional as an analysis of con
ditionals in English is that it simply is not plausible that a condi
tional be true just because its antecedent is false or its consequent
true. Something more is called for, and one of the first things to try
is to formalize the notion that some connection obtains between the an
tecedent and consequent. Now we have seen that this is not enough, but
it is a place to start.
In an effort to provide an alternative to the material condi
tional for the analysis of "if . then; . a number of modalized
conditionals have been developed, either as primitives in a logical sys
tem, or defined in terms of the modalities of possibility and necessity.
The insight which these systems are formalizing is that the antecedent
of a conditional somehow necessitates the consequent: if the antecedent
is true, then the consequent is true, of Necessity. Though the debate
over the modal nature of the conditional dates back to antiquity, the
genesis of both modem modal logic and modalized conditionals can be
traced to C. I. Lewis, as we indicated in Section 2.1. C. I. Lewis
developed several systems of modal logic incorporating what he called
"strict implication." (For a survey of Lewis-type modal systems one
should read J. Jay Zeman's Modal Logic [104]. f These systems are of
interest for our purposes only in their failure to provide an analysis
for counterfactuals, for while the material conditional is too weak to
serve as a counterfactual conditional, the strict conditional is too
strong, as we shall see.
The idea behind the strict conditional, expressed in terms of
possible worlds, is that, virile neither the antecedent nor consequent


6
from Pollock [80]:
K1.1.8: If the witch doctor were to do a rain dance, then it would
(still) not rain.
El. 1.4: Even if the witch doctor were to do a rain dance, it would
(still) not rain.
Finally, we can "reverse" sequentiality:
El.1.5: If the engine were to seize up, then the oil would not have
been changed.
I have included another version of El.1.3 because it might be
thought that the El. 1.4 is a more natural way of expressing the lack
of a connection between antecedent and consequent which appears to
underlie such a conditional. Such conditionals were called by Goodman
[27, p. 5] "semifactuals" and by Pollock [80, p. 29] "even if'-condi-
tionals. The former appellation comes from the fact that in the typi
cal cases of semifactuals the consequent is already true and the ante
cedent s being true cannot alter that. A modification of our engine ex
ample may raise some doubts about whether the truth of the consequent
is essential to such conditionals.
Suppose that upon checking your oil and finding it excessively
dirty I also notice the reprehensible shape you have allowed your
fifteen-year-old engine to get into. In fact, I am convinced that your
engine would seize up whether or not you changed the oil. You notice
the dirty oil, but are ignorant of the engine's sad state generally. I
tell you:
El.1.6: Even if the oil were changed, the engine would (still) seize
up.


Normal
Lawful
Modal/
Partial
Semiconnected
(Weak)
Substitutional
Order
Partial
Total
Order
Order
C2
G
CG
SS
CK + MP
B
CO
CA
CK
CK + ID
CP
Materiality
Comparison *
EXTENSIONS OF CK
Figure 5.1.1
SS + CB
CA + CB
CP + CB
VC
W
K>
N3
O
V


53
C2.3.4: If the nonempty set Y is a subset of $ then its intersection
is an element of $ .
u
C2.3.5: For each u,v in U the union of equals the union of $v<
Following Lewis' terminology, we shall call these conditions
respectively (strong) centering, nesting, closure under unions, closure
under (nonempty) intersections, and uniformity [51, pp. 14, 117]. Lewis
calls $ a system of spheres. We shall depart from Lewis slightly by
calling $ a sphere function, and by calling $u the system of spheres
about u. A picture suggestive of a system of spheres about u which we
shall have occasion to use repeatedly is that of Figure 2.3.1.
Figure 2.3.1
Each circle represents the boundary of one of the spheres of acces
sibility about u. In what follows for brevity we will use the following
symbols:
e df
"is an element of"
- = df
"is a subset of"
U = df
"the union of"
ii
<
"the intersection of
0 ~ df
"the empty set"
In this notation the conditions listed above may be more briefly stated
as


151
frames, which we will define precisely below, are called canonical
frames, and the models canonical models. The following definition is
adapted from Segerberg [91, p. 16].
D4.2.6: Let L be a classical logic, and
(a) denote the set of all maximally L-consistent sets of
formulas.
(b) |a|^ denote the maximally L-consistent sets of which formula
a is a member (so 11= U).
(c) N^:U -> P(P(U)) be a function such that La e u iff |a|^ e N^dJ)
(d) For each p e P, VL(p) = |p|L
,V^ >, is called the neighborhood canonical model for L.
It is easy to show that in D4.2.6 is well-defined and unique.
Because L is classical, is unambiguous with respect to representatives
of a set of maximally L-consistent sets:
P42.1: in D4.2.6 is unambiguous. That is, if |a|^ = |b[^, then
lal, e N iff Ib IT e N .
1 'L u L u
Proof: (Herein we drop the superscript and subscript L, and will do so
in further proofs as well.) Assume |a| = |b|. Then by the properties of
maximally L-consistent sets, | Eab. Hence by RE, | ELaLb. So for all
u, La e u iff Lb e u. Therefore, by definition of N, |a| e N iff |b| e
N .
u
Segerberg's "Fundamental Theorem for Classical Logics" [91, p. 17]
is an iirmediate consequence:
T4.2.1: Let be a canonical neighborhood model for classical logic L.
Then for all formulas a and all u e U:
Then M = <
M y
|= a iff a e u.


107
with one of the ideals of mechanical explanation: that all effects
were to be explained in terms of corpuscular motion and impact. The
notion of attraction at a distance was "occult" in the perjorative
jargon of the time, and reminiscent of the rejected "Aristotlean"
types of explanation. The result was that until after the end of the
nineteenth century, among those who accepted Newton's laws, there were
two distinct camps. These camps differed not in their acceptance of
Newton's laws and the widening applications of them, but in their in
terpretation of the laws.
On the one hand were those who clung to the ideal of a mechanical
explanation as the ultimate explanatory tool. For them, attraction at
a distance was a way station in explanation, to be superseded eventually
by a more properly mechanical explanation. On the other hand were those
who accepted forces (and later fields) as fundamental constituents of
nature. For them gravity required no further explanation in terms of
mechanical principles. In effect it became one of the mechanical prin
ciples .
The distinction between these two groups can be seen as a distinc
tion in metaphysical cornnitment. Those in the first group would admit
the inmense heuristic value of Newton's law of gravity, but avoid the
comnitment to a force of gravity constitutive of nature. The others
would commit themselves to a fundamental force of gravity in the absence
of acceptable alternatives. However, this divergence in metaphysical
comnitment did not carry with it a divergence in views on what counted
as an explanation in mechanics itself. To reduce an effect to (among
other things) the force of gravity was itself a sufficient explanation.


140
rules of inference in terms of which certain logics and families of
logics may be classified.2 We assume the language is CL.
E4.1.1: Modal rules of inference:
RE: From Eab infer ELaLb
EM: From Cab infer CLalb
RR: From CKabc infer CKLalbLc
RN: Frcm a infer La
RK: Infer CKK . KLa^I^ . LanL, n £ 0
from CKK . Kana0 . a b
12 n
(Conventionally, RK for n = 0 is RN and RK for n = 1 is RM.)
E4.1.2: Modal axioms:
M: CLKpqKLpLq
R: CKLpLqLKpq
K: CLCpqCLpLq
N: LI
Q: Lp
S: Mp
T: CLpp
U: CLpLLp
B: CMLpp
E: (MLpLp
A basis for a system is the set of axioms and rules of inference
for it. Hence A1 through A3 plus MP and US is a basis for PC. We may
indicate this as PC = Al + A2 + A3 + MP + US. Where L is a logic and
L' an extension of L, we may indicate a basis for L' by L' = L + R, where
adding R to the basis for L produces a basis for L'. The ambiguity where
by PC denotes both system and logic will be extended, so that the expression


172
The following lemma is immediate:
L4.5.1: PC is consistent wrt the class of all sentential (propositional)
frames.
It would normally be our intention to classify conditional logics in terms
of classes of frames as we did with modal logics. This will not generally
be possible in the case of half-classical logics larger than Ce^ (see
Challas [11, p. 149nl4]). We shall see below why this is the case. Partly
for this reason, and partly for reasons of uniformity, Nute [74] classifies
conditional logics in terms of classes of models, preferring to incorporate
the valuation function, V, directly into the definition of the set of pos
sible worlds [74, pp. 21, 56]. We prefer to retain the more algebraic
classification in terms of frames whenever possible. As most conditional
logics intended to represent the counterfactual conditional are classical,
including Lewis', this will not be overly restrictive.
As in Section 4.2 we will show first consistency results and then
completeness results. In view of L4.5.1 it will be necessary to check
only the conditional axioms and rules of inference for validity in a par
ticular class of frames (or, rarely, models).
L4.5.2: Ce is consistent wrt the class of all sentential frames (or
models).
Proof: We need only show RCEC preserves validity. Suppose Eab is valid.
Then, for any sentiential model, ||a|| = ||b||. So for each u e U, we have
either both |= Wca and |= Web or neither, by D4.5.3. Hence we have
EWcaWcb. QED
Note that this proof goes through because the set of worlds where a (or b)
is true appears in the truth definition as a member of N(u,c).


185
T4.6.1: CR is determined by the class of all propositional accessi
bility function frames, and CK is determined by the class of all such
frames for which Q = 0.
T4.6.2: CR is determined by the class of all propositional relational
frames, and CK is determined by the class of all such frames for which
Q = 0.
Similar results hold for the sentential versions, though in terms of
classes of models. It is important to note that Nute denotes by
"propositional relational model" only those models whose frame is such
that no singularities are present [74, p. 61], that is, normal proposi
tional relational frames.
In investigating the relationship between sphere function frames
and alternative semantics for conditional logics it will be useful to
have a formal definition of a sphere function frame and associated models
and truth definitions. Several definitions toward this end are stated
below:
D4.6.6: A sphere function frame F = < U, $ > is an ordered pair such that
(a) U is a set of possible worlds, and
(b) $: U * P(P(U)) is a function such that
X,Y e $u implies Xc Yor Ye X.
Any of our previous definitions of model will do, as will previous defi
nitions of truth for nonconditional connectives.
D4.6.7: Let F be a sphere function frame and M a model on P.
Then truth in M at world u for conditional formulas is defined by:
Ijjwab iff if ||a|| A U$u + 0,
then for some X e $ 0 (||a|| AX) c ||b||.



PAGE 1

&2817(5)$&78$/6 %< -2+1 &/<'( 0$<(5 $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( &281&,/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$

PAGE 2

&RS\ULJKW E\ -RKQ &O\GH 0D\HU

PAGE 3

$&.12:/('*(0(176 DP LQGHEWHG WR P\ DGYLVRU -D\ =HPDQ IRU LQWURGXFLQJ PH WR PRGDO ORJLF FRQGLWLRQDO IXQFWLRQV SRVVLEOH ZRUOGV DQG D PRUH HPSLUn LFDO YLHZ RI ORJLF WKDQ PLJKW RWKHUZLVH KDYH KDG )RU EHWWHU RU ZRUVH DP RQO\ VOLJKWO\f OHVV D 3ODWRQLVW DV D UHVXOW 7R P\ W\SLVW -R\FH 3DQGH OLV RZH D SDUWLFXODU GHEW RI WKDQNV IRU WKH WLPH DQG HIIRUW VKH KDV VSHQW LQ ZRUNLQJ ZLWK PH WKURXJK UHYLn VLRQV FRUUHFWLRQV DQG UHUHYLVLRQV $V D SKLORVRSKHU LQ KHU RZQ ULJKW KHU FRQPHQWV DQG VXJJHVWLRQV KDYH EHHQ YDOXDEOH DQG RQO\ RFFDVLRQDOO\ PLVFKLHYRXV LQ

PAGE 4

7$%/( 2) &217(176 $&.12:/('*(0(176 LLL $%675$&7 YL 21( :+$7 $5( 22817(5)$&78$/6" $ &HQWUDO &RQFHSW RI &RQGLWLRQDOLW\ 7KH 0HWDOLQJXLVWLF $QDO\VLV 1RWHV 7:2 3266,%/( :25/'6 $1$/<6,6 2) &2817(5)$&78$/6 3RVVLEOH :RUOGV +LVWRU\ 3RVVLEOH :RUOGV 0RGDOLW\ DQG WKH 6WULFW &RQGLWLRQDO /HZLVn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

PAGE 5

),9( 2281,(5)$&,8$/6 $1' &203$5,621 2) :25/'6 $Q $GHTXDWH &RXQWHUIDFWXDO /RJLF &RPSDUDWLYH 2UGHU $QDO\VLV 5()(5(1&(6 %,2*5$3+,&$/ 6.(7&+ Y

PAGE 6

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n WLFV 9DULRXV IDPLOLHV RI QRUPDO FRQGLWLRQDO ORJLFV DUH WKHUHE\ LGHQWLILHG YL

PAGE 7

7KHVH IDPLOLHV DUH FODVVLILHG LQ WHUPV RI WZR GLPHQVLRQV RQH RI LQn FUHDVLQJ PDWHULDOLW\ RI WKH FRQGLWLRQDO FRQQHFWLYH DQG WKH RWKHU RI LQFUHDVLQJ VWUHQJWK LQ WKH FRPSDULVRQ RI SRVVLEOH ZRUOGV LPSOLFLW LQ DQ\ VHPDQWLFV IRU WKH ORJLFV 0DFK RI WKLV ZRUN LV D FRQWLQXDWLRQ RI WKDW RI %ULDQ &KHOODV DQG 'RQDOG 1XWH &RPSDUDWLYH RUGHU VHPDQWLFV D JHQHUDOL]DWLRQ RI /HZLVn FRPSDUDWLYH VLPLODULW\ VHPDQWLFV LV GHYHORSHG ,Q FRPSDUDWLYH RUGHU VHPDQWLFV SRVn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f DQG FHUWDLQ RI WKH 9ORJLFV RI /HZLV ZKRVH FRPSDUDWLYH RUGHU VHPDQWLFV DUH D ZHDN WRWDO RUGHU UHODWLYH WR HDFK EDVH ZRUOGf 7KH VPDOOHVW PHPEHU RI WKLV LQWHUYHQLQJ IDPLO\ LV FKDUDFWHUL]HG E\ D VHPLFRQQHFWHG SDUWLDO RUGHU 7KH YLHZ WKDW VRPH YHUVLRQ RI FRPSDUDWLYH VLPLODULW\ RUGHULQJ LV FDSDEOH RI VXSSRUWLQJ DQ DQDO\VLV RI WKH FRXQWHUIDFWXDO FRQGLWLRQDO LV GHIHQGHG YLL

PAGE 8

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n GLWLRQDO WLPH DQG H[SHQVH P\ LQWHUIHUHQFH KDV SXW \RX WR DQG \RX EHn JLQ WR ZRQGHU DORXG LI UHDOO\ NQHZ ZKDW ZDV WDONLQJ DERXW ,Q WKH SURFHVV RI WKH FRQYHUVDWLRQ DVVHUW ,I \RX KDG QRW FKDQJHG WKH RLO \RXU HQJLQH ZRXOG KDYH VHL]HG XS 2I FRXUVH WKLV SUHVHQW DVVHUWLRQ LV QRWKLQJ QHZ DP PDNLQJ WKH VDPH DVVHUWLRQ QRZ DV GLG LQ WKH SDVW 0\ ORFXWLRQ KDV FKDQJHG WR UHIOHFW RXU QHZ SHUVSHFWLYH %XLOGLQJ D OLWWOH PRUH RQ WKLV H[DPSOH VXSSRVH WKDW \RXU GRXEWV DERXW P\ H[SHUWLVH DULVH HYHQ DV \RX DUH GUDLQLQJ WKH FUDQNFDVH %\ ZD\ RI UHDVVXULQJ \RX VD\ ,I WKH RLO ZHUH QRW FKDQJHG WKH HQJLQH ZRXOG VHL]H XS $JDLQ VHHP WR KDYH VDLG WKH VDPH WKLQJ 8SRQ DUn ULYLQJ DW RXU GHVWLQDWLRQ \RXU GRXEWV DQG P\ FRQILGHQFH XQVZHUYLQJ ZH FRQVXOW D WKLUG SDUW\ GHVFULELQJ WKH VWDWH RI WKH RLO EHIRUH ZH VHW RXW 7KH WKLUG SDUW\ VD\V ,I WKDW RLO ZDV QRW FKDQJHG WKHQ \RXU

PAGE 9

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n JDUGLQJ WKH DVVHUWDELOLW\ FRQGLWLRQV DQG WKH WUXWK FRQGLWLRQV RI VXFK FRQGLWLRQDOV ZKHWKHU RU QRW WKH LQGLFDWLYH RU VXEMXQFWLYH PRRG LV HVn VHQWLDOO\ LQYROYHG LQ ZKDW VHQVH WKH FRQGLWLRQDO LV FRXQWHUIDFWXDO ZKHWKHU WKH FRQGLWLRQDO LV D PDWHULDO FRQGLWLRQDO DQG ZKDW RWKHU FRQGLWLRQDOW\SHV PD\ EH VXIILFLHQWO\ FORVHO\ UHODWHG WR WKLV SDUDGLJn PDWLF RQH VR DV WR EH VXEVXPDEOH WR D FRPPRQ DQDO\VLV &RQVLGHU WKH FRQGLWLRQDO ,I WKH RLO ZHUH QRW FKDQJH WKH HQJLQH ZRXOG QRW VHL]H XS ,Q WKH FLUFXPVWDQFH GHVFULEHG VLQFH WKH RLO LV FKDQJHG ERWK RI WKHVH FRQGLWLRQDOV KDYH IDOVH DQWHFHGHQWV +HQFH LI WKH\ DUH PDWHULDO FRQGLWLRQDOV WKH\ DUH ERWK WUXH ,I WKLV LV WKH FDVH WKHQ ZH PXVW ORRN HOVHZKHUH WKDQ WR WKHLU WUXWK FRQGLWLRQV IRU ZK\ RXU EHKDYLRU LV VR GLIIHUHQW GHSHQGLQJ XSRQ ZKLFK ZH EDVH RXU DFn WLRQV RQ 7KLV LV KLJKO\ LPSODXVLEOH DQG IOLHV LQ WKH IDFH RI WKH IDFW WKDW \RX FKDQJH WKH RLO MXVW EHFDXVH \RX EHOLHYH P\ DVVHUWLRQ WR EH WUXH

PAGE 10

2QH PLJKW VWLOO FKDUJH WKDW WKH FRQGLWLRQDO LV D PDWHULDO FRQn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n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n TXHQW GLVFRYHU\ WKDW WKH RLO ZDV QRW FKDQJHG DQG WKH HQJLQH GLG VHL]H XS 6LQFH WKH ILUVW RI WKH DERYH DVVHUWLRQV LV LQGLFDWLYH DQG WKH

PAGE 11

VHFRQG VXEMXQFWLYH WKH VXEMXQFWLYH PRRG LV QRW HVVHQWLDO WR VXFK FRQn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n GHQW DQG FRQVHTXHQW FRXQW DJDLQVW WKH WUXWK RI WKH FRQGLWLRQDO ,Qn GHHG /HZLVn DFFRXQW >@ LQFRUSRUDWHV WKH FRQWUDU\ YLHZf 7KH WKLUG SDUW\nV FKRLFH RI D SDVW LQGLFDWLYH RU VXEMXQFWLYH VHQWHQFH WR H[SUHVV KLPVHOI GRHV QRW EHOLHYH UHIOHFW D FKRLFH EHn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

PAGE 12

7U\LQJ WR LGHQWLI\ D SDUWLFXODU NLQG RI FRQGLWLRQDO QRW RQ WKH EDVLV RI LWV JUDPPDWLFDO IRUP DQG YDULDQWV EXW E\ DSSHDO WR H[DPSOHV KDV WKH GUDZEDFN RI QHHGLQJ WR VSHFLI\ D VHW RI IDLUO\ FOHDUFXW H[n DPSOHV FRYHULQJ DOO VHULRXV SRVVLELOLWLHV 2QH H[DPSOH LV QRW VXIILn FLHQW VLQFH VSHFLDO IHDWXUHV WKDW LW SRVVHVVHV PD\ FDXVH WKH NLQG RI FRQGLWLRQDO WR EH FLUFXPVFULEHG WRR QDUURZO\ WKXV QDUURZLQJ WKH VFRSH RI DQ\ VXEVHTXHQW DQDO\VLV 7KHUH DUH WZR IHDWXUHV RI WKH H[DPSOH ZH KDYH FRQVLGHUHG WKDW EHDU PHQWLRQLQJ LQ WKLV FRQQHFWLRQ WKH H[DPSOH JLYHQ LV HVVHQWLDOO\ D FRQGLWLRQDO SUHGLFWLRQ VHH (OOLV >@f RU D VHTXHQWLDO FRQGLWLRQDO 6HTXHQWLDO FRXQWHUIDFWXDO LV -DFNVRQnV WHUP LQ >@f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n GHQW DQG FRQVHTXHQW $V D VHFRQG H[DPSOH OHW XV UHWDLQ FRQQHFWLRQ EXW DW OHDVW FORXG VHTXHQWLDOLW\ DV LQ WKLV H[DPSOH IURP /HZLV >@ (O ,I NDQJDURRV KDG QR WDLOV WKH\ ZRXOG WRSSOH RYHU 1H[W OHW XV UHWDLQ VHTXHQWLDOLW\ EXW RPLW FRQQHFWLRQ DV LQ WKLV H[DPSOH

PAGE 13

IURP 3ROORFN >@ ,I WKH ZLWFK GRFWRU ZHUH WR GR D UDLQ GDQFH WKHQ LW ZRXOG VWLOOf QRW UDLQ (O (YHQ LI WKH ZLWFK GRFWRU ZHUH WR GR D UDLQ GDQFH LW ZRXOG VWLOOf QRW UDLQ )LQDOO\ ZH FDQ UHYHUVH VHTXHQWLDOLW\ (O ,I WKH HQJLQH ZHUH WR VHL]H XS WKHQ WKH RLO ZRXOG QRW KDYH EHHQ FKDQJHG KDYH LQFOXGHG DQRWKHU YHUVLRQ RI (O EHFDXVH LW PLJKW EH WKRXJKW WKDW WKH (O LV D PRUH QDWXUDO ZD\ RI H[SUHVVLQJ WKH ODFN RI D FRQQHFWLRQ EHWZHHQ DQWHFHGHQW DQG FRQVHTXHQW ZKLFK DSSHDUV WR XQGHUOLH VXFK D FRQGLWLRQDO 6XFK FRQGLWLRQDOV ZHUH FDOOHG E\ *RRGPDQ > S @ VHPLIDFWXDOV DQG E\ 3ROORFN > S @ HYHQ LInFRQGL WLRQDOV 7KH IRUPHU DSSHOODWLRQ FRPHV IURP WKH IDFW WKDW LQ WKH W\SLn FDO FDVHV RI VHPLIDFWXDOV WKH FRQVHTXHQW LV DOUHDG\ WUXH DQG WKH DQWHn FHGHQW n V EHLQJ WUXH FDQQRW DOWHU WKDW $ PRGLILFDWLRQ RI RXU HQJLQH H[n DPSOH PD\ UDLVH VRPH GRXEWV DERXW ZKHWKHU WKH WUXWK RI WKH FRQVHTXHQW LV HVVHQWLDO WR VXFK FRQGLWLRQDOV 6XSSRVH WKDW XSRQ FKHFNLQJ \RXU RLO DQG ILQGLQJ LW H[FHVVLYHO\ GLUW\ DOVR QRWLFH WKH UHSUHKHQVLEOH VKDSH \RX KDYH DOORZHG \RXU ILIWHHQ\HDUROG HQJLQH WR JHW LQWR ,Q IDFW DP FRQYLQFHG WKDW \RXU HQJLQH ZRXOG VHL]H XS ZKHWKHU RU QRW \RX FKDQJHG WKH RLO
PAGE 14

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n SUHVV WKH VDPH SURSRVLWLRQ RQH SURVSHFWLYHO\ DQG WKH RWKHU UHWURVSHFn WLYHO\ WKHQ FDQQRW PDLQWDLQ WKDW LQ WKH ILUVW WKH FRQVHTXHQW LV WUXH PDNLQJ LW D VHPLIDFWXDOf DQG LQ WKH VHFRQG LW LV IDOVH PDNLQJ LW D FRXQWHUIDFWXDOf 7KH GLIILFXOW\ RI FRXUVH LV WKDW WKH H[DPSOHV DUH HYDOXDWHG ZLWK UHVSHFW WR WKH VDPH K\SRWKHWLFDO VLWXDWLRQ EXW WKH MXGJPHQW DV WR WKHLU IDFWXDO VWDWXV VHPL RU FRXQWHUf LV PDGH ZLWK UHVSHFW WR GLIIHUHQW DFWXDO VLWXDWLRQV +RZHYHU WKLV GLOHPQD ZLOO QRW SUHYHQW PH IURP EHLQJ DEOH WR DFn FHSW RU UHMHFW WKH FRQGLWLRQDOVf LQ TXHVWLRQ VLQFH GR WKLV XSRQ WKH EDVLV RI WKH DFWXDO VLWXDWLRQ DW WKH WLPH WKH HQJLQH ZDV LQVSHFWHG ZLWK WKH DGGLWLRQDO DVVXPSWLRQ WKDW WKH RLO LV WKHUHXSRQ FKDQJHG DQG P\ NQRZOHGJH RI ZKDW JHQHUDOO\ KDSSHQV WR VXFK PHVVHG XS HQJLQHV $QG LW VHHPV FDQ GR WKLV ZKHWKHU DP LQ WKH SRVLWLRQ RI PDNLQJ D FRQGLn WLRQDO SUHGLFWLRQ EHIRUH ZH VHW RXW D FRQWHPSRUDU\ ODPHQW DV ZH VLW EHVLGH WKH URDG ZLWK D VHL]HG XS HQJLQH RU D UHWURVSHFWLYH UHPLQGHU DIWHU ZH KDYH VDIHO\ DUULYHG ZLWKRXW D VHL]HG XS HQJLQH EHFDXVH WKH RYHUKDXO WRRN SODFH

PAGE 15

, WKLQN WKDW DQ HYHQ LInFRQGLWLRQDO FDQ EHVW EH YLHZHG DV GHQ\LQJ D FRQQHFWLRQ EHWZHHQ DQWHFHGHQW DQG FRQVHTXHQW EXW WKLV QHHG QRW PDNH LW OHVV RI D FRQGLWLRQDO QRU GR ZH QHFHVVDULO\ QHHG D VHSDn UDWH DQDO\VLV IRU VXFK 7KH HYHQ LI LV QRW LQYDULDEO\ D VLJQDO WKDW WKH FRQVHTXHQW LV WUXH DQG WKH DQWHFHGHQW FDQQRW FKDQJH WKDW EXW UDWKHU D VLJQDO WKDW WKLV FRQGLWLRQDO LV QRW JURXQGHG RQ D FRQQHFWLRQ EHWZHHQ DQWHFHGHQW DQG FRQVHTXHQW EXW UDWKHU D ODFN WKHUHRI :H PD\ GLVWLQJXLVK EHWZHHQ WKRVH FDVHV ZKHUH DQ HYHQ LInFRQGLWLRQDO SUHGLFn WLRQ LV DVVHUWHG DV RSSRVHG WR D VWDQGDUG FRQGLWLRQDO SUHGLFWLRQ ,I ZH EHOLHYH WKDW D FHUWDLQ FRQGLWLRQnV EHLQJ IXOILOOHG ZLOO HLWKHU QRW FKDQJH VRPHWKLQJ WKDW LV DOUHDG\ WKH FDVH RU QRW LQ LWVHOI SUHn YHQW VRPHWKLQJ WKDW LV JRLQJ WR FRPH DERXW WKHQ DQ HYHQ LIn FRQGLn WLRQDO SUHGLFWLRQ LV DSSURSULDWH 2Q WKH RWKHU KDQG LI D FHUWDLQ FRQGLWLRQnV EHLQJ IXOILOOHG ZLOO EULQJ VRPHWKLQJ DERXW WKHQ D VGXSOH FRQGLWLRQDO SUHGLFWLRQ LV DSSURSULDWH +RZHYHU WKHVH DUH FRQGLWLRQV RI DVVHUWLELOLW\ QRW WUXWK FRQGLWLRQV ,W UHPDLQV WR EH VHHQ ZKHWKHU D VLQJOH VHW RI WUXWK FRQGLWLRQV FDQ KDQGOH ERWK FRQGLWLRQDOV 8S WR WKLV SRLQW ZH KDYH FRQVLGHUHG FRQGLWLRQDOV ZKHWKHU VXEn MXQFWLYH RU LQGLFDWLYH FRXQWHUIDFWXDO RU VHPLIDFWXDO WKDW DUH DW OHDVW FORVHO\ DVVRFLDWHG ZLWK FRQGLWLRQDO SUHGLFWLRQV ([DPSOH (O DERYH DQG WKH IROORZLQJ (O ,I NDQJDURRV KDG QR WDLOV WKHQ WKH\ ZRXOG VWLOOf EH YHJHWDUn LDQV IDLO WR KDYH DQ REYLRXV VHTXHQWLDO FKDUDFWHU WKRXJK (O LV SUHn VXPDEO\ EDVHG XSRQ WKH SUHVHQFH RI D FRQQHFWLRQ EHWZHHQ DQWHFHGHQW DQG FRQVHTXHQW DQG (O XSRQ D ODFN WKHUHRI

PAGE 16

:KDW VXJJHVWV WKDW WKHVH FRQGLWLRQDOV DUH DFFHVVLEOH WR WKH VDPH DQDO\VLV DV FRQGLWLRQDO SUHGLFWLRQV LV YHU\ URXJKO\f WKH VLPLODULW\ LQ WKH FRQVLGHUDWLRQV WKDW JR LQWR RXU MXGJPHQW WR DFFHSW RU UHMHFW FRQGLWLRQDOV RI HLWKHU W\SH 7KH IROORZLQJ LV WKH FRPPRQ VWDUWLQJ SRLQW RI PDQ\ DQDO\VHV RI FRXQWHUIDFWXDOV > @ 5HFDOO WKDW D FRXQWHUI DFWXDO LQ RXU YLHZ LV RIWHQ D FRQGLWLRQDO SUHGLFWLRQ YLHZHG UHWURVSHFWLYHO\ DJDLQVW WKH NQRZOHGJH WKDW WKH FRQGLWLRQ GLG QRW REWDLQ DW WKH WLPH WKH SUHGLFWLRQ ZDV DSn SURSULDWH 7R RXU LQIRUPDWLRQ DERXW WKH DFWXDO VLWXDWLRQ DW WKH WLPH RI WKH FRQGLWLRQDO SUHGLFWLRQ ZH DGG WKH DVVXPSWLRQ WKDW WKH DQWHFHGHQW FRQn GLWLRQ LV IXOILOOHG FKDQJLQJ ZKDWHYHU LV UHTXLUHG LQ RXU DVVXPSWLRQV DERXW WKH DFWXDO VLWXDWLRQ WR ILW WKLV DGGHG DVVXPSWLRQ :H WKHQ FRQVLGHU ZKDW KDV RFFXUUHG LQ VLPLODU VLWXDWLRQV ZKLFK NQRZOHGJH PD\ EH SUHVHQW IRU XV LQ WKH IRUP RI YDULRXV ODZV FDXVDO DQG RWKHUZLVH 2Q WKLV EDVLV ZH GHWHUPLQH ZKHWKHU RU QRW WKH FRQVHTXHQW ZRXOG EH UHn DOL]HG LQ VXFK D VLWXDWLRQ 6RPHWKLQJ URXJKO\ OLNH WKLV PHWKRG LV ZKDW PLJKW DSSO\ LQ WKH HQJLQH H[DPSOH LQ PDNLQJ WKH SUHGLFWLRQ WKDW GR %DVHG XSRQ P\ SULRU NQRZOHGJH RI VLPLODU VLWXDWLRQV WKH DFWXDO VWDWH RI WKH HQJLQH QRZ XQGHU LQVSHFWLRQ DQG WKH DVVXPSWLRQ WKDW WKH RLO LV QRW FKDQJHG SUHGLFW WKDW LQ WKH QHDU IXWXUH XQGHU QRUPDO GULYLQJ WKH HQJLQH ZLOO VHL]H XS 7R HYDOXDWH (O RQ WKH RWKHU KDQG PD\ WDNH LQWR DFFRXQW P\ NQRZOHGJH RI WKH SK\VLRJQRP\ RI NDQJURRV WKHLU VNHOHWDO VWUXFWXUH DQG ODZV FRQFHUQLQJ EDODQFH DQG FHQWHU RI JUDYLW\ 7R WKLV DGG WKH DVn VXPSWLRQ WKDW NDQJDURRV KDYH QR WDLOV FKDQJLQJ WKH NQRZQ IDFWV DERXW NDQJDURRV QR PRUH WKDQ QHFHVVDU\ WR DFFRPPRGDWH WKLV DVVXPSWLRQ QRZ

PAGE 17

KDYH D VHW RI IDFWV DQG ODZV DQG LI LW LV D FRQVHTXHQFH RI WKLV VHW WKDW NDQJDURRV WRSSOH RYHU WKHQ WKH FRQGLWLRQDO LQ TXHVWLRQ LV DFn FHSWHG 7KH RQO\ GLIIHUHQFH LQ WKH WZR FDVHV LV WKH VSHFLILF WHPSRUDO RUGHU LQ WKH IRUPHU QRW LQ WKH ODWWHU 3UHVXPDEO\ WKLV GLIIHUHQFH LV LQFRUSRUDWHG LQ ODUJH SDUW LQ WKH ODZV DSSOLFDEOH WR WKH GLIIHULQJ VLWXDWLRQV 2WKHUZLVH LW VHHPV FDQ KDQGOH WKHP TXLWH VLPLODUO\ ,Q DSSO\LQJ D VLPLODU SURFHGXUH WR (O ILQG WKDW WKH FKDQJHV PDNH LQ WKH NQRZQ IDFWV DQG ODZV LQ RUGHU WR DFFRPPRGDWH WKH DVVXPSn WLRQ WKDW NDQJDURRV KDYH QR WDLOV OHDYH XQFKDQJHG WKDW IDFW WKDW NDQn JDURRV DUH YHJHWDULDQV +HQFH WKLV IDFW DSSHDUV LQ WKH VHW RI IDFWV DQG ODZV VR DV D FRQVHTXHQFH RI LW +HQFH WKH HYHQ LIFRQGLWLRQDO LV DFFHSWHG 1RWH DOVR WKDW DQ DVHTXHQWLDO FRQGLWLRQDO VHH -DFNVRQ >@f LV VWLOO QRW WKDW IDU UHPRYHG IURP D VHTXHQWLDO FRQGLWLRQDO ,I (O LV DFFHSWDEOH DQG ZH ZHUH WR VRPHKRZ EULQJ LW DERXW WKDW NDQJDURRV EHFDPH WDLOOHVV ZH ZRXOG H[SHFW WKHP WR WRSSOH RYHU +HQFH ZH FDQ DOVR PDNH WKH SUHGLFWLRQ WKDW ,I NDQJDURRV DUH GHWDLOHG WKHQ WKH\ ZLOO WRSSOH RYHU 7KH VLPLODULW\ LQ WKH LQIRUPDOO\ VNHWFKHG PHWKRGV DERYH VXJJHVWV WKDW ERWK VHTXHQWLDO DQG DVHTXHQWLDO FRQGLWLRQDOV PD\ EH DFFHVVLEOH WR WKH VDPH DQDO\VLV LQ WHUPV RI WUXWK FRQGLWLRQV 7KH UHYHUVH RU EDFNn ZDUGV VHTXHQWLDO VHH -DFNVRQ >@f RI (O FDQ DOVR EH VHHQ DV VLPLODU :KHUH ZLWK D IRUZDUGV VHTXHQWLDO ZH FRQVLGHU ZKHWKHU WKH DQWHFHGHQW LV VXIILFLHQW IRU WKH FRQVHTXHQW LQ WHUPV RI WKH ODZV IRU WKH EDFNZDUGV VHTXHQWLDO ZH FRQVLGHU ZKHWKHU WKH FRQVHTXHQW LV QHFHVn VDU\ IRU WKH DQWHFHGHQW WR EH VXEVHTXHQWO\ UHDOL]HG LQ WHUPV RI WKH ODZV LQYROYHG

PAGE 18

:KLOH WLH WKUXVW RI WKH DERYH UHPDUNV LV WR EURDGHQ WKH VFRSH RI WKH FRQGLWLRQDOV ZLWK ZKLFK ZH ZLOO EH FRQFHUQHG DQG WR LQGLFDWH WKDW FRXQWHUIDFWXDO RU VXEMXQFWLYH LV QRW D QHFHVVDU\ PDUN RI VXFK FRQn GLWLRQDOV QHYHUWKHOHVV PRVW RI WKH FRQGLWLRQDOV ZH DUH FRQFHUQHG ZLWK FDQ EH H[SUHVVHG DV VXEMXQFWLYH FRQGLWLRQDOV ZLWK SURSRVLWLRQDO FRQn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f EH WKH FDVH WKDW T 8ST U ,I LW ZHUH WKH FDVH WKDW S WKHQ LW FRXOG QRW EH A IDOVH WKDW T 7KH IRUPHU LV LQWHQGHG WR V\PEROL]H WKRVH FRQGLWLRQDOV ZKHUH WKHUH LV DQ DEVHQFH RI D FRQQHFWLRQ EHWZHHQ DQWHFHGHQW DQG FRQVHTXHQW HYHQ LI FRQGLWLRQDOVf DQG WKH ODWWHU WKRVH FRQGLWLRQDOV ZKHUH WKHUH LV D FRQn QHFWLRQ EHWZHHQ DQWHFHGHQW DQG FRQVHTXHQW LH ZKHUH WKH DQWHFHGHQW EULQJV DERXW WKH FRQVHTXHQW ZKDW 3ROORFN > S @ FDOOV QHFHVVL WDWLRQFRQGLWLRQDOVf :H VKDOO DOVR DGRSW WKH VWDQGDUG 3ROLVK RU SUHIL[ QRWDWLRQ IRU WKH XVXDO ORJLFDO RSHUDWLRQV RI PDWHULDO FRQGLn WLRQDOLW\ PDWHULDO ELRFRQGLWLRQDOLW\ QHJDWLRQ FRQMXQFWLRQ DQG LQn FOXVLYHf GLVMXQFWLRQ 7KHVH DQG WKH V\PEROL]DWLRQ IRU VWULFW FRQGLn WLRQDOLW\ QHFHVVLW\ DQG SRVVLELOLW\ LQ PRGDO ORJLFV DUH OLVWHG EHORZ

PAGE 19

FST GI ,I S WKHQ T (ST GI S LI DQG RQO\ LI T 13 GI 1RW S .ST r GI %RWK S DQG T $ST GI (LWKHU S RU T rGI S LV QHFHVVDU\ ,, e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n FHSW ERWK RI WKH IROORZLQJ (O ,I %L]HW DQG 9HUGL ZHUH FRPSDWULRWV WKH\ PLJKW ERWK EH )UHQFK (O ,I %L]HW DQG 9HUGL ZHUH FRPSDWULRWV WKH\ PLJKW ERWK EH ,WDOLDQ 7R V\PEROL]H WKH PLJKWFRQGLWLRQDO ZH LQWURGXFH 9ST ,I LW ZHUH WKH FDVH WKDW S WKHQ LW PLJKW EH WKH FDVH WKDW T

PAGE 20

$W WKLV SRLQW ZH KDYH PDGH UHIHUHQFH WR VL[ FRQGLWLRQDOV :ST 7ST DQG 8ST ZKHUH ZH H[SHFW RQH VHW RI WUXWK FRQGLWLRQV WKH ODWWHU WZR FRQGLWLRQDOV SUHVXPDEO\ EHLQJ VXEFODVVHV RI WKH IRUPHU 9ST GLVn WLQFW IURP WKH DERYH WKUHH DQG &ST DQG /&ST ZKLFK DUH WKH WUDGLWLRQDO PDWHULDO DQG VWULFW FRQGLWLRQDOV UHVSHFWLYHO\ %\ ZD\ RI WHUPLQRORJ\ ZH ZLOO UHIHU WR WKH ILUVW IRXU LQGLVFULPLQDWHO\ DQG VRPHZKDW LQDFn FXUDWHO\f DV FRXQWHUIDFWXDOV WKH ILUVW WKUHH DV ZRXOGFRUQWHUIDFWXDOV WKH IRXUWK DV WKH PLJKWFRXQWHUIDFWXDO WKH VHFRQG DV WKH HYHQ LI FRXQWHUIDFWXDO WKH WKLUG DV WKH QHFHVVLWDWLRQFRXQWHUIDFWXDO 7KH VDPH SUHIL[HV ZLWK WKH VXIIL[ FRQGLWLRQDO ZLOO DOVR EH XVHG :KHQ WKH WHUP FRXQWHUIDFWXDO DORQH LV XVHG WKLV ZLOO XVXDOO\ UHIHU WR WKH n fZRXOGn FRQGLWLRQDO ,W ZRXOG EH DSSURSULDWH DW WKLV SRLQW WR FRQVLGHU WKH YDULRXV LQn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f ,W LV ZHOO NQRZQ WKDW ERWK RI WKH IROORZLQJ LQIHUHQFHV DUH YDOLG LQ FODVVLFDO SURSRVLWLRQDO DQG PRGDO ORJLFV &ST /&ST f f &.SUT /&.SUT +RZHYHU FRQVLGHUDWLRQ RI D VLQJOH H[DPSOH ZLOO VKRZ WKDW WKH FRUUHVSRQGn LQJ SDWWHUQ IRU FRXQWHUIDFWXDOV

PAGE 21

:ST :.SUT LV QRW YDOLG 6SHDNLQJ RI D FHUWDLQ GU\ PDWFK LQ IDYRUDEOH FRQGLWLRQV HQRXJK R[\JHQ HWFf PD\ VD\,I WKLV PDWFK ZHUH VWUXFN LW ZRXOG OLJKW %XW LW GRHV QRW IROORZ IUDQ WKLV WKDW ,I WKLV PDWFK ZHUH VRDNHG LQ ZDWHU DQG VWUXFN LW ZRXOG OLJKW 7KH IDLOXUH RI 6WUHQJWKn HQLQJ WKH $QWHFHGHQW LV RQH RI WKH VWULNLQJ SHFXOLDULWLHV RI FRXQWHU IDFWXDO V DQG WKH VLQJOH VWURQJHVW DUJXPHQW DJDLQVW WKH FRXQWHUIDFWXDO FRQGLWLRQDO EHLQJ D VWULFW FRQGLWLRQDO 7KH 0HWDOLQJXLVWLF $QDO\VLV UHIHUUHG HDUOLHU WR D FHUWDLQ SURFHGXUH ZKHUHE\ D FRXQWHUIDF WXDO FRXOG EH HYDOXDWHG DV WKH VWDUWLQJ SRLQW RI D QXPEHU RI DQDO\VHV RI FRXQWHUIDFWXDOV 7KLV LV ZKDW KDV EHHQ FDOOHG WKH OLQJXLVWLF RU PHWDOLQJXLVWLF DFFRXQW E\ 3ROORFN >@ DQG /HZLV >@ UHVSHFWLYHO\f %HFDXVH PRVW KROG WKDW WKH FRQVHTXHQW RI D FRXQWHUIDFWXDO LV D ORJLFDO FRQVHTXHQFH RI WKH DQWHFHGHQW FRQMRLQHG ZLWK RWKHU VWDWHPHQWV WKH\ DUH DOVR FDOOHG FRQVHTXHQFH WKHRULHV $FFRUGLQJ WR VXFK DFFRXQWV WKH WUXWK RI D FRXQWHUIDFWXDO FRQGLWLRQDO LV ODUJHO\ EDVHG XSRQ WKH UHODn WLRQV DPRQJ FHUWDLQ OLQJXLVWLF HQWLWLHV VXFK DV VHQWHQFHV RU LQ VDQH FDVHV EHOLHIV 6XFK DFFRXQWV KDYH EHHQ RIIHUHG E\ *RRGPDQ >@ &KLVKROP >@ 0DFNLH >@ 5HVFKHU >@ -DFNVRQ >@ (OOLV >@ DQG RWKHUVf VKDOO VNHWFK D JHQHUDO RXWOLQH RI VXFK DQ DFFRXQW ZKLFK GRHV QRW GR IXOO MXVWLFH WR DQ\ RI WKRVH WKDW KDYH DFWXDOO\ EHHQ RIn IHUHG EXW LV VXIILFLHQW WR IRUP D VWDUWLQJ SRLQW IRU FULWLFLVP &RQVLGHU WKH ZHOOZRUNHGRYHU H[DPSOH FRQFHUQLQJ D FHUWDLQ SUHVn HQWO\ XQOLW PDWFK

PAGE 22

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n FLILF IDFWV ) LV QRZ WUDQVIRUPHG LQWR WKH SUREOHP RI GHWHUPLQLQJ WKH VSHn FLILF FRQGLWLRQV & XQGHU ZKLFK WKH ODZ KROGV )XUWKHUPRUH WKLV DSSURDFK ZRXOG QRW ZRUN IRU HYHQ LInFRQGLWLRQDOV ZKHUH WKHUH LV QR FRYHULQJ ODZ

PAGE 23

FRQQHFWLQJ WKH DQWHFHGHQW DQG UHOHYDQW FRQGLWLRQVf WR WKH FRQVHn TXHQW :H FDQ UHWUHDW WR WKH RULJLQDO GHILQLWLRQ OHW WKH ODZV EH RI UHDVRQDEOH JHQHUDOLW\ DQG FRQFHQWUDWH RQ WKH SUREOHP RI VSHFLI\LQJ WKH IDFWV ) DQG ODZV / IRU D JLYHQ FRXQWHUIDFWXDO %XW SHUKDSV ERWK WKHVH LVVXHV FDQ EH VLGHVWHSSHG SUHVXPDEO\ RXU ODZV DUH FRQVLVWHQW DV D VHW OLNHZLVH WKH IDFWV HPERGLHG LQ D GHVFULSWLRQ RI DOO WKH FLUn FXPVWDQFHV VXUURXQGLQJ WKH DQWHFHGHQW :K\ QRW WDNH DOO ODZV DQG DOO WUXH IDFWV REWDLQLQJ DQG FRQMRLQ WKHP ZLWK WKH DQWHFHGHQW 7KH SURn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n GLQJ WKDW LW LV VWUXFN LPSOLHV E\ D VXLWDEOH ODZ WKDW LW ZDV QRW GU\ 6R ZH PD\ LQFOXGH LQ ) RQO\ VXFK VWDWHPHQWV U ZKLFK DUH QRW RQO\ WUXH EXW ZRXOG QRW EH IDOVH LI S ZHUH WUXH LH IRU ZKLFK 1:S1U LV WUXH *RRGPDQ > S @ FDOOV VXFK VWDWHPHQWV WKRVH FRWHQDEOH ZLWK S DQG ULJKWO\ REVHUYHV WKDW QRZ ZH DUH DQDO\]LQJ D FRXQWHUI DFWXDO LQ WHUPV RI RWKHU FRXQWHUIDFWXDOV VR RXU DFFRXQW LV LUUHGHHPDEO\ FLUFXODU

PAGE 24

3ROORFN >@ REVHUYHV WKDW LI DOO WKDW LV UHTXLUHG IRU LQFOXVLRQ LQ ) DV *RRGPDQ DSSHDUV WR EHOLHYH LV WUXWK DQG FRWHQDELOLW\ WKHQ WKLV LPSOLHV DQ HYHQ VWURQJHU UHTXLUHPHQW RQ WKH WUXWKV LQ ) 7R VKRZ WKLV ZH UHTXLUH DFFHSWDQFH RI WZR REYLRXV SULQFLSOHV UHJDUGLQJ FRXQWHU IDFWXDO V $f ,I :ST LV WUXH DQG /&TU LV WUXH WKHQ :SU LV WUXH %f ,I :S&ST LV WUXH WKHQ :ST LV WUXH :H SRVWXODWH WKH IROORZLQJ LQ DFFRUGDQFH ZLWK *RRGQDQnV SURSRVDOV Df :ST LV D FRXQWHUIDFWXDO WR EH HYDOXDWHG DQG S LV IDOVH Ef &RWHQDELOLW\ DQG WUXWK DUH VXIILFLHQW IRU LQFOXVLRQ RI U LQ VHW ) LH U LV WUXH DQG 1:S1U LV WUXH 1RWLQJ WKDW /&.S1U1U LV WUXH DQG XVLQJ $f FRQWUDSRVLWLYHO\ LW IROORZV WKDW 1:S.S1U LV WUXH 1RWLQJ WKDW /(.S1U1&SU LV WUXH DQG XVLQJ $f FRQWUDSRVLWLYHO\ DJDLQ LW IROORZV WKDW 1:S1&SU LV WUXH 6LQFH S LV IDOVH &SU LV WUXH KHQFH E\ Ef &SU LV LQFOXGHG LQ ) DV LW LV WUXH DQG FRWHQDEOH ZLWK S 6LQFH ) WRJHWKHU ZLWK S ORJLFDOO\ LPSOLHV DQ\n WKLQJ LQFOXGHG LQ ) *RRGQDQnV SURSRVDO YDOLGDWHV :S&SU +HQFH E\ %f :SU LV WUXH 6R U LV FRWHQDEOH ZLWK S DPRXQWV WR :SU LV WUXH RQ *RRGPDQnV DVVXPSWLRQV > S @ ,I ) LV WR LQFOXGH HYHU\WKLQJ FRWHQDEOH ZLWK S WKHQ ) LQFOXGHV WKH FRQVHTXHQWV RI DOO WUXH VWDWHPHQWV RI WKH IRUP :SU 7KDW LV ) LQn FOXGHV HYHU\WKLQJ WKDW ZRXOG EH WKH FDVH LI S ZHUH WUXH &RXQWHUI DFn WXDOV DUH DQDO\]HG LQ WHUPV RI FRWHQDELOLW\ DQG RWKHU HOHPHQWVf EXW WKHQ FRWHQDELOLW\ LV DQDO\]HG LQ WHUPV RI FRXQWHU IDFWXDO V DQG VR RXU DQDO\VLV LV FLUFXODU %HFDXVH KH GLG QRW VHH D ZD\ RXW RI WKLV YLFLRXV FLUFOH LQ DQDO\]LQJ FRXQWHUIDFWXDOV *RRGPDQ VKLIWHG KLV FRQFHUQ WR D ZHDNHU QRWLRQ WKDW RI GLVSRVLWLRQV

PAGE 25

6LQFH DQDO\]LQJ FRXQWHUIDFWXDOV LQ WHUPV RI FRXQW HUI DFWXDOO\ GHILQHG FRWHQDELOLW\ LV VR REYLRXVO\ FLUFXODU LW LV FXULRXV WKDW D UHFHQW WUHDWPHQW RI FRXQWHUIDFWXDOV VHHPV WR PDNH D YLUWXH RI LW (OOLV SURYLGHV D XQLILHG DFFRXQW RI WKUHH NLQGV RI FRQGLWLRQDOV LQ WHUPV RI KLV QRWLRQ RI D UDWLRQDO EHOLHI V\VWHP > S @ 6HH >@ DOVRf 2QH RI WKHVH FRQGLWLRQDOV LV WKDW ZKLFK ZH KDYH EHHQ FDOOLQJ FRXQWHU IDFWXDO :KLOH DP LQ FRPSOHWH DJUHHPHQW ZLWK (OOLVn FRQFOXVLRQ WKDW LQGLFDWLYH DQG VXEMXQFWLYH FRQGLWLRQDOV DUH XVXDOO\ YDULDQW ORFXn WLRQV IRU WKH RQH NLQG RI FRQGLWLRQDO ZKLFK LV YDULDEO\ VWULFW > S @ DQG KDYH VR DUJXHG LQ WKH ILUVW VHFWLRQ GR QRW VHH KRZ KLV DFFRXQW FDQ EH FRQVWUXHG DV DQ DQDO\VLV RI FRQGLWLRQDOV SDUWLFXODUO\ RI WKH YDULDEO\ VWULFW FRQGLWLRQDO ZKLFK ZH VKDOO VHH ODWHU LV DQ DSSURn SULDWH ZD\ WR UHIHU WR WKH FRXQWHUIDFWXDO FRQGLWLRQDO 0\ UHDVRQ IRU WKLV UHVHUYDWLRQ LV WKDW KLV DFFRXQW XVHV WKH FRXQWHUIDFWXDO FRQGLWLRQDO WR JLYH WKH WUXWK FRQGLWLRQV IRU WKH FRXQWHUIDFWXDO FRQGLWLRQDO LQ PXFK WKH VDPH ZD\ DV *RRGPDQnV VHOIDGPLWWHGO\ IDLOHG DFFRXQW (OOLVn WUXWK FRQGLWLRQ IRU WKH FRQGLWLRQDO PD\ EH SDUDSKUDVHG DV IROORZV > S @ 'O :ST LV KHOG WUXH LQ EHOLHI V\VWHP % MXVW LQ FDVH LQ DOO FRPn SOHWHG H[WHQVLRQV RI D FHUWDLQ PRGLILFDWLRQ RI % %A 1T QRZKHUH RFFXUV $ UDWLRQDOf EHOLHI V\VWHP LV HVVHQWLDOO\ D SDUWLDO HYDOXDWLRQ RQ DOO WKH VHQWHQFHV RI D ODQJXDJH FHUWDLQ VHQWHQFHV DUH KHOG WUXH RWKHUV IDOVH DQG RWKHUV ZLWKKHOG LH QR ILUP EHOLHI RQH ZD\ RU WKH RWKHUf 7KHUH DUH D QXPEHU RI UDWLRQDOLW\ UHTXLUHPHQWV RQ D EHOLHI V\VWHP DPRQJ ZKLFK LV 'O DERYH $ FRPSOHWHG H[WHQVLRQ RI D EHOLHI V\VWHP LV WKH UHSODFHPHQW RI DOO ZLWKKHOG HYDOXDWLRQV E\ WUXH RU IDOVH HYDOXn DWLRQV ZLWKRXW YLRODWLQJ DQ\ RI WKH UDWLRQDOLW\ UHTXLUHPHQWV 7KHVH

PAGE 26

QRWLRQV DUH DOO XQSUREOHPDWLF DV DUH WKH UDWLRQDOLW\ UHTXLUHPHQWV QRW SUHVHQWHG KHUH :KDW LV RI FRQFHUQ LV WKH GHILQLWLRQ RI WKH PRGLILHG EHOLHI V\VWHP LI FRQVWUXFWHG IURP % ZKLFK VHUYHV WR FKDUDFWHUL]H WKH FRQGLWLRQDO :ST $FFRUGLQJ WR (OOLV EHOLHI V\VWHP %A FDQ EH WKRXJKW RI DV WKH DVVXPHG EDVLV RI UHDVRQLQJ IURP WKH VXSSRVLWLRQ WKDW S > S @ )RU WKH FRXQWHUIDFWXDO FRQGLWLRQDO (OOLVn GHILQLWLRQ RI %A PD\ EH SDUDSKUDVHG DV IROORZV > S @ 'O f U LV KHOG WUXH LQ %A LI HLWKHU /U RU :SU LV KHOG WUXH LQ % f S LV KHOG WUXH LQ %? f 2WKHUZLVH %A LV DJQRVWLF ,Q (OOLVn ZRUGV %A LQFOXGHV QRW RQO\ ZKDW ZH VKRXOG WDNH WR EH QHFHVn VDULO\ WUXH EXW DOVR ZKDW ZH WKLQN ZRXOG VWLOO EH RU KDYH EHHQ WKH FDVH LI nSn ZHUH > S @ ,Q YLHZ RI FRQGLWLRQ 'Of DV DQ DQDO\VLV RI WKH FRXQWHUIDF WXDO FRQGLWLRQ WKLV DFFRXQW LV FLUFXODU 'Of UHSODFHV *RRGPDQnV QRWLRQ RI FRWHQDELOLW\ DQG 'O UHSODFHV WKH UHTXLUHPHQW WKDW WKH FRQMXQFWLRQ RI ) / DQG S ORJLFDOO\ LPSOLHV T :KDW WKHQ GRHV (OOLVn DFFRXQW DFFRPSOLVK" $QG ZK\ FRQVLGHULQJ WKH IDPLO\ UHVHPEODQFH RI 'Of WR FRWHQDELOLW\ VKRXOG WKLV DFFRXQW EH WKRXJKW WR DGYDQFH WKH WKHRU\ RI FRQGLWLRQDOV" 7R DQVZHU WKHVH TXHVWLRQV UHTXLUHV D GLJUHVVLRQ RQ WKH VXEMHFW RI ZKDW FRQVWLWXWHV DQ DQDO\VLV $QDO\VLV FDQ PHDQ RQH RI WZR WKLQJV QRW QHFHVVDULO\ H[FOXVLYH %RWK DUH URXWHV IRU WKH FODULILFDWLRQ RI D FRQFHSW 2QH LV WR H[SOLFDWH RU DUWLFXODWH WKH FRQFHSW LQ WHUPV RI RWKHU SUHVXPDEO\ EHWWHU XQGHUn VWRRG FRQFHSWV ,Q WKLV FRQWH[W DQ DQDO\VLV LV PXFK OLNH D GHILQLWLRQ

PAGE 27

IRU D FRPSOHWH DQDO\VLV WKH DQDO\VDQGXP RIIHUHG LV D GHILQLWLRQDO HTXLYn DOHQW IRU WKH DQDO\VDQV ,W LV RI FRXUVH D VHULRXV VKRUWFRPLQJ LQ D GHILQLWLRQ IRU WKH WHUP GHILQHG WR DSSHDU LQ WKH GHILQLWLRQ LWVHOI RQ WKH DQDO\VDQGXP VLGH :H DUH QRW VSHDNLQJ KHUH RI D UHFXUVLYH GHILQLn WLRQf 7KHUH DUH RWKHU FRQVWUDLQWV RQ DQDO\VLV :KHUH FOHDU XVDJH LV HYLGHQW LQ WKH SUHDQDO\WLF FRQFHSW WKLV XVDJH VKRXOG EH SUHVHUYHG XQGHU WKH DQDO\VLV $ FRQFHSW ZLWK QR SX]]OLQJ FDVHV LV LQ QHHG RI QR FODULILFDWLRQ VR QR DQDO\VLV KHQFH DQ DQDO\VLV VKRXOG JR VRPH ZD\ WRn ZDUG UHVROYLQJ WKH SX]]OLQJ FDVHV 3X]]OLQJ FDVHV IRU FRXQWHUIDFWXDOV LQYROYH WKH %L]HW DQG 9HUGL H[DPSOHV RI 6HFWLRQ FRXQWHUOHJ£LV FRXQWHULGHQWLFDOV DQG RWKHUV ZKHUH WKHUH VHHPV WR EH VHPH TXHVWLRQ DV WR KRZ WR LQWHUSUHWH WKH DQWHFHGHQW $W WLPHV HYHQ D IDLOXUH WR FRYHU DOO SUHDQDO\WLF FDVHV RI FOHDU XVDJH LV IRUJLYHDEOH LI WKH DQDO\VLV RIn IHUV DGYDQWDJHV LQ RWKHU UHVSHFWV 2I FRXUVH RQH LV WKHQ ULJKWO\ VXEn MHFW WR WKH FKDUJH RI DGYRFDWLQJ D FKDQJH LQ WKH FRQFHSW $ VHFRQG PHWKRG RI DQDO\VLV LV WR FRGLI\ WKH UXOHV JRYHUQLQJ WKH RSHUDWLRQ RI D FRQFHSW 7KLV LV RIWHQ H[SUHVVHG DV PDNLQJ H[SOLFLW WKH ORJLF RI WKH FRQFHSW ,Q WKLV FRQWH[W DQ DQDO\VLV LV PXFK OLNH WKH QRWLRQ RI V\QWDFWLF PHDQLQJ ZKHUH WKH PHDQLQJ RI VD\ D ORJLFDO FRQn QHFWLYH LV VDLG WR EH LPSOLFLWO\ JLYHQ E\ WKH D[LRPV DQG UXOHV RI LQn IHUHQFH WKDW IRUPDOL]H LWV RSHUDWLRQ )RU WHUPV WKDW DSSHDU DV SULPLWLYHV LQ D WKHRU\ VXFK D QRWLRQ LV YDOXDEOH $Q H[DPSOH RI D UHODWLYHO\ SXUH FDVH RI WKH ILUVW W\SH RI DQDOn \VLV ZRXOG EH WKH DQDO\VLV RI NQRZOHGJH DV MXVWLILHG WUXH EHOLHI RU PRUH DFFXUDWHO\ QRQGHIHFWLYHO\ MXVWLILHG WUXH EHOLHIf 0RUH UHOHYDQW WR RXU VXEMHFW ZRXOG EH /HZLVn DQDO\VLV RI D ODZ RI QDWXUH

PAGE 28

D FRQWLQJHQW JHQHUDOL]DWLRQ LV D ODZ RI QDWXUH LI DQG RQO\ LI LW DSSHDUV DV D WKHRUHP RU D[LRPf LQ HDFK RI WKH WUXH GHGXFWLYH V\VWHPV WKDW DFKLHYHV D EHVW FRPn ELQDWLRQ RI VLPSOLFLW\ DQG VWUHQJWK > S @ $Q H[DPSOH RI D UHODWLYHO\ SXUH FDVH RI WKH ODWWHU NLQG RI DQDOn \VLV LV IRXQG LQ :DVVHUPDQ >@ ZKHUHLQ KH SUHVHQWV D VRFDOOHG ORJn LFDO DQDO\VLV RI WKH FRXQWHUIDFWXDO FRQGLWLRQDO :KDW :DVVHUPDQ GRHV LV SURYLGH D ODQJXDJH FRQWDLQLQJ D ELQDU\ FRQQHFWLYH LQWHQGHG WR UHSUHn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f 7KLV UHIOHFWV D SUDFWLFH ZKLFK KDV EHFRPH VWDQGDUG LQ DQDO\WLFDO SKLORVn RSK\ WKH PHDQLQJ RI D FRQFHSW FDQ EH JLYHQ LQ WHUPV RI LWV WUXWK FRQn GLWLRQV 7KXV /HZLV VWDWHV WKDW WKH WDVN LQYROYHG LQ JLYLQJ DQ DQDO\VLV RI FRXQWHUIDFWXDOV LV WR JLYH D FOHDU DFFRXQW RI WKHLU WUXWK FRQGLn WLRQV > S @ )RU 6WDOQDNHU WKH WDVN LV WR ILQG D VHW RI WUXWK FRQGLWLRQV IRU VWDWHPHQWV KDYLQJ FRQGLWLRQDO IRUP ZKLFK H[SODLQV ZK\ ZH XVH WKH PHWKRG ZH GR XVH WR HYDOXDWH WKHP > S @ :KLOH WKH FODLP PD\ QRW EH WKDW WKH WUXWK FRQGLWLRQV FRQVWLWXWH WKH PHDQLQJ RI WKH

PAGE 29

FRQFHSW WKH FODLP FHUWDLQO\ LV WKDW RQFH RQH KDV JUDVSHG WKH WUXWK FRQGLWLRQV RQH KDV JUDVSHG WKH PHDQLQJ RI WKH FRQFHSW %XW KHUH ZH PXVW EH FDUHIXO 1RW MXVW DQ\ VHW RI WUXWK FRQn GLWLRQV ZLOO GR DV 3ROORFN SRLQWV RXW LQ D GLIIHUHQW FRQQHFWLRQ LQ > S @f 6WDOQDNHU LQGLFDWHV WKLV ZKHQ KH VD\V DERYH WKDW WKH WUXWK FRQGLWLRQV PXVW H[SODLQ VRPHWKLQJ -XGJLQJ E\ KLV DQDO\VLV /HZLV KDV VRPHWKLQJ VLPLODU LQ PLQG 7KH TXHVWLRQ WR EH DQVZHUHG LV :KHQ VKRXOG ZH EH VDWLVILHG ZLWK D SXUSRUWHG DQDO\VLV" *RRGPDQ ULJKWO\ UHMHFWV KLV RZQ DQDO\VLV DV FLUFXODU EXW (OOLV RIIHUV DQ DQDO\VLV FRQn WDLQLQJ IRUPDO HOHPHQWV ZLWK SUHFLVHO\ WKH VDPH FKDUDFWHULVWLFV :DVVHUPDQnV DQDO\VLV SURYLGHV D ORJLF IRU WKH FRQGLWLRQDO EXW ZRXOG ZH EH MXVWLILHG LQ FODLPLQJ RQ WKDW EDVLV WR KDYH JUDVSHG LWV PHDQLQJ" 6WDOQDNHU ZKR VKDUHV ZLWK (OOLV WKH FRQYLFWLRQ WKDW WKH PRRG RU IDFWXDO VWDWXV RI D FRQGLWLRQDO LV D VHFRQGDU\ FRQVLGHUDWLRQ GLVWLQn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n FRQGLWLRQDO VWDWHPHQWV > SS @ 7KH GHYHORSPHQW RI D VHPDQWLF WKHRU\ IRU FRXQWHUIDFWXDOV 6WDOQDNHU UHn JDUGV DV SDUW RI WKH ORJLFDO SUREOHP 7KH VHPDQWLF WKHRU\ WKDW KH GRHV GHYHORS VKHGV OLJKW RQ WKH VHFRQG SUREOHP DV ZHOO LQ KLV YLHZ

PAGE 30

EHFDXVH LW VKRZV ZKHUH WKH VHPDQWLF FRPSRQHQW RI WKH FRQFHSW OHDYHV RII DQG WKH SUDJPDWLF FRPSRQHQW EHJLQV > S @ VKRXOG WKLQN KRZHYHU WKDW WKH ORJLFDO SUREOHP DFWXDOO\ LQn YROYHV WZR SUREOHPV WKH WDVN RI GHVFULELQJ WKH IRUPDO ORJLFDO SURn SHUWLHV RI WKH FRQGLWLRQDO DQG WKH WDVN RI GHYLVLQJ D VDWLVIDFWRU\ VHPDQWLFV 7KHVH WZR WDVNV DUH GLIIHUHQW 2QH FRXOG GHVFULEH WKH IRUPDO SURSHUWLHV RI WKH FRQGLWLRQDO LQ WHUPV RI D SURRIWKHRUHWLF V\VWHP D VHW RI D[LRPV DQG UXOHV RI LQIHUHQFH LQ ZKLFK D FRQGLWLRQDO FRQQHFWLYH RFFXUV DQG LQ ZKLFK WKRVH VHQWHQFHV DQG UXOHV RI LQIHUHQFH RXU SUHDQDO\WLF LQWXLWLRQV KROG YDOLG RFFXU ZKLOH WKRVH ZH UHJDUG DV LQYDOLG GR QRW :H ZRXOG EH UHPLVV WR DFFHSW VXFK DQ DQDO\VLV DV FRPn SOHWH IRU LW LV SRVVLEOH WR XQGHUVWDQG WKH ORJLF RI D FRQFHSW ZLWKRXW XQGHUVWDQGLQJ WKH FRQFHSW LWVHOI )RU H[DPSOH LQ >@ &KLVKROP PDNHV XVH RI D UHODWLRQ PRUH UHDVRQDEOH WKDQ KROGLQJ EHWZHHQ SURSRVLWLRQV 7KDW LV D FHUWDLQ SURSRVLWLRQ S PD\ EH PRUH UHDVRQDEOH IRU VXEMHFW 6 DW WLPH W WKDQ DQRWKHU SURSRVLWLRQ T 7KLV DSSHDUV LQ KLV IRUPDO GHILQLWLRQV DV DQ XQGHILQHG UHODWLRQ EXW WR H[SOLFDWH LW KH RIIHUV FHUWDLQ EDVLF SULQn FLSOHV DV D[LRPV RI WKH FRQFHSW LQWHQGHG WR PDNH H[SOLFLW LWV ORJLFDO VWUXFWXUH > S @ ,I OHIW DW WKLV SRLQW ZKLFK &KLVKROP GRHV QRW GRf ZH PD\ KDYH LQ RXU JUDVS WKH ORJLF RI PRUH UHDVRQDEOH WKDQ ZLWKn RXW XQGHUVWDQGLQJ ZKDW LW LV IRU RQH SURSRVLWLRQ WR EH PRUH UHDVRQDEOH WKDQ DQRWKHU :H GR QRW NQRZ KRZ WR DSSO\ WKH UHODWLRQ WR SURSRVLWLRQV RQO\ KRZ WR PDQLSXODWH LWV SUHYLRXV DSSOLFDWLRQ 'HYLVLQJ D VHPDQWLFV WR YDOLGDWH WKLV D[LRPDWLF V\VWHP PD\ QRW LQ LWVHOI EH VXIILFLHQW WR FRQYH\ DQ XQGHUVWDQGLQJ RI WKH FRQFHSW 7KH

PAGE 31

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n VLEO\ SUHVHQWHG LQ WKH IRUP RI D VHPDQWLFV IRU VHQWHQFHV HPSOR\LQJ WKH FRQFHSW PXVW H[SODLQ ZK\ WKH PHWKRG ZRUNV DV LW GRHV 7KXV WKHUH DUH WZR FRQVWUDLQWV RQ D ORJLFDO DQDO\VLV RI D FRQFHSW HYHQ ZKHQ FRQVWUXHG DV D VHDUFK IRU WUXWK FRQGLWLRQV 7KH WUXWK FRQGLWLRQV RU VHPDQWLFVf PXVW EH DSSOLFDEOH DQG XQGHUn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n V DQDO\VLV UHPDLQV LQFRPSOHWH ,Q IDFW VLQFH 6 LV D WUXWK VHW PD[LPDO ZLWK UHVSHFW WR MRLQW VDWLVILDELOLW\ ZLWK S > S @ 6 ZLOO EH XQGHUn FRQVWUDLQHG LQ DQ\ FDVH DV *RRGPDQ KDV SRLQWHG RXW IRU DOO WKDW LV UHTXLUHG RI 6 LV WKDW LW EH WUXH DQG FRQVLVWHQW ZLWK S +HQFH :DVVHUPDQnV DQDO\VLV ZLOO IDLO WR H[SODLQ KRZ RXU SUHDQDO\WLF

PAGE 32

HPSOR\PHQW RI WKH FRQFHSW ZRUNV LI RQO\ EHFDXVH LW ZLOO IDLO WR GLVn WLQJXLVK EHWZHHQ FRUUHFW DQG LQFRUUHFW DSSOLFDWLRQV RI LW (OOLVn DQDO\VLV IDLOV IRU MXVW WKH UHDVRQ ZH VXSSRVHG WKH FRQFHSWV LQ WHUPV RI ZKLFK WKH WUXWK FRQGLWLRQV IRU WKH FRQGLWLRQDO DUH VWDWHG UHTXLUH LQ SDUW WKDW ZH DOUHDG\ XQGHUVWDQG WKH FRQFHSW RI WKH FRQGLWLRQDO :H FDQQRW FRQVWUXFW WKH EHOLHI V\VWHP XQOHVV ZH DQWHFHGHQWO\ XQGHUVWDQG XQGHU ZKDW FRQGLWLRQV VWDWHPHQWV RI WKH IRUP :SU DUH KHOG WUXH LQ V\VWHP % 7KLV LV QRW WR VD\ WKDW (OOLVn VHPDQWLFV LV LQ D IRUPDO VHQVH LOOGHILQHG EXW UDWKHU WKDW WKHUH DUH QRQIRUPDO FULWHULD WKDW WUXWK FRQGLWLRQV PXVW PHHW LQ RUGHU WR TXDOLI\ DV DQ DQDO\VLV 7KXV DQ DQDOn \VLV RI WKH FRQGLWLRQDO PXVW EH LQ WHUPV RI QRQFRQGLWLRQDO QRWLRQV MXVW EHFDXVH ZH DUH UHJDUGLQJ WKH FRQGLWLRQDO QRWLRQV DV SUREOHPDWLF (OOLVn GHILQLWLRQV DPRXQW WR D UHFDVWLQJ RI FRWHQDELOLW\ LQ WHUPV RI EHOLHI V\VWHPV 8QWLO ZH KDYH DQ DQDO\VLV RI FRWHQDELOLW\ LQGHSHQGHQW RI WKH FRQFHSW RI WKH FRQGLWLRQDO RXU DQDO\VLV ZLOO IDOO VKRUW $ JHQHUDO VKRUWFRPLQJ RI ERWK OLQJXLVWLF DFFRXQWV DQG UHODWHG EHOLHIEDVHG DFFRXQWV LV WKDW WKH\ DWWHPSW WR PRGHO RXU LQIRUPDO SURn FHGXUH IRU HYDOXDWLQJ FRQGLWLRQDOV UDWKHU WKDQ H[SODLQ LW 2I FRXUVH ZH GR WDNH DV RXU DVVXPHG EDVLV RI UHDVRQLQJ RQ WKH DVVXLSWLRQ WKDW S LV WUXH ZKDW ZH EHOLHYH ZRXOG VWLOO EH WKH FDVH RQ WKDW DVVXPSWLRQ EXW WKLV LV MXVW WR VD\ ZKDW LW LV ZH GR QRW WR H[SODLQ KRZ RU ZK\ LW ZRUNV 5HWXUQLQJ RQFH DJDLQ WR *RRGPDQnV DQDO\VLV VXSSRVH ZH FRXOG VDWLVIDFWRULO\ VHWWOH WKH SUREOHP RI ZKDW WR LQFOXGH LQ WKH VHW RI VSHFLILF IDFWV ) /HW XV WXUQ RXU DWWHQWLRQ WR WKH VHW RI ODZV / 7KUHH SUREOHPV LPPHGLDWHO\ DULVH

PAGE 33

:KDW DUH ZH WR PDNH RI FRXQWHUIDFWXDOV ZKRVH DQWHFHGHQWV GHQ\ DFFHSWHG ODZV VRFDOOHG FRXQWHUOHJ£LV" +RZ GR ZH GHWHUPLQH ZKLFK ODZV DUH UHOHYDQW RU DOWHUQDWHO\ ZKLFK ODZV DUH LUUHOHYDQW DQG ZRXOG OHDG WR LQFRUUHFW HYDOXDWLRQ RI WKH FRXQWHUIDFWXDO" ,V QRW WKH FRQFHSW RI ODZ LWVHOI SUREOHPDWLF WR SHUKDSV DV JUHDW DQ H[WHQW DV WKH FRQFHSW RI WKH FRQGLWLRQDO LW LV EHLQJ WDNHQ WR FODULI\" ,Q UHIHUHQFH WR WKH ILUVW SUREOHP ZH FRXOG UHIXVH WR FRXQWHQDQFH FRXQWHU OHJ£LV EXW WKLV ZRXOG EH EODWDQWO\ DG KRF %XW LI ZH SHUPLW FRXQWHU OHJ£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n QRW EH WKDW RQH ODZ LV WUXH DQG WKH RWKHU IDOVH IRU ERWK DUH WUXH 5HVFKHU > S @ FRQVLGHUV D VLPLODU H[DPSOH LQ H[SOLFDWLQJ FRXQWHUIDFWXDOV LQ WHUPV RI KLV EHOLHIFRQWUDYHQLQJ VXSSRVLWLRQV :H

PAGE 34

KDYH D FRYHULQJ ODZ /,f WKH EHOLHIV WKDW WKH PDWFK LV QRW VWUXFN DQG QRW OLW DQG WKH EHOLHIV WKDW WKH DX[LOLDU\ FRQGLWLRQV DUH PHW 5HVFKHUn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nV WR WKLV DQDO\VLV ZKLFK SDUDOOHOV RXUV $FFRUGLQJ WR 5HVFKHU /, DQG / DUH UHSUHVHQWHG E\ *RRGPDQ DV ORJn LFDOO\ HTXLYDOHQW DV ZRXOG VHYHUDO RWKHU SDUWLDO FRQWUDSRVLWLYHV RI /, EH (DFK ZRXOG YDOLGDWH D GLIIHULQJ FRXQWHUIDFWXDO 5HVFKHUnV UHSO\ WDNHV WKH IRUP WKDW WKH RWKHU HTXLYDOHQWV WR WKH FRYHULQJ ODZ /O PD\ EH GHGXFWLYHO\ HTXLYDOHQW WR /O EXW DUH QRW HTXLYDOHQW LQ WKH FRQWH[W RI LQGXFWLYH ORJLF 7KLV FODLP LV UHODWHG WR D VROXWLRQ RI +HPSHOnV UDYHQ SDUDGR[f 5HVFKHU FODLPV WKH FRYHULQJ ODZ /O KDV SULPDF\ LQ WKH HYDOXn DWLRQ RI FRXQWHUIDFWXDOV RYHU LWV HTXLYDOHQWV 5HVFKHUnV UHVSRQVH PLVVHV RQH SRLQW DQG UDLVHV DQRWKHU RI UHOHn YDQFH WR RXU WKLUG SUREOHP &RQWUD 5HVFKHU *RRGPDQ QHHG QRW FODLP WKDW /O DQG / DUH HTXLYDOHQW ZLWK /O EHLQJ WKH IDYRUHG IRUPXODWLRQ RI WKH FRYHULQJ ODZ UDWKHU *RRGPDQ FDQ PDLQWDLQ WKDW /O DQG / DUH ERWK LQGXFWLYHO\ FRQILUPHG ODZV 7KHQ WKH TXHVWLRQ LV LQGHHG ZKDW UHOHYDQW

PAGE 35

ODZ GR ZH FKRRVH QRW ZKHWKHU ZH UHMHFW WKH FRQVHTXHQW RU DQ DX[LOLDU\ K\SRWKHVLV RI WKH IDYRUHGn IRUPXODWLRQ RI WKH ODZ 7KH SRLQW RI UHOHYDQFH WR WKH WKLUG SUREOHP LV WKDW ZH GR IDYRU /O QRW EHFDXVH LW LV GLUHFWO\ LQGXFWLYHO\ FRQILUPHG DV / LV QRW ZKLFK LV D IDOVH FODLPf EXW UDWKHU EHFDXVH LW KDV WKH IRUP RI D FDXVDO ODZ ZLWK D GLUHFWLRQ 7KLV DPRXQWV WR LWV EHLQJ FRQGLWLRQDO LQ QDWXUH DQG QRW PDWHULDO DV ZH VKDOO VHH ,Q PRUH GLUHFW UHIHUHQFH WR WKH WKLUG SUREOHP RQH PLJKW IHHO WKDW DV ODZV DQG FRXQWHUIDFWXDOV DUH ERWK SUREOHPDWLF WR DQDO\]H RQH LQ WHUPV RI WKH RWKHU LV QRW WR VROYH WKH SUREOHP 7KH LPQHGLDWH UHn MRLQGHU ZRXOG EH EHWWHU RQH SUREOHPDWLF FRQFHSW WKDQ WZR ,I FRXQWHUn IDFWXDOV FDQ EH DQDO\]HG LQ WHUPV RI ODZV WKHQ ZH VLPSO\ KDYH WR JR RQ WR DQDO\]H ODZV 5HVFKHU DSSDUHQWO\ KROGV WKLV YLHZ DQG UHJDUGV WKH DQDO\VLV RI FRXQWHUIDFWXDOV WR EH ODLG DW UHVW ZKLOH PRUH VWXG\ LV QHHGHG RI ODZV DQG FRQILUPDWLRQ WKHRU\ > S @ ,Q WKLV FRQn QHFWLRQ VHH DOVR WKH UHVW RI *RRGPDQ >@f 7KLV LV D SUREOHP RI PHWDDQDO\VLV DQG LWV DSSHDUDQFH LV QRW QHZ WR SKLORVRSK\ 2QH LV UHPLQGHG RI 4XLQHnV DWWDFNV RQ WKH FRQFHSWV RI DQDO\WLFLW\ PHDQLQJ DQG V\QRQRQ\ :KHQ ZH KDYH D VHW RI V\VWHPDWLn FDOO\ LQWHUUHODWHG FRQFHSWV DOO RI D SUREOHPDWLF QDWXUH WKH UHGXFWLRQ RI DOO WKH RWKHUV WR RQH PD\ RQO\ EH DQ DSSDUHQW QRW DQ DFWXDO DGYDQFH ,W LV P\ IHHOLQJ WKDW WKH ODFN RI DGYDQFH LV PRVW SRLQWHGO\ IHOW DV D IDLOXUH WR H[SODLQ DQ\ RI WKH FRQFHSWV DW LVVXH 5HSHDWHG IDLOXUH WR H[SODLQ DQ\ RQH RI WKH LQWHUUHODWHG FRQFHSWV OHDGV WR RQH RI WZR RXWn FRPHV 6RXU JUDSHV LQ ZKLFK WKH ZKROH FRPSOH[ LV JLYHQ XS DV D EDG LGHD

PAGE 36

6ZHHW OHPRQ LQ ZKLFK LW LV EOLVVIXOO\ DJUHHG WKDW WKH FRQFHSWV PXVW EH DFTXLUHG DV D VHW DOO DUH EDVLF DQG QRQH KDV SULRULW\ RYHU WKH RWKHUV 2WKHUZLVH LW UHPDLQV RSHQ VHDVRQ RQ WKH VHW RI LQWHUUHODWHG FRQFHSWV ZLWK UHSHDWHG HIIRUWV WR H[SOLFDWH RQH RI WKHP DQG WKHQ WKH RWKHUV ZLOO IDOO LQWR OLQH :H KDYH QRW VKRZQ WKDW ODZV IDOO LQWR WKLV DQDO\WLF FLUFOH QRU FRQVLGHUHG RWKHU HVFDSHV VXFK DV WKH PRYH WR GLVSRVLWLRQV *RRGPDQ KDYLQJ DQDO\]HG FRXQWHUIDFWXDOV LQ WHUPV RI FRWHQDELOLW\ DQG ODZV DQG QRWLQJ WKH FLUFOH LQWR ZKLFK FRWHQDELOLW\ DQG FRXQWHUIDFWXDOV IDOO DQG WKH SUREOHPV ZLWK ODZV WKHPVHOYHV VKLIWV WKH SUREOHP OLNH 5HVFKHU WR ODZV DQG FRQILUPDWLRQ WKHRU\ 'LVSRVLWLRQV DV D ZHDNHU EXW UHn ODWHG QRWLRQ WR FRXQWHUIDFWXDOV DUH SLFNHG XS DORQJ WKH ZD\ (OOLV UHn GXFHV FRXQWHUIDFWXDOV WR FRWHQDELOLW\ EXW WKLV LV IODWO\ FLUFXODU :LWK WKH H[FHSWLRQ RI (OOLV WKH PHWDOLQJXLVWLF DFFRXQWV LQFOXGLQJ EHn OLHI DFFRXQWV VXFK DV 5HVFKHUn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f

PAGE 37

,W LV FRPPRQSODFH WKDW ODZV RU PRUH SURSHUO\ ODZOLNH VWDWHn PHQWV RI ZKLFK WKH WUXH RQHV DUH ODZVf DUH JHQHUDOL]DWLRQV ,W LV DSSDUHQWO\ HTXDOO\ FRPPRQSODFH WKDW WKH\ DUH QRW PDWHULDO JHQHUDOL]Dn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f $ PDWHULDO JHQHUDOL]DWLRQ LV FRQFOXVLYHO\ FRQILUPHG LQ YLUWXH RI WKH YDFXLW\ RI LWV DQWHFHGHQW RU E\ H[KDXVWLYH HQXPHUDWLRQ 6XFK LV QRW WKH FDVH ZLWK ODZV $ ODZ IRU H[DPSOH 1HZWRQnV VHFRQG ODZ RI PRn WLRQ PD\ LQGHHG KDYH D YDFXRXV DQWHFHGHQW EXW LW LV QRW WUXH LQ YLUWXH RI WKDW 2WKHU H[DPSOHV FRXOG EH FLWHG EXW WKLV PD\ EH EHDWLQJ D GHDG KRUVH ,W LV JHQHUDOO\ DGPLWWHG WKDW ODZV DUH QRW PDWHULDO JHQn HUDOL]DWLRQV +RZHYHU LW LV HTXDOO\ REYLRXV WKDW ODZV DUH JHQHUDOO\ FRQGLWLRQDO VWDWHPHQWV RI VRPH VRUW DV ZHOO DV JHQHUDOL]DWLRQV RI VRPH VRUW SDUWLFn XODUO\ WKH FDXVDO ODZV XVXDOO\ WDNHQ WR EH LQWLPDWHO\ UHODWHG WR FRXQn WHUIDFWXDOV ,Q IDFW FRQVLGHULQJ WKH PDWFK H[DPSOH DJDLQ WKH IDn YRUHG ODZ ,, LV D JHQHUDOL]HG FRQGLWLRQDO SUHGLFWLRQ ZKLOH LWV DEHUn UDQW UHODWLYHV DUH FRQGLWLRQDO SRVWGLFWLRQV ,QGHHG 6WDOQDNHU VXJJHVWV

PAGE 38

WKDW ODZV DUH MXVW XQLYHUVDOO\ TXDQWLILHG FRXQWHUIDFWXDO FRQGLWLRQDOV > S @ ,I WKLV EH DGPLWWHG WKHQ ODZV VKDUH ZLWK FRXQWHUIDFn WXDO V WKH SURSHUW\ RI EHLQJ FRQGLWLRQDO LQ QDWXUH EXW QRW PDWHULDO LQ QDWXUH 2Q WKH RWKHU KDQG ODZV H[WHQG WR FRQWUDU\WRIDFW VLWXDWLRQV ZKHUH PDWHULDO VWDWHPHQWV GR QRW 7KLV LV DPSO\ LOOXVWUDWHG E\ (O ZKHUH QRW RQO\ LV LW KHOG WKDW HDFK DFWXDO SXOVDU LV D QHXWURQ VWDU EXW WKDW DQ\WKLQJ HOVH ZKLFK FRXOG EH D SXOVDU EXW LV QRWf ZRXOG EH D QHXWURQ VWDU 3ROORFN QRWHV WKLV DV WKH VXEMXQFWLYH QDWXUH RI ODZV ZKLFK KH WKHQ FDOOV VXEMXQFWLYH JHQHUDOL]DWLRQV DV RSSRVHG WR PDWHULDO JHQHUDOL]DWLRQV > SS @ +RZHYHU DW WKLV SRLQW ZKHWKHU ZH KDYH WZR FRQFHSWV WR DQDO\]H FRQGLWLRQDOLW\ DQG VXEMXQFWLYLW\ RU RQH FRQGLWLRQDOLW\ LV EHVLGH WKH SRLQW ,Q HLWKHU FDVH ODZV ZLOO VKDUH ZLWK FRXQWHUIDFWXDOV D FKDUDFWHULVWLF ZKLFK RXU DQDO\VLV RI HLWKHU RU ERWKf PXVW H[SODLQ )RU FRQYHQLHQFH ZLOO FRQWLQXH WR UHIHU WR WKH FRQFHSW RI WKH FRQGLWLRQDO DV ZKDW LV WR EH H[SOLFDWHG 7KLV SODFHV ODZV VTXDUHO\ LQ WKH DQDO\WLF FLUFOH ZLWK FRQGLWLRQDOV $QG WR EUHDN RXW RI WKH FLUFOH DQG DYRLG VRXU JUDSHV RU VZHHW OHPRQV VRPH RQH RI WKH SUREOHPDWLF FRQFHSWV PXVW EH JLYHQ DQ H[SODQDWRU\ DQDOn \VLV 7KHUH VHHP WR EH WKUHH DSSURDFKHV WR WKH UHVXOWLQJ SUREOHP RI EUHDNLQJ RXW RI WKH FLUFOH HDFK ZLWK LWV DWWHQGDQW SUREOHPV DQG YLUWXHV 2QH DSSURDFK LV WR DFFHSW WKDW FRXQWHUIDFWXDOV FDQ EH DQDO\]HG LQ WHUPV RI ODZV DQG FRWHQDELOLW\ DQG WKHQ WR SURYLGH D PRUH EDVLF H[SODQn DWRU\ DQDO\VLV RI ODZV DQG D UHVROXWLRQ RI WKH FLUFXODULW\ LQ FRWHQn DELOLW\ 7KLV LV IXQGDPHQWDOO\ DQ DWWDFN RQ WKH ODZ SUREOHP 7KH DQDOn \VLV RI ODZV WDNHV WKH IRUP LQ 3ROORFNnV DSSURDFK RI DQDO\]LQJ WKHP LQ WHUPV QRW RI WKHLU WUXWK FRQGLWLRQV EXW RI WKHLU MXVWLILFDWLRQ

PAGE 39

FRQGLWLRQV DQG LV WKXV DQ H[HUFLVH LQ FRQILUPDWLRQ WKHRU\ 7KH FRQn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n DELOLW\ SUREOHP DV IDOORXW IURP WKH DQDO\VLV RI ODZV > S @ $Q DFFRXQW VXFK DV -DFNVRQnV >@ RU %DUNHUnV >@ LQ WHUPV RI FDXVDO ODZV ZKLOH KDYLQJ YLUWXHV DQG GHIHFWV RI WKHLU RZQ PRVW FRXQWHU OHJ£LV EHFRPH LUUHGHHPDEO\ DPELJXRXVf FHUWDLQO\ DUH SUHGLFDWHG XSRQ DQ DQDOn \VLV RI FDXVDO ODZV LI WKH\ DUH WR KDYH DQ\ H[SODQDWRU\ IRUFH 7KH VDPH FDQ EH VDLG RI 5HVFKHUnV >@ DQDO\VLV RI QRPRORJLFDO FRXQWHUIDFn WXDOV DOO RWKHUV DUH LUUHGHHPDEO\ DPELJXRXVf LQ WHUPV RI ODZV :LWK WKH H[FHSWLRQ RI 3ROORFNn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n YLGLQJ DQ H[SODQDWRU\ WUXWK FRQGLWLRQ DFFRXQW RI WKHP ,W LV KHUH WKDW

PAGE 40

SRVVLEOH ZRUOGV VHPDQWLFV PDNHV LWV DSSHDUDQFH DV DQ H[SODQDWLRQ RI ZK\ DQG KRZ RXU LQIRUPDO SURFHGXUH IRU HYDOXDWLQJ FRXQWHUIDFWXDOV ZRUNV DV LW GRHV ,I 6WDOQDNHU LV WDNHQ DV H[HPSOLI\LQJ WKLV DSSURDFK WKHQ ODZV DUH DQDO\]HG LQ WHUPV RI FRXQWHUIDFWXDOV VSHFLILFDOO\ DV TXDQWLn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n DSSURDFK KRZHYHU ZKLOH VKDULQJ ZLWK 6WDOQDNHUnV WKH DVn VXPSWLRQ WKDW WKH FRQGLWLRQDO LV SULRU GLIIHUV LQ LWV WUHDWPHQW RI ODZ DQG WKXV UHSUHVHQWV WKH WKLUG DSSURDFK %RWK WKH VHFRQG DQG ILUVW DSn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n WXH LQ WKDW DQDO\VHV RI WKH RWKHU FRQFHSWV LPPHGLDWHO\ IROORZ &RPn SHWLQJ DQDO\VHV ZLOO WKHQ GLIIHU LQ D FRPELQDWLRQ RI UHVSHFWV WKH\

PAGE 41

PD\ EH DQDO\VHV RI WKH VDPH RU GLIIHUHQW VRUWV RI WKH VDPH RU GLIIHUHQW FRQFHSWV ZLWKLQ WKH FLUFOH ,I WKH DQDO\WLF FLUFOH LV RQO\ VXJJHVWHG E\ WKH IDLOXUH RI SULRU DWWHPSWV WR DUULYH DW D VDWLVIDFWRU\ DQDO\VLV QHYHUWKHOHVV PXFK WKH VDPH VLWXDWLRQ REWDLQV +H ZKR PDLQWDLQV WKDW WKH FLUFOH LV QRW RQO\ EURNHQ EXW UHVROYHV LWVHOI LQWR WZR OLQHV KDV RI FRXUVH DQ DGGLWLRQDO WDVN WR XQGHUPLQH WKH HYLGHQFH IRU FLUFXODUn LW\ 7KH PDWDOLQJXLVWLF DQDO\VHV LQFOXGLQJ EHOLHI DQDO\VHV OHDG LQn H[RUDEO\ WR ODZV FRQILUPDWLRQ WKHRU\ DQG DQDO\VLV LQ WHUPV RI MXVWLn ILFDWLRQ FRQGLWLRQV ,W ZLOO EH PDLQWDLQHG WKDW VXFK DQDO\VHV DUH VXEn MHFW WR WKH FKDUJH RI IDLOLQJ WR H[SODLQ WKH FRQFHSWV WKH\ WDNH DV SURn EOHPDWLF 2Q WKH RWKHU KDQG SRVVLEOH ZRUOG DFFRXQWV ZLWK WKH H[FHSn WLRQ RI 3ROORFNnV PL[HG DFFRXQWf KDYH WKH SULPD IDFLH YLUWXH RI SURn YLGLQJ DQ H[SODQDWLRQ RI ZK\ WKH FRQFHSW ZRUNV WKH ZD\ LW GRHV 1RWHV $ VXUYH\ RI HDUO\ DFFRXQWV RI WKLV VRUW PD\ EH IRXQG LQ 6FKQHLGHU >@ )RU RWKHU ORJLFDO DQDO\VHV DQG FULWLFLVPV WKHUHRI VHH %RGH >@ )XPHUWRQ >@ /HKPDQQ >@ DQG 1XWH >@ :H GLVFXVV :DVVHUPDQ >@ DV DQ H[DPSOH 7KLV VXJJHVWLRQ ZDV PDGH WR PH LQ FRQYHUVDWLRQ E\ *DU\ )XOOHU )RU PRUH GLVFXVVLRQ WKDQ ZH VKDOO KDYH VSDFH IRU RI WKH UHODWLRQn VKLS DPRQJ ODZV QHFHVVLW\ FRQGLWLRQDOV DQG FDXVDWLRQ VHH %DUNHU >@ &KLVKROP >@ )LQH >@ *RRVHQV >@ +RQGHULFN >@ -DFNVRQ >@ .LP >@ .QHDOH >@ /HZLV >@ /RHE >@ /\RQ >@ 0DFNLH > @ 1XWH >@ 6KRUWHU >@ 6RVD >@ 6ZDLQ >@ 7HPSOH >@ 9HQGOHU >@ DQG @

PAGE 42

&+$37(5 7:2 3266,%/( :25/'6 $1$/<6,6 2) 22817(5)$&78$/6 3RVVLEOH :RUOGV +LVWRU\ ,Q WKH SHULRG EHWZHHQ DQG VHYHUDO DQDO\VHV RI WKH FRXQWHU IDFWXDO FRQGLWLRQDO DSSHDUHG WKDW GLYHUJHG VKDUSO\ IURP WKH PHWDOLQJXLVWLF DFFRXQWV WKDW KDG EHHQ SURGXFHG LQ WKH SUHFHGLQJ WZR GHFDGHV 7KH GLYHUn JHQFH ZDV LQ WKH XQLIRUP UHOLDQFH RI WKHVH QHZ DSSURDFKHV XSRQ WKH SRVn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n ERRN &RXQWHUIDFWXDOV DSSHDUHG DV HDUO\ DV 2 DQG $TYLVWnV DUWLFOH ZDV HDUOLHUf SXEOLVKHG E\ 8SVDOOD 1XWHnV DUWLFOH ZDV GHOD\HG WY} \HDUV LQ SXEOLFDWLRQ VR WKH LQLWLDO GHn YHORSPHQWV LQ WKLV ILHOG ZHUH JURXSHG LQWR WKH \HDUV QRWHG DERYH 7KH DSSHDUDQFH RI WKHVH HIIRUWV VSDUNHG D UHVXUJHQFH RI FRPSHWLQJ DFFRXQWV RI FRXQWHUIDFWXDOV WRR QXPHURXV WR PHQWLRQ DV ZHOO DV VSLULWHG GHIHQVHV RI WKH SRVVLEOH ZRUOGV DFFRXQW $ QRWDEOH HIIRUW LQWHQGHG WR FRYHU WKH HQWLUH UDQJH RI VXEMXQFWLYH FRQVWUXFWLRQV LV WKDW RI 3ROORFN >@ ZKLFK

PAGE 43

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n VLRQ RI D SURSRVLWLRQ >@ LQ DQ DWWHPSW WR H[SOLFDWH PHDQLQJ 7KH VDPH WDFWLF ZDV IROORZHG E\ &DUQDS >@ ZKR VSHFLILFDOO\ LGHQWLILHG SURn SRVLWLRQV DV VHWV RI /HLEQLW] SRVVLEOH ZRUOGV RU :LWWJHQVWHLQLDQ SRVn VLEOH VWDWHV RI DIIDLUV 2QH VWULNLQJ DQWLFLSDWLRQ RI SRVVLEOH ZRUOG VHPDQWLFV IRU PRGDO ORJLF RFFXUV HDUOLHU WKDQ DQ\ RI WKH ZRUNV FLWHG DERYH UHIHU WR WKH ([LVWHQWLDO *UDSKV RI & 6 3HLUFH 7KH LQFRPSOHWH GHYHORSPHQW E\ 3HLUFH RI KLV V\VWHP RI ([LVWHQWLDO *UDSKV LV WUDFHG LQ 5REHUWV >@ 2I LQWHUHVW WR XV LV WKH UHYLHZ RI 5REHUWVn ERRN E\ =HPDQ >@ ZKLFK

PAGE 44

EULQJV RXW WKH VLJQLILFDQW SDUDOOHOV EHWZHHQ 3HLUFHnV JDUUPD V\VWHP DQG SRVVLEOH ZRUOG VHPDQWLFV (YLGHQWO\ KDYLQJ GHYHORSHG JUDSKLFDO V\VWHPV IRU SURSRVLWLRQDO DQG TXDQWLILFDWLRQDO FDOFXOXV WKH DOSKD DQG EHWD V\VWHPVf 3HLUFH H[SHULPHQWHG ZLWK D WKLUG V\VWHP RU IUDJn PHQWV RI VHYHUDO V\VWHPVf LQ ZKLFK KH HQGHDYRUHG WR PDNH SRVVLEOH WKH UHSUHVHQWDWLRQ RI XQLYHUVHV RI GLVFRXUVH RWKHU WKDQ WKH DFWXDO WKHVH ZRXOG EH ZRUOGV RI SRVVLELOLW\ +H SURSRVHG WKDW LQVWHDG RI FRQVLGHULQJ MXVW RQH 6$ ZH WKLQN RI RXUVHOYHV DV ZRUNLQJ ZLWK D ERRN RI VXFK VKHHWV ZLWK HDFK VKHHW LQ WKH ERRN UHSUHn VHQWLQJ D SRVVLEOH ZRUOG PXFK DV .ULSNHDQ VHPDQWLFV FRUUHODWHV D VHPDQWLF WDEOHDX ZLWK HDFK SRVVLEOH ZRUOG > S @ ,Q WKH DERYH TXRWDWLRQ IURP =HPDQ 6$ UHIHUV WR WKH 6KHHW RI $VVHUWLRQ XSRQ ZKLFK JUDSKLFDO VLJQV DUH ZULWWHQ DV DVVHUWLRQV DERXW WKH XQLYHUVH RI GLVFRXUVH 3HLUFH GLG QRW TXLWH UHDFK FRQFHSWXDO FORVXUH RQ WKLV LGHD GXH WR WKH IDFW WKDW KH GLG QRW KDYH DQ DGHTXDWH ZD\ WR UHSUHVHQW WKH DFn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n SOLFDWLRQ RI QHLJKERUKRRG VHPDQWLFV WR FRQGLWLRQDOV LV GHYHORSHG E\ &KHOODV >@ DQG D V\VWHPDWLF FRPSDULVRQ RI WKH YDULHWLHV RI UHODWLRQDO

PAGE 45

DQG QHLJKERUKRRG VHPDQWLFV KDV EHHQ FDUULHG RXW E\ 1XWH >@ 6HFWLRQV DUH FRQVLGHUDEO\ LQGHEWHG WR WKHVH ODVW WZR PHQWLRQHG ZRUNV 7KH ORJLFLDQ TXD ORJLFLDQ LV LQWHUHVWHG LQ WKH DGHTXDF\ RI D IRUPDO VHPDQWLFV TXLWH DSDUW IURP ZKHWKHU RU QRW LW DIIHFWV DQ DQDO\VLV RI WKH FRQFHSWV WKH V\VWHP LV LQWHQGHG WR IRUPDOL]H ,W LV SRVVLEOH WR UHJDUG UHODWLRQDO SRVVLEOH ZRUOG VHPDQWLFV DV SURYLGLQJ DQ DQDO\VLV RI LPSRUWDQW FRQFHSWV RI PRGDOLW\ VHH %UDGOH\ DQG 6ZDUW] >@ )RXOLV DQG 5DQGDOO > @ DQG =HPDQnV DSSOLFDWLRQ DQG GHYHORSPHQW RI WKH ODWWHU > @f 7KLV LV RI LQWHUHVW WR WKH ORJLFLDQ TXD SKLORVRSKHU 7KH VHPDQWLFV IRU FRQGLWLRQDO ORJLF GHYHORSHG E\ /HZLV 6WDOQDNHU 1XWH DQG RWKHUV DUH LQWHQGHG DV DQDO\VHV DQG PXVW WKHUHIRUH PHHW FRQVWUDLQWV ZH VXJJHVWHG LQ 6HFWLRQ DQG ZLOO H[SORUH IXUWKHU LQ 6HFWLRQ DQG &+$37(5 7+5(( :KDW LV ODFNLQJ LQ WKH DSSOLFDWLRQ RI QHLJKERUKRRG VHPDQWLFV WR FRQGLWLRQDO ORJLF LV WKH LGHD RI DQ DQDO\VLV DV RSSRVHG WR D IRUPDOL]DWLRQ 7KRXJK 1XWH >@ FRQSDUHV YDULRXV VHPDQWLFV IRU FRQn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

PAGE 46

FRPSRVHG RI PDQ\ ZRUOGV > S Y@ )RU WKH GLIILFXOWLHV LQYROYHG LQ GUDZLQJ DQ\ VLJQLILFDQW SRVLWLYH SKLORVRSKLFDO FRQFOXVLRQV IUFP WKLV WKHRU\ RQH VKRXOG VHH 6N\UPVn >@ FULWLFLVP RI UHDOLVWLF SRVn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n HPSOR\PHQW RI FRPSDUDWLYH RYHUDOO VLPLODULW\ DV D ZD\ RI RUGHULQJ SRVn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nV

PAGE 47

VWRU\ RXU ZRUOG LV D OLQH RI VXFK SRLQWV DQG LW KDV QHLJKERUV RWKHU ZRUOGOLQHV O\LQJ SDUDOOHO WR RXUV GHILQLQJ D SODQH RI SRLQWV ZLWK WZR GLPHQVLRQV RI VLPLODULW\ (LWKHU LQ WKH QRUPDO WHPSRUDO GLUHFn WLRQV RU DW ULJKW DQJOHV WR WKHP ZRUOGV IDUWKHU IURP RXUV DFWXDOO\ IURP DQ LQVWDQW RI RXUVf ZLOO EH OHVV VLPLODU WR WKLV LQVWDQW WKDQ ZRUOGLQVWDQWV FORVHU E\ ,I ZH WUDYHO DORQJ D ZRUOGOLQH WKH LQn VWDQWV JUDGXDOO\ EHFRPH OHVV VLPLODU WR RXU VWDUWLQJ SRLQW OLNHZLVH LI ZH WUDYHO DORQJ D ULJKW DQJOH OLQH WR RXU ZRUOGOLQH D JUDGXDO VHTXHQFH RI DOWHUDWLRQV ZLOO REWDLQ /DXPHU GHVFULEHV VHYHUDO VXFK LPDJLQDU\ MRXUQH\V DQG RWKHUV FRPELQLQJ ERWK GLUHFWLRQV $VVXPLQJ WKDW ZH FRXOG FRPH XS ZLWK D XQLIRUP PHWULF IRU WKLV SODQH RI ZRUOG LQVWDQWV WKHQ LW VHHPV REYLRXV WKDW WKHUH DUH ERWK GHJUHHV RI VLPn LODULW\ WR D JLYHQ ZRUOGLQVWDQW DQG QXPEHUV RI ZRUOGV WKDW DUH HTXDOO\ VLPLODU WR RXU SUHVHQW ZRU,GLQVWDQW WKRXJK GLIIHULQJ IURP LW LQ GLIIHUHQW UHVSHFWV 2I FRXUVH /HZLV GRHV QRW VXJJHVW WKDW ZH FDQ DFWXDOO\ SXW D PHWULF RQ WKH VSDFH RI SRVVLEOH ZRUOGV (YHQ LQ WKH VFLHQFH ILFWLRQ WDOH DERYH LW LV KDUG WR VHH KRZ WKDW FRXOG EH GRQH %XW KH GRHV SXW D FHUWDLQ RUJDQL]DWLRQ RQ WKDW VSDFH D WRSRORJ\ RI VRUWV LI QRW D PHWULF ,Q 6DEHUKDJHQnV 0DVN RI WKH 6XQ >@ ZKLOH SDUDOOHO ZRUOGV DUH QRW DFFHVVLEOH LQ WHUPV RI SK\VLFDO WUDQVIHUHQFH RI WKH SURWDJRQLVW WR WKHP WKH ZHDULQJ RI D FHUWDLQ PDVN HQDEOHV WKH ZHDUHU WR YLHZ IXWXUH SRVVLEOHV 7KH DXWKRU GHYHORSV WKLV LGHD LQ D FRQWH[W RI EUDQFKLQJ WLPH WKH ZHDUHU LV QRW VHHLQJ WKH IXWXUH EXW RQH RI PDQ\ SRVVLEOH IXWXUHV EUDQFKLQJ RXW IURP WKH PDVNnV WHPSRUDO SRLQW RI YLHZ 7KH H[SODQDWLRQ RI WKLV FDSDFLW\ VXJJHVWHG LQ WKH QRYHO LV WKDW WKH

PAGE 48

PDVN FRPSXWHV WKH SRVVLELOLWLHV EDVHG XSRQ D FRPSUHKHQVLYH DFFHVV WR IDFWV DERXW WKH SUHVHQW 7KLV LV UHPLQLVFHQW RI 6WDOQDNHUn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n GXFLQJ SRVVLEOH ZRUOGV ZKHQ ZH KDYH HQRXJK GLIILFXOW\ GHWHUPLQLQJ ZKDW WKH DFWXDO ZRUOG LV LV WR FRPSRXQG RXU SUREOHPV WR QR SXUSRVH 2XU SUREOHPV KRZHYHU DUH DOUHDG\ FRPSRXQGHG WKH GLIILFXOW\ LQ GHWHUPLQLQJ ZKDW WKH DFWXDO ZRUOG LV OLHV LQ WKH IDFW WKDW WKH H[WHQW RI RXU NQRZOHGJH DQG EHOLHI RU WUXH EHOLHIf VHULRXVO\ XQGHUGHWHUn PLQHV LW 7KH FULWLF ZRXOG H[SUHVV WKLV XQGHU GHWHUPLQDWLRQ E\ VD\LQJ ZH GR QRW RU FDQQRW NQRZ HYHU\WKLQJ DERXW WKH DFWXDO ZRUOG ZKLOH ZRXOG H[SUHVV LW E\ VD\LQJ ZH GR QRW RU FDQQRW NQRZ ZKLFK ZRUOG LV DFWXDO H[FHSW ZLWKLQ FHUWDLQ OLPLWV :KLOH WKH FULWLF FDQ RQO\ VD\ WKDW RXU NQRZOHGJH XQGHUGHWHUPLQHV WKH ZRUOG WKDW LV FDQ PDNH VHQVH RI D SRVLWLYH DVVHUWLRQ DV WR ZKDW LW GRHV GHWHUPLQH WKH VHW RI ZRUOGV WKDW IRU DOO ZH NQRZ DQ\ RQH RI ZKLFK PLJKW EH WKH DFWXDO ZRUOG ,I

PAGE 49

ZH PXVW RSHUDWH ZLWKLQ D FRQWH[W RI LQGHWHUPLQDF\ WKHQ SOXUDOLW\ DPRQJ ZKDW LV LQGHWHUPLQDWH DOORZV URRP IRU JUHDWHU IXWXUH GHWHUPLQDF\ $V WKH SRVLWLYH EHQHILWV RI WKLV ZD\ RI ORRNLQJ DW WKLQJV DFFXPXODWH WKH FULWLF ZLOO RI FRXUVH DGRSW D SRLQW RI YLHZ FORVHU WR PLQH 3RVVLEOH ZRUOGV PDNH WKHLU DSSHDUDQFH LQ UHFHQW HIIRUWV WR SURn YLGH D VHPDQWLFV IRU PRGDO ORJLF $V D PDWKHPDWLFDO WRRO RI IRUPDO ORJLF WKHUH LV QR VHULRXV TXHVWLRQ DV WR LWV XWLOLW\ +RZHYHU DV D GHYLFH IRU WKH DQDO\VLV RI FRQFHSWV WKHUH LV QHLWKHU D VKRUWDJH RI ODVHUV QRU RI FULWLFV 7KH RQO\ GHIHQVH WKDW FDQ WKLQN RI IRU XWLOL]LQJ WKH FRQFHSW RI SRVVLEOH ZRUOGV IRU DQDO\]LQJ RWKHU FRQFHSWV LV WKDW LW XQOLNH VRPH RI LWV DOWHUQDWLYHV SURYLGHV DQ H[SODQDWLRQ IRU KRZ DQG ZK\ WKH FRQFHSWV DQDO\]HG ZRUN WKH ZD\ WKH\ GR WKRXJK IRU D GHWDLOHG GHIHQVH VHH >@f %XW WKLV GHIHQVH PXVW ZDLW XQWLO 6HFWLRQ ,W ZLOO EH PRUH DSSURSULDWH DQ\ZD\ RQFH ZH KDYH D SXUSRUWHG DQDOn \VLV LQ WHUPV RI SRVVLEOH ZRUOGV DV D FRQFUHWH H[DPSOH :H PD\ WDNH PRGDO ORJLF WR EH WKH ORJLF RI SRVVLELOLW\ DQG QHFHVn VLW\ 9DULRXV V\VWHPV RI PRGDO ORJLF PD\ KDYH DSSOLFDWLRQ RU EH GHn VLJQHG WR KDYH DSSOLFDWLRQ RXWVLGH WKH ERXQGV RI WKHVH QRWLRQV VXFK DV WHQVH ORJLF GHRQWLF ORJLF HSLVWHPLF ORJLF HWF +RZHYHU ZH VHHP WR KDYH VXIILFLHQW RSSRUWXQLW\ IRU YDULDWLRQ ZLWKLQ WKH ERXQGV RI SRVn VLELOLW\ DORQH WKHUH LV WKH ORJLFDOO\ SRVVLEOH WKH SK\VLFDOO\ SRVn VLEOH WKH WHFKQRORJLFDOO\ SRVVLEOH DQG WKH DFWXDOO\ SRVVLEOH WR QDPH EXW D IHZ 7KHUH DUH WKHVH NLQGV RI QHFHVVLW\ DV ZHOO LQ DGn GLWLRQ WR QHFHVVLW\ LQ WHUPV RI QHHG RU LQ WHUPV RI NHHSLQJ FHUWDLQ WKLQJV IL[HG :KDW ZLOO NLOO WKH DSKLGV ZLWKRXW GRLQJ LQ WKH URVHV"f :H VKRXOG EHJLQ ZLWK RXU ZLGHVW VHQVH RI SRVVLELOLW\ DQG RXU QDUURZVVW VHQVH RI QHFHVVLW\ WKDW ZKLFK LV LQ VHPH ZD\ SRVVLEOH DQG

PAGE 50

WKDW ZKLFK LV QHFHVVDU\ QR PDWWHU ZKDW )RFXVLQJ RQ WKH ODWWHU FKDUDFWHUL]DWLRQ WKH IROORZLQJ GHILQLWLRQ VHHPV DSSURSULDWH /S LV WUXH LII S LV WUXH LQ HYHU\ SRVVLEOH ZRUOG &RUUHVSRQGLQJO\ IRU SRVVLELOLW\ ZH KDYH 0S LV WUXH LII S LV WUXH LQ VRPH SRVVLEOH ZRUOG 7KHVH GHILQLWLRQV KDYH WKH YLUWXH RI PDNLQJ ZKDW LV QRW SRVVLEO\ IDOVH HTXLYDOHQW WR ZKDW LV QHFHVVDULO\ WUXH 7R KDQGOH RXU QRWLRQV RI SK\VLFDOO\ SRVVLEOH WHFKQRORJLFDOO\ SRVn VLEOH HWF ZH FRXOG VLPSO\ VXEVWLWXWH WKHVH WHUPV IRU SRVVLEOH LQ WKH DERYH GHILQLWLRQV 7KHUH DUH VHYHUDO GUDZEDFNV WR WKLV FKLHI DPRQJ ZKLFK LV WKDW RXU YDULRXV QRWLRQV RI SRVVLELOLW\ DSSHDU LUUHGXFn LEOH LQ RXU GHILQLWLRQV ZKLOH LQ IDFW WKH YDULRXV QRWLRQV RI SRVVLn ELOLW\ PD\ EH V\VWHPDWLFDOO\ LQWHUUHODWHG 6XUHO\ WKH SK\VLFDOO\ SRVn VLEOH ZRUOGV DUH D VXEVHW RI WKH ORJLFDOO\ SRVVLEOH DQG WKH WHFKQRn ORJLFDOO\ SRVVLEOH D VXEVHW RI WKH SK\VLFDOO\ SRVVLEOH )RU VRPH NLQGV RI SRVVLELOLW\ QRW DOO WKH SRVVLEOH ZRUOGV LQ WKH EURDGHVW VHQVH QHHG WR EH WDNHQ LQWR FRQVLGHUDWLRQ $OVR WKHUH DUH FLUFXPVWDQFHV XQGHU ZKLFK WKH RSHUDWLYH FRQFHSW RI SRVVLELOLW\ GRHV QRW GHWHUPLQH D VWDWLF VHW RI SRVVLELOLWLHV &RQn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

PAGE 51

QRW EH SRVVLEOH UHODWLYH WR DQRWKHU )XUWKHU DV ZH VDZ DERYH QRW HYHU\ SRVVLEOH ZRUOG QHHG EH UHOHYDQW WR ZKDW LV SRVVLEOH LQ VRPH UHVWULFWHG VHQVH 7KH WHFKQLTXH IRU KDQGOLQJ WKHVH FRQVLGHUDWLRQV LQ SRVVLEOH ZRUOG VHPDQWLFV DQG ZKDW JLYHV LW LWV FRQVLGHUDEOH IOH[LELOLW\ IRU SHUPLWn WLQJ WKH UHSUHVHQWDWLRQ RI D YDULHW\ RI FRQFHSWLRQV RI SRVVLELOLW\ LV WKH QRWLRQ RI D IUDPH $ IUDPH FRQVLVWV QRW RQO\ RI D VHW RI SRVVLEOH ZRUOGV EXW DOVR RI D UHODWLRQ DPRQJ WKHVH SRVVLEOH ZRUOGV WKDW UHSUHn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n WLYH DQG V\PPHWULF ZLWK HYHU\ ZRUOG DFFHVVLQJ DW OHDVW RQH RWKHU 7KH SRVVLEOH IXWXUHV VLWXDWLRQ FRXOG EH PRGHOHG E\ DQ 5 ZKLFK LV MXVW UHIOH[n LYH DQG WUDQVLWLYH 7KHVH DQG RWKHU DOWHUQDWLYHV IRUP WKH EDVLV RI D JUHDW YDULHW\ RI V\VWHPV RI PRGDO ORJLF 'HWDLOV QHHG QRW FRQFHUQ XV QRZ ,I ZH DVVXPH WKDW WKH DFFHVVLQJ UHODWLRQ LV XQLYHUVDO WKHQ HYHU\ ZRUOG KDV DFFHVV WR HYHU\ RWKHU

PAGE 52

$ SLFWRULDO UHSUHVHQWDWLRQ PXFK XVHG E\ /HZLV IRU WKH DFFHVVLELOLW\ UHODWLRQ LV WKDW RI D FLUFOH ZKHUH WKH FHQWHU UHSUHVHQWV WKH JLYHQ ZRUOG X DQG DOO WKH SRLQWV ERXQGHG E\ WKH FLUFOH UHSUHVHQW WKH ZRUOGV DFFHVn VLEOH IURP X 7KH FLUFOH DQG LWV LQWHULRU LV WKH VSKHUH RI DFFHVn VLELOLW\ DURXQG X )LJXUH 7KLV VXJJHVWV WKDW ZH PD\ GHILQH RXU IUDPH LQ D GLIIHUHQW EXW HTXLYDn OHQW IDVKLRQ ) 86 ZKHUH 8 LV DJDLQ WKH VHW RI SRVVLEOH ZRUOGV 6 LV D IXQFWLRQ IURP WKH VHW RI SRVVLEOH ZRUOGV 8 WR WKH SRZHU VHW RI 8 38f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

PAGE 53

:KDW \RX PD\ DVN KDV WKLV WR GR ZLWK FRQGLWLRQDOV" 5HFDOO WKDW D SULQFLSDO REMHFWLRQ WR WKH PDWHULDO FRQGLWLRQDO DV DQ DQDO\VLV RI FRQn GLWLRQDOV LQ (QJOLVK LV WKDW LW VLPSO\ LV QRW SODXVLEOH WKDW D FRQGLn WLRQDO EH WUXH MXVW EHFDXVH LWV DQWHFHGHQW LV IDOVH RU LWV FRQVHTXHQW WUXH 6RPHWKLQJ PRUH LV FDOOHG IRU DQG RQH RI WKH ILUVW WKLQJV WR WU\ LV WR IRUPDOL]H WKH QRWLRQ WKDW VRPH FRQQHFWLRQ REWDLQV EHWZHHQ WKH DQn WHFHGHQW DQG FRQVHTXHQW 1RZ ZH KDYH VHHQ WKDW WKLV LV QRW HQRXJK EXW LW LV D SODFH WR VWDUW ,Q DQ HIIRUW WR SURYLGH DQ DOWHUQDWLYH WR WKH PDWHULDO FRQGLn WLRQDO IRU WKH DQDO\VLV RI LI WKHQ D QXPEHU RI PRGDOL]HG FRQGLWLRQDOV KDYH EHHQ GHYHORSHG HLWKHU DV SULPLWLYHV LQ D ORJLFDO V\Vn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nV 0RGDO /RJLF >@ I 7KHVH V\VWHPV DUH RI LQWHUHVW IRU RXU SXUSRVHV RQO\ LQ WKHLU IDLOXUH WR SURYLGH DQ DQDO\VLV IRU FRXQWHUIDFWXDOV IRU ZKLOH WKH PDWHULDO FRQGLWLRQDO LV WRR ZHDN WR VHUYH DV D FRXQWHUIDFWXDO FRQGLWLRQDO WKH VWULFW FRQGLWLRQDO LV WRR VWURQJ DV ZH VKDOO VHH 7KH LGHD EHKLQG WKH VWULFW FRQGLWLRQDO H[SUHVVHG LQ WHUPV RI SRVVLEOH ZRUOGV LV WKDW YLULOH QHLWKHU WKH DQWHFHGHQW QRU FRQVHTXHQW

PAGE 54

QHHG EH QHFHVVDU\ LQ WKHPVHOYHV UHODWLYH WR WKRVH ZRUOGV ZKHUH WKH DQWHFHGHQW LV WUXH WKH FRQVHTXHQW LV QHFHVVDU\ LH WUXH LQ DOO RI WKHP 7KHUH DUH YDULDWLRQV RQ WKLV RI FRXUVH EXW PDQ\ RI WKHP DUH DPHQDEOH WR UHSKUDVLQJ WKH FODLP LQ WHUPV RI DFFHVVLELOLW\ 5HWXUQLQJ WR RXU IXWXUH SRVVLEOHV H[DPSOH D FHUWDLQ FRQGLWLRQDO SUHVHQWO\ IDOVH VLQFH LQ WKRVH IXWXUHV ZKHUH WKH DQWHFHGHQW LV WUXH LW LV QRW LQ DOO WKH FDVH WKDW WKH FRQVHTXHQW LV WUXHf PD\ EH WUXH IURP WKH SRLQW RI YLHZ RI RQH RI WKRVH IXWXUHV VLQFH E\ WKHQ FHUWDLQ SRVVLELOLWLHV PD\ QR ORQJHU EH DFFHVVLEOH SHUKDSV LQFOXGLQJ WKRVH LQ ZKLFK WKH DQWHFHn GHQW ZDV WUXH DQG WKH FRQVHTXHQW IDOVHf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n VLELOLW\ DQG QHFHVVLW\ E\ SODFLQJ YDULRXV FRQGLWLRQV RQ WKH DFFHVVLn ELOLW\ UHODWLRQ RU LQ RXU SUHVHQW SDUODQFH RQ WKH VSKHUH RI DFFHVn VLELOLW\ 7KH VSKHUH RI DFFHVVLELOLW\ IRU D JLYHQ ZRUOG FRXOG UDQJH IURP WKH HPSW\ VHW WR WKH HQWLUH VHW RI SRVVLEOH ZRUOGV RU DQ\ZKHUH LQ EHWZHHQ 6XSSRVH WKDW IRU D FHUWDLQ ZRUOG X ZH KDYH D FKRLFH RI WZR GLIIHUHQW VSKHUHV DQG HDFK GHWHUPLQLQJ D VRPHZKDW GLIIHUHQW

PAGE 55

VHQVH RI ZKDW LV SRVVLEOH UHODWLYH WR X &RUUHVSRQGLQJ WR WKHVH ZH KDYH WZR QHFHVVLW\ RSHUDWRUV / DQG / 1RZ LI DQG DUH GLVMRLQW RU SURSHUO\ LQWHUVHFW WKHQ / DQG / DUH QRW LQ DQ\ REYLRXV ZD\ FRUQSD UDELH LI KRZHYHU LV D VXEVHW RI WKHQ / S ZLOO LPSO\ / S IRU DQ\ SURSRVLWLRQ S ,I WKH FRQWDLQPHQW LV SURSHU LW ZLOO QRW JHQHUDOO\ EH WKH FDVH WKDW / S LPSOLHV /AS +HQFH RXU WZR QHFHVVLW\ RSHUDWRUV ZLOO EH RUGHUHG ,Q WKLV FRQWH[W /HZLV > S @ GHVFULEHV RQH RSHUDWRU /Af DV VWULFWHU WKDQ WKH RWKHU DQG KHQFH D FRQGLWLRQDO GHn ILQHG LQ WHUPV RI RQH DV D VWULFWHU FRQGLWLRQDO WKDQ WKH RWKHU 7KH GLIILFXOW\ RI WDNLQJ WKH FRXQWHUIDFWXDO WR EH D VWULFW FRQn GLWLRQDO OLHV LQ WKH YDULDWLRQV RQ VWULFWQHVV RI WKH FRQGLWLRQDO )RU IRU DQ\ IL[HG GHJUHH RI VWULFWQHVV RI WKH FRQGLWLRQDO LW LV DOZD\V SRVn VLEOH WR VWUHQJWKHQ WKH DQWHFHGHQW ( /&ST /&.SUT LV YDOLG IRU DQ\ RSHUDWRU RI IL[HG VWULFWQHVV / DV WKH IROORZLQJ DUJXn PHQW VKRZV /&ST LV WUXH DW X LII DW HYHU\ Z H &ST LV WUXH %XW LI &ST LV WUXH DW DQ\ ZRUOG Z WKHQ &.SUT LV WUXH VLQFH VWUHQJWKHQLQJ WKH DQWHFHn GHQW LV YDOLG IRU WKH PDWHULDO FRQGLWLRQDO +HQFH WKH WUXWK RI /&ST OHDGV LQH[RUDEO\ WR WKH WUXWK RI /&.SUT ZLWK QR SDUWLFXODU FRQGLWLRQV RI WKH IXQFWLRQ 6 VLQFH ZLWK &.STU WUXH DW HYHU\ ZRUOG LQ 6 /&.SUT ZLOO EH WUXH DW X +RZHYHU IRU DQ\ JLYHQ FRXQWHUIDFWXDO RU DW OHDVW WKRVH ZLWK FRQWLQJHQW DQWHFHGHQWVf LW LV SRVVLEOH WR nnXQGHUPLQHnn WKH DQWHFHGHQW

PAGE 56

E\ FRQMRLQLQJ DQRWKHU SURSRVLWLRQ WR LW )RU H[DPSOH WKH IROORZLQJ LQIHUHQFH LV FHUWDLQO\ LQYDOLG ( ,I WKLV PDWFK ZHUH VWUXFN WKHQ LW ZRXOG OLJMKW ,I WKLV PDWFK ZHUH VRDNHG LQ ZDWHU DQG VWUXFN WKHQ LW ZRXOG OLJKW +HQFH LQ JHQHUDO WKH LQIHUHQFH IURP :ST WR :.SUT LV LQYDOLG &RQVHn TXHQWO\ DV /HZLV FRQFOXGHV :ST FDQQRW EH D FRQGLWLRQDO RI D IL[HG GHJUHH RI VWULFWQHVV 9DULDELOLW\ PXVW EH EXLOW LQWR WKH WUXWK FRQn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n WXDO LV HOOLSWLFDO IRU VRPHWKLQJ PXFK PRUH FRPSOH[ WKHQ LW VWURQJO\ GHn SHQGV XSRQ WKH H[DFW FRQWH[W RI XWWHUDQFH IRU LWV LQWHUSUHWDWLRQ :KLFK PHDQV WKH FRXQWHUIDFWXDO LV SUDJPDWLFDOO\ DPELJXRXV WR D KLJK GHJUHH 2Q /HZLVn YLHZ

PAGE 57

,W FRQVLJQV WR WKH ZDVWHEDVNHW RI FRQWH[WXDOO\ UHVROYHG YDJXHQHVV VRPHWKLQJ PXFK PRUH DPHQDEOH WR V\VWHPDWLF DQDO\VLV WKDQ PRVW RI WKH UHVW RI WKH PHVV LQ WKDW ZDVWHn EDVNHW > S @ +HQFH LQ WKH VXEVHTXHQW VHFWLRQ ZH ZLOO SUHVHQW /HZLVn DQDO\VLV RI WKH FRXQWHU IDFWXDO FRQGLWLRQDO DV D YDULDEO\ VWULFW FRQGLWLRQDO ZLWK WKH H[SHFWDWLRQ WKDW DPELJXLW\ ZLOO EH NHSW ZLWKLQ PRUH DFFHSWDEOH ERXQGV /HZLVn $QDO\VLV RI WKH &RXQWHUIDFWXDO &RQGLWLRQDO 7KDW WKH FRXQWHUIDFWXDO FRQGLWLRQDO LV D YDULDEO\ VWULFW FRQGLn WLRQDO LV JHQHUDOO\ DGPLWWHG E\ DGYRFDWHV RI ERWK WKH PHWDOLQJXLVWLF DQG SRVVLEOH ZRUOGV DSSURDFKHV 7KH GLIIHUHQFH LQ WUHDWPHQW LQYROYHV LQ SDUW IL[LQJ WKH ERXQGDU\ EHWZHHQ VHPDQWLF DQG SUDJSQDWLF DPELJXLW\ WR ZKLFK HIIHFW ZH TXRWHG 6WDOQDNHU HDUOLHU 7KH HVVHQWLDO UHTXLUHPHQW LV WR KDYH D V\VWHP IRU UHVROYLQJ DV PXFK RI WKH DSSDUHQW DPELJXLW\ LQ FRQGLWLRQDOV DV SRVVLEOH 2QFH DPELJXLW\ LV VHHQ DV V\VWHPDWLF LW LV QR ORQJHU D EDUULHU WR DQDO\VLV 6HH /HZLV >@f 7R PRWLYDWH KLV FRQVWUXFWLRQ RI WKH WUXWK FRQGLWLRQV IRU FRXQWHU IDFWXDOV /HZLV > S @ LQYLWHV XV WR FRQVLGHU ZKDW KDV EHFRPH P\ IDYRULWH H[DPSOH ,I NDQJDURRV KDG QR WDLOV WKH\ ZRXOG WRSSOH RYHU 3UHYLRXVO\ ZH VDLG WKDW LQ HYDOXDWLQJ WKLV FRQGLWLRQDO ZH FRQVWUXFW IRU RXUVHOYHV D VLWXDWLRQ DOWHULQJ ZKDW ZH PXVW LQ ZKDW ZH WDNH WR EH DFWXDOO\ WUXH LQ ZKLFK NDQJDURRV KDYH QR WDLOV ,I LW LV WKH FDVH LQ VXFK D VLWXDWLRQ WKDW NDQJDURRV WRSSOH RYHU WKHQ WKH FRQGLWLRQDO LV WUXH +RZHYHU DV ZH VDZ D FULWLFDO HOHPHQW RI WKLV SURFHGXUH LV WKH DOWHULQJ ZKDW ZH PXVW IRU LW LV VXUHO\ SRVVLEOH WR LPDJLQH VLWXDWLRQV LQ ZKLFK NDQJDURRV KDYH QR WDLOV EXW HYROYHG WKDW ZD\ VR DOVR HYROYHG D VWUXFWXUH ZKLFK RWKHUZLVH SHUPLWV WKHP WR EH EDODQFHG 2U D VLWXDWLRQ LQ ZKLFK NDQJDURRV KDYH QR WDLOV EXW KDYH OHDUQHG WR XVH FUXWFKHV VR

PAGE 58

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n YLGHV LV DQ DQDO\VLV ZKLFK H[SODLQV YK\ WKDW LV WKH FDVH :H DUH FRQFHUQHG RQO\ ZQ/WK ZRUOGV YHU\ PXFK OLNH RXUV WKDW DUH VLPLODU WR D FHUWDLQ GHJUHH WR WKH DFWXDO ZRUOG 7KH PRUH LPDJLQDWLYH ZRUOGV DERYH DUH OHVV VLPLODU WR WKH DFWXDO ZRUOG WKDQ DUH ZRUOGV ZKHUH OHVV KDV FKDQJHG /HZLVn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

PAGE 59

ZH DUH FRQFHUQHG ZLWK ZRUOGV VLPLODU WR RXUV WR D FHUWDLQ IL[HG WKRXJK VRPHZKDW YDJXH GHJUHH 7KHUH DUH D QXPEHU RI HTXLYDOHQW ZD\V LQ ZKLFK /HZLVn IRUPDO VHPDQWLFV FDQ EH VHW XS VR DV WR FDUU\ LQIRUPDWLRQ DERXW DFFHVVLELOLW\ DQG VLPLODULW\ FKRRVH WKDW ZKLFK LV DSSDUHQWO\ FRPn SDWLEOH ZLWK WKH QHLJKERUKRRGV VHPDQWLFV WR EH GLVFXVVHG ODWHU )RU D VWULFW FRQGLWLRQDO ZH QHHG RQH VSKHUH RI DFFHVVLELOLW\ IRU HDFK ZRUOG JLYHQ E\ WKH IXQFWLRQ 6 8 r 38f ZKHUH GHQRWHV WKH VXEVHW RI 8 ZKLFK LV WKH VSKHUH DERXW X )RU D YDULDEO\ VWULFW FRQn GLWLRQDO ZH ZLOO LQ JHQHUDO UHTXLUH PRUH WKDQ RQH VSKHUH DERXW HDFK ZRUOG X RU DV ZH PLJKW VD\ X ZLOO KDYH PDQ\ QHLJKERUKRRGV 7KRXJK ZH VKDOO VHH WKHVH DUH QRW TXLWH WKH QHLJKERUKRRGV RI QHLJKERUKRRG VHPDQWLFVf /HW 8 UHSUHVHQW WKH VHW RI SRVVLEOH ZRUOGV DQG OHW 8 338ff EH D IXQFWLRQ IURP 8 WR WKH SRZHU VHW RI WKH SRZHU VHW RI 8 7KDW LV DVVLJQV WR HDFK X LQ 8 QRW D VLQJOH VXEVHW RI 8 D VLQJOH VSKHUH DERXW Xf EXW D VHW RI VXEVHWV RI 8 D VHULHV RI VSKHUHV DERXW Xf :H VKDOO GHVLJQDWH WKH LPDJH RI X XQGHU X ZKHUH HDFK LQ X LV D VLQJOH VSKHUH DERXW X /HZLV SODFHV IRXU FRQGLWLRQV RQ LQ RUGHU WKDW LW SODXVLEO\ FDUU\ LQn IRUPDWLRQ DERXW VLPLODULW\ > S @ 7R WKHVH ZH DGG D ILIWK ZKLFK LV RSWLRQDO IRU /HZLV DQG GHWHUPLQHV LQ SDUW WKH NLQG RI PRGDO ORJLF WKDW LV YDOLGDWHG E\ WKLV IUDPHZRUN 7KH FRQGLWLRQV RQ DUH & Xf LV DQ HOHPHQW RI X & )RU DOO $% LQ HLWKHU $ LV D VXEVHW RI % RU % LV D VXEVHW RI $ 8 & ,I ; LV D VXEVHW RI WKHQ WKH XQLRQ RI ; LV DQ HOHPHQW RI 8 X

PAGE 60

& ,I WKH QRQHPSW\ VHW < LV D VXEVHW RI WKHQ LWV LQWHUVHFWLRQ LV DQ HOHPHQW RI X & )RU HDFK XY LQ 8 WKH XQLRQ RI HTXDOV WKH XQLRQ RI Y )ROORZLQJ /HZLVn WHUPLQRORJ\ ZH VKDOO FDOO WKHVH FRQGLWLRQV UHVSHFWLYHO\ VWURQJf FHQWHULQJ QHVWLQJ FORVXUH XQGHU XQLRQV FORVXUH XQGHU QRQHPSW\f LQWHUVHFWLRQV DQG XQLIRUPLW\ > SS @ /HZLV FDOOV D V\VWHP RI VSKHUHV :H VKDOO GHSDUW IURP /HZLV VOLJKWO\ E\ FDOOLQJ D VSKHUH IXQFWLRQ DQG E\ FDOOLQJ X WKH V\VWHP RI VSKHUHV DERXW X $ SLFWXUH VXJJHVWLYH RI D V\VWHP RI VSKHUHV DERXW X ZKLFK ZH VKDOO KDYH RFFDVLRQ WR XVH UHSHDWHGO\ LV WKDW RI )LJXUH )LJXUH (DFK FLUFOH UHSUHVHQWV WKH ERXQGDU\ RI RQH RI WKH VSKHUHV RI DFFHVn VLELOLW\ DERXW X ,Q ZKDW IROORZV IRU EUHYLW\ ZH ZLOO XVH WKH IROORZLQJ V\PEROV H GI LV DQ HOHPHQW RI GI LV D VXEVHW RI 8 GI WKH XQLRQ RI LL WKH LQWHUVHFWLRQ RI a GI WKH HPSW\ VHW ,Q WKLV QRWDWLRQ WKH FRQGLWLRQV OLVWHG DERYH PD\ EH PRUH EULHIO\ VWDWHG DV

PAGE 61

& ^X`eX & ,I $% H WKHQ $ F % RU % F $ & ,I ; F WKHQ 8; H & ,I < F WKHQ $< H X & )RU DOO XY H 8 8X 8Y ,Q /HZLVn YLHZ WKHVH FRQGLWLRQV RU UDWKHU WKH ILUVW IRXU DUH QHFHVVDU\ IRU WKH V\VWHP RI VSKHUHV WR EH SODXVLEO\ FRQVLGHUHG WR FRQYH\ LQIRUPDWLRQ DERXW FRPSDUDWLYH VLPLODULW\ ,Q ZKDW IROORZV ZH FRQWLQXH WR DGKHUH WR /HZLVn SUHVHQWDWLRQ H[FHSW ZKHUH QRWHG 6HH > SS @f ,W LV UHDVRQDEOH WKDW WKH DFWXDO ZRUOG RU DQ\ JLYHQ ZRUOG LV PRUH VLPLODU WR LWVHOI WKDQ DQ\ RWKHU SRVVLEOH ZRUOG KHQFH WKH FHQWHULQJ UHTXLUHPHQW 7KH VLQJOHWRQ VHW ^X`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

PAGE 62

6LQFH HDFK VHW LQ X LV D VHW RI ZRUOGV PRUH VLPLODU WR X WKDQ DQ\ ZRUOGV RXWVLGH WKH VHW LW IROORZV WKDW Y LV PRUH VLPLODU WR X WKDQ Z IURP Y H $ DQG Z O $f EXW DOVR WKDW Z LV PRUH VLPLODU WR X WKDQ Y IURP Z H % DQG Y L Ef +HQFH QHVWLQJ LV UHTXLUHG LI LV WR FDUU\ LQIRUPDn WLRQ DERXW FRPSDUDWLYH RYHUDOO VLPLODULW\ 2I FRXUVH LI ZH DUH FRQn FHUQHG RQO\ ZLWK VLPLDULW\LQFHUWDLQUHVSHFWV WKHQ VLPLODU KDV GLIn IHUHQW WR EH VSHFLILHGf VHQVHV LQ WKH DSSDUHQWO\ LQFRQVLVWHQW VWDWHn PHQWV DERYH VR LQ WKDW FDVH WKH\ ZRXOG EH FRPSDWLEOH +RZHYHU LQ /HZLVn DQDO\VLV ZRUOGV DUH FRPSDUHG LQ WHUPV RI RYHUDOO VLPLODULW\ WR WKH JLYHQ ZRUOG X IRU HDFK V\VWHP RI VSKHUHV VR QHVWLQJ LV UHTXLUHG 7KH MXVWLILFDWLRQ RI FORVXUH XQGHU XQLRQV DQG LQWHUVHFWLRQV LV EDVHG RQ WKH IROORZLQJ FRQVLGHUDWLRQ VXSSRVH WKHUH LV D VHW RI ZRUOGV VXFK WKDW DQ\ ZRUOG LQVLGH LW LV PRUH VLPLODU WR WKH JLYHQ ZRUOG X WKDQ DQ\ ZRUOG RXWVLGH LW 7KHQ WKLV VHW VKRXOG EH D VSKHUH DERXW X LQ YLUn WXH RI EHLQJ VLPLODU WR X WR DW OHDVW D FHUWDLQ GHJUHH %XW 8; ZKHUH ; F X LV MXVW VXFK D VHW VLQFH DQ\ ZRUOG Z H OO;ALV DQ HOHPHQW RI VRPH e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

PAGE 63

VHWV WKHQ ^X` ZRXOG EH WKH VPDOOHVW VSKHUH DERXW X $V ZH VKDOO VHH EHORZ WKH ODUJHVW VSKHUH LQ PD\ EH LGHQWLILHG ZLWK 8 LQ YLUWXH RI FRQGLWLRQ XQLIRUPLW\ /HZLV FDUHIXOO\ SRLQWV RXW D FRQVHTXHQFH WKDW PLJKW EH RYHUORRNHG > S @ :KLOH FORVXUH XQGHU XQLRQV DQG LQWHUVHFWLRQV JXDUDQWHHV DQ XSSHU ERXQG DQG D ORZHU ERXQG RQ HDFK VXEVHW RI A LW LV QRW QHFHVn VDULO\ WKH FDVH WKDW WKHVH ERXQGV PXVW EH LQ WKH VXEVHW RI X XQGHU FRQn VLGHUDWLRQ 7KDW LV IRU ; F X DVVXPH QRQHPSW\f ZKLOH 8; H X DQG $; e X LW GRHV QRW IROORZ WKDW 8; H ; RU WKDW $; H [ 7KLV LV SUHn FLVHO\ DQDORJRXV WR WKH VHW RI UDWLRQDO QXPEHUV OHVV WKDQ DQG PRUH WKDQ WKHUH LV QHLWKHU D JUHDWHVW QRU D OHDVW HOHPHQW RI WKDW VHW RI UDWLRQDO QXPEHUV EXW WKH VHW LV ERXQGHG DERYH DQG EHORZ 7KLV LV RI LPSRUWDQFH LQ FRQQHFWLRQ ZLWK WKH OLPLW DVVXPSWLRQ ZKLFK ZH VKDOO GLVn FXVV LQ 6HFWLRQ &RQGLWLRQV DQG WRJHWKHU LPSO\ WKDW WKH ODUJHVW VSKHUH LQ X LV 8 &RQVLGHU DQ\ SDLU RI ZRUOGV XY H 8 %\ FHQWHULQJ ^X` H X DQG ^Y` H Y KHQFH X H 8X DQG Y H 8Y %XW E\ XQLIRUPLW\ 8X 8Y KHQFH X H 8Y DQG Y H 8A %XW X DQG Y ZHUH DUELWUDULO\ FKRVHQ HOHPHQWV RI 8 VR IRU DOO X H 8 IRU DOO Y H 8 Y H 8 ,IHQFH IRU DOO X H 8 8 8 X X 7KXV LV XQLYHUVDO LQ WKH VHQVH WKDW HYHU\ ZRUOG KDV DFFHVV WR HYHU\ RWKHU DW WKH OHYHO RI WKH ODUJHVW VSKHUHV DERXW HDFK /HZLV GRHV QRW LPn SRVH XQLIRUPLW\ LQ JHQHUDO RQ WKH VSKHUH IXQFWLRQ DQG DOORZV IRU WKH SRVn VLELOLW\ WKDW 8X PD\ QRW H[KDXVW 8 IRU VRPH RU DOO X H 8 :H GR VR LQ RUGHU WR SURYLGH IRU D VLPSOHU FKDUDFWHUL]DWLRQ RI WKH PRGDO ORJLF WKLV VHPDQWLFV YDOLGDWHV $V ZH VKDOO VHH LW LV 6f

PAGE 64

7KHUH DUH PDQ\ VSKHUH IXQFWLRQV ZKLFK ZRXOG VDWLVI\ WKHVH FRQn GLWLRQV $Q\ SDUWLFXODU IXQFWLRQ ZLOO EH GHWHUPLQHG E\ QRQIRUPDO FRQVLGHUDWLRQV :H ZLOO FRQVLGHU VRPH RI WKHVH ZKHQ ZH ORRN DW VLPLn ODULW\ DJDLQ LQ &+$37(5 7+5(( *LYHQ WKH VSKHUH IXQFWLRQ DQG WKH UHVXOWLQJ V\VWHP RI VSKHUHV IRU HDFK ZRUOG ZH FDQ QRZ VWDWH WKH WUXWK FRQGLWLRQV IRU WKH FRXQWHU IDFWXDO FRQGLWLRQDO :ST )LUVW ZH ZLOO IROORZ /HZLV LQ DGRSWLQJ WKH FRQYHQWLRQ WKDW D ZRUOG DW ZKLFK SURSRVLWLRQ S LV WUXH ZLOO EH FDOOHG D SZRUOG DQG WKH FRQYHQWLRQ WKDW DQ\ VSKHUH FRQWDLQLQJ D SZRUOG ZLOO EH FDOOHG D SSHUPLWWLQJ VSKHUH :H PD\ WKHQ VWDWH WKH WUXWK FRQn GLWLRQV IRU WKH FRXQWHUIDFWXDO FRQGLWLRQDO > S @ :ST # X LII HLWKHU f WKHUH LV QR SSHUPLWWLQJ VSKHUH LQ A RU f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f LV GLDJUDQPHG LQ )LJXUH

PAGE 65

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

PAGE 66

LQWHQGHG WR DSSO\ WR WKH PLJKWFRXQWHUIDFWXDO DOVR GLVFXVVHG LQ 6HFWLRQ )RU WKLV FRQGLWLRQDO ZH PD\ XVH WKH VDPH EDVLF VHPDQWLFV EXW ZLOO UHTXLUH D GLIIHUHQW VHW RI WUXWK FRQGLWLRQV )LJXUH

PAGE 67

)LJXUH UHSUHVHQWV YDULRXV GLVWULEXWLRQV RI WKH WUXWK YDOXHV RI SURSRVLWLRQV S DQG T RYHU WKH V\VWHP RI VSKHUHV f :H VKDOO UHIHU WR WKHP LQ ZKDW LPPHGLDWH IROORZV )LJXUH DQG ILJXUHV Df DQG Gf DOO UHSUHVHQW FDVHV ZKHUH :ST LV WUXH $V ZH VKDOO VHH LQ 6HFWLRQ Df UHSUHVHQWV D FDVH ZKHUH /&ST LV WUXH DV ZHOO ,Q Gf T LV WUXH DW X DQG S EHLQJ WUXH ZLOO QRW FKDQJH WKLV WKDW LV HYHQ LI S ZHUH WUXH T ZRXOG EH 7KLV LV D FDVH ZKHUH DQ HYHQLIFRQGLWLRQDO LV DSSURSULDWH EXW QR IXUWKHU GHILQLWLRQ RI WUXWK IRU WKH FRQGLWLRQDO LQ TXHVWLRQ WKH ZRXOG FRQGLWLRQDO LV UHTXLUHG 7KH FRQGLWLRQ WKDW HYHU\ SZRUOG EH D TZDUOG LQ VRPH SSHUPLWWLQJ VSKHUH VXIILFHV ,Q ERWK Ef DQG Ff WKH FRQGLWLRQDO :ST LV QRW WUXH 7KH VLWXDWLRQV GLIIHU LQ WKDW LQ Ef WKHUH LV VHPH SSHUPLWWLQJ VSKHUH ZKHUH VRPH RI WKH SZRUOGV DUH TZRUOGV EXW QRW DOO ZKLOH LQ Ff WKHUH LV D SSHUPLWWLQJ VSKHUH ZKHUH QRQH RI WKH SZRUOGV DUH TZRUOGV ,Q WKH ODWWHU FDVH Ff :S1T LV WUXH EXW LQ WKH IRUPHU FDVH Ef QHLWKHU :ST QRU :S1T LV WUXH )RU HQWHUWDLQDEOH DQWHFHn GHQWV :ST DQG :S1T EHKDYH DV FRQWUDULHV WKH\ PD\ QRW ERWK EH WUXH EXW WKH\ PD\ ERWK EH IDOVH ,Q WUDGLWLRQDO TXDQW LI LFDWLRQDO ORJLF WR HDFK FRQWUDU\ FRUUHVSRQGV D VXEFRQWUDU\ 6LPLODUO\ LQ /HZLVn DQDO\VLV WR HDFK RI WKH FRQWUDULHV :ST DQG :S1T FRUUHVSRQGV D VXEFRQWUDU\ 9ST DQG 9S1T 7KHVH DUH WKH DIRUHPHQWLRQHG PLJKWFRQGLWLRQDOV ,Q WKRVH FDVHV ZKHUH S LV HQWHUWDLQDEOH EXW QHLWKHU :ST QRU :S1T DUH WUXH WKHQ ERWK 9ST DQG 9S1T DUH WUXH 5HFDOO WKH SDLU RI FRQGLWLRQDOV FRQFHUQLQJ %L]HW DQG 9HUGL DV DQ H[DPSOH

PAGE 68

7KH GHILQLWLRQ RI WUXWK IRU WKH PLJKWFRQGLWLRQDO LV WKHQ JLYHQ DV IROORZV > S @ 9ST # X LII ERWK f WKHUH LV VRPH SSHUPLWWLQJ VSKHUH LQ DQG f HYHU\ SSHUPLWWLQJ VSKHUH FRQWDLQV DW OHDVW RQH .STZRUOG 1RWH WKDW WKH WUDGLWLRQDO GHEDWH RYHU ZKHWKHU RU QRW XQLYHUDOO\ TXDQWLn ILHG SURSRVLWLRQV SUHVXQH H[LVWHQFH DULVHV DQHZ LQ WKH FDVH RI WKH FRXQWHU IDFWXDO FRQGLWLRQDO ZKHUH LW UHDSSHDUV DV D GHEDWH RYHU ZKHWKHU WKH FRQn GLWLRQDO SUHVXSSRVHV WKDW LWV DQWHFHGHQW LV HQWHUWDLQDEOH $V ZH KDYH GHILQHG WKH ZRXOGFRQGLWLRQDO LW LV YDFXRXVO\ WUXH ZKHQ WKH DQWHFHn GHQW LV QRW HQWHUWDLQDEOH LH ZKHQ WKHUH LV QR SSHUPLWWLQJ VSKHUH ,Q VXFK D FDVH WKH VXEDOWUQDWH PLJKWFRQGLWLRQDO ZLOO EH IDOVH WKH FRQWUDU\ ZRXOGFRQGLWLRQDO WUXH DQG LWV VXEDOWUQDWH PLJKWFRQGLWLRQDO IDOVH +HQFH WKHVH GHILQLWLRQV GR QRW VXSSRUW WKH FRQGLWLRQDO DQDORJ RI WKH WUDGLWLRQDO VTXDUH RI RSSRVLWLRQ *LYHQ WKH GHILQLWLRQV DQG WKH RQO\ UHODWLRQ WKDW GRHV REWDLQ LV WKH FRQWUDGLFWRU\ UHODWLRQ DORQJ WKH GLDJRQDOV LH ERWK RI WKH IROORZLQJ DUH YDOLGDWHG (:ST19S1T (:S1T19ST

PAGE 69

,Q YLHZ RI WKHVH HTXLYDOHQFH WKH ZRXOG DQG PLJKWFRQGLWLRQDOV DUH LQWHUGHILQDEOH RQ /HZLVn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n VHPDQWLFV QHFHVVDULO\ IDLOV WR YDOLGDWH WKH SULQFLSOH RI FRQGLWLRQDO H[FOXGHG PLGGOH &(0f ,Q FODVVLFDO SURSRVLWLRQDO ORJLF $&ST&S1T LV D WKHRUHP 7KH FRUUHVSRQGLQJ $/&ST/&S1T LV QRW JHQHUDOO\ D WKHRUHP RI PRGDO ORJLF LH *(0 IDLOV IRU WKH VWULFW FRQGLWLRQDO 7KH FRXQWHU IDFWXDO FRQGLWLRQDO VKDUHV WKLV SURSHUW\ ZLWK WKH VWULFW FRQGLWLRQDO 7KH SULQFLSOH RI &(0 LV YDOLGDWHG E\ 6WDOQDNHUnV VHPDQWLFV IRU UHDVRQV ZKLFK ZH VKDOO GLVFXVV LQ &+$37(5 7+5((f

PAGE 70

,I IRU VRPH UHDVRQ LW LV GHVLUDEOH WR SUHVHUYH DOO RI WKH UHODWLRQV RI WKH WUDGLWLRQDO VTXDUH RI RSSRVLWLRQ WKHQ ZH PXVW H[FOXGH YDFXRXV WUXWK IRU WKH FRQGLWLRQDO :ST 7KDW LV LI WKH DQWHFHGHQW RI WKH FRQn GLWLRQDO LV LPSRVVLEOH ZH UHTXLUH LW WR EH IDOVH )RU WKLV SXUSRVH WKH IROORZLQJ WZR GHILQLWLRQV VXIILFH :ST # X LII WKHUH LV VRPH SSHUPLWWLQJ VSKHUH LQ DQG &ST # HYHU\ ZRUOG LQ WKDW VSKHUH 9ST # X LII HYHU\ SSHUPLWWLQJ VSKHUH LQ X FRQWDLQV DW OHDVW RQH .STZRUOG 7KHVH KDYH WKH YLUWXH RI VWLOO SUHVHUYLQJ WKH LQWHUGHILQDELOLW\ RI WKH ZRXOG DQG PLJKWFRQGLWLRQDOV /HZLVn GHILQLWLRQ HLWKHU RQHf RI WKH PLJKWFRQGLWLRQDO LV RI FRQVLGHUDEOH LPSRUWDQFH RIIHULQJ DV LW GRHV D ZD\ RI UHVROYLQJ SUREOHPV ZKLFK KDG KHUWRIRUH EHHQ ODEHOHG LUUHGHHPDEOH 5HVFKHU JLYHV WKH IROn ORZLQJ H[DPSOHV RI SXUHO\ K\SRWKHWLFDO FRXQWHUIDFWXDOV LH WKRVH QRW WKRURXJKO\ JURXQGHG LQ ODZV > S @ ( ,I %L]HW DQG 9HUGL ZHUH FRPSDWULRWV %L]HW ZRXOG EH DQ ,WDOLDQ ( ,I %L]HW DQG 9HUGL ZHUH FRPSDWULRWV 9HUGL YURXOG EH D )UHQFKPDQ ( ,I *HRUJLD LQFOXGHG 1HZ S @ ,W LV FOHDU WKDW WKHVH DUH UHEXWWLQJ RQO\ LI RQH KROGV WKDW FRQn GLWLRQDO H[FOXGHG PLGGOH LV YDOLG IRU WKH FRXQWHUI DFWXDO FRQGLWLRQDO

PAGE 71

+RZHYHU /HZLVn DQDO\VLV DOORZV XV WR DYRLG KDYLQJ WR WRVV WKHVH LQWR WKH LUUHGHHPDEO\ DPELJXRXV ELQ %RWK ( DQG ( DUH IDOVH KHQFH QRW UHEXWWLQJ ZKLOH WKH FRUUHVSRQGLQJ PLJKWFRQGLWLRQDOV DV DOUHDG\ QRWHG DUH WUXH 6LPLODU FRQVLGHUDWLRQV PD\ EH DSSOLFDEOH WR ( DQG ( KRZHYHU`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n DQDO\VLV SURYLGHV IRU WKLV LV D SRLQW LQ LWV IDYRU 0RGDO ORJLF DQG WKH 6\VWHP RI 6SKHUHV ,Q WKH SUHFHGLQJ VHFWLRQV ZH LQGLFDWHG WKDW /HZLVn DQDO\VLV RI WKH FRXQWHUIDFWXDO FRQGLWLRQDO LV DQ H[WHQVLRQ RU DSSOLFDWLRQ RI SRVVLEOH ZRUOGV VHPDQWLFV GHYHORSHG IRU PRGDO ORJLF ,Q WKH YHUVLRQ RI /HZLVn DQDO\VLV WKDW ZH KDYH JLYHQ WKH ORJLFDO PRGDOLWLHV LQFOXGLQJ WKH VWULFW FRQGLWLRQDO FDQ EH H[SUHVVHG

PAGE 72

:LWK PRUH WKDQ RQH VSKHUH RI DFFHVVLELOLW\ DVVLJQHG WR HDFK ZRUOG X LQ 8 WKH FRQFHSWV RI SRVVLELOLW\ DQG QHFHVVLW\ LQ WKHLU ZLGHVW VHQVH QHHG WR EH FRUUHODWHG ZLWK WKH ODUJHVW VSKHUH 7KH FRQGLWLRQV ZH KDYH SODFHG RQ WKH QHLJKERUKRRG IXQFWLRQ UHTXLUH WKDW WKH ODUJHVW VSKHUH EH 8 IRU HDFK X LQ 8 DQG IXUWKHUPRUH WKDW 8 8 IRU DOO X LQ 8 :KDW X X /HZLV FDOOV WKH RXWHU PRGDOLWLHV > S @ DUH GHILQHG DV IROORZV /S # X LII HYHU\ ZRUOG LQ 8X LV D SZRUOG 0S # X LII VRPH ZRUOG LQ 8X LV D SZRUOG ,Q YLHZ RI WKH IDFW WKDW 8X 8 IRU DOO X LQ 8 WKHVH RXWHU PRGDOLWLHV FRUUHVSRQG WR WKH ORJLFDO PRGDOLWLHV RI 6 +HQFH WKH UHTXLUHPHQW WKDW S EH HQWHUWDLQDEOH FDQ EH H[SUHVVHG DV WKH UHTXLUHPHQW WKDW /S EH WUXH *LYHQ WKH GHILQLWLRQV DQG RI :ST DQG 9ST LW WKHQ IROORZV WKDW :ST GRHV QRW ZKLOH 9ST GRHV HQWDLO WKDW /S LV WUXH :H PD\ DOVR JLYHQ WKH DERYH GHILQLWLRQ RI QHFHVVLW\ GHILQH D VWULFW FRQGLWLRQDO ZKLFK ZLOO EH WKH VWULFW FRQGLWLRQDO RI 6 /&ST # X LII HYHU\ ZRUOG LQ 8X LV D &STZRUOG 5HIHUULQJ EDFN WR )LJXUH Df WKDW GLDJUDPV D VLWXDWLRQ LQ ZKLFK /&ST LV WUXH ,W LV WKHQ UHDGLO\ VHHQ WKDW /&ST HQWDLOV :ST RQ /HZLVn DQDO\VLV VLQFH LI &ST LV WUXH DW DOO ZRUOGV LQ 8X LW PXVW WKHQ EH WUXH DW HYHU\ ZRUOG LQ VRPH SSHUPLWWLQJ VSKHUH 7KH FRQYHUVH KRZHYHU GRHV QRW KROG DV )LJXUHV DQG Gf LOOXVWUDWH :ST PD\ EH WUXH WKRXJK WKHUH DUH 1&STZRUOGV WKDW LV .S1TZRUOGVf 2QH PD\ HDVLO\ FRQILUP IURP WKH GHILQLWLRQ RI :ST WKDW LW HQWDLOV &ST KHQFH ZH KDYH D KLHUDUFK\ RI FRQGLWLRQDOV /FST HQWLDOV :ST HQWDLOV &ST +RZHYHU LQ QR FDVH GRHV WKH FRQYHUVH HQWDLOPHQW KROG

PAGE 73

,I IURP WKH FRQGLWLRQV RQ ZH ZHUH WR GURS & WKH XQLIRUPLW\ FRQGLWLRQ WKHQ LW ZRXOG QRW JHQHUDOO\ EH WKH FDVH WKDW 8X 8 IRU DOO X LQ 8 ,Q VXFK D FDVH WKH RXWHU PRGDOLWLHV DV DERYH GHILQHG ZRXOG QRW FRUUHVSRQG WR WKH PRGDOLWLHV RI 6 EXW UDWKHU WR WKH V\VWHP NQRZQ JHQHUDOO\ DV 7 7KH RQO\ FRQGLWLRQ RQ WKH DFFHVVLELOLW\ UHODWLRQ EHLQJ WKH UHIOH[LYLW\ FRQGLWLRQ LPSOLHG E\ FHQWHULQJf $GRSWLQJ D FRQGLWLRQ VXFK DV & )RU DOO XYZ H 8 LI X H 8Y DQG Y e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n H[DPS OHG )LJXUH

PAGE 74

,Q WKH VLWXDWLRQ GLDJUDPPHG LQ )LJXUH 0.ST LV WUXH EHFDXVH WKHUH LV D .STZRUOG LQ 8X KRZHYHU 9ST LV IDOVH EHFDXVH WKRXJK WKHUH LV D SSHUPLWWLQJ VSKHUH LW LV QRW WKH FDVH WKDW HYHU\ SSHUPLWWLQJ VSKHUH FRQWDLQV D .STZRUOG WKH VHFRQG QRQWULYLDO VSKHUH RXW GRHV QRW LXf DUH WULYLDO VSKHUHVf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n DQDO\VLV E\ FRQVLGHULQJ PRGDO ORJLF LQ 6HFWLRQ DQG ZH ZLOO UHWXUQ WR PRGDO ORJLF DJDLQ ZKHQ ZV FRQVLGHU QHLJKERUKRRG VHPDQWLFV LQ &+$37(5 )285 ,Q WKH QH[W VHFWLRQ ZKHUH ZH GLVFXVV FRXQWHU IDFWXDO LQIHUHQFHV DQG IDOODFLHV ZH ZLOO KDYH RFFDVLRQ WR FRPSDUH LQIHUn HQFHV YDOLG LQ FODVVLFDO SURSRVLWLRQDO DQG PRGDO ORJLFV ZLWK WKRVH LQ FRXQWHUIDFWXDO ORJLF 5DWKHU WKDQ VHW IRUWK WKRVH SURSHU WR PRGDO ORJLF KHUH ZH VKDOO PHQWLRQ WKHP DV ZH FRPH WR WKHP LQ WKH QH[W VHFWLRQ &RXQWHUIDFWXDO ,QIHUHQFHV DQG )DOODFLHV 2QH WHVW RI DGHTXDF\ IRU DQ DQDO\VLV RI WKH FRXQWHUIDFWXDO FRQGLn WLRQDO LV WR VHH LI LW YDOLGDWHV LQIHUHQFH SDWWHUQV UHFRJQL]HG DV YDOLG

PAGE 75

DQG LQYDOLGDWHV LQIHUHQFH SDWWHUQV UHFRJQL]HG DV LQYDOLG WKDW LV JHQHUDOO\ SUHVHUYHV RXU SUHDQDO\WLF LQWXLWLRQV DV WR WKH ORJLF RI WKH FRQFHSW LQYROYHG /HZLVn DQDO\VLV ZLWKVWDQGV WKLV WHVW DGPLUDEO\ DV ZH VKDOO VKRZ LQ WKLV VHFWLRQ )LUVW ZLVK WR GHILQH D QRWLRQ RI VHPDQWLF HQWDLOPHQW IRU WKH DQDO\VLV VR IDU SUHVHQWHG 7R GLVWLQJXLVK LW IURP RXU V\QWDFWLF IRUPXn ODWLRQV ZH VKDOO XVH LQIL[ QRWDWLRQ WKH V\PERO LV LQWHQGHG WR GHQRWH VHPDQWLF HQWDLOPHQW ZKLFK LV GHILQHG DV IROORZV S_ TLII __S>>F_,T__ ZKHUH __U__ GHQRWHV WKH VHW RI ZRUOGV ZKHUH U LV WUXH 7KLV QRWLRQ ZLOO KDYH WR EH UHODWLYL]HG WR D PRGHO ZKHQ ZH VKLIW WR IRUPDO VHPDQWLFV LQ &+$37(5 3285f &RQVLGHU DJDLQ )LJXUH Df :LWK WKH KHOS RI WKLV ILJXUH DQG GHILQLWLRQ LW LV FOHDU WKDW S_ TLII /&ST LV WUXH ,Q ZKDW IROORZV ZKHUH /&ST LV XVHG DV D SUHPLVH VXEVWLWXWLRQ RI S_ T ZLOO QRW DOWHU RXU FRQFOXVLRQV ZLWK UHJDUG WR WKH LQIHUHQFH SDWWHUQ 'LIILFXOW\ DULVHV RQO\ LI ZH GHILQH VHPDQWLF HQWDLOPHQW IRU VHWV RI SURSRVLWLRQV *_ T LII ^Z LI S H WKHQ S # Z` F __T_c :H FDQQRW WDNH /&*T DV HTXLYDOHQW WR *_ TDV WKH VWULFW FRQGLWLRQDO KROGV RQO\ EHWZHHQ SURSRVLWLRQV 1RU ZLOO WKH FRQMXQFWLRQ RI DOO SURSRVLWLRQV LQ ZRUN VLQFH FRXOG EH LQILQLWH DQG LW LV QRW RXU LQWHQWLRQ WR UHSUHn VHQW LQILQLWH FRQMXQFWLRQV LQ RXU REMHFW ODQJXDJH 6WULFWO\ VSHDNLQJ ZH KDYH QRW UHDOO\ LQGLFDWHG ZKDW RXU REMHFW ODQJXDJH LV H[FHSW LQIRUPDOO\ 7KLV ZQOO EH GRQH LQ &+$37(5 )285 EHOLHYH LW ZURXOG EH GLVWUDFWLQJ DW WKLV SRLQW ,Q 6HFWLRQ ZH ZLOO KDYH WR PDNH XVH RI DQG VR LW LV VWDWHG DW WKLV WLPHf

PAGE 76

:H PD\ GLYLGH RXU FRQFHUQV LQWR WKRVH LQIHUHQFH SDWWHUQV WKDW RXJKW WR EH YDOLG IRU WKH FRXQWHUIDFWXDO FRQGLWLRQDO DQG WKRVH ZKLFK RXJKW QRW WR EH YDOLG $PRQJ WKH ILUVW ZLOO EH WKRVH SDWWHUQV ZH ZRXOG H[SHFW DQ\ FRQGLWLRQDO WR DGKHUH WR 0DQ\ DXWKRUV LQFOXGLQJ =HPDQ >@ +DUGHJUHH >@f FRQVLGHU WZR UHTXLUHPHQWV DEVROXWHO\ PLQLPDO IRU D FRQGLWLRQDO IXQFWLRQ & ,I S> T WKHQ :ST LV WUXH & ,I :ST DQG S DUH WUXH WKHQ T LV WUXH 7KH ILUVW LV D VLPSOLILHG VHPDQWLF YHUVLRQ RI WKH GHGXFWLRQ WKHRUHP DQG WKH VHFRQG LV PRGXV SRQHQV RU GHWDFKPHQW %RWK DUH VDWLVILHG E\ WKH PDWHULDO FRQGLWLRQDO DV ZHOO DV WKH VWULFW FRQGLWLRQDO DV ZH KDYH GHILQHG LWf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f DQG FRQWUDSRVLWLRQ 7KH ILUVW WZR DUH LPPHGLDWH FRUROn ODULHV WR D VWURQJHU SULQFLSOH LGHQWLILHG E\ -D\ =HPDQ WKDW RI VHPL VXEVWLXWLYLW\ RI LPSOLFDWLRQ UHVSHFWLYHO\ VWULFW LPSOLFDWLRQf ZKLFK LV YDOLG IRU WKH PDWHULDO UHVSHFWLYHO\ VWULFWf FRQGLWLRQDO > SS

PAGE 77

@ WKH VWURQJHVW YHUVLRQ RI 66,666f KROGV RQO\ IRU WKH PDWHULDO FRQGLWLRQDO DQG IRU VWULFW FRQGLWLRQDOV DW OHDVW DV VWURQJ DV WKDW RI 6 :H KDYH DOUHDG\ FRQVLGHUHG WKH LQIHUHQFH RI VWUHQJWKHQLQJ WKH DQWHFHGHQW WKH FRXQWHUH[DPSOH DQG FRUUHVSRQGLQJ LQIHUHQFH SDWWHUQ DUH UHSHDWHG EHORZ ( :ST ,I WKLV PDWFK ZHUH VWUXFN LW ZRXOG OLJKW f f :.SUT ,I WKLV PDWFK ZHUH VRDNHG LQ ZDWHU DQG VWUXFN LW ZRXOG OLJKW 7KH FRQMRLQLQJ RI U ZLWK S UHPRYHV XV WR PRUH UHPRWH ZRUOGV ZKHUH WKH FRQVHTXHQW LV QR ORQJHU WUXH DV )LJXUH Df LOOXVWUDWHV )LJXUH 7KH VDPH ILJXUH DOVR VHUYHV DV D FRXQWHUPRGHO WR WUDQVLWLYLW\ D FRXQWHUH[DPSOH WR ZKLFK LV JLYHQ RQ WKH IROORZLQJ SDJH

PAGE 78

( :US :ST :UT ,I (GJDU +RRYHU KDG EHHQ ERP 5XVVLDQ KH ZRXOG KDYH EHHQ D &RQPPLVW ,I KH KDG EHHQ D &RPQXQLVW KH ZRXOG KDYH EHHQ D WUDLWRU ,I KH KDG EHHQ ERP 5XVVLDQ KH ZRXOG KDYH EHHQ D WUDLWRU 7KLV SDUWLFXODU H[DPSOH LV GXH WR 6WDOQDNHU > S @f ,I RQH LV LQFOLQHG WR WU\ WR UHWDLQ WUDQVLWLYLW\ LQ VSLWH RI VXFK FRXQWHUH[DPSOHV EHFDXVH LW LV IHOW LW LV HVVHQWLDO WR DQ\ FRQGLWLRQDO IXQFWLRQ WKH IROORZLQJ REVHUYDWLRQ VKRXOG EH SHUVXDVLYH RI DEDQGRQLQJ WKH DWWHPSW 5HFDOO WKDW /&ST HQWDLOV :ST /&.SUS LV YDOLG KHQFH :.SUS LV YDOLG 7R DEDQGRQ WKLV ZRXOG EH HLWKHU WR DEDQGRQ WKH YDOLGLW\ RI /&.SUS RU WKH HQWDLOPHQW VR :.SUS LV YDOLG ,I WUDQVLWLYLW\ LV DFn FHSWHG WKHQ IURP :ST DQG WKH YDOLG :.SUS :.SUT IROORZV WKXV DJDLQ YDOLGDWLQJ VWUHQJWKHQLQJ WKH DQWHFHGHQW ,W ZLOO QRW EH VXIILFLHQW WR VWUHQJWKHQ WKH ILUVW SUHPLVH RI WUDQVLWLYLW\ WR D VWULFW FRQGLWLRQDO RU HQWDLOPHQWf DV WKH IROORZLQJ FRXQWHUH[DPSOH VKRZV GXH WR /HZLV > S @f ( /&ST :TU 1HFHVVDULO\ LI VWDUWHG DW DP VWDUWHG EHIRUH DP ,I KDG VWDUWHG EHIRUH DP ZRXOG KDYH DUULYHG EHIRUH QRRQ ,I KDG VWDUWHG DW DP ZRXOG KDYH DUULYHG EHIRUH QRRQ )LJXUH Ef LV D FRXQWHUPRGHO WR WKLV LQIHUHQFH SDWWHUQ 7KH LQIHUn HQFH IDLOV LQ WKH IROORZLQJ VLWXDWLRQ VXSSRVH WKDW DFWXDOO\ VWDUWHG MXVW D IHZ PLQXWHV DIWHU DP DQG DFWXDOO\ DUULYHG MXVW DIWHU QRRQ VR

PAGE 79

WKH VHFRQG SUHPLVH LV WUXH +RZHYHU LI KDG VWDUWHG DW DP ZRXOG KDYH EHHQ YHU\ WLUHG DQG VR ZRXOG KDYH IRUJRWWHQ WR WDNH WKH VKRUWFXW WKDW DFWXDOO\ GLG WDNH WKXV OHQJWKHQLQJ P\ WULS E\ RYHU DQ KRXU 7KHQ WKH FRQFOXVLRQ IDLOV 7KHUH DUH VHYHUDO SDWWHUQV UHODWHG WR WUDQVLWLYLW\ WKDW DUH YDOLG IRU WKH FRXQWHUIDFWXDO FRQGLWLRQDO Df KDOI RI VXEVWLWXWLRQ XQGHU VWULFW HTXLYDOHQFH 66(f Ef VXEVWLWXWLRQ XQGHU FRXQWHUIDFWXDO HTXLYDOHQFH 6&(f Ff WKH RWKHU KDOI RI 66( WKH FRQVHTXHQFH SULQFLn SOH &3f DQG Gf D VWUHQJWKHQHG YHUVLRQ RI WUDQVLWLYLW\ 55,f 7KH SDWWHUQV DQG GLDJUDPV FRUUHVSRQGLQJ ZKLFK VXJJHVW KRZ RQH PLJKW DUJXH IRU WKHLU YDOLGLW\ RQ /HZLVn VHPDQWLFV DUH UHSUHVHQWHG LQ ( DQG LQ )LJXUH ( /(ST RU :TU /(ST :SU :SU Df 66( :TU )ST )ST RU :TU :SU :SU :TU Ef 6&( :ST /&TU :SU Ff &3 :ST :.STU f :SU Gf 557

PAGE 80

)LJXUH ( DQG (Ff SUHVHQW DQ LQWHUHVWLQJ FRQWUDVW (DFK FDQ EH FRQVLGHUHG KDOI RI D SULQFLSOH RI VHPLVXEVWLWXWLYLW\ RI WKH FRXQWHU IDFWXDO FRQGLWLRQDO ZLWK UHVSHFW WR WKH VWULFW FRQGLWLRQDO 7KH LQn YDOLGLW\ RI WKH ILUVW LV HVVHQWLDO LI ZH DUH WR DYRLG WKH IDOODF\ RI VWUHQJWKHQLQJ WKH DQWHFHGHQW VLQFH IURP /&.SUS DQG :ST :.SUT IROORZV

PAGE 81

LI ZH DFFHSW WKH SDWWHUQ RI ( WKXV RQFH DJDLQ YDOLGDWLQJ VWUHQJWKn HQLQJ WKH DQWHFHGHQW 7KH IDFW WKDW WKH IROORZLQJ LQIHUHQFH LV YDOLG PD\ SURYLGH SDXVH ( 3 /4MT :TU U +RZHYHU FRQVLGHUDWLRQ RI WKH IDFW WKDW DW WKH ZRUOG ZKHUH S LV WUXH :TU PD\ QRW EH WUXH DV ZRXOG EH WKH FDVH LQ WKH H[DPSOH FRQVLGHUHG VKRZV ZH KDYH QRWKLQJ WR IHDU RQ WKDW DFFRXQW 5HMHFWLRQ RI (Ff RQ WKH RWKHU KDQG ZRXOG EH H[WUHPHO\ LPn SODXVLEOH IRU WKHQ ZH ZRXOG EH LQ WKH SRVLWLRQ RI KROGLQJ WKDW T ZRXOG EH WUXH LI S ZHUH EXW WKDW VRPHWKLQJ HQWDLOHG E\ T ZRXOG QRW EH WUXH $Q DUJXPHQW IRU WKH YDOLGLW\ RI 6&( (Eff DQG FRQVHTXHQWO\ IRU 5(7 (Gff ZKLFK IROORZV IURP LW PD\ EH IRXQG LQ /HZLV > SS @ :H PD\ QRWH WKDW 66( (Dff DOVR IROORZV IURP 6&( (Eff VLQFH /(ST HQWDLOV )ST LH .:ST:TSf 7KH FRQVHTXHQFH SULQFLSOH &3 (Fff LV RI VSHFLDO QRWH VLQFH D UHODWHG SULQFLSOH ZKLFK VHHPV WR KDYH WKH VDPH SODXVLELOLW\ DV WKH FRQVHTXHQFH SULQFLSOH IDLOV RQ /HZLVn VHPDQWLFV 7KLV LV LQWLn PDWHO\ WLHG XS ZLWK WKH OLPLW DVVXPSWLRQ VR ZH VKDOO SRVWSRQH FRQVLGn HUDWLRQ RI LW XQWLO 6HFWLRQ 7KH WKLUG LQIHUHQFH SDWWHUQ YDOLG IRU ERWK WKH PDWHULDO DQG VWULFW FRQGLWLRQDO LV WKDW RI FRQWUDSRVLWLRQ ,W RXJKW QRW EH YDOLG IRU WKH FRXQWHU IDFWXDO FRQGLWLRQDO DV WKH IROORZLQJ H[DPSOH VKRZV

PAGE 82

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

PAGE 83

IROORZLQJ SDLU RI H[DPSOHV ZDV VXJJHVWHG E\ UHPDUNV RI 'RQDOG 1XWH >@ WKH RUGHU RI WKH SUHPLVHV KDV EHHQ UHYHUVHG IRU D UHDVRQ WR EH H[SODLQHGf ( Df :TU :ST :SU Ef :TU :VT ,I 3URI ; ZHUH WR ZRUN OHVV VKH ZRXOG EH OHVV WHQVH ,I 3URI ; ZHUH WR GHOHJDWH KHU DXWKRULW\ VKH ZRXOG ZRUN OHVV ,I 3URI ; ZHUH WR GHOHJDWH KHU DXWKRULW\ VKH ZRXOG EH OHVV WHQVH ,I 3URI ; ZHUH WR ZRUN OHVV VKH ZRXOG EH OHVV WHQVH ,I 3URI ; ZHUH WR EH FDQQHG VKH ZRXOG ZRUN OHVV :VU ,I 3URI ; ZHUH WR EH FDQQHG VKH ZRXOG EH OHVV WHQVH ,I \RXU LQWXLWLRQ LV WR DFFHSW Df DQG UHMHFW Ef WKHQ ZRXOG DJUHH :KDW VHQVH RU V\VWHP FDQ ZH PDNH RI WKLV RQ /HZLVn DQDO\VLV" )LJXUH GLDJUDPV D VLWXDWLRQ ZKHUH WKH SUHPLVHV RI WKH WZR DUJXPHQWV DERYH DUH WUXH Df VXFFHHGV DQG Ef IDLOV LH :SU LV WUXH DQG :VU LV IDOVHf ,W LV LQVWUXFWLYH WR FRPSDUH WKDW ILJXUH ZLWK Ef DQG Gf )ST UHTXLUHV WKDW WKHUH EH VRPH SSHUPLWWLQJ VSKHUH ZKHUH SZRUOGV H[DFWO\ FRLQFLGH ZLWK TZRUOGV DV GLDJUDPPHG LQ Ef %XW LV ZHDNHU VLQFH WKHUH WKH SZRUOGV QHHG PHUHO\ EH D VXEVHW RI WKH TZRUOGV LQ VRPH SSHUPLWWLQJ VSKHUH 2Q WKH RWKHU KDQG LQ Gf ZKLOH WKH VHW RI ZRUOGV ZKHUH U LV WUXH DQG WKH VHW RI ZRUOGV ZKHUH T LV WUXH PXVW LQWHUVHFW LQ VXFK D ZD\ DV WR FRQWDLQ DOO WKH SZRUOGV LQ VRPH SSHUPLWWLQJ VSKHUH LW LV QRW QHFHVVDULO\ WKH FDVH WKDW WKLV LQWHUVHFWLRQ H[KDXVWV WKH TZRUOGV LQ WKDW VSKHUH DV LW PXVW WR PDNH :TU WUXH 7KRVH FDVHV ZKHUH D

PAGE 84

K\SRWKHWLFDO V\OORJLVP ZRUNV EXW IRU ZKLFK 6&( LV WRR VWURQJ PD\ EH OLNH WKDW GLDJUDPHG LQ )LJXUH ZKLOH WKRVH ZKHUH 6&( IDLOV PXVW EH ([FHSW WKDW WKHUH FRXOG EH VRPH .UVZRUOGV LQ WKH VSHUPLWWLQJ VSKHUH EXW WKHQ DOVR VRPH .U1VZRUOGV HOVH :VU LV WUXH FRQWUDU\ WR WKH DVVXPSWLRQ WKDW WKH K\SRWKHWLFDO V\OORJLVP IDLOHGf )LJXUH DOVR UHSUHVHQWV D FDVH LQ ZKLFK K\SRWKHWLFDO V\OORJLVP ZRUNV EXW VKDOO DUJXH WKDW WKLV LV OHVV XVXDO )LJXUH

PAGE 85

(Df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n WLYH WR WKH VHOHFWLRQ RI D VLQJOH VSKHUH IRU HYDOXDWLQJ ERWK FRQGLWLRQDOV (Ef IDLOV EHFDXVH WKH TSHUPLWWLQJ VSKHUH ZKHUH DOO TZRUOGV DUH UZRUOGV LV QRW WKH VDPH DV WKH SSHUPLWWLQJ VSKHUH ZKHUH DOO VZRUOGV DUH SZRUOGV DQG LQ WKH ODWWHU VSKHUH U LV QRW VWLOO WUXH DW WKRVH V ZRUOGV ,I ZH LPDJLQH WKHVH FRQGLWLRQDOV DV XWWHUHG LQ WKH RUGHU JLYHQ WKHQ DJUHHPHQW RQ WKH WUXWK RI :TU IROORZHG E\ WKH XWWHUDQFH RI :VU LQn YLWHV WKH UHVSRQVH %XW ZDV QRW WKLQNLQJ RI ZRUNLQJ OHVV WKDW ZD\ WKXV UHVHUYLQJ WKH ULJKW QRW WR DFFHSW WKH LQIHUHQFH 7KH ILUVW FRQGLn WLRQDO HVWDEOLVKHG WKH ERXQGDULHV RI WKH VWULFWQHVV UHTXLUHG WR YDOLGDWH LW 7KH VHFRQG FRQGLWLRQDO WDFLWO\ YLRODWHV WKRVH ERXQGDULHV ,W LV WKHVH FRQVLGHUDWLRQV WKDW OHDG 1XWH >@ WR UHJDUG K\SRn WKHWLFDO V\OORJLVP DV SUDJPDWLFDOO\ YDOLG EXW QRW VHPDQWLFDOO\ YDOLG GHSHQGLQJ DV LW GRHV XSRQ WKH FRQWH[W RI XWWHUDQFH +RZHYHU LW LV WR EH QRWHG WKDW WKH GHSHQGHQFH LV V\VWHPDWLF UDWKHU WKDQ PHUHO\ DPELJXRXV 7KH VLWXDWLRQ GLDJUDPPHG LQ )LJXUH LV WKLQN H[HPSOLILHG E\ WKH IROORZLQJ

PAGE 86

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n IHUHQFH LV SUDJPDWLFDOO\ YDOLG LI LW LV LPSRVVLEOH IRU WKH SUHPLVHV WR EH WUXH ZLWK UHVSHFW WR WKH VDPH VSKHUH DQG WKH FRQFOXVLRQ IDOVH (Df XQOLNH HLWKHU (Ef RU ( LV SUDJPDWLFDOO\ YDOLG $GRSWLQJ VXFK D YLHZ RI K\SRWKHWLFDO V\OORJLVP DQG KHQFH RI WUDQVLWLYLW\ RI FRXQWHUIDFWXDO LPSOLFDWLRQ DOORZV XV WR PDNH VHQVH ERWK RI WKRVH LQVWDQFHV ZKHUH LW VHHPV WR KROG DQG WKRVH ZKHUH LW FHUWDLQO\ IDLOV $ VLPLODU VLWXDWLRQ DULVHV LQ WKH FDVH RI WKH LQIHUHQFH SDWWHUQ VLPSOLILFDWLRQ RI GLVMXQFWLYH DQWHFHGHQWV 6'$f $ FRQVLGHUDEOH OLWHUDn WXUH KDV DULVHQ LQ UHFHQW SDSHUV RQ WKH WRSLF RI WKH FRXQWHUIDFWXDO FRQn GLWLRQDO ZLWK UHVSHFW WR WKLV LQIHUHQFH DORQH > @ 7KH LQIHUHQFH SDWWHUQ DQ H[DPSOH DQG WKH DSSURSULDWH GLDJUDP IROORZ DGDSWHG IURP 1XWH >@f

PAGE 87

( :$STU ,I WKH VXQ ZHUH WR JURZ FROG RU ZH ZHUH WR KDYH D PLOG ZLQWHU ZH ZRXOG KDYH D EXPSHU FURS .:SU:TU ,I WKH VXQ ZHUH WR JURZ FROG ZH ZRXOG KDYH D EXPSHU FURS DQG LI ZH ZHUH WR KDYH D PLOG ZLQWHU ZH ZRXOG KDYH D EXPSHU FURS )LJXUH :KLOH LW LV FOHDU WKDW )LJXUH LV D FRXQWHUPRGHO WR 6'$ LW LV QRW FOHDU WKDW ( FRQWDLQV D FRXQWHUH[DPSOH 2QH FRXOG DUJXH WKDW /HZLVn VHPDQWLFV LV LQDGHTXDWH MXVW EHFDXVH LW SHUPLWV RXU H[DPSOH RI :$STU WR EH WUXH 5DWKHU RQH PXVW WDNH ERWK SZRUOGV DQG TZRUOGV LQWR DFFRXQW LQ HYDOXDWLQJ FRQGLWLRQDOV ZLWK GLVMXQFWLYH DQWHFHGHQWV )LJXUH LOOXVWUDWHV WKDW LI ZH PXVW ILQG ERWK D SSHUPLWWLQJ DQG D T SHUPLWWLQJ VSKHUH WKHQ :$STU LV QRW WUXH DW X 2QH ZRXOG DUJXH WKLV ZD\ LI RQH ZDQWHG WR UHWDLQ 6'$ DV D YDOLG LQIHUHQFH SDWWHUQ IRU FRXQWHU IDFWXDOV

PAGE 88

“XWH > @ DUJXHV IRU WKH UHWHQWLRQ RI 6'$ EDVHG XSRQ LWV LQLWLDO LQWXLWLYH SODXVLELOLW\ +RZHYHU WKLV KDV WKH FRQVHTXHQFH WKDW 66( DQG WKH VWURQJHU 6&( PXVW WKHQ EH UHMHFWHG VLQFH WRJHWKHU WKH\ LPSO\ WKDW WKH FRXQWHU IDFWXDO LV D VWULFW FRQGLWLRQDO 7R SURYH WKLV ZH QHHG WKH IROORZLQJ REYLRXVO\ YDOLG LQIHUHQFH SDWWHUQV DQG VHQWHQFHV ( Df :ST :S$TU :HDNHQLQJ WKH FRQVHTXHQW Ef /($.ST.S1TS Ff /(.S1T$1ST 'H0EUJDQnV /DZV Gf /(&ST$1ST Hf :1SS /S :H DOUHDG\ KDYH WKDW /&ST LPSOLHV :ST 7KH IROORZLQJ VXIILFHV WR SURYH WKH FRQYHUVH WKXV SURYLQJ WKH HTXLYDOHQFH GHVLUHG :ST :$.ST.S1TT :.S1TT :.S1T$1ST :1$1ST$1ST :1&ST&ST /&ST $VVXPHG 66( t Ef 6'$ Df 66( t Ff 66( 6F Gf Hf 1XWH UHMHFWV 6&( ZKLFK LPSOLHV 66( LQ RUGHU WR UHWDLQ 6'$ ,W KDV EHHQ DUJXHG WKLQN VXFFHVV IXOO\ E\ /RHZHU >@ DQG RWKHUV > @ WKDW WKLV LV WRR KLJK D SULFH WR SD\ IRU 6'$ %XW SHUKDSV ZH FDQ VDYH ERWK RXU UHOXFWDQFH WR DEDQGRQ 66( DQG RXU LQWXLWLRQV DERXW 6'$ ,W LV P\ XQGHUVWDQGLQJ WKDW 1XWH KDV VLQFH FRPH WR WKLV SRVLWLRQ WKURXJK DSn SO\LQJ WKH FDWHJRU\ RI SUDJPDWLF YDOLGLW\ WR 6'$ ZKLOH UHFRJQL]LQJ WKDW LW LV VHPDQWLFDOO\ LQYDOLG

PAGE 89

$V /RHZHU SRLQW RXW > S @ WKH FRQGLWLRQV XQGHU ZKLFK D FRXQWHUIDFWXDO ZLWK D GLVMXQFWLYH DQWHFHGHQW LV XWWHUHG DUH XVXDOO\ VXFK WKDW ZH ZRXOG EH SUHSDUHG WR GHIHQG HLWKHU 6'$ FRQMXQFW RXU LQWHQWLRQ LV WR PDNH D PRUH LQFOXVLYH VWDWHPHQW WKDQ HLWKHU FRXQWHU IDFWXDO ZLWK VLQJOH DQWHFHGHQW DORQH 7KH IROORZLQJ H[DPSOHV DQG DFFRPSDQ\LQJ GLDJUDPV ZLOO LOOXVWUDWH P\ SRLQW ( Df :$STU ,I 3URI ; ZHUH WR ZRUN OHVV RU ZHUH XQGHU OHVV SUHVVXUH WR SXEOLVK VKH ZRXOG EH OHVV WHQVH .:SU:ST ,I 3URI ; ZHUH WR ZRUN OHVV VKH ZRXOG EH OHVV WHQVH DQG LI VKH ZHUH XQGHU OHVV SUHVVXUH WR SXEOLVK VKH ZRXOG EH OHVV WHQVH Ef :$SVU ,I 3URI ; ZHUH WR ZRUN OHVV RU WR GLH VKH ZRXOG EH OHVV WHQVH .:SU:VU ,I 3URI ; ZHUH WR ZRUN OHVV VKH ZRXOG EH OHVV WHQVH DQG LI VKH ZHUH WR GLH VKH ZRXOG EH OHVV WHQVH )LJXUH

PAGE 90

,I WKH SUHPLVH RI (Df LV XWWHUHG LQ FRQYHUVDWLRQ ZH HYDOXn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f LV SUDJn PDWLFDOO\ YDOLG DQG QRQH DUH VHPDQWLFDOO\ YDOLG QR LQVWDQFH RI 6'$ LV 2XU FKDUDFWHUL]DWLRQ RI SUDJPDWLF YDOLGLW\ DV UHTXLULQJ HYDOXDWLRQ RI SUHPLVHV DQG FRQFOXVLRQ ZLWK UHVSHFW WR WKH VDPH VSKHUH LV XQIRUWXn QDWHO\ WRR ZHDN 3DUW RI RXU LQWXLWLYH QRWLRQ LV WKDW WKH RUGHU RI XWn WHUDQFH RI WKH SUHPLVHV VKRXOG QRW UHTXLUH XV WR FKDQJH VSKHUHV LI ZH KDYH DOUHDG\ IRFXVHG RQ RQH WKDW VXIILFHV WR PDNH WKH ILUVW SUHPLVH WUXH +RZHYHU WKH LQIHUHQFHV GLDJUDPPHG LQ )LJXUH VDWLVI\ RXU GHILQLWLRQ RI SUDJPDWLF YDOLGLW\ EXW UXQ FRXQWHU WR WKLV LQWXLWLRQ )LJXUH

PAGE 91

2QH VKRXOG QRWH WKDW WKHVH ILJXUHV DUH RQO\ VOLJKW YDULDWLRQV RI )LJXUHV DQG Ef ZKHUH ZH DJUHHG WKH LQIHUHQFH VKRXOG QRW EH FRQVLGHUHG SUDJPDWLFDOO\ YDOLG %XW ERWK RI WKH DERYH PHHW WKH FRQn GLWLRQ WKDW DOO VWDWHPHQWV EH HYDOXDWHG ZLWK UHVSHFW WR WKH VDPH VSKHUH WKH WKLUG QRQWULYLDO RQHf )RU WKH 6'$ H[DPSOH ERWK GLVMXQFWV RFFXU LQ WKH VDPH ODUJHU VSKHUH +RZHYHU LQ ERWK FDVHV WKHUH LV D VPDOOHU VSKHUH WKDW ZRXOG PDNH VRPH SUHPLVH WUXH ,W ZRXOG WKHUHIRUH VHHP WKDW ZH VKRXOG GHILQH SUDJPDWLF YDOLGLW\ DV IROORZV $Q LQIHUHQFH LV SUDJPDWLFDOO\ YDOLG LII LW LV LPSRVVLEOH IRU WKH SUHPLVHV WR EH WUXH DQG WKH FRQFOXVLRQ IDOVH XQGHU WKH FRQGLWLRQV WKDW f DOO SUHPLVHV DUH HYDOXDWHG DV WUXH ZLWK UHVSHFW WR WKH VDPH VSKHUH DQG f QR SUHPLVH LV WUXH ZLWK UHVSHFW WR DQ\ VPDOOHU VSKHUH 7KLV KDV WKH HIIHFW RI UHTXLULQJ WKDW DOO YDULDEO\ VWULFW FRQGLWLRQDOV LQ WKH LQIHUHQFH EH RI WKH VDPH GHJUHH RI VWULFWQHVV DQG VTXDUHV ZLWK RXU LQWXLWLRQ WKDW ZH QHHG FRQVLGHU QR ZRUOGV PRUH GLVVLPLODU WR WKH DFWXDO ZRUOG WKDQ WR VRPH IL[HG GHJUHH 7KLV ZRXOG HOLPLQDWH WKH VLWXn DWLRQV RI )LJXUH EXW LW KDV WZR FRQVHTXHQFHV ZKLFK PD\ EH XQZHOFRPH )LUVW LW LPSRVHV RQ WKH V\VWHP RI VSKHUHV WKH OLPLW DVVXPSWLRQ & ,I ; F X! WKHQ $; H ; 7KDW LV WKHUH LV VRPH VPDOOHVW VSKHUH LQ WKH VHW RI VSKHUHV PDNLQJ WKH SUHPLVHV WUXH /HZLV IRU UHDVRQV ZH VKDOO GLVFXVV LQ 6HFWLRQ ZDQWV WR UHMHFW WKH OLPLW DVVXPSWLRQ 6HFRQG LW PDNHV WKH IROORZLQJ LQIHUHQFH DSSHDU SUDJPDWLFDOO\ LQn YDOLG

PAGE 92

:ST /&TU :SU )LJXUH %XW WKLV LQIHUHQFH LV VHPDQWLFDOO\ YDOLG DV DQ LQVWDQFH RI WKH FRQVHn TXHQFH SULQFLSOH DQG H[DPSOHV RI LW DOO VHHP TXLWH REMHFWLRQOHVV :KDW RXU GHILQLWLRQ RYHUORRNV LV WKDW RXU FRQFHUQ LQ WKH H[DPSOHV FLWHG ZDV ZLWK HVVHQWLDOO\ FRXQWHUIDFWXDO FRQGLWLRQDOV WKRVH IRU ZKLFK WKH FRUUHVSRQGLQJ VWULFW FRQGLWLRQDO QHHG QRW EH WUXH $OVR RXU FRQn FHUQ LQ IL[LQJ D VSKHUH IRU HYDOXDWLRQ ZDV WR HQVXUH WKDW QR VPDOOHU VSKHUH PDGH RQH SUHPLVH WUXH ZLWKRXW PDNLQJ WKH RWKHUV WUXH DV ZHOO +HQFH WKH IROORZLQJ DPPHQGHG GHILQLWLRQ VKRXOG PHHW ERWK REMHFWLRQV $Q LQIHUHQFH LV SUDJPDWLFDOO\ YDOLG LII LW LV LPSRVVLEOH IRU WKH SUHPLVHV WR EH WUXH DQG WKH FRQFOXVLRQ WR EH IDOVH XQGHU WKH FRQn GLWLRQV WKDW f DOO SUHPLVHV DUH HYDOXDWHG DV WUXH ZLWK UHVSHFW WR WKH VDPH VSKHUH DQG f QR QRQQHFHVVDU\ SUHPLVH LV WUXH ZLWK UHVSHFW WR DQ\ VPDOOHU VSKHUH ZKLFK GRHV QRW PDNH HYHU\ SUHPLVH WUXH DQG f QR QRQQHFHVVDU\ VWDWHPHQW LV YDFXRXVO\ WUXH

PAGE 93

7KHUH LV ZLGHVSUHDG DJUHHPHQW RQ WKH VHPDQWLF LQYDOLGLW\ RI WUDQVLWLYLW\ IRU WKH FRXQWHUIDFWXDO FRQGLWLRQDO DQG QHDUO\ HTXDOO\ ZLGHVSUHDG DJUHHPHQW RQ WKH LQYDOLGLW\ RI 6'$ KRZHYHU WKHUH DUH VHYHUDO RWKHU LQIHUHQFH SDWWHUQV XSRQ ZKLFK DJUHHPHQW LV QRW DV HDV\ WR ILQG )RUWXQDWHO\ DGRSWLQJ GLIIHUHQW SRVLWLRQV RQ WKHVH GRHV QRW JUHDWO\ DOWHU WKH QDWXUH RI WKH FRQGLWLRQDO LQ TXHVWLRQ 2I WKH IROn ORZLQJ LQIHUHQFHV WKH ILUVW DQG ODVW DUH LQYDOLG RQ /HZLVn SUHIHUUHG VHPDQWLFV DV ZHOO DV RQ WKDW YHUVLRQ ZKLFK ZH KDYH SUHVHQWHG ( :ST .ST .ST /1S /1S 9ST 9ST 9ST 9ST 9ST Df Ef Ff Gf Hf 7KH LQYDOLGLW\ RI Df DQG Hf DQG WKH YDOLGLW\ RI Gf UHVW XSRQ WKH IDFW WKDW WKH FRQGLWLRQDO :ST FDQ EH YDFXRXVO\ WUXH ZKLOH 9ST LV QHYHU YDFXRXVO\ WUXH JLYHQ WKH SUHIHUUHG GHILQLWLRQV ,I D JLYHQ SURSRVLWLRQ S LV QRW HQWHUWDLQDEOH WKHQ +LS LV WUXH DV LV :ST EXW 9ST LV WKHQ IDOVH VLQFH WKHUH LV QR SSHUPLWWLQJ VSKHUH DW DOO 7KH YDOLGLW\ RI Ef DQG Ff IROORZV IURP WKH GHILQLWLRQV GLUHFWO\ DV ^X` LV WKHQ D SSHUPLWWLQJ VSKHUH LQ ZKLFK HYHU\ SZRUOG LV DOVR D TZRUOG 'HSHQGLQJ XSRQ RQHn V WDVWHV WKH YDOLGLW\ RU LQYDOLGLW\ RI WKHVH LQIHUHQFHV FDQ EH DOWHUHG E\ PLQRU DOWHUDWLRQV LQ WKH FRQGLWLRQV XSRQ WKH IXQFWLRQ RU WKH WUXWK FRQGLWLRQV IRU WKH ZRXOG DQG PLJKW FRQGLWLRQDOV %\ VZLWFKLQJ WR WKH DOWHUQDWH GHILQLWLRQV RI :ST DQG 9ST DQG 'f ZH SUHVHUYH WKHLU LQWHUGHILQDELOLW\ EXW PDNH Df DQG Hf YDOLG DQG Gf LQYDOLG ,I ZH DUH DJUHHDEOH WR DEDQGRQLQJ WKH LQWHUGHILQDELOLW\ RI :ST DQG 9SZ WKHQ WKH FRPELQDWLRQ RI DQG

PAGE 94

' ZLOO UHQGHU DOO YDFXRXV FRXQWHUIDFWXDOV IDOVH WKXV LQYDOLGDWLQJ ERWK Gf DQG Hf EXW UHQGHULQJ Df YDOLG :H FDQ LQYDOLGDWH ERWK Ef DQG Ff E\ DEDQGRQLQJ WKH FHQWHULQJ FRQGLWLRQ RQ &f VLQFH WKHQ WKH VPDOOHVW VSKHUH DERXW X PD\ QRW HYHQ FRQWDLQ X VR LW ZRXOG EH SRVVLEOH IRU .ST WR EH WUXH DW X EXW HLWKHU :ST RU ERWK :ST DQG 9ST WR EH IDOVH ,I Ff VHHPV GHVLUDEOH EXW Ef QRW WKHQ ZH FDQ HPSOR\ /HZLVn FRQGLWLRQ RI ZHDN FHQWHULQJ LQ SODFH RI FHQWHULQJ & )RU DOO A$HXH$ X ,Q VXFK D FDVH ZH KDYH D VPDOOHVW QRQHPSW\ VSKHUH DERXW X $X RI ZRUOGV LQGLVWLQJXLVKDEOH IURP X LQ WHUPV RI RXU VLPLODULW\ RUGHULQJ $V /HZLV VXJJHVWV ZH PD\ ZDQW WR YDU\ WKH FRQGLWLRQV DQ\ZD\ IRU GLIIHUHQW DSSOLFDWLRQV RI WKH DQDO\VLV RI FRQGLWLRQDOV +RZHYHU KHUHLQ ZH ZLOO FRQWLQXH ZLWK WKH DQDO\VLV SUHVHQWHG ZLWKRXW UHPDUNLQJ RQ WKH RWKHUZLVH GHVLUDEOH IOH[LELOLW\ RI /HZLVn IXOO DQDO\VLV DP VDWLVILHG ZLWK WKH SUHVHQW DVVLJQPHQW RI YDOLGLW\ DQG LQYDOLGLW\ WR DOO RI WKH LQn IHUHQFHV LQ ( H[FHSW Df DQG Gf +RZHYHU P\ GLVVDWLVIDFWLRQ LV DOO EXW HYDSRUDWHG E\ WKH UHDOL]DWLRQ WKDW ZKLOH Df LV VHPDQWLFDOO\ LQn YDOLG LW LV DOZD\V SUDJPDWLFDOO\ YDOLG RQ RXU GHILQLWLRQ 'f $QG Gf WKRXJK VHPDQWLFDOO\ YDOLG LV QHYHU SUDJPDWLFDOO\ YDOLG &ODXVH f RI RXU GHILQLWLRQ UXOHV RXW YDFXRXV SUHPLVHV RU FRQFOXVLRQV VR HYHU\ WLPH :ST LV QRQYDFXRXVO\ WUXH 9ST PXVW EH DQG WKRXJK ZKHQ /1S LV WUXH VR PXVW :ST EH WKH ODWWHUnV WUXWK LV YDFXRXV 2QH FRXOG DUJXH WKDW (Gf VKRXOG EH UHWDLQHG EHFDXVH RI WKH IROORZLQJ LQWXLWLYHO\ YDOLG DUJXPHQW 6XSSRVH LW LV QRW WKH FDVH WKDW LI S ZHUH WUXH WKHQ T ZRXOG EH WUXH 7KHQ LW VHHPV WR IROORZ WKDW WKHUH DUH FLUFXPVWDQFHV XQGHU ZKLFK LI S KHOG WKHQ 1S PLJKW KROG 2WKHUZLVH

PAGE 95

ZH FRXOG KDUGO\ GHIHQG WKH VXSSRVLWLRQ ,Q IDFW RXU EHVW GHIHQVH LV WKDW ERWK S DQG 1T DUH SRVVLEOH VLPXOWDQHRXVO\ 6R RI FRXUVH S LV SRVn VLEOH DQG KHQFH QRW LPSRVVLEOH 7KH IROORZLQJ FKDLQ RI LQIHUHQFHV VXPn PDUL]HV WKH DERYH DUJXPHQW ( 1:ST 9S1T 0.ST 1/1S ( LV VHPDQWLFDOO\ YDOLG RQ WKH DQDO\VLV ZH KDYH JLYHQ DV EHOLHYH LW VKRXOG EH EXW LW LV DOVR SUDJPDWLFDOO\ YDOLG LQWXLWLYHO\ DQG LQ WHUPV RI RXU GHILQLWLRQ %XW ZH FDQQRW KROG LW VHPDQWLFDOO\ YDOLG ZLWKn RXW KROGLQJ (Gf YDOLG DV ZHOO VLQFH LW LV WKH FRQWUDSRVLWLYH RI WKH DERYH DUJXPHQW 7KHUH DUH WZR DSSDUHQW RGGLWLHV DERXW SUDJPDWLF YDOLGLW\ DV ZH KDYH GHILQHG LW ILUVW VHPDQWLF YDOLGLW\ GRHV QRW JXDUDQWHH SUDJPDWLF YDOLGn LW\ DV ZLWQHVV (Gf VHFRQG WKH FRQWUDSRVLWLYH DUJXPHQW WR D SUDJPDWLFDOO\ YDOLG RQH PD\ QRW LWVHOI EH SUDJPDWLFDOO\ YDOLG VDPH H[n DPSOHf 7KHVH VLWXDWLRQV GHSHQG XSRQ WKH SUHVHQFH RI FODXVH 'f VR PD\ EH DYRLGHG LI WKDW FODXVH LV GURSSHG DP UHOXFWDQW WR FDOO DQ\ DUJXPHQW SUDJPDWLFDOO\ YDOLG ZKHQ LW FRQWDLQV YDFXRXVO\ WUXH FRXQWHU IDFWXDOV VR DP ZLOOLQJ WR SXW XS ZLWK WKHVH RGGLWLHV DP LQIOXHQFHG SHUKDSV E\ P\ IHHOLQJ WKDW WKRXJK WKH LQIHUHQFH IURP $OO XQLFRUQV DUH IXUU\ WR ,W LV QRW WKH FDVH WKDW VRPH XQLFRUQV DUH EDOG LV VHPDQWLFDOO\ YDOLG LW LV GHFLGHGO\ RGG LQ YLHZ RI WKH QRQH[LVWHQFH RI XQLFRUQV

PAGE 96

$V IRU WKH YDOLGLW\ RI (Ef DQG Ff ZH UHPDUNHG LQ 6HFWLRQ WKDW WKH DQWHFHGHQW DQG FRQVHTXHQW EHLQJ WUXH ZDV QR EDU WR WKH WUXWK RI WKH FRQGLWLRQDO WKRXJK DVVHUWLRQ RI D FRQGLWLRQDO XVXDOO\ SUHVXSSRVHV WKH XWWHUHU GRHV QRW NQRZ WKH DQWHFHGHQW WR EH WUXH 1XWH > @ DUJXHV WKDW Ef LV FRXQWHULQWXLWLYH RQ WKH JURXQGV WKDW LQ PDQ\ VLWXDn WLRQV ZKHUH ERWK S DQG T KDSSHQ WR EH WUXH ZH ZRXOG GHQ\ WKDW LI S ZHUH WUXH WKHQ T ZRXOG EH EHFDXVH ZH GHQ\ WKDW WKH FRQQHFWLRQ EHWZHHQ S DQG T LI DQ\ JXDUDQWHHV WKH WUXWK RI T MXVW EHFDXVH S LV WUXH 7KDW LV ZH KROG .ST DQG 9S1T DV FRPSDWLEOH %XW WKH\ DUH QRW RQ /HZLVn VHPDQn WLFV DV ZH KDYH SUHVHQWHG LW EHFDXVH ^X` LV WKH VPDOOHVW QRQHPSW\ VSKHUH DERXW X 6R QRW HYHU\ SSHUPLWWLQJ VSKHUH FRQWDLQV DQ 1TZRUOG 7KLV DUJXPHQW KDV PHULW DQG VR RQH PLJKW ZDQW WR UHWUHDW WR ZHDN FHQWHULQJ WKRXJK ZLOO DUJXH WKDW SHUKDSV WKH LQWXLWLRQ 1XWH FDOOV XSRQ LV DFWXDOO\ VRPHWKLQJ HOVH 7R (Ff NQRZ RI QR REMHFWLRQ LW ZRXOG VHHP WKDW WKH IDFW WKDW S DQG T DUH ERWK WUXH LV VXIILFLHQW SULPD IDFLH HYLGHQFH IRU LI S ZHUH WUXH WKHQ T PLJKW EH :H PD\ REVHUYH WKDW LI 19ST LV DVVHUWHG WKHQ .ST LV DQ HQWLUHO\ VXIILFLHQW UHEXWWDO EHOLHYH 1XWHnV DUJXPHQW UHDOO\ UHVWV XSRQ WKH IROORZLQJ ZH KROG .ST DQG 0.S1T DV FRPSDWLEOH WKDW LV WKRXJK S DQG T DUH ERWK WUXH LW LV SRVVLEOH ZH WKLQN WKDW S FRXOG EH WUXH DQG T IDOVH %XW 0.S1T LV DV ZH REVHUYHG LQ 6HFWLRQ ZHDNHU WKDQ 9S1T ,W LV UHDOO\ WKH IRUPHU ZH KROG FRPSDWLEOH ZLWK .ST UDWKHU WKDQ WKH ODWWHU /HZLVn DQDO\VLV JHQHUDOO\ SUHVHUYHV RXU SUHDQDO\WLF QRWLRQV FRQn FHUQLQJ WKH YDOLGLW\ DQG LQYDOLGLW\ RI YDULRXV FRXQWHU IDFWXDO LQIHUHQFHV &RXSOHG ZLWK WKH FRQFHSW RI SUDJPDWLF YDOLGLW\ GXH LQ SDUW WR /HZLV KLPVHOI >@ EXW PRUH WR 1XWH >@f LW DOVR H[SODLQV ZK\ ZH DFFHSW PDQ\ FRXQWHUIDFWXDO LQIHUHQFHV WR ZKLFK WKHUH VHHP WR EH FRXQWHUH[DPSOHV :H

PAGE 97

GR VR ZKHQ WKH VWULFWQHVV RI WKH YDULDEO\ VWULFW FRQGLWLRQDOV LQYROYHG FRLQFLGH 7KDW WKH DQDO\VLV KHUH SUHVHQWHG GRHV KHOS WR H[SODLQ WKLV LV WKLQN D VWURQJ SRLQW LQ LWV IDYRU ,Q WKH QH[W WZR VHFWLRQV DQG LQ &+$37(5 7+5(( ZH VKDOO FRQVLGHU DVSHFWV RI /HZLVn DQDO\VLV WKDW DUH PRUH SUREOHPDWLF KLV UHMHFWLRQ RI WKH OLPLW DVVXPSWLRQ KLV SRVVLEOH ZRUOGV UHDOLVP DQG WKH QRWLRQ RI FRPSDUDWLYH RYHUDOO VLPLODULW\ LWVHOI 7KH /LPLW $VVXPSWLRQ ,Q 6HFWLRQ ZH REVHUYHG WKDW FORVXUH XQGHU XQLRQV DQG LQWHUn VHFWLRQV LPSRVHV D FHUWDLQ NLQG RI ERXQGLQJ FRQGLWLRQ RQ X DQG VXEVHWV WKHUHRI 7KDW LV WKHUH LV D VPDOOHVW DQG ODUJHVW VSKHUH LQ $X DQG 8X UHVSHFWLYHO\ $OVR DQ\ QRQHPSW\ VXEVHW ; RI X LV ERXQGHG ERWK DERYH DQG EHORZ E\ VSKHUHV 8; DQG $; UHVSHFWLYHO\ +RZHYHU WKH FORVXUH FRQGLWLRQV DUH QRW DV VWURQJ DV WKH\ PLJKW EH LW LV QRW WKH FDVH WKDW IRU DOO ; F HLWKHU 8; H ; RU WKDW $; H ; :H UHVWDWH KHUH ZKDW /HZLV FDOOV WKH /LPLW $VVXPSWLRQ /$ DFWXDOO\ LW LV D ORZHUOLPLW DVVXPSWLRQf & ,I ; F WKHQ $; H ; ,I ZH UHSODFHG WKH FORVXUH XQGHU QRQHPSW\f LQWHUVHFWLRQV FRQGLWLRQ &f RQ E\ WKLV ZH ZRXOG KDYH D VSKHUH IXQFWLRQ ZKLFK VDWLVILHG WKH OLPLW DVVXPSWLRQ $ VSKHUH IXQFWLRQ ZKLFK VDWLVILHG /$ ZRXOG DOORZ XV WR VSHDN RI D FORVHVW VSKHUH VDWLVI\LQJ DQ\ JLYHQ FRQGLWLRQ VLQFH LW ZRXOG EH WKH LQWHUVHFWLRQ RI DOO VSKHUHV VDWLVI\LQJ WKDW FRQGLWLRQ )RU LQVWDQFH IRU SURSRVLWLRQ S ZH ZRXOG EH DVVXUHG RI D FORVHVW SSHUPLWWLQJ VSKHUH :H FRXOG WKHQ GHILQH WUXWK IRU WKH ZRXOGFRQGLWLRQDO PRUH VLPSO\

PAGE 98

' :ST # X LII HYHU\ SZRUOG LQ WKH FORVHVW SSHUPLWWLQJ VSKHUH LQ LV D TZRUOG X Q ZKLFK ZRXOG EH QRQYDFXRXVO\ WUXH LQ FDVH WKHUH ZDV D SSHUPLWWLQJ VSKHUH DQG YDFXRXVO\ WUXH RWKHUZLVH 7KH VSKHUHV WKRXJK QRW WKH ZRUOGV WKH\ FRQWDLQf ZRXOG EH ZHOORUGHUHG ZLWK UHVSHFW WR VXEVHW FRQWDLQPHQW 7KDW LV HYHU\ VXEVHW RI X ZRXOG KDYH D OHDVW HOHPHQW %XW /HZLV TXHVWLRQV ZKHWKHU WKLV ZRXOG LQ JHQHUDO EH D VXLWDEOH UHVWULFWLRQ RQ +H DUJXHV 6XSSRVH ZH HQWHUWDLQ WKH FRXQWHUIDFWXDO VXSSRVLWLRQ WKDW DW WKLV SRLQW WKHUH DSSHDUV D OLQH PRUH WKDQ DQ LQFK ORQJ $FWXDOO\ LW LV MXVW XQGHU DQ LQFKf 7KHUH DUH ZRUOGV ZLWK D OLQH ORQJ ZRUOGV SUHVXPDEO\ FORVHU WR RXUV ZLWK D OLQH bf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n QL]HV DQ HYHQ VWURQJHU YHUVLRQ RI /$ WKH XQLTXHQHVV DVVXPSWLRQ /$ &DOOHG 6WDOQDNHUnV DVVXPSWLRQ E\ /HZLV EHFDXVH LW LV DVVRFLDWHG ZLWK 6WDOQDNHUnV VHPDQWLFV IRU FRQGLWLRQDOV :H XVH 1XWHnV > S @ PRUH GHVFULSWLYH WHUPf :H ZLOO GLVFXVV 6WDOQDNHUn V VHPDQWLFV LQ &+$37(5 7+5(( DV SDUW RI D JHQHUDO GLVFXVVLRQ RI VLPLODULW\ )RU QRZ LW VXIILFHV WR LQGLFDWH WKDW WKH XQLTXHQHVV DVVXPSWLRQ UHTXLUHV WKDW WKH FORVHVW SSHUPLWWLQJ VSKHUH IRU DQ\ SURSRVLWLRQ S FRQWDLQV EXW RQH SZRUOG 7KLV PHDQV WKH VSKHUH

PAGE 99

IXQFWLRQ XQGHU WKLV FRQGLWLRQ FDQ EH YLHZHG DV SODFLQJ D ZHOORUGHU RQ WKH ZRUOGV WKHPVHOYHV 6LQFH DW VXFK D FORVHVW SZRUOG HLWKHU T RU 1T LV WUXH IRU DQ\ SURSRVLWLRQ T 8$ YDOLGDWHV FRQGLWLRQDO H[FOXGHG PLGGOH &(0f IRU FRXQWHUIDFWXDOV DQG UHQGHUV /HZLVn GHILQLWLRQ RI WKH PLJKWFRQGLWLRQDO VXSHUIOXRXV /HZLV GRHV GLVFXVV RWKHU ZD\V WR GHn ILQH WKLV FRQGLWLRQDO RQ WKH XQLTXHQHVV DV VLPS W LRQ DOO RI ZKLFK KH GHHPV XQVDWLVIDFWRU\ > SS @f $V ERWK RI WKHVH FRQVHTXHQFHV DUH XQZHOFRPH ZH VKDOO DJUHH ZLWK /HZLV LQ UHMHFWLQJ WKH XQLTXHQHVV DVVXPSn WLRQ $GRSWLQJ /$ WKRXJK QRW 8$ GRHV QRW KDYH WKH FRQVHTXHQFH RI YDOLGDWLQJ &(0 VLQFH WKH FORVHVW SSHUPLWWLQJ VSKHUH PD\ ZHOO FRQWDLQ PRUH WKDQ RQH SZRUOG 8QGHU /HZLVn DQDO\VLV LW LV HQWLUHO\ SRVVLEOH IRU ZRUOGV WR WLH LQ FRPSDUDWLYH RYHUDOO VLPLODULW\ WR D JLYHQ ZRUOG 7KLV LV WKH UDWLRQDOH IRU GHILQLQJ WKH PLJKWFRQGLWLRQDO $FFHSWLQJ /$ GRHV KDYH WKH FRQVHTXHQFH RI JRLQJ DJDLQVW WKH LQWXLWLRQ /HZLV GUDZV RQ LQ KLV H[DPSOH KRZHYHU WKH JDPH RI LQWXLWLRQV KDV WZR VLGHV DQG 3ROORFN >@ DQG +HU]EHUJHU >@ EULQJ RXW WKH RWKHU VLGH ZLWK UHVSHFW WR /$ 6WDUWLQJ ZLWK DQ DGDSWDWLRQ RI /HZLVn H[DPSOH TXRWHG DERYH VXSn SRUWLQJ WKH UHMHFWLRQ RI /$ 3ROORFN VKRZV WKDW WKLV UHMHFWLRQ DOVR UHn TXLUHV WKH UHMHFWLRQ RI D JHQHUDOL]DWLRQ RI WKH FRQVHTXHQFH SULQFLSOH &3 (Fff WKDW VHHPV WR KDYH DV PXFK FODLP IRU YDOLGLW\ DV WKH RULJLQDO FRQVHTXHQFH SULQFLSOH ,Q D UHODWHG IDVKLRQ +HU]EHUJHU DUJXHV WKDW UHMHFWLRQ RI /$ LQWURGXFHV FRXQWHUIDFWXDO LQFRQVLVWHQFLHV RQ /HZLVn DQDO\VLV :H ZLOO FRQVLGHU HDFK DUJXPHQW LQ WXUQ ,Q WKH SDVVDJH TXRWHG DERYH /HZLV FODLPV WKDW LW LV UHDVRQDEOH WR VXSSRVH WKDW IRU HDFK ZRUOG ZKHUH WKH OLQH LV [ ORQJ WKHUH LV

PAGE 100

D FORVHU ZRUOG ZKHUH LW LV EHWZHHQ DQG [ ORQJ IRU HDFK SRVLWLYH YDOXH RI [ *UDQWHG WKLV DVVXPSWLRQ 3ROORFN WKHQ FODLPV WKDW WKH IROn ORZLQJ VHQWHQFH LV WUXH RQ /HZLVn DQDO\VLV IRU HDFK SRVLWLYH YDOXH RI [ > S @ ( ,I WKH OLQH ZHUH PRUH WKDQ DQ LQFK ORQJ LW ZRXOG QRW EH [ ORQJ )RU WKLV WR EH WUXH LW PXVW EH WKH FDVH WKDW LQ VDQH DQWHFHGHQW SHUPLWWLQJ VSKHUH HYHU\ DQWHFHGHQWZRUOG LV D FRQVHTXHQWZRUOG $QG WKLV IRU HDFK [ 7KDW LV IRU HDFK [ WKHUH LV D VSKHUH ZKHUHnWKH OLQH LV PRUH WKDQ DQ LQFK ORQJ DW VRPH ZRUOG DQG DW HYHU\ ZRUOG ZKHUH LW LV PRUH WKDQ DQ LQFK ORQJ LQ WKDW VSKHUH LW LV QRW [ ORQJ /HW XV DVn VXUH WKLV FRQGLWLRQ LV PHW WKRXJK LW LV QRW FOHDU WKDW /HZLV PHDQW LW DQG FHUWDLQO\ GRHV QRW QHHG LW WR PDNH KLV SRLQW :LWK [ JRLQJ WR ]HUR LW IROORZV WKDW WKH OLQH ZRXOG QRW EH [ ORQJ IRU DOO SRVLWLYH YDOXHV RI [ KHQFH WKH OLQH ZRXOG QRW EH PRUH WKDQ RQH LQFK ORQJ IRU LI LW LV LW LV E\ VRPH SRVLWLYH DPRXQW 6R 3ROORFN FRQFOXGHV LI WKH OLQH ZHUH PRUH WKDQ RQH LQFK ORQJ LW ZRXOG QRW EH PRUH WKDQ RQH LQFK ORQJ D IODW FRQWUDGLFWLRQ > S @ $OO WKDW VDYHV /HZLVn VHPDQWLFV IURP HYLGHQW LQFRQVLVWHQF\ LV WKDW WKH NH\ SULQFLSOH XVHG DERYH LV QRW YDOLG RQ WKDW VHPDQWLFV > S @ ( 7KH *HQHUDOL]HG &RQVHTXHQFH 3ULQFLSOH *&3f ,I LV D VHW RI VHQWHQFHV DQG IRU HDFK T H :ST LV WUXH DQG U WKHQ :SU LV WUXH *&3 LV WKH YHUVLRQ RI &3 JHQHUDOL]HG WR DOO VHWV RI VHQWHQFHV LQFOXGLQJ DV LQ 3ROORFNnV H[DPSOH LQILQLWH VHWV :KLOH &3 LV DV ZH KDYH QRWHG YDOLG DQG LWV ILQLWH JHQHUDOL]DWLRQ LV YDOLG VLQFH ZH FDQ WKHQ WDNH *

PAGE 101

WR EH D ILQLWH FRQMXQFWLRQ WKH VLWXDWLRQ GHVFULEHG DERYH LV JUDQWLQJ 3ROORFNnV DVVXPSWLRQV D FRXQWHUPRGHO WR *&3 RQ /HZLVn VHPDQWLFV 7KLV UHIOHFWV WKH UHMHFWLRQ RI WKH OLPLW DVVXPSWLRQ WKH LQWHUVHFWLRQ RI DOO WKH SSHUPLWWLQJ VSKHUHV LV QRW LWVHOI SSHUPLWWLQJ %\ WKH UHDVRQLQJ DERYH 3ROORFN FRQFOXGHV WKDW /HZLVn VHPDQWLFV LV LQDGHTXDWH DV *&3 VKRXOG EH D YDOLG LQIHUHQFH SULQFLSOH IRU FRXQWHU IDFWXDOV RQ WKH JURXQGV WKDW LW LV DV LQWXLWLYHO\ YDOLG DV &3 > S @ 7KHUH DUH WZR SUREOHPV ZLWK 3ROORFNnV VXJJHVWLRQ ILUVW LW LV QRW FOHDU WR PH WKDW *&3 LV DV LQWXLWLYHO\ YDOLG DV &3 VHFRQG WKH DVVXPSWLRQ ZH JUDQWHG DERXW WKH RUGHULQJ RI ZRUOGV LV VWURQJHU WKDQ WKDW ZKLFK /HZLV PDNHV LQ ILUVW RIIHULQJ WKH H[DPSOH DQG WKH DGGLWLRQDO VWUHQJWK LV XQn MXVWLILDEOH +HU]EHUJHU SRLQWV WKLV RXW FUHGLWLQJ ,VDDF /HYL > S Q@f 0\ LQWXLWLRQV KDYH EHHQ VKRFNHG VXIILFLHQWO\ RIWHQ LQ WKH LPRYHPHQW IURP WKH ILQLWH WR WKH LQILQLWH WKDW DP KDELWXDOO\ VXVSLFLRXV RI LQn ILQLWH VHWV :KDW LV WUXH RI WKHLU ILQLWH FRXQWHUSDUWV LV RIWHQ QRW WUXH RI WKH LQILQLWH 6R WKDW ZKLOH D ILQLWH LQWHUVHFWLRQ RI WRZHUHG RSHQ LQWHUYDOV RI WKH QXPEHU OLQH LV QRQHPSW\ LW LV HDV\ WR FRQVWUXFW LQILQLWH WRZHUV RI RSHQ LQWHUYDOV ZKRVH LQWHUVHFWLRQ LV HPSW\ ,Q IDFW /HZLVn OLQH H[DPSOH VXJJHVWV RQH 7KH DVVXPSWLRQ WKDW LI HYHU\ ILQLWH VXEVHW RI D FHUWDLQ LQILQLWH VHW KDV D FHUWDLQ SURSHUW\ WKHQ HYHU\ VXEVHW GRHV LV LWVHOI D SRZHUIXO DVVXPSWLRQ LQ PDWKHPDWLFV DQG QRW DOZD\V MXVWLILHG +HQFH P\ LQWXLWLRQV FRQFHUQLQJ *&3 DUH DW EHVW QR OHVV WUXVWZRUWK\ WKDQ P\ LQWXLWLRQV DERXW VLPLODULW\ RUGHULQJV RI ZRUOGV :K\ VKRXOG QRW RXU QRWLRQV RI VLPLODULW\ UHVXOW LQ RSHQ QRQFRPSDFWf UDWKHU WKDQ FORVHG FRPSDFWf VHWV" *&3 DQG WKH UHMHFWLRQ RI /$ DUH RQ

PAGE 102

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b ORQJ EXW LQ WKH VDPH VSKHUH ZLWK D ZRUOG ZKHUH LW LV b ORQJ WKHUH PD\ ZHOO EH ZRUOGV ZKHUH LW LV DOO VRUWV RI OHQJWKV LQFOXGLQJ PRUH RU OHVV 7KH OHQJWK RI OLQHV LV QRW WKH RQO\ HOHPHQW RI FRPSDUDWLYH VLPLODULW\ 6R ZKDW GRHV IROORZ IURP /HZLVn DVVXPSWLRQ LV ( ,I WKH OLQH ZHUH PRUH WKDQ DQ LQFK ORQJ LW PLJKW QRW EH [ ORQJ DQG LW PLJKW EH [ ORQJ IRU HDFK SRVLWLYH YDOXH RI [ ,W LV HDV\ WR LPDJLQH WKDW LQ HDFK VSKHUH WKHUH DUH ZRUOGV HTXDOO\ VLPLODU WR WKH DFWXDO YURUOG ZLWK ZLGHO\ YDU\LQJ OHQJWKV RI OLQH %XW RI FRXUVH ( GRHV QRW OHDG WR WKH FRXQWHUn H[DPSOH WR *&3 EHFDXVH WKH WUXWK RI ERWK SDUWV RI ( SUHFOXGHV WKH WUXWK RI WKH FRUUHVSRQGLQJ ZDXOGFRQGLWLRQDOV +HU]EHUJHU LQ DQ DVLGH > S @ VXJJHVWV D ZHDNHQLQJ RI 3ROORFNnV VFKHPD ( ,I WKH OLQH ZHUH PRUH WKDQ DQ LQFK ORQJ LW ZRXOG EH OHVV WKDQ [ ORQJ ,W LV IRUWXQDWH WKDW KLV SXUSRVH LQ GRLQJ VR LV QRW WR VWUHQJWKHQ 3ROORFNnV DUJXPHQW EXW WR PDNH DQRWKHU SRLQW ZKLFK ZH VKDOO GLVFXVV VKRUWO\ IRU WKH IROORZLQJ LV FHUWDLQO\ WUXH

PAGE 103

( ,I WKH OLQH ZHUH PRUH WKDQ DQ LQFK ORQJ LW PLJKW EH OHVV WKDQ [ ORQJ DQG LW PLJKW QRW EH OHVV WKDQ [ ORQJ DQG WKLV SUHFOXGHV WKH WUXWK RI ( WKLQN WKDW 3ROORFNnV LQIRUPDO DUJXPHQW LV XQUHSDLUDEOH EDVHG DV LW LV XSRQ WKH LQGHIHQVLEOH ( RU ( ( DQG ( UDLVH WKH SRVVLELOLW\ RI DQRWKHU DUJXPHQW FRQn FHUQLQJ D VRUW RI FRQVHTXHQFH SULQFLSOH 7KH IROORZLQJ PLJKW FRQVHTXHQFH SULQFLSOH 9&3f LV YDOLG RQ /HZLVn VHPDQWLFV ( 9ST /&TU 9SU DV )LJXUH VXJJHVWV )LJXUH

PAGE 104

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nnZRXOGnnFRQGLWLRQDO LV XVXDOO\ VHHQ DV D FDVH RI QHFHVVLW\ :KLOH HYHU\WKLQJ WKDW PLJKW EH WKH FDVH VKRXOG VXFKDQGVXFK EH WUXH LV QRW QHFHVVDULO\ RU HYHQ OLNHO\ FRQVLVWHQW MXVW DV WKH FRPELQDWLRQ RI HYHU\WKLQJ WKDW LV SRVVLEOH LV QRW FRQVLVWHQW ZH PD\ IHHO WKDW WKH FRPELQDWLRQ RI HYHUWKLQJ WKDW LV QHFHVVDU\ EHLQJ

PAGE 105

FRQVLVWHQW GHPDQGV WKDW HYHU\WKLQJ WKDW ZRXOG EH WKH FDVH ZHUH VXFKDQG VXFK WUXH PXVW OLNHZLVH EH FRQVLVWHQW ,W LV SUHFLVHO\ WKLV WKDW +HU]EHUJHU ZLVKHV WR FODLP IRU WKH FRXQWHU IDFWXDO FRQGLWLRQDO DOO WKLQJV WKDW ZRXOG EH WUXH XQGHU DQ\ SURn SHUO\ HQWHUWDLQDEOH K\SRWKHVLV DUH WKLQJV WKDW DW OHDVW FRXOG EH MRLQWO\ WUXH > S @ $QG LW LV WKLV WKDW KH VKRZV WKH UHMHFWLRQ RI ,$ E\ /HZLV YLRODWHV 0RUH SUHFLVHO\ KH VKRZV WKDW WKH FROOHFWLRQ RI DOO FRXQWHUIDFWXDO FRQVHTXHQWV RI D JLYHQ HQWHUWDLQDEOH DQWHFHGHQW LV QRW DOZD\V FRQVLVWHQW RQ /HZLVn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f WKDW ZRXOG EH WUXH LI S ZHUH ,W LV FOHDU WKDW LI /1S LV WUXH WKHQ E\ /HZLVn WUXWK FRQGLWLRQV IRU YDFXRXV WUXWKf :ST LV WUXH IRU DQ\ VHQWHQFH T ZKDWVRHYHU 6R HS LV FHUWDLQO\ LQFRQVLVWHQW IRU QRQHQWHUWDLQDEOH VHQWHQFH S :KDW DERXW HQWHUWDLQDEOH VHQWHQFH S" 8VLQJ 3ROORFNnV VFKHPD OHW S EH WKH HQWHUWDLQDEOH VHQWHQFH 7KH OLQH LV PRUH WKDQ DQ LQFK ORQJ 7KHQ S LQFOXGHV RQ 3ROORFNnV

PAGE 106

DVVXPSWLRQA DOO FRQVHTXHQWV RI WKH IRUP 7KH OLQH LV QRW KF ORQJ IRU HDFK SRVLWLYH [ %XW DV :SS LV WUXH IRU DOO VHQWHQFHV S S DOVR LQFOXGHV S :H KDYH DOUHDG\ REVHUYHG WKDW DOO LQVWDQFHV RI T[ DQG S DUH QRW VLPXOWDQHRXVO\ VDWLVILDEOH VR S LV LQFRQVLVWHQW 2I FRXUVH WKLV FRQFOXVLRQ UHVWV RQ 3ROORFNn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f§f§ IRU HDFK SRVLWLYH LQWHJHU Q DV D Q GHQXPHUDEOH VHW RI VXFK VHQWHQFHV TA )RU HDFK SZRUOG HYHQ VKRXOG WKHUH EH QRQGHQXPHUDEO\ PDQ\ RI WKHP VRPH LQVWDQFH RI TA ZLOO PDNH :STA WUXH 6LQFH WKLV VWLOO UHVWV RQ 3ROORFNnV DVVXPSWLRQV ZH KDYH QRW VKRZQ /HZLVn VHPDQWLFV WR SHUPLW LQFRQVLVWHQW H[SUHVVLEOH FRXQWHUIDFWXDO WKHRULHV WKRXJK WKH SRVVLELOLW\ LV WKHUH +RZHYHU +HU]EHUJHU VKRZV WKDW RQ WKH OHYHO RI SURSRVLWLRQV FRXQWHUIDFWXDO LQFRQVLVWHQF\ LV XQn DYRLGDEOH ,Q FRQVLGHULQJ SURSRVLWLRQV ZH DUH QRW ERXQG E\ FRQVLGHUDWLRQV RI H[SUHVVLELOLW\ 7KH IROORZLQJ GHILQLWLRQV LQWURGXFH WKH WHUPLQRORJ\ QHHGHG WR PDNH +HU]EHUJHUnV SRLQW > S @ )RU SURSRVLWLRQ S DQG ZRUOG Z TA LV D FULWLFDO FRQVHTXHQW IRU SZf LII ERWK

PAGE 107

Df LV D FRXQWHUIDFWXDO FRQVHTXHQW RI S DQG Ef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f ZKHWKHU WKDW SURSRVLWLRQ LV H[SUHVVLEOH RU QRW 8VLQJ GHILQLWLRQ IRU HDFK HQWHUWDLQDEOH SURSRVLWLRQ S LGHQWLILHG ZLWK D QRQHPSW\ VHW RI ZRUOGVf WKHUH LV D SURSRVLWLRQDO FRXQWHUIDFWXDO WKHRU\ S FRQVLVWLQJ RI WKH FRXQWHUIDFWXDO FRQVHTXHQWV RI S LH WKRVH SURSRVLWLRQV T IRU ZKLFK :ST KROGV 6XSSRVH WKDW S YLRODWHV WKH OLPLW DVVXPSWLRQ 7KHQ WKHUH LV QR PD[LPDOO\ FORVH SZRUOG RU FORVHVW VHW RI SZRUOGVf )RU HDFK SZRUOG WKHUH LV DW OHDVW RQH FORVHU SZRUOG /HW Z EH DQ\ SZRUOG DQG OHW TA EH WKDW SURSRVLWLRQ LGHQWLILHG ZLWK WKH VHW RI SZRUOGV FORVHU WR X WKDQ Z %\ WKH YLRODWLRQ RI WKH OLPLW DVVXPSWLRQ ZH DUH JXDUDQWHHG WKDW IRU HDFK Z WKLV VHW LV QRQHPSW\ 6LQFH TZ LV D VHW RI SZRUOGV DOO FORVHU WKDQ Z 'Df LV VDWLVILHG DQG VLQFH Z LV QRW LQ TA 'Ef LV VDWLVILHG DV ZHOO 6R HDFK S ZRUOG KDV D FULWLFDO FRQVHTXHQW DQG KHQFH S KDV D FRPSOHWH VHW RI FULWLFDO FRQVHTXHQWV ZKLFK LV D VXEVHW RI S $V S LV DOVR LQ S S LV XQVDWLVILDEOH 7R SURYLGH D VOLJKWO\ GLIIHUHQW SURRI S LV VDWLVILDEOH ZKHUH HDFK HOHPHQW LV D VHW RI ZRUOGVf SURYLGHG $4S I /HW 4 EH WKH VHW

PAGE 108

RI DOO TA GHILQHG DERYH (DFK LV D VXEVHW RI WKH VHW RI ZRUOGV LGHQn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n VLVWHQF\ DQG /$ LPSO\ HDFK RWKHU 7KH OLPLW DVVXPSWLRQ FDQ QRZ EH VKRZQ WR EH HTXLYDOHQW WR *&3 EHLQJ YDOLG %HORZ ZH UHVWDWH WKH GHILQLWLRQ IRU HQWDLOPHQW RI D SURSRVLWLRQ E\ D VHW RI SURSRVLWLRQV LQ D VOLJKWO\ GLIIHUHQW WKRXJK HTXLYDOHQW IRUP WR ' S LII $* F __S__ +HUH ZH DUH LGHQWLI\LQJ S ZLWK D VHW RI ZRUOGV DQG ZLWK D VHW RI VHWV RI ZRUOGV 7KH VHW RI ZRUOGV DW ZKLFK HYHU\ SURSRVLWLRQ LQ LV WUXH LV $* )LUVW ZH VKRZ WKDW WKH OLPLW DVVXPSWLRQ LPSOLHV WKH YDOLGLW\ RI *&3 VXSSRVH /$ KROGV :ST LV WUXH IRU HDFK T LQ 4 DQG 4 U :H PXVW VKRZ :SU LV WUXH %\ GHILQLWLRQ 4 F S KHQFH $S F $4 6LQFH $4 LV LGHQWLILHG ZLWK D SURSRVLWLRQ OHW LW EH V :H FODLP :SV LV WUXH IRU LI QRW WKHQ V LV QRW LQ S VR E\ /$ $ V %XW WKHQ $S $4 D FRQWUDGLFWLRQ 6R :SV LV WUXH %XW 4_ U VR $4 F __U__ KHQFH __V__F __U__

PAGE 109

VR /&VU HTXLYDOHQWO\ V_ Uf LV WUXH %XW E\ &3 :SV DQG /&VU EHLQJ WUXH UHTXLUHV :SU EH WUXH 7KDW WKH YDOLGLW\ RI *&3 LPSOLHV /$ LV VKRZQ DV IROORZV 6XSSRVH *&3 LV YDOLG DQG E\ ZD\ RI FRQWUDGLFWLRQf ,$ GRHV QRW KROG IRU HQWHUWDLQDEOH DQWHFHGHQW S 7KHQ $S VLQFH /$ GRHV QRW KROG %XW S LV D VHW VXFK WKDW :ST LV WUXH IRU HDFK T H S 6LQFH $S DQG F __1S__ LH WKH HPSW\ VHW LV D VXEVHW RI WKH VHW RI ZRUOGV ZKHUH 1S LV WUXH E\ GHILQLWLRQ S_ 1S +HQFH E\ *&3 :S1S LV WUXH 7KHQ /1S LV WUXH %XW LI VR S LV QRW HQWHUWDLQDEOH FRQWUDU\ WR K\n SRWKHVLV 7KXV *&3 WKH OLPLW DVVXPSWLRQ DQG FRXQWHUIDFWXDO FRQVLVWHQF\ DUH DOO HTXLYDOHQW RQ /HZLVn VHPDQWLFV 7R UHMHFW RQH LV WR UHMHFW DOO RWKHU WKLQJV EHLQJ HTXDO 6R UDWKHU WKDQ GHEDWH WKH LQWXLWLYH PHULWV RI *&3 YHUVXV WKH OLPLW DVVXPSWLRQ ZH FDQ FRQVLGHU WKH PHULWV RI FRXQWHUIDFWXDO LQFRQVLVWHQF\ YHUVXV ,$ +HU]EHUJHU FODLPLQJ WKDW FRXQWHUIDFWXDO FRQVLVWHQF\ LV HPLQHQWO\ GHVLUDEOH IRU FRXQWHUIDFWXDO ODQJXDJHV FRQFOXGHV RQH FDQ QRZ DUJXH UDWKHU DVVXPH WKDW WKHUH PXVW EH D FORVHVW >DQWHFHGHQW@ZRUOG > S @ 7KRXJK KH VHHPV KHUH WR KDYH VOLSSHG IURP WKH OLPLW DVVXPSWLRQ WR WKH XQLTXHQHVV DVVXPSWLRQ KLV SRLQW LV FOHDU LI FRXQWHUIDFWXDO LQFRQVLVWHQF\ LV XQGHVLUDEOH DQG HTXLYDOHQW WR UHMHFWLQJ WKH OLPLW DVn VXPSWLRQ WKHQ WKH OLPLW DVVXPSWLRQ VKRXOG QRW EH UHMHFWHG 7KH TXHVWLRQ WKHQ LV KRZ XQGHVLUDEOH LV FRXQWHUIDFWXDO LQn FRQVLVWHQF\" 2U KRZ GHVLUDEOH FRXQWHUIDFWXDO FRQVLVWHQF\" 6KRXOG WKH VWRU\ RI KRZ WKLQJV ZRXOG EH EH D FRQVLVWHQW VWRU\" 3ROORFN QRWHV WKDW /HZLV FRQVLGHUHG *&3 DQG UHMHFWHG LW EHFDXVH RI WKH FRQIOLFW ZLWK

PAGE 110

/$ +HU]EHUJHU QRWHV WKDW /HZLV LQ D GHRQWLF DSSOLFDWLRQ RI KLV DQDOn \VLV OLQNHG /$ ZLWK FRXQWHUIDFWXDO FRQVLVWHQF\ 7KHUHLQ /HZLV UHPDUNV 6HPDQWLFDOO\ D OLPLWHG YDOXH VWUXFWXUH LV RQH WKDW JXDUDQWHHV H[FHSW LQ WKH FDVH RI YDFXLW\f WKDW WKH IXOO VWRU\ RI KRZ WKLQJV RXJKW WR EH JLYHQ VRPH FLUn FXPVWDQFH LV D SRVVLEOH VWRU\ 7KDW LV QRW DOZD\V VR 6HPDQWLF $QDO\VLV IRU '\DGLF 'HRQWLF /RJLF S TXRWHG LQ > S Q@f $ GHIHQVH WKHQ RI UHMHFWLQJ WKH OLPLW DVVXPSWLRQ WR ZKLFK SUHVXPDEO\ /HZLV ZRXOG VXEVFULEH LV WKDW LQ VRPH DSSOLFDWLRQV WKH IXOO JHQHUDOLW\ RI KLV DQDO\VLV ZLWKRXW WKH OLPLW DVVXPSWLRQ LV SUHIHUDEOH HJ GHRQWLF DSSOLFDWLRQV +RZHYHU RXU SULPDU\ FRQFHUQ LV QRW ZLWK FDVHV RI ZKDW RXJKW WR EH EXW UDWKHU ZKDW ZRXOG EH 7KHUHIRUH LW LV RQ WKLV OHYHO WKDW /$ LV WR EH DFFHSWHG RU UHMHFWHG ,I RQH LV LQFOLQHG RQ WKH EDVLV RI /HZLVn VXJn JHVWLYH DUJXPHQW WR UHMHFW /$ WKHQ RQH PXVW EH SUHSDUHG WR GHIHQG FRXQn WHUIDFWXDO LQFRQVLVWHQF\ IRU WKH RUGLQDU\ ZRXOGFRQGLWLRQDO ZKLFK WKHUHE\ IROORZV ,I ZH JUDQW /HZLVn SRLQW DERYH WKDW WKH IXOO VWRU\ RI ZKDW RXJKW WR EH WKH FDVH QHHG QRW DOZD\V EH FRQVLVWHQW GRHV WKLV H[WHQG WR WKH IXOO VWRU\ RI ZKDW ZRXOG EH WKH FDVH" /HW XV FRQVLGHU DJDLQ /HYVn H[DPSOH RI WKH FRXQWHUIDFWXDO VXSn SRVLWLRQ DERXW WKH OLQH ,I ZH DJUHH ZLWK /HZLV WKDW WKHUH LV QR FORVHVW VSKHUH FRQWDLQLQJ ZRUOGV ZLWK WKH OLQH PRUH WKDQ RQH LQFK ORQJ WKHQ WKH FRXQWHUIDFWXDO WKHRU\ IRU WKLV VXSSRVLWLRQ LV SURSRVLWLRQDOO\ LQFRQVLVWHQW VKDOO DUJXH WKDW WKLV LV SUHFLVHO\ ZKDW ZH VKRXOG H[n SHFW DQG DFFHSW IRU VXFK D VXSSRVLWLRQ FRQVLGHUHG LQ LVRODWLRQ IURP DQ\ SDUWLFXODU FRQVHTXHQW &RQVLGHU WKH IR+HZLQJ H[DPSOH RI D FRXQWHUIDFWXDO LQFRUSRUDWLQJ /HZLVn VXSSRVLWLRQ

PAGE 111

( ,I WKH OLQH ZHUH PRUH WKDQ DQ LQFK ORQJ WKH SULQWHU ZRXOG KDYH PDGH D PLVWDNH /HW XV V\PEROL]H WKH FRQVHTXHQW RI ( E\ U UHWDLQLQJ S IRU WKH DQWHFHGHQW DQG IRU WKH FULWLFDO FRQVHTXHQWV $Q\ UHDVRQDEOH VSKHUH IXQFWLRQ ZKHUHLQ S IDLOHG WKH OLPLW DVVXPSWLRQ ZRXOG QHYHUWKHOHVV PDNH ( WUXH WKDW LV WKHUH LV VHPH SSHUPLWWLQJ VSKHUH DQG HYHU\ SZRUOG LQ WKDW VSKHUH LV DQ UZRUOG 'HVLJQDWH D SSHUPLWWLQJ VSKHUH VDWLVI\LQJ WKLV WUXWK FRQGLWLRQ IRU :SU DV 6A /HW 4 UHSUHVHQW WKH VXEn VHW RI WKH VHW RI FULWLFDO FRQVHTXHQWV VXFK WKDW HDFK FULWLFDO FRQVHn TXHQW FRQVLGHUHG DV D VHW RI ZRUOGVf LV D VXEVHW RI 6A ,W LV HDV\ WR VHH WKDW 4 LV VWLOO LQFRQVLVWHQW +RZHYHU IRU HDFK T\ H 4 U LV WUXH DW HDFK ZRUOG LQ TA ,I LW ZHUH QRW WKHQ WKH WUXWK FRQGLWLRQ IRU :SU ZRXOG QRW EH VDWLVILHG DV ZDV DVVXPHG 5HFDOO HDFK ZRUOG LQ TA LV D SZRUOGf 6R IRU HDFK TA H 4 TZ e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

PAGE 112

ZD\ LQ ZKLFK WKH SULQWHU PDGH D PLVWDNH LV LUUHOHYDQW VR DUH WKH FRQn VLGHUDWLRQV WKDW OHG XV WR WDNH S WR EH D /$YLRODWLQJ VXSSRVLWLRQ %XW WKLV DUJXPHQW GHSHQGV XSRQ RXU WDNLQJ ERWK DQWHFHGHQW DQG FRQVHTXHQW LQWR DFFRXQW LQ RXU RUGHULQJ RI SRVVLEOH ZRUOGV 7KLV ZRXOG UHTXLUH FRQn VLGHUDEO\ PRUH RI D GHSDUWXUH IURP /HZLVn WUXWK FRQGLWLRQV WKDQ HLWKHU 3ROORFN RU +HU]EHUJHU VXJJHVW 1XWH UHYLHZV VXFK DQ DFFRXQW LQ > SS @ EXW XOWLPDWHO\ UHMHFWV LW %XWFKHU >@ FRQVWLWXWHV VXFK DQ DFFRXQWf 3ROORFN FRQFOXGHV WKDW /HZLV LV ULJKW DERXW WKH IDLOXUH RI WKH OLPLW DVVXPSWLRQ LQ WKH VLPLODULW\ RUGHULQJ RI SRVVLEOH ZRUOGV EXW ZURQJ LQ WDNLQJ RYHUDOO VLPLODULW\ RUGHULQJ WR EH WKH DSSURSULDWH EDVLV IRU MXGJLQJ WKH WUXWK YDOXH RI FRXQWHUIDFWXDOV > S @ :H VKDOO GLVFXVV 3ROORFNnV YLHZ PRUH IXOO\ LQ &+$37(5 7+5(( ZKHQ ZH GLVFXVV LQ JHQHUDO WKH WRSLFV RI VLPLODULW\ DQG RUGHULQJ RI SRVVLEOH ZRUOGV IRU WKH SXUSRVH RI FRXQWHUIDFWXDO GHOLEHUDWLRQ 7KH RUGHU WKDW 3ROORFN DGRSWV LQ KLV DQDO\VLV LV VWLOO EDVHG XSRQ WKH DQWHFHGHQW DORQH 5HMHFWLQJ /$ JXDUGV DJDLQVW RXU PDNLQJ WKH PLVWDNH RI DFFHSWLQJ 3ROORFNnV VFKHPD ( EHFDXVH ZH FDQ SRLQW WR /HZLVn H[DPSOH WR MXVWLI\ WKH PLJKWFRQGLWLRQDO ( ZKLFK FRQWUDGLFWV WKH IRUPHU 3ROORFN RGGO\ HQRXJK EHOLHYHV WKH UHOHYDQW PLJKWFRQGLWLRQDOV DUH WUXH > S @ EXW GRHV QRW VHH WKDW WKLV XQGHUPLQHV KLV VFKHPD ( 1HLWKHU WKH IDLOXUH RI *&3 QRU WKH DSSHDUDQFH RI FRXQWHUI DFWXDO LQFRQVLVWHQF\ LV D GHFLVLYH REMHFWLRQ WR /HZLVn UHMHFWLRQ RI /$ 7KH IXOO VWRU\ RI ZKDW ZRXOG EH WKH FDVH JLYHQ DQ\ DQWHFHGHQW LV QRW DOZD\V D FRQVLVWHQW VWRU\ QRU VKRXOG ZH H[SHFW LW WR EH

PAGE 113

3RVVLEOH :RUOGV 5HDOLVP DQG ([SODQDWLRQ 2QH PD\ REMHFW WR /HZLVn DQDO\VLV RQ WKUHH OHYHOV WKH QRWLRQ RI SRVVLEOH ZRUOGV LV LWVHOI VXVSHFW DQG VR FDQQRW VHUYH WR FODULI\ VRPHn WKLQJ HOVH VHH > @f WKH QRWLRQ RI FRPSDUDn WLYH VLPLODULW\ LV HLWKHU WRR YDJXH RU LQDSSURSULDWH IRU DQDO\]LQJ FRXQWHUIDFWXDOV VHH > @f SRVVLEOH ZRUOGV DQG VLPLODULW\ DUH DFn FHSWDEOH EXW WKH SDUWLFXODU DQDO\VLV LV IODZHG VHH > f@ ,Q WKLV VHFWLRQ ZLOO FRPPHQW RQO\ RQ WKH ILUVW OHYHO RI REMHFn WLRQ 5HDOL]LQJ WKH VXVSHFW QDWXUH RI KLV IRXQGDWLRQV /HZLV XQGHUWDNHV D GHIHQVH RI SRVVLEOH ZRUOGV UHDOLVP ZKLFK LV HVVHQWLDOO\ WKH YLHZ WKDW SRVVLEOH ZRUOGV DUH HQWLWLHV VXL JHQHULV QRW UHGXFLEOH WR VRPH RWKHU VRUWV RI WKLQJV DQG IXUWKHUPRUH DUH PRUH HQWLWLHV RI WKH VDPH NLQG DV WKH DFWXDO ZRUOG > SS @ +H VSHFLILFDOO\ UHMHFWV WDNLQJ SRVVLEOH ZRUOGV WR EH D GLVSHQVDEOH ORFXWLRQ IRU PD[LPDOO\ FRQn VLVWHQW VHWV RI VHQWHQFHV VHWV RI EHOLHIV PD[LPDO VWDWHV RI DIIDLUV RU PDWKHPDWLFDO HQWLWLHV RI VRPH VRUW +H ZRXOG SUHVXPDEO\ DOVR UHMHFW WKH YLHZ WKDW WKH\ DUH WKH PDQ\ ZRUOGV RI WKH PDQ\ ZRUOGV LQWHUSUHWDWLRQ RI TXDQWXP PHFKDQLFV VHH >@ DQG >@f 5DWKHU WKDQ UHSHDW RU PRGLI\ /HZLVn DUJXPHQWV ZRXOG OLNH WR SRVH DQ DQDORJ\ EHWZHHQ VFLHQWLILF H[SODQWLRQ DQG WKH NLQG RI H[SODQDn WRU\ DQDO\VLV VHH DV HVVHQWLDO WR FODULI\LQJ D FRQFHSW 7KH VLWXDWLRQ ZLVK WR FRQVLGHU LV WKH VWDWXV RI SK\VLFV DQG ZKDW FRXQWHG DV D SK\VLFDO H[SODQDWLRQ LQ WKH GHFDGH IROORZLQJ WKH JHQHUDO DFFHSWDQFH RI 1HZWRQnV ODZV RI PRWLRQ DQG ODZ RI JUDYLW\ (YHQ ZKLOH 1HZWRQnV DFKLHYHPHQW ZDV JHQHUDOO\ DFFHSWHG LW ZDV UHFRJQL]HG QRWDEO\ E\ 1HZWRQ KLPVHOI WKDW WKH ODZ RI JUDYLW\ FRQIOLFWHG

PAGE 114

ZLWK RQH RI WKH LGHDOV RI PHFKDQLFDO H[SODQDWLRQ WKDW DOO HIIHFWV ZHUH WR EH H[SODLQHG LQ WHUPV RI FRUSXVFXODU PRWLRQ DQG LPSDFW 7KH QRWLRQ RI DWWUDFWLRQ DW D GLVWDQFH ZDV RFFXOW LQ WKH SHUMRUDWLYH MDUJRQ RI WKH WLPH DQG UHPLQLVFHQW RI WKH UHMHFWHG $ULVWRWOHDQ W\SHV RI H[SODQDWLRQ 7KH UHVXOW ZDV WKDW XQWLO DIWHU WKH HQG RI WKH QLQHWHHQWK FHQWXU\ DPRQJ WKRVH ZKR DFFHSWHG 1HZWRQnV ODZV WKHUH ZHUH WZR GLVWLQFW FDPSV 7KHVH FDPSV GLIIHUHG QRW LQ WKHLU DFFHSWDQFH RI 1HZWRQnV ODZV DQG WKH ZLGHQLQJ DSSOLFDWLRQV RI WKHP EXW LQ WKHLU LQn WHUSUHWDWLRQ RI WKH ODZV 2Q WKH RQH KDQG ZHUH WKRVH ZKR FOXQJ WR WKH LGHDO RI D PHFKDQLFDO H[SODQDWLRQ DV WKH XOWLPDWH H[SODQDWRU\ WRRO )RU WKHP DWWUDFWLRQ DW D GLVWDQFH ZDV D ZD\ VWDWLRQ LQ H[SODQDWLRQ WR EH VXSHUVHGHG HYHQWXDOO\ E\ D PRUH SURSHUO\ PHFKDQLFDO H[SODQDWLRQ 2Q WKH RWKHU KDQG ZHUH WKRVH ZKR DFFHSWHG IRUFHV DQG ODWHU ILHOGVf DV IXQGDPHQWDO FRQVWLWXHQWV RI QDWXUH )RU WKHP JUDYLW\ UHTXLUHG QR IXUWKHU H[SODQDWLRQ LQ WHUPV RI PHFKDQLFDO SULQFLSOHV ,Q HIIHFW LW EHFDPH RQH RI WKH PHFKDQLFDO SULQn FLSOHV 7KH GLVWLQFWLRQ EHWZHHQ WKHVH WZR JURXSV FDQ EH VHHQ DV D GLVWLQFn WLRQ LQ PHWDSK\VLFDO FRUQQLWPHQW 7KRVH LQ WKH ILUVW JURXS ZRXOG DGPLW WKH LQPHQVH KHXULVWLF YDOXH RI 1HZWRQnV ODZ RI JUDYLW\ EXW DYRLG WKH FRPQLWPHQW WR D IRUFH RI JUDYLW\ FRQVWLWXWLYH RI QDWXUH 7KH RWKHUV ZRXOG FRPPLW WKHPVHOYHV WR D IXQGDPHQWDO IRUFH RI JUDYLW\ LQ WKH DEVHQFH RI DFFHSWDEOH DOWHUQDWLYHV +RZHYHU WKLV GLYHUJHQFH LQ PHWDSK\VLFDO FRPQLWPHQW GLG QRW FDUU\ ZLWK LW D GLYHUJHQFH LQ YLHZV RQ ZKDW FRXQWHG DV DQ H[SODQDWLRQ LQ PHFKDQLFV LWVHOI 7R UHGXFH DQ HIIHFW WR DPRQJ RWKHU WKLQJVf WKH IRUFH RI JUDYLW\ ZDV LWVHOI D VXIILFLHQW H[SODQDWLRQ

PAGE 115

, FRQVLGHU WKH LVVXH RI SRVVLEOH ZRUOGV UHDOLVP WR EH RQ D SDU ZLWK WKH LVVXH RI WKH UHDOLW\ RI JUDYLWDWLRQDO IRUFH DV LW ZRXOG KDYH DSSHDUHG WR D 1HZWRQLDQ SK\VLFLVW $ GLYHUJHQFH RI YLHZV RQ WKH UHGXF LELOLW\ RI SRVVLEOH ZRUOGV WR RWKHU HQWLWLHV LV QRW LQ LWVHOI D EDU WR WKHLU VHUYLQJ DV D EDVLV IRU H[SODQDWLRQ RI RWKHU FRQFHSWV -XVW DV 1HZWRQnV ODZV VHUYHG WR XQLI\ YDULRXV DUHDV RI PHFKDQLFV DQG SHUPLW WKH DUWLFXODWLRQ RI VSHFLILF DSSOLFDWLRQV LQ WHUPV RI D VLQJOH WKHRU\ VR SRVVLEOH ZRUOGV VHPDQWLFV VHUYHV WR XQLI\ YDULRXV DUHDV RI ORJLF LQGHHG RI SKLORVRSK\ PRUH JHQHUDOO\ DQG WR SHUPLW WKH DUWLFXODWLRQ RI YDULRXV DSSOLFDWLRQV LQ WHUPV RI D VLQJOH WKHRU\ 7KH KHXULVWLF YDOXH RI SRVn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n ILFLHQW IRU WKHLU XVH LQ H[SODQDWLRQV RI RWKHU FRQFHSWV $ PRUH VXEWOH DWWDFN ZRXOG WKHQ EH WR FKDOOHQJH WKH KHXULVWLF YDOXH WKDW KDV EHHQ DVn VXPHG DERYH

PAGE 116

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n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

PAGE 117

DFWXDO LV DQDORJRXV WR SUHVHQWf LV WR FRQYH\ XQGHUVWDQGLQJ RI ZKDW D SRVVLEOH ZRUOG LV WKURXJK DQDORJ\ WR VRPHWKLQJ ZH DOUHDG\ XQGHU VWDQG UDWKHU WKDQ UHGXFWLRQ WR VRPHWKLQJ ZH DOUHDG\ XQGHUVWDQG 7R JR RQ WR H[SODLQ ZK\ SRVVLEOH ZRUOGV DUH QRW PRPHQWV RI WLPH LV WR GHHSHQ RXU XQGHUVWDQGLQJ E\ H[KLELWLQJ WKH OLPLWV RI WKH DQDORJ\ ,I WKH FRQFHSW RI D SRVVLEOH ZRUOG FDQ EH JUDVSHG E\ DQDORJ\ DV WKLQN LW FDQ WKHQ FHUWDLQ DSSOLFDWLRQV VD\ WR DQ DQDO\VLV RI SRVn VLELOLW\ DQG QHFHVVLW\ RU WR FRQVLVWHQF\ FDQ VHUYH DV SDUDGLJPDWLF DSSOLFDWLRQV ZKLFK H[WHQG RXU XQGHUVWDQGLQJ 7KDW LV ZH FRPH WR VHH SRVVLEOH ZRUOGV DV WKH NLQG RI WKLQJ ZKLFK FDQ VHUYH LQ DQ H[SODQDWLRQ 7KH SDUDOOHO LQ VFLHQWLILF H[SODQDWLRQ VKRXOG QRW JR XQUHPDUNHG +HQFH WKLQN WKH YLHZ WKDW SRVVLEOH ZRUOGV DQDO\VLV FRQYH\V QR JHQXLQH XQGHUVWDQGLQJ LV GXH WR D PLVWDNHQ YLHZ RI ZKDW FRQVWLWXWHV DQ H[SODQDWLRQ UHGXFWLRQ WR WKH IDPLOLDU /HZLVn SRVVLEOH ZRUOGV UHDOLVP LV D GLVSHQVDEOH SDUW RI KLV DQDOn \VLV EXW LW LV QRW GLVSHQVDEOH LQ IDYRU RI VRPH LGHQWLILFDWLRQ RI SRVn VLEOH ZRUOGV ZLWK RWKHU PRUH IDPLOLDU HQWLWLHV 7KHVH LGHQWLILFDWLRQV UHVXOW LQ WRR QDUURZ D YLHZ SRLQWV ZKLFK /HZLV PDNHV VXFFHVVIXOO\ %XW WR GLVSHQVH ZLWK /HZLVn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

PAGE 118

,OO 0XFK RI WKH PDWHULDO RQ PRGDO ORJLF LQ WKLV VHFWLRQ DQG HOVHZKHUH LQ WKLV HVVD\ LV GUDZQ IURP WKH FLWHG ZRUN DQG FODVV QRWHV IURP -D\ =HPDQn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

PAGE 119

&+$37(5 7+5(( 25'(5,1*6 2) 3266,%/( :25/'6 &RPSDUDWLYH 6LPLODULW\ 7KH WZR QRWLRQV FHQWUDO WR /HZLVn DQDO\VLV RI FRXQWHUIDFWXDOV DUH WKRVH RI SRVVLEOH ZRUOGV DQG FRPSDUDWLYH RYHUDOO VLPLODULW\ ,Q &+$37(5 7:2 ZH SRVWSRQHG DQ\ FULWLFDO H[DPLQDWLRQ RI FRPSDUDWLYH RYHUDOO VLPLODUn LW\ EXW DW VDQH SRLQW WKH TXHVWLRQV UDLVHG E\ WKH XVH RI VXFK D FRQFHSW PXVW EH DQVZHUHG $V ZH LQGLFDWHG DW WKH EHJLQQLQJ RI 6HFWLRQ REMHFn WLRQV WR FRPSDUDWLYH VLPLODULW\ PD\ WDNH WKH IRUP RI TXHVWLRQLQJ ZKHWKHU VLPLODULW\ LV WKH DSSURSULDWH SULQFLSOH IRU DQDO\]LQJ ZKDW JRHV RQ LQ FRXQWHUIDFWXDO GHOLEHUDWLRQ RU DOWHUQDWLYHO\ PD\ JUDQW WKH DSSURSULDWHn QHVV EXW TXHVWLRQ /HZLVn SDUWLFXODU DQDO\VLV $V KLV GHIHQVH RI KLV IRXQGDWLRQV LQGLFDWHV >@ SS f /HZLV EHOLHYHV WKH DSSDUHQW ZHDNQHVV RI FRPSDUDWLYH RYHUDOO VLPLODULW\ DV D WRRO RI DQDO\VLV LV LWV HYLGHQW YDJXHQHVV ,Q LWV GHIHQVH KH VKRZV DQG WKLQN FRUUHFWO\ WKDW WKH LOOXQGHUVWRRG YDJXHQHVV RI FRXQWHUIDFn WXDOV WKHPVHOYHV LV DSSURSULDWHO\ PDWFKHG E\ WKH ZHOOXQGHUVWRRG YDJXHQHVV RI FRPSDUDWLYH VLPLODULW\ $ YDJXH WKRXJK IDPLOLDU FRQFHSW LV MXVWLILn DEO\ HPSOR\HG LQ H[SOLFDWLQJ D YDJXH EXW XQIDPLOLDU FRQFHSW +RZHYHU /HZLVn GHIHQVH LV PLVDLPHG 7KH PRVW WHOOLQJ REMHFWLRQV WR FRPSDUDWLYH RYHUDOO VLPLODULW\ DULVH QRW IURP LWV YDJXHQHVV DQG WKHUHn IRUH WKH SRVVLELOLW\ RI RXU EHLQJ PLVOHG LQ DQ XQV\VWHPDWLF DQG UDQGRP

PAGE 120

IDVKLRQ EXW UDWKHU IURP WKH SRVVLELOLW\ WKDW ZH PD\ EH V\VWHPDWLFDOO\ PLVOHG 2EMHFWLRQV RI WKLV ODWWHU VRUW DUH UDLVHG IRU H[DPSOH E\ %DUNHU >@ -DFNVRQ >@ DQG 3ROORFN >@ 7KHLU REMHFWLRQV SDYH WKH ZD\ IRU DQDO\VHV E\ HDFK RI FRXQWHUIDFWXDOV LQ WHUPV RI ODZV SDUWLFXn ODUO\ FDXVDO ODZV LQ WKH IRUPHU WZR FDVHV WKRXJK 3ROORFN DOVR HPSOR\V SRVVLEOH ZRUOGV 7R GLVFXVV WKHVH DOWHUQDWLYH FDXVDO WKHRULHV RI FRXQWHUIDFWXDOV LQ DQ\ GHWDLO ZRXOG JR EH\RQG WKH VFRSH RI RXU SUHVHQW FRQFHUQV *HQHUDOO\ VXFK WKHRULHV DGRSW WKH YLHZ WKDW FRXQWHUIDFWXDOV DUH E\ DQG ODUJH QRPR ORJLFDO WKDW LV HVVHQWLDOO\ K\SRWKHWLFDO LQVWDQWLDWLRQV RI ODZV RI QDWXUH RU FDXVDO ODZV 3ROORFNnV LV SHUKDSV WKH PRVW WKRURXJKO\ ZRUNHG RXW RI VXFK DFFRXQWV EXW DV KLV WKHRU\ LV DW OHDVW LQ SDUW DOVR D SRVn VLEOH ZRUOGV WKHRU\ ZH VKDOO FRQVLGHU LW LQ 6HFWLRQ 7KRVH WKHRULHV ZKLFK YLHZ FRXQWHUIDFWXDOV DV QRPRORJLFDO FDQQRW KRSH WR GR PRUH WKDQ DVn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n WHUIDFWXDOV 7KDW ODZV DUH JHQHUDOL]HG FRXQWHUI DFWXDO FRQGLWLRQDOV ZDV VXJJHVWHG LQ 6WDOQDNHUnV DFFRXQW 8WLOL]LQJ D VLPLODULW\ FRPSDULVRQ UXOHV RXW WKLV VLPSOH VROXWLRQ IRU WKH IROORZLQJ UHDVRQ LW ZRXOG UHLQWURGXFH WKH YHU\ FLUFOH ZH DUH WU\LQJ WR DYRLG

PAGE 121

,Q WKH PDNLQJ RI MXGJPHQWV RI FRPSDUDWLYH RYHUDOO VLPLODULW\ RI SRVVLEOH ZRUOGV WR D JLYHQ SRVVLEOH ZRUOG ZKDW NLQGV RI WKLQJV ZRXOG ZH WDNH LQWR DFFRXQW" &HUWDLQO\ ZH ZRXOG QRW EH FRQFHUQHG RQO\ ZLWK PDWWHUV RI SDUWLFXODU IDFW EXW DOVR ZLWK ZKDW ODZV KHOG DW WKH ZRUOGV FRQFHUQHG ,Q JHQHUDO D ZRUOG ZKRVH ODZV DUH LGHQWLFDO WR RXUV LV PRUH VLPLODU WR WKH DFWXDO ZRUOG WKDQ D ZRUOG OLNH RXUV LQ PDWWHUV RI SDUWLFn XODU IDFW EXW ZLWK UDGLFDOO\ GLIIHUHQW ODZV /HZLV VXJJHVWV WKDW VLPLn ODULWLHV RI IDFW DQG ODZ DUH EDODQFHG RQH DJDLQVW WKH RWKHU LQ GHWHUn PLQLQJ FRPSDUDWLYH RYHUDOO VLPLODULW\ ZLWK VLPLODULWLHV LQ ODZV EHLQJ JHQHUDOO\ RI PRUH LPSRUWDQFH > S @ 6XSSRVH ZH ZHUH WR DQDO\]H ODZV LQ WHUPV RI FRXQWHUIDFWXDOV 2XU DQDO\VLV ZRXOG FOHDUO\ EH FLUFXODU VLQFH ZH ZRXOG KDYH WR HPSOR\ ODZV WR DQDO\]H FRXQWHUIDFWXDOV LQ WKH ILUVW SODFH ,I FRPSDUDWLYH VLPLODULW\ LQYROYHV LQ SDUW FRPSDULVRQ RI SRVVLEOH ZRUOGV RQ WKH EDVLV RI ODZV WKHQ /HZLV LV XQGHU VRPH REOLJDWLRQ WR GHn YHORS DQ DQDO\VLV RI ODZV WKHPVHOYHV WKDW GRHV QRW UHGXFH KLV ODUJHU DQDOn \VLV WR MXVW DQRWKHU FLUFOH /HZLV DGRSWV WKH IROORZLQJ VOLJKWO\ UHZRUGHGf GHILQLWLRQ RI D ODZ RI QDWXUH VHH > S @f $ FRQWLQJHQW JHQHUDOL]DWLRQ LV D ODZ RI QDWXUH DW ZRUOG X LII LW DSSHDUV DV D WKHRUHP RU D[LRPf LQ HYHU\ GHGXFWLYH V\VWHP WUXH DW X WKDW DFKLHYHV D EHVW FRPELQDWLRQ RI VLPSOLFLW\ DQG VWUHQJWK %\ WKLV GHILQLWLRQ D ODZ LV MXVW D PDWHULDO JHQHUDOL]DWLRQ UDWKHU WKDQ VRPH RWKHU VRUW RI JHQHUDOL]DWLRQ VD\ D VXEMXQFWLYH RQH 7KDW ODZV WHQG WR KDYH VXEMXQFWLYH IRUFH LV D FRQVHTXHQFH RI WKH VLPLODULW\ RUGHULQJ VLQFH ODZV DUH ZHLJKHG KHDYLO\ LQ VLPLODULW\ WKH FORVHU VSKHUHV DERXW D ZRUOG X ZLOO WHQG WR EH RFFXSLHG E\ ZRUOGV ZLWK WKH VDPH ODZV DV DW X

PAGE 122

> S @ 7KXV /HZLVn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n DQDO\VLV LV V\PSDWKHWLF WR WKH &DUWHVLDQ LGHDO RI GHYHORSLQJ DOO VFLHQFH DV D GHn GXFWLYH V\VWHP FDQQRW VHUYH DV D ZRUNLQJ FULWHULRQ RI ODZKRRG VLQFH RXU ODPHQWDEOH IDLOXUH WR EH RPQLVFLHQW EDUV XV IURP DFKLHYLQJ RU NQRZLQJ ZH KDYH DFKLHYHGf HYHQ RQH FRPSOHWH GHGXFWLYH ZRUOGV\VWHP 7KH GHOLQHDWLRQ RI ODYV IURP QRQODZV LQ WKH DFWXDO ZRUOG LV KRZHYHU D PDWWHU RI MXVWLILFDWLRQ FRQGLWLRQV UDWKHU WKDQ WUXWK FRQGLWLRQV DQG /HZLV LV RIIHULQJ KLV DQDO\VLV DV LQ WHUPV RI WUXWK FRQGLWLRQV :RXOG /HZLVn DQDO\VLV RI FRXQWHUIDFWXDOV EH JUHDWO\ LPSDULHG LI ZH DEDQGRQHG KLV GHILQLWLRQ RI ODZ" WKLQN QRW IRU RQH DOWHUQDWLYH RSHQ WR XV LV WKDW SXUVXHG E\ 3ROORFN WR SURYLGH DQ DQDO\VLV RI ODZV LQ WHUPV RI WKHLU MXVWLILFDWLRQ FRQGLWLRQV KDYH RWKHU REMHFWLRQV WR VXFK DQ DOWHUQDWLYH IRUHPRVW EHLQJ WKDW DP GRXEWIXO WKDW MXVWLILFDWLRQ FRQn GLWLRQV HYHU FRQVWLWXWH DQ DQDO\VLV $Q DQDO\VLV PXVW VXUHO\ EH WUXWK SUHVHUYLQJ EXW ZKDW LV MXVWLILHG LV QRW LQYDULDEO\ WUXH )XUWKHUPRUH HYHQ LI ZH ZHUH WR OHDYH WKH FRQFHSW RI ODZ XQDQDO\]HG ZH ZRXOG EH EHWn WHU RII ZLWK /HZLVn DQDO\VLV RI FRXQWHUI DFWXDOV RU RQH OLNH LW WKDQ DQ DQDO\VLV GLUHFWO\ LQ WHUPV RI ODZV IRU WKH ODWWHU FDQ SURYLGH QR H[n SODQDWLRQ RI RXU LQWXLWLRQV UHJDUGLQJ QRQQRPR ORJLFDO FRXQWHUI DFWXDOV

PAGE 123

:H ZRXOG EH MXVWLILHG LQ VHWWOLQJ IRU D PRUH OLPLWHG DQDO\VLV RI FRXQWHUIDFWXDOV OLPLWHG WR WKH QRPRORJLFDO RQHV VD\ LI /HZLVn DSn SURDFK ZHUH WR V\VWHPDWLFDOO\ \LHOG LQFRUUHFW WUXWK YDOXHV IRU VRPH FODVV RI FRXQWHUIDFWXDOV 6XEVWDQWLDOO\ WKLV LV WKH SRLQW DUJXHG E\ %DUNHU DQG -DFNVRQ VKDOO LOOXVWUDWH ZLWK RQH RI -DFNVRQnV FRXQWHUn H[DPSOHV WR /HZLVn DQDO\VLV > SS @f :H DUH WR DVVXPH WKDW DQ HYHQW RI W\SH $ FDWDVHV WKH RFFXUUHQFH RI HLWKHU DQ HYHQW RI W\SH & RU W\SH EXW UDQGRPO\ DQG ZLWK HTXDO SUREn DELOLW\ DV LV RIWHQ WKH FDVH LQ TXDQWXP PHFKDQLFV 1RZ VXSSRVH WKDW LQ IDFW DW WLPH W QR $HYHQW RFFXUUHG &RQVLGHU WKH IROORZLQJ FRXQWHUn IDFWXDOV ( Df ,I DQ $ KDG RFFXUUHG DW W WKHQ D & RU D ZRXOG KDYH RFFXUUHG Ef ,I DQ $ KDG RFFXUUHG DW W WKHQ D & ZRXOG KDYH RFFXUUHG Ff ,I DQ $ KDG RFFXUUHG DW W WKHQ D ZRXOG KDYH RFFXUUHG -DFNVRQ UHFRJQL]HV WKDW WKLV SURGXFHV D VLWXDWLRQ VLPLODU WR WKDW RI WKH %L]HW DQG 9HUGL H[DPSOHV GLVFXVVHG SUHYLRXVO\ 7KDW LV Df LV WUXH EXW Ef DQG Ff DUH IDOVH ,W VKRXOG EH IXUWKHU REVHUYHG WKDW LI PLJKW LV VXEVWLWXWHG IRU ZRXOG LQ Ef DQG Ff WKHQ WKH UHVXOWLQJ FRXQWHUIDFWXDOV DUH WUXH -DFNVRQ WKHQ DUJXHV WKDW XQGHU FHUWDLQ FLUFXPVWDQFHV VLPLODULW\ FRQVLGHUDWLRQV ZRXOG OHDG WR DQ HUURQHRXV HYDOXDWLRQ RI Ef DV WUXH 6XSSRVH WKDW LQ IDFW $HYHQWV KDYH IUHTXHQWO\ EHHQ IROORZHG E\ &HYHQWV LQ WKH DFWXDO ZRUOG ZKLFK LV LPSUREDEOH EXW SRVVLEOH 7KH ZRUOGV PRVW VLPLODU WR WKH DFWXDO ZRUOG ZLOO KDYH WKH VDPH ODZV DV WKH DFWXDO ZRUOG DQG PD[LPL]LQJ VLPLODULW\ LQ WHUPV RI SDUWLFXODU IDFW $HYHQWV ZLOO KDYH

PAGE 124

EHHQ PQUS IUHTXHQWO\ IROORZHG E\ &HYHQWV WKDQ 'HYHQWV LQ WKH PRVW VLPLODU ZRUOGV LQFOXGLQJ WLPH W VR Ef ZLOO EH MXGJHG WUXH HUURQn HRXVO\ -DFNVRQ LV FHUWDLQO\ ULJKW LQ VXSSRVLQJ WKH FORVHVW ZRUOGV ZLOO KDYH WKH VDQH ODZV DQG WKDW $HYHQWV ZLOO KDYH EHHQ PRUH IUHTXHQWO\ IROORZHG E\ &HYHQWV WKDQ E\ 'HYHQWV LQ WKH PRVW VLPLODU ZRUOGV EXW KH LV FHUWDLQO\ ZURQJ LQ VXSSRVLQJ WKDW LQ WKH PRVW VLPLODU ZRUOGV & HYHQWV ZLOO KDYH RFFXUUHG DW WLPH W /HW XV VXSSRVH WKDW WKH PRVW VLPLODU ZRUOGV KDYH H[DFWO\ RQH PRUH $HYHQW WKDQ WKH DFWXDO ZRUOG DQG LW RFFXUV DW WLPH W 2WKHUZLVH WKH PRVW VLPLODU ZRUOGV KDYH SUHn FLVHO\ WKH VDPH $& DQG $' FRPELQDWLRQV DW HYHU\ WLPH DQ $HYHQW RFn FXUUHG -DFNVRQ ZRXOG KDYH LW WKDW EHFDXVH $HYHQWV DUH DFWXDOO\ PRUH IUHTXHQWO\ IROORZHG E\ &HYHQWV WKDW WKLV H[WUD $HYHQW ZLOO EH IROORZHG E\ D &HYHQW LQ WKH FORVHVW ZRUOGV WR PD[LPL]H VLPLODULW\ %XW LI WKH FORVHVW ZRUOGV GR KDYH WKH VDPH ODZV DQG QR KLGGHQ RQHV WKDW ZH GR QRW NQRZ DERXW WKHQ WKLV DVVXPSWLRQ LV XQZDUUDQWHG 7R VHH WKDW WKLV LV WKH FDVH FRQVLGHU D PRUH SURVDLF EXW VXIn ILFLHQWO\ DQDORJRXV H[DPSOH 6XSSRVH WKDW LQ IDLUO\ IOLSSLQJ D IDLU FRLQ SURGXFH D VWULQJ RI WHQ KHDGV DQG WKHQ TXLW ,I KDG IOLSSHG WKH FRLQ DJDLQ WKHQ FHUWDLQO\ LW ZRXOG KDYH EHHQ HLWKHU KHDGV RU WDLOV EXW ZRXOG LW KDYH EHHQ KHDGV" 8QOHVV LUUDWLRQDOO\ EHOLHYH LQ UXQV RI OXFN ZRXOG QRW EHW RQ LW /HW XV DSSO\ VLPLODULW\ FRQVLGHUDWLRQV WR WKLV H[DPSOH 3UHVXPDEO\ WKH FORVHVW ZRUOGV DOO KDYH WKH VDPH ODZV LQn FOXGLQJ WKH ODZV JRYHUQLQJ IDLU FRLQ IOLSSLQJ $PRQJ WKH FORVHVW ZRUOGV ZLOO EH WKH ZRUOGV ZKHUH WHQ EHDGV RFFXUUHG LQ WKH ILUVW WHQ IOLSV 1RZ FRQVLGHU WZR RI WKRVH ZRUOGV LQ RQH WKH HOHYHQWK IOLS QRW PDGH LQ WKH DFWXDO ZRUOGf LV KHDGV DQG LQ WKH RWKHU WDLOV 2WKHUZLVH WKH\ DUH DV

PAGE 125

VLPLODU WR HDFK RWKHU DV SRVVLEOH :KLFK LV PRUH VLPLODU WR WKH DFWXDO ‘ZRUOG" $V LQ WKH %L]HW DQG 9HUGL H[DPSOH WKH TXHVWLRQ LQ WKLV IRUP LV XQDQVZHUDEOH 1HLWKHU ZRUOG LV PRUH VLPLODU WR WKH DFWXDO ZRUOG EXW ERWK DUH HTXDOO\ VLPLODU 7KLV LV SUHFLVHO\ ZKDW YDOLGDWHV WKH PLJKW FRXQWHUIDFWXDOV 1RZ LI ZH DVVXPHG WKDW UXQV WHQG WR EH FRQWLQXHG WKLV ZRXOG EH WDQWDPRXQW WR DQ DGGLWLRQDO ODZ RI SUREDELOLW\f DQG ZRXOG GLVSODFH VXFK ZRUOGV IXUWKHU IURP WKH DFWXDO ZRUOG VLQFH WKHUH LV SUHn VXPDEO\ QR VXFK ODZ LQ WKH DFWXDO ZRUOG 6LPLODUO\ ZH FDQQRW DVVXPH WKDW UXQV WHQG WR EUHDN 7R DUJXH DV -DFNVRQ GRHV UHJDUGLQJ WKH $& IUHTXHQF\ LV WR FRQWUDGLFW WKH UDQGRPQHVV RI WKH YHU\ ODZ KH KDV DVVXPHG LV FRQVWDQW LQ WKH FORVHVW ZRUOGV 2WKHU FRXQWHUH[DPSOHV WR /HZLVn DFFRXQW UHYHDO QRW D EDVLF IODZ LQ VLPLODULW\ RUGHULQJ EXW UDWKHU WKH YDJXH QDWXUH RI RXU VWDQGDUGV RI FRPn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n ODULW\ WR RXU FDXVDO LQWXLWLRQV WKHQ WKLV PD\ GLFWDWH FHUWDLQ VWDQGDUGV WR EH REVHUYHG LQ FRPSDULQJ SRVVLEOH ZRUOGV )RU H[DPSOH DV LV ZLGHO\ UHFRJQL]HG VHH -DFNVRQ >@ /HZLV >@ (OOLV >@f VLPLODULW\ EHIRUH WKH HYHQW K\SRWKHVL]HG LQ WKH DQWHFHGHQW RI D VHTXHQWLDO FRXQWHUIDFWXDO LV JHQHUDOO\ PRUH VLJQLILFDQW WKDQ VLPLODULW\ DIWHU WKH HYHQW LQ FRPSDULQJ

PAGE 126

SRVVLEOH ZRUOGV 7KLV LV QRW DQ DG KRF DVVXPSWLRQ EXW D QHFHVVLW\ RI H[WHQGLQJ WKH FRQFHSW RI FRPSDUDWLYH VLPLODULW\ WR D QHZ GRPDLQ SRVn VLEOH ZRUOGV VWDWHG SUHYLRXVO\ WKDW DQ DFFRXQW PXVW EH LQ WHUPV RI SUHYLRXVO\ XQGHUVWRRG FRQFHSWV LQ RUGHU WR TXDOLI\ DV DQ DQDO\VLV DV RSSRVHG WR D IRUPDOL]DWLRQ RI WKH ORJLF RI WKH FRQFHSW +RZHYHU SUHYLRXVO\ XQGHUn VWRRG VKRXOG QRW EH WDNHQ WR PHDQ RUGLQDU\ RU FRLUPRQ EXW UDWKHU LQGHn SHQGHQW %HFDXVH ZH FDQ DFKLHYH DQ XQGHUVWDQGLQJ RI SRVVLEOH ZRUOGV LQn GHSHQGHQWO\ RI FRXQWHUIDFWXDOV SRVVLEOH ZRUOGV DUH DFFHSWDEOH LQ DQ DQDO\VLV RI FRXQWHUIDFWXDOV EXW WKH FRQFHSW RI SRVVLEOH ZRUOGV LV VXUHO\ QRW DQ RUGLQDU\ RU FRPPRQ RQH 2Q WKH RWKHU KDQG FRPSDUDWLYH VLPLODULW\ LV DQ RUGLQDU\ FRQFHSW EXW LV DSSOLHG WR DQ H[WUDRUGLQDU\ GRPDLQ LQ /HZLVn DQDO\VLV 2XU SUHYLRXV XQGHUVWDQGLQJ KHUH FDQQRW EH E\ ZD\ RI VHPH LQGHSHQGHQW JUDVS RI FRPSDUDn WLYH VLPLODULW\ DV DSSOLHG WR SRVVLEOH ZRUOGV VLQFH WKH DSSOLFDWLRQ ZRXOG QRW KDYH DULVHQ EXW IRU /HZLVn DFFRXQW $QG WKH IDFW WKDW FRPSDUDn WLYH VLPLODULW\ RI SRVVLEOH ZRUOGV LQYROYHV GLIIHUHQW VWDQGDUGV WKDQ FRPn SDUDWLYH VLPLODULW\ RI PRUH SURVDLF WKLQJV VXJJHVWV WKDW LQGHSHQGHQW XQGHUn VWDQGLQJ LV LPSRVVLEOH $V LQ RXU GLVFXVVLRQ RI SRVVLEOH ZRUOGV UHDOLVP D VFLHQWLILF DQDORJ\ PD\ EH RI VRPH KHOS LQ VHHLQJ /HZLVn DFFRXQW DV DQ DQDO\VLV 7R XQGHUn VWDQG WKH NLQHWLF PROHFXODU WKHRU\ RI JDVHV UHTXLUHV WKDW ZH XQGHUVWDQG I WKH DSSOLFDWLRQ RI WKH FRQFHSW RI PRWLRQ RXWVLGH WKH GRPDLQ RI WKH FRQFHSW RI PRWLRQ LQ RXU RUGLQDU\ SK\VLFDOREMHFW ODQJXDJH 7KLV GRHV QRW PDNH IRU D QHZ FRQFHSW RI PRWLRQ SULPLWLYH WR WKH NLQHWLF PROHFXODU WKHRU\ EXW UDWKHU DQ H[WHQVLRQ E\ ZD\ RI DQDORJ\ RI RXU FRQFHSW RI PRWLRQ WR D QHZ GRPDLQ :H JUDVS WKH PLFURVFRSLF WKHRU\ WKURXJK D PDFURVFRSLF

PAGE 127

DQDORJ\ RU QR GHO 7KDW WKH PRWLRQ RI PROHFXOHV KDV SURSHUWLHV WKDW WKH PRWLRQ RI ELOOLDUG EDOOV GRHV QRW KDYH LV OHDUQHG LQ WKH FRQWH[W RI WKH QHZ WKHRU\ DIWHU WKH EULGJLQJ DQDORJ\ KDV EHHQ PDGH /RHZHU > S @ FKDUJHV WKDW HLWKHU FRPSDUDWLYH VLPLODULW\ LV QRW ZHOO HQRXJK XQGHUVWRRG WR VXSSRUW DQ DQDO\VLV RI FRXQWHUIDFWXDOV RU HOVH LW DSSHDUV LQ /HZLVn DFFRXQW DV D SULPLWLYH WHFKQLFDO FRQFHSW DQG VR FDQQRW EH DQWHFHGHQWO\ XQGHUVWRRG +H FKDUJHV WKDW /HZLV LQFRQVLVn WHQWO\ VHHPV WR ZDQW WR KDYH LW ERWK ZD\V FRPSDUDWLYH VLPLODULW\ LV DQWHFHGHQWO\ XQGHUVWRRG DQG DOVR LV D QHZ WHFKQLFDO FRQFHSW SULPLWLYH WR /HZLVn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n WHU EXW WKHQ ZKDW KH RIIHUV LV QRW DQ DQDO\VLV +RZHYHU WKHUH ZRXOG DSn SHDU WR EH QR RUGLQDU\ DSSOLFDWLRQV RI FRPSDUDWLYH VLPLODULW\ WKDW ZRXOG VXJJHVW WKH QHHG IRU WKH VSHFLDO SULQFLSOHV LQYROYHG LQ DSSO\LQJ LW WR SRVVLEOH ZRUOGV WKXV WKHVH VSHFLDO SULQFLSOHV FDQQRW EH DQWHFHGHQWO\ XQGHUVWRRG 2Q WKH FRQWUDU\ DP VXJJHVWLQJ WKDW /HZLV FDQ KDYH LW ERWK ZD\V ZLWKRXW LQFRQVLVWHQF\ 7KH FRQFHSW RI FRPSDUDWLYH RYHUDOO VLPLODULW\ FDQ EH LQWURGXFHG WKURXJK DQDORJ\ WR LWV DQWHFHGHQWO\ XQGHUVWRRG DSSOLn FDWLRQV 7R WKDW H[WHQW LW LV QRW VLPSO\ D SULUPLWLYH WHFKQLFDO FRQFHSW

PAGE 128

+RZHYHU LQ D QHZ GRPDLQ FHUWDLQ IDFWRUV LQYROYHG LQ FRPSDULVRQ DUH PRUH LPSRUWDQW WKDQ RWKHUV LQFOXGLQJ IDFWRUV WKDW PLJKW QRW KDYH DSn SHDUHG LQ WKH DQDORJRXV XVDJHV DSSOLFDWLRQ WR FLWLHV RU IDFHV 7KH QHZ IDFWRUV FDQ EH LQWURGXFHG WKURXJK SDUDGLJPDWLF H[DPSOHV LQ WKH FRQWH[W RI WKH WKHRU\ /HZLV RIIHUV 7KDW WKH UHVXOWLQJ DQDO\VLV EHDUV PRUH UHVHPEODQFH WR D VFLHQn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n DFFRXQW D WKHRU\ RI FRXQWHU IDFWXDOV UDWKHU WKDQ DQ DQDO\VLV %XW WKHQ LV WKHUH UHDOO\ PXFK GLIn IHUHQFH EHWZHHQ WKH WZR" 7UDGLWLRQDO DQDO\VHV VXFK DV 5XVVHOOnV WKHRU\ RI GHILQLWH GHVFULSWLRQV DUH SKLORVRSKLFDO WKHRULHV RQ DQDORJ\ ZLWK VFLHQWLILF WKHRULHV :H WHVW VXFK WKHRULHV DJDLQVW RXU LQWXLWLRQV UDWKHU WKDQ ZLWK SK\VLFDO H[SHULPHQWV ,W LV LQ WKLV WUDGLWLRQ WKDW /HZLV LV ZULWLQJ (YHQ WKRXJK ZH DFFHSW SRVVLEOH ZRUOGV DQG VRPH NLQG RI VLPLODULW\ RUGHULQJ RI WKHP DV DSSURSULDWH IRU DQ DQDO\VLV RI FRXQWHUIDFWXDOV DQG DQ H[SODQDWLRQ RI FRXQWHUIDFWXDO GHOLEHUDWLRQ WKHUH LV URRP IRU GLVn DJUHHPHQW RQ WKH W\SH RI RUGHULQJ LQYROYHG 7KH FRQWURYHUV\ RYHU WKH /LPLW $VVXPSWLRQ SURYLGHV RQH LOOXVWUDWLRQ RI WKLV ,Q 6HFWLRQ ZH GLVFXVV VRPH RI WKH YDULHWLHV RI RUGHU WKDW KDYH DSSHDUHG LQ SRVVLEOH ZRUOGV DQDO\VHV RI FRXQWHUIDFWXDOV DOWHUQDWLYH WR /HZLVn

PAGE 129

9DULHWLHV RI 2UGHU $V GHYHORSHG E\ /HZLV FRPSDUDWLYH RYHUDOO VLPLODULW\ LV D IDPLO\ RI RUGHULQJ UHODWLRQV RQ WKH VHW RI SRVVLEOH ZRUOGV :H LQGLFDWHG LQ 6HFWLRQ KRZ /HZLVn VSKHUH IXQFWLRQ SURGXFHG IRU HDFK ZRUOG D V\VWHP RI VSKHUHV DERXW WKDW ZRUOG ZKLFK FDUULHG LQIRUPDWLRQ DERXW FRPSDUDWLYH VLPLODULW\ 6XSSRVH $ DQG % DUH WZR RI WKH VSKHUHV LQ X ZLWK $F % )XUWKHU VXSSRVH ZRUOGV YZ DUH VXFK WKDW Y H $ DQG Y L % ZKLOH Z H % DQG Z L $ 7KHQ ZH VDLG WKDW Y ZDV PRUH VLPLODU WR X WKDQ Z 2Q WKH RWKHU KDQG LI HYHU\ VSKHUH WKDW FRQWDLQV ZRUOG Z DOVR FRQWDLQV ZRUOG Y WKHQ Y LV DW OHDVW DV VLPLODU WR X DV Z (LWKHU RI WKH XQGHUOLQHG SKUDVHV DERYH FRQVWLWXWHV DQ RUGHULQJ UHODWLRQ ZLWK UHVSHFW WR X RQH VWULFW WKH RWKHU ZHDN ,Q KLV FKDSWHU RQ UHIRUPXODWLRQV /HZLV VKRZV WKDW RQH FDQ FRQVWUXFW D VHPDQWLFV HTXLYDOHQW WR KLV VSKHUH IXQFWLRQ VHPDQWLFV GLUHFWO\ LQ WHUPV RI D WKUHHSODFH FRPSDUDWLYH VLPLODULW\ UHODWLRQ > SS @ :H PD\ GHILQH D UHODWLRQ 5RQ8[8[8DV IROORZV /HW 8 EH D VHW RI SRVVLEOH ZRUOGV DQG D VSKHUH IXQFWLRQ :H GHILQH 5 e 8 [ 8 [ 8 E\ Y5AZ LII IRU DOO $ H A LI Z H $ WKHQ Y H $ )RU XYZ H 8 Y5AZ LV WR EH UHDG DV Y LV DW OHDVW DV VLPLODU WR X DV Z :H ZLOO FRQWLQXH WR DVVXPH WKDW 8X 8 VR WKDW ZH GR QRW QHHG D VSHFLILn FDWLRQ RI VRPH DFFHVVLELOLW\ IXQFWLRQ RU UHODWLRQ IRU HDFK X H 8 LQ DGGLn WLRQ WR WKH DERYH FRPSDUDWLYH VLPLODULW\ UHODWLRQ /HW XV IL[ RXU DWWHQWLRQ RQ RQH X H 8 IRU FRQYHQLHQFH WKH DFWXDO ZRUOG 5 WKHQ LQSRVHV D ZHDN WRWDO RUGHU RQ WKH ZRUOGV LQ 8 ZLWK X WKH VWULFWO\ 5APLQLPDO HOHPHQW 7KDW LV 5 VDWLVILHV WKH IROORZLQJ FRQGLWLRQV

PAGE 130

& )RU DOO [\] H 8 Df [5A[ UHIOH[LYHf Ef LI [5A\ DQG \5A] WKHQ [5A] WUDQVLWLYHf Ff HLWKHU [5A\ RU \5A[ FRQQHFWHGf Gf QRW [5 X IRU DOO [ I X X &RQGLWLRQV Df DQG Ef IROORZ LLUPHGLDWHO\ IURP ZKLOH WKH WRZHULQJ RI WKH VSKHUHV OHDGV WR Ff DQG WKH FHQWHULQJ FRQGLWLRQ WR Gf 7KDW 5 LV D ZHDN RUGHU IROORZV IURP WKH IDFW WKDW ZH PD\ KDYH ERWK [5A\ DQG \5 [ ZLWKRXW [ \ +RZHYHU LI ZH GHILQH >[@ ^\ ERWK [5A\ DQG \5X[` WKHQ >[@ LV DQ HTXLYDOHQFH FODVV RI ZRUOGV DOO HTXDOO\ VLPLODU WR X DQG WKH VHW RI HTXLYDOHQFH FODVVHV LV WRWDOO\ RUGHUHG E\ WKH LQGXFHG UHODWLRQ 5 GHILQHG DV 5 RQ DQ\ UHSUHVHQWDWLYHV RI WZR HTXLYDOHQFH FODVVHV 7KHQ >X@ FRQVLVWV RI D VLQJOH HOHPHQW X DQG LV 5 OHDVW 6LQFH /HZLV H[n SUHVVO\ ZLVKHV WR SHUPLW WLHV LQ FRPSDUDWLYH VLPLODULW\ LQ JHQHUDO IRU ZRUOG Z >Z@ ZLOO QRW EH D VLQJOHWRQ VHW 8VLQJ WKH FRPSDUDWLYH VLPLODULW\ UHODWLRQ ZH PD\ WKHQ VWDWH WKH WUXWK FRQGLWLRQ IRU D FRXQWHUIDFWXDO FRQGLWLRQDO DV :ST # X H 8 LII HLWKHU Df WKHUH LV QR SZRUOG LQ 8 RU Ef IRU VRPH Z H 8 Z LV D SZRUOG DQG IRU DOO Y VXFK WKDW Y5AZ Y LV D &STZRUOG 7KLV LV /HZLVn GHILQLWLRQ > S @ WKRXJK LQFRUSRUDWLQJ WKH UHVWULFn WLRQ WKDW 8X 8 /HZLV VKRZV WKDW WKH UHVXOWLQJ VHPDQWLFV LV HTXLYDn OHQW WR WKH VSKHUH IXQFWLRQ VHPDQWLFV

PAGE 131

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f WKHUH LV QR SSHUPLWWLQJ FODVV RU Ef IRU VRPH FODVV >Z@ >Z@ LV SSHUPLWWLQJ DQG HYHU\ ZRUOG LQ >Z@ LV D &STZRUOG 9ST # X LII ERWK Df WKHUH LV D SSHUPLWWLQJ FODVV DQG Ef IRU WKH 5AOHDVW SSHUPLWWLQJ FODVV >Z@ VRPH ZRUOG LQ >Z@ LV D .STZRUOG 1RWH WKDW 5 LV WKHQ D ZHOORUGHU RI HTXLYDOHQFH FODVVHV WKDW LV HYHU\ VHW RI HTXLYDOHQFH FODVVHV KDV D OHDVW HOHPHQW +RZHYHU 5A LWVHOI LV QRW D ZHOORUGHU VLQFH VHYHUDO ZRUOGV LQ D VHW PD\ WLH IRU PLQLPDOLW\ RU WKHUH PD\ EH QR 5 PLQLPDO ZRUOGV DV LV WKH FDVH LI D VHW RI ZRUOGV YLRODWHV WKH OLPLW DVVXPSWLRQ ,Q VXPPDU\ ZH PD\ FRQFOXGH WKDW /HZLVn DFFRXQW UHTXLUHV D ZHDN WRWDO RUGHU 5A ZLWK X VWULFWO\ 5 PLQLPDO IRU HDFK ZRUOG X 7KLV DPRXQWV WR D VWURQJ WRWDO RUGHU RQ HTXLYDOHQFH FODVVHV RI ZRUOGV HTXDOO\ VLPLODU WR X :LWK WKH OLPLW DVVXPSWLRQ WKH ODWWHU EHFRPHV D ZHOORUGHU RI HTXLYDOHQFH FODVVHV 6RPH RI WKH DFFRXQWV RI FRXQWHUIDFWXDOV LQ WHUPV

PAGE 132

RI SRVVLEOH ZRUOGV DOWHUQDWLYH WR /HZLVn DFFRXQW PD\ EH VHHQ DV YDULDn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f RI S 7KH FODVVLFDO SUREOHP KDG EHHQ WR GHWHUPLQH MXVW ZKDW FKDQJHV ZHUH UHOHYDQW WKH SUREOHP RI FRWHQDELOLW\ DV *RRGPDQ > @ ZRXOG KDYH LW $OO WKH DERYH PHQWLRQHG DXWKRUV H[FHSWLQJ *RRGPDQf DJUHH WKDW SRVVLEOH ZRUOGV FDQ KHOS XV SURGXFH D PRUH SUHFLVH DFFRXQW RI ZKDW LV LQYROYHG LQ WKLV URXJK PRGHO ,Q GRLQJ VR WKH\ DOO LQSRVH RQ WKH VHW RI SRVVLEOH ZRUOGV VRPH YDULHW\ RI RUGHU 6WDOQDNHUnV DFFRXQW LV WKH VLPSOHVW DQG LPSRVHV WKH VWURQJHVW RUGHU UHTXLUHPHQWV 7KH UHILQHPHQW RI WKH URXJK PRGHO RQ ZKLFK KH EDVHV KLV IRUPDO DFFRXQW LV WKDW ZH VHOHFW D SRVVLEOH ZRUOG Z ZKLFK GLIIHUV PLQLPDOO\ IURP WKH DFWXDO ZRUOG P\ HPSKDVLVf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

PAGE 133

7KH NH\ HOHPHQW RI 6WDOQDNHUnV DFFRXQW LV ZKDW KH FDOOV D VHOHFn WLRQ IXQFWLRQ EXW ZKDW ZH VKDOO FDOO D ZRUOGVHOHFWLRQ IXQFWLRQ VHH > S @f /HW 8 EH D VHW RI SRVVLEOH ZRUOGV DQG WKH DEVXUG ZRUOG DW ZKLFK HYHU\ SURSRVLWLRQ LV WUXH $ ZRUOGVHOHFWLRQ IXQFWLRQ LV DQ\ IXQFWLRQ I 8 [ 38f r‘ 8 8 .f VXFK WKDW IRU DQ\ SURSRVLWLRQV ST H 38f ZH LGHQWLI\ D SURSRVLWLRQ ZLWK D VHW RI SRVVLEOH ZRUOGVf DQG DQ\ ZRUOG Z H 8 Df S # IZSf Ef ,I WKHUH LV QR X H 8 DW ZKLFK S LV WUXH WKHQ IZSf Ff ,I S # Z WKHQ IZSf Z Gf ,I S # IZTf DQG T # IZSf WKHQ IZSf IZTf 7KH WUXWK FRQGLWLRQ IRU FRQGLWLRQDO :ST FDQ WKHQ EH VWDWHG LQ WHUPV RI I DV IROORZV :ST # Z LII T # I ZSf 1RZ DV DW DQ\ ZRUOG Z D JLYHQ SURSRVLWLRQ T LV HLWKHU WUXH RU IDOVH H[FOXVLYHO\ 6WDOQDNHUnV VHPDQWLFV YDOLGDWHV FRQGLWLRQDO H[FOXGHG PLGGOH &0 $:ST:S1T DV ZH KDYH QRWHG SUHYLRXVO\ 7KXV WKH PLJKWFRQGLWLRQDO FDQQRW EH GHn ILQHG DV LQ /HZLVn DFFRXQW :H PD\ IXUWKHU FRQSDUH 6WDOQDNHUnV DFFRXQW WR /HZLVn E\ QRWLQJ ZKDW YDULHW\ RI RUGHU 6WDOQDNHUnV V\VWHP UHTXLUHV /HW 8.I EH DV LQ :H GHILQH 6 F 8 [ 8 [ 88.f DV Y6XZ LII HLWKHU Z RU IRU VRPH S VXFK WKDW S # Y DQG S # Z I XSf Y ,I ZH IL[ RXU DWWHQWLRQ RQ RQH ZRUOG X H 8 WKHQ 6 HVWDEOLVKHV D ZHOO X RUGHU RI 8 8 ZLWK X WKH 6AOHDVW DQG WKH 6AJUHDWHVW HOHPHQWV

PAGE 134

&RQGLWLRQV 'Ff DQG Gf LQ SDUWLFXODU DUH UHTXLUHG IRU VKRZLQJ WKLV DQG WKH GHILQLWLRQ RI I UHYHDOV WKDW I VHOHFWV WKH 6AOHDVW SZRUOG IRU DQ\ SURSRVLWLRQ S 7KDW LV D FRXQWHUIDFWXDO :ST LV WUXH DW X SURYLGHG WKH 6AOHDVW SZRUOG LV D TZRUOG 6WDOQDNHUnV DFFRXQW LPSRVHV WKH OLPLW DVVXPSWLRQ SOXV WKH HYHQ VWURQJHU DVVXPSWLRQ WKDW IRU HQWHUWDLQDEOH DQWHFHGHQW S WKHUH LV D PRVW VLPLODU SZRUOG 7KRXJK 6WDOQDNHU DSSURDFKHV WKH SUREOHP IURP WKH SRLQW RI YLHZ RI ORFDWLQJ WKH OHDVW GLIIHUHQW ZRUOG WKDW PDNHV WKH DQWHFHGHQW WUXH WKLV DPRXQWV WR ORFDWLQJ WKH PRVW VLPLODU ZRUOG WKDW PDNHV WKH DQWHFHGHQW WUXH 7KXV 6WDOQDNHUnV DFFRXQW LV VLPSO\ D PRUH UHVWULFWHG YHUVLRQ RI /HZLVn 2QH FDQ DUULYH DW /HZLVn SRVLWLRQ E\ DVNLQJ WZR TXHVWLRQV DERXW 6WDOQDNHUnV DVVXPSWLRQV IRU D JLYHQ DQWHFHGHQW S ZK\ VKRXOG WKHUH EH MXVW RQH PLQLPDOO\ GLIIHUHQW ZRUOG" )XUWKHUPRUH ZK\ VKRXOG WKHUH EH DQ\ PLQLPDOO\ GLIIHUHQW ZRUOG" /HZLVn DUJXPHQW IRU WKHUH EHLQJ PRUH DQG PRUH VLPLODU ZRUOGV ZLWKRXW HQG LV DQ DUJXPHQW IRU WKHUH EHLQJ OHVV DQG OHVV GLIIHUHQW ZRUOGV ZLWKRXW HQG 7KHUH PD\ IRU FHUWDLQ DQWHFHGHQWV EH QR PLQLPDOO\ GLIIHUHQW ZRUOG %XW HYHQ DVVXPLQJ WKDW WKHUH DUH PLQLPDOO\ GLIIHUHQW ZRUOGV WKHUH FRXOG EH PRUH WKDQ RQH 7KH %L]HW DQG 9HUGL H[DPSOH RU -DFNVRQnV H[DPSOH GLVFXVVHG LQ WKH SUHYLRXV VHFWLRQ LOOXVWUDWH WKLV SRVn VLELOLW\ +RZHYHU /HZLVn DQG 6WDOQDNHUnV DFFRXQWV VKDUH D VLJQLILFDQW DV VXQSWLRQ WKDW WKH RUGHU RI WKH SRVVLEOH ZRUOGV LV FRQQHFWHG %RWK 3ROORFN DQG 1XWH DUJXH IRU DQDO\VHV WKDW LQFRUSRUDWH D SDUWLDO RUGHU UDWKHU WKDQ D WRWDO RUGHU :H VKDOO ILUVW GLVFXVV WKHLU LQIRUPDO DUJXn PHQWV DQG WKHQ IRU 3ROORFN RQO\ GHILQH DQ RUGHU UHODWLRQ LQ WHUPV RI D VHPDQWLFV DGHTXDWH WR KLV ORJLF RI WKH VLPSOH VXEMXQFWLYH DV KH FDOOV RXU ZRXOGFRXQWHUIDFWXDO > S @

PAGE 135

,Q >@ “XWH DUJXHV WKDW LW LV QRW VXIILFLHQW WR FRQVLGHU MXVW WKH ZRUOGVf PRVW VLPLODU WR WKH DFWXDO ZRUOG LQ ZKLFK WKH DQWHFHGHQW LV WUXH QRU ZRUOGV PRUH VLPLODU WR WKH DFWXDO ZRUOG LQ ZKLFK DQWHFHGHQW DQG FRQVHTXHQW DUH WUXH WKDQ DQ\ ZRUOGV LQ ZKLFK WKH DQWHFHGHQW DQG FRQVHn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n FHGHQWV 6'$ &:$STU.:SU:TU DOVR GLVFXVVHG LQ 6HFWLRQ ,I ( LV HYDOXDWHG DV WUXH DV ZRXOG VHHP UHDVRQDEOH RQ /HZQVn VHPDQWLFV WKHQ 6'$ OLFHQVHV WKH LQIHUHQFH RI ( ,I WKH VXQ ZHUH WR JURZ FROG EHIRUH WKH HQG RI WKH VXPPHU ZH ZRXOG KDYH D EXPSHU FURS %XW WKLV LV FHUWDLQO\ IDOVH 7R DYRLG WKLV LQIHUHQFH RQH PXVW HLWKHU UHn MHFW 6'$ RU EDU WKH HYDOXDWLRQ RI ( DV WUXH :H KDYH LQGLFDWHG WKDW RXU SUHIHUHQFH LV WR UHMHFW 6'$ VLQFH DFn FHSWLQJ LW UHTXLUHV ZH JLYH XS VXEVWLWXWLRQ RI HTXLYDOHQWV 1XWH SD\V WKLV SULFH DQG RIIHUV D VHPDQWLFV LQ ZKLFK JHQHUDOO\ ZH PXVW FRQVLGHU ZRUOGV DW ZKLFK HDFK GLVMXQFW LV WUXH LQ HYDOXDWLQJ FRXQWHUIDFWXDOV ZLWK GLVMXQFWLYH DQWHFHGHQWV > S @ +RZHYHU WKH ZRUOGV ZH WKHQ VLGHU PD\ GLIIHU LQ VLPLODULW\ WR WKH DFWXDO ZRUOG 7KRXJK 1XWHnV FRQ

PAGE 136

DUJXPHQW KHUH LV OLQNHG WR 6'$ ZKLFK LQ P\ RSLQLRQ JUHDWO\ ZHDNHQV LW D VLPLODU SRLQW FDQ EH PDGH ZLWKRXW EULQJLQJ LQ 6'$ RU GLVMXQFWLYH DQWHn FHGHQWV ,Q >@ 1XWH DUJXHV DJDLQVW WKH WRWDO VLPLODULW\ RUGHULQJ RI /HZLVn DQDO\VLV LQ D ZD\ WKDW LV QRW OLQNHG WR WKH DFFHSWDQFH RI 6'$ EXW UDWKHU WR WKH NLQGV RI GLVFULPLQDWLRQV ZH DUH FDSDEOH RI PDNLQJ LQ MXGJn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n IHDWHG IRU WKH 6HQDWH KH ZLQV D 6HQWDWH ELG EXW HVWDEOLVKHV D SRRU UHFRUG 2QO\ LQ WKH VHFRQG FDVH GRHV &DUWHU VWLOO EHFRPH 3UHVLGHQW 1XWH WKHQ FODLPV WKDW LW LV GLIILFXOW HQRXJK WR GHFLGH ZKLFK VLWXDWLRQV DUH VXIn ILFLHQWO\ OLNH WKH DFWXDO VLWXDWLRQ IRU FRPSDULVRQ DW DOO ZLWKRXW KDYLQJ WR UDQN WKHP LQ VLPLODULW\ RUGHU DV ZHOO 7KH ZHDNQHVV RI WKLV DUJXPHQW LV WKDW ZH FRXOG HDVLO\ FRQVLGHU DOO WKH VLWXDWLRQV GHVFULEHG WR EH HTXDOO\ VLPLODU WR WKH DFWXDO ZRUOG RQ D VXIILFLHQWO\ ORRVH VHQVH RI FRPSDUDWLYH VLPLODULW\ 7KH FROOHFWLRQ ZRXOG WKHQ FRQVWLWXWH DQ HTXLYDOHQFH FODVV VR WKLV ZRXOG QRW GLIIHU IURP /HZLVn

PAGE 137

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f WKH DFWXDO ZRUOG /HZLV VXJn JHVWHG WKDW SHUKDSV QR FKDQJHV ZHUH PLQLPDO EXW HYHQ LI WKHUH ZHUH PRUH WKDQ RQH FKDQJH PLJKW EH PLQLPDO EXW WKHVH ZRXOG DOO UHVXOW LQ ZRUOGV RI HTXDO VLPLODULW\ WR WKH DFWXDO ZRUOG 7KH LVVXH UDLVHG E\ 1XWH LV WKDW WKH PDNLQJ RI GLIIHUHQW FKDQJHV PD\ UHVXOW LQ ZRUOGV ZKLFK GLIIHU LQ WKHLU VLPLODULW\ WR WKH DFWXDO ZRUOG VR WKDW D PLQLPDO FKDQJH RUGHU PD\ QRW EH WKH VDPH DV D FRPSDUDWLYH VLPLODULW\ RUGHU 7KLV QRWLRQ RI GLIIHUHQW EXW LQ VRPH VHQVH HTXDOO\ VPDOO FKDQJHV LV FHQWUDO WR 3ROORFNnV DFFRXQW LQ >@ RI FRXQWHUIDFWXDOV ,Q IDFW 3ROORFN VSHFLILFDOO\ FODLPV WKDW WKH RUGHU EDVHG XSRQ PLQLPDO FKDQJH LV D SDUWLDO UDWKHU WKDQ D WRWDO RUGHU > S @ 7ZR GLIIHUHQW FKDQJHV LQ WKH DFWXDO ZRUOG PD\ EH VXFK WKDW QHLWKHU FRQWDLQV D VPDOOHU FKDQJH \HW QHLWKHU FRQWDLQV WKH RWKHU 6XFK DQ RUGHU LV QRW FRQQHFWHG 3ROORFNnV DUJXPHQW IRU WKH OLPLW DVVXPSWLRQ GLVFXVVHG LQ 6HFWLRQ LV XOWLPDWHO\ DLPHG DW GLVFUHGLWLQJ FRPSDUDWLYH VLPLODULW\ LQ IDYRU RI PLQLPDO FKDQJH DV WKH RSHUDWLYH SULQFLSOH LQ FRXQWHUIDFWXDO HYDOXDWLRQ

PAGE 138

:LWK UHIHUHQFH WR /HZLVn PXFK GLVSXWHG OLQH H[DPSOH 3ROORFN FODLPV WKDW FKDQJLQJ WKH OLQH WR b ORQJ RU WR ORQJ DUH GLVWLQFW PLQLPDO FKDQJHV QHLWKHU FRQWDLQLQJ D VPDOOHU FKDQJH QRU WKH RWKHU EXW ZKLFK OLNHO\ UHVXOW LQ ZRUOGV RI GLIIHULQJ VLPLODULW\ WR WKH DFWXDO ZRUOG > S @ 3ROORFNnV IXOO DQDO\VLV QHHG QRW FRQFHUQ XV QRZ +H GHYRWHV PXFK HIIRUW WR DQDO\]LQJ WKH QRWLRQ RI PLQLPDO FKDQJH LWVHOI ZKLFK /HZLV GRHV QRW DQG SHUKDSV FDQQRW GR IRU FRPSDUDWLYH VLPLODULW\ :H ZLOO WDNH D VLPSOHU URXWH DQG FRQVWUXFW D VHPDQWLFV LQ WHUPV RI DQ XQH[SOLFDWHG QRWLRQ RI PLQLPDO FKDQJH DSSURSULDWH WR 3ROORFNnV ORJLF RI WKH VLPSOH VXEMXQFWLYH 1XWH > @ DQG &KHOODV >@ XVH D VHPDQWLFV IRU FRQn GLWLRQDOV EDVHG XSRQ VHOHFWLRQ IXQFWLRQV :H VKDOO GLVFXVV WKHVH LQ PRUH GHWDLO LQ &+$37(5 3285 /HZLVn DFFRXQW FDQ DOVR EH UHIRUPXODWHG LQ WHUPV RI VHOHFWLRQ IXQFWLRQV WKRXJK RQO\ LQ WUXWKSUHVHUYLQJ HTXLYDOHQFH WR VSKHUH IXQFWLRQV VDWLVI\LQJ WKH OLPLW DVVXPSWLRQ > SS @ %DVLFDOO\ D VHOHFWLRQ IXQFWLRQ LV LQWHQGHG WR SLFN RXW WKRVH ZRUOGV WKDW PXVW EH FRQVLGHUHG LQ HYDOXDWLQJ D SDUWLFXODU FRXQWHUIDFWXDO DW D SDUWLFXODU ZRUOG :H KDYH DOUHDG\ VHHQ DQ H[DPSOH RI VXFK D IXQFWLRQ LQ 6WDOQDNHUnV ZRUOGVHOHFWLRQ IXQFWLRQ ,Q PRUH JHQHUDO WHUPV ZH PD\ GHILQH D VHOHFWLRQ IXQFWLRQ DV IROORZV /HW 8 EH D VHW RI SRVVLEOH ZRUOGV $ VHOHFWLRQ IXQFWLRQ LV DQ\ IXQFWLRQ I 8 [ S8f 38f :H DVVXPH WKDW DOO :'U,GV KDYH DFFHVV WR DOO RWKHUV LQ 2WKHUZLVH ZH ZRXOG QHHG WR DGG DQ DFFHVVLELOLW\ FRQGLWLRQ WR WKH GHILQLWLRQ RI I :H FDQ JHW /HZLVn VHPDQWLFV IRU VLWXDWLRQV VDWLVI\LQJ WKH OLPLW DVn VXPSWLRQ E\ DGGLQJ WKH ILUVW IRXU RI WKH IROORZLQJ FRQGLWLRQV WR RXU GHILn QLWLRQ RI I

PAGE 139

& )RU DOO X H 8 DQG SURSRVLWLRQV ST H 38f DJDLQ ZH LGHQWLI\ D SURSRVLWLRQ ZLWK D VHW RI SRVVLEOH ZRUOGVf Df I XSf F S Ef LI S # X WKHQ IXSf ^X` Ff LI S F T DQG IXSf A WKHQ IXTf A Gf LI S F T DQG S $ IXTf A WKHQ IXSf S $ IXTf Hf I XSf F IX$STf RU IXTf F IX$STf If IX$STf F IXSf 8 I XTf /HZLV VKRZV WKDW D VHOHFWLRQ IXQFWLRQ VDWLVI\LQJ FRQGLWLRQV Df WKURXJK Gf ZKLFK KH FDOOV D FHQWHUHG VHWVHOHFWLRQ IXQFWLRQ LV HTXLYDOHQW WR VRPH OLPLWHG VSKHUH IXQFWLRQ > SS @ 7KH WUXWK FRQGLWLRQV IRU WKH ZRXOGFRQGLWLRQDO DQG WKH PLJKWFRQGLWLRQDO FDQ WKHQ EH VWDWHG DV :ST # X LII IRU DOO Z H IXSf T # Z 9ST # X LII IRU VRPH Z H I XSf T # Z )RU D JLYHQ VSKHUH IXQFWLRQ VDWLVI\LQJ WKH OLPLW DVVXPSWLRQ WKH HTXLYDOHQW VHWVHOHFWLRQ IXQFWLRQ SLFNV RXW WKH SZRUOGV LQ WKH LQWHUn VHFWLRQ RI DOO SSHUPLWWLQJ VSKHUHV WKDW LV WKH VHW RI FORVHVW SZRUOGV &RPSDULQJ WKLV WR RXU GHILQLWLRQ RI WKH RUGHU UHODWLRQ LQ DQG WKH VXEVHTXHQW LQGXFHG UHODWLRQ 5 RQ HTXLYDOHQFH FODVVHV RI ZRUOGV ZH VHH WKDW I SLFNV RXW WKH SZRUOGV LQ WKH 5 OHDVW SSHUPLWWLQJ HTXLYDOHQFH FODVV &RQVHTXHQWO\ ZH PD\ GHILQH WKH RUGHU UHODWLRQ IRU /HZLVn DFFRXQW E\ /HW 8I EH Z LQ DQG &DfGf :H GHILQH 5 F 8 [ 8 [ 8 E\ Y5XZ LII HLWKHU IX^YZ`f ^Y` RU IX^YZ`f ^YZ`

PAGE 140

7KLV GHILQLWLRQ LV XVHG E\ /RHZHU > S @ DQG LV HTXLYDOHQW WR WKDW JLYHQ HDUOLHU IRU OLPLWHG VSKHUH IXQFWLRQV ,W VKRXOG EH QRWHG WKDW /RHZHU PLVVWDWHV FRQGLWLRQ Gf XVLQJ S F IXTf DV WKH VHFRQG FRQGLWLRQ RI WKH DQWHFHGHQW RI Gf ,W LV FRQGLWLRQ Gf WKDW PDNHV IRU WKH WRZHULQJ RI WKH V\VWHP RI VSKHUHV DERXW X LQ WKH VSKHUH IXQFWLRQ HTXLYDOHQW WR I :LWKRXW FRQGLWLRQ Gf WKH UHPDLQLQJ FRQGLWLRQV ZRXOG QRW JHQHUDOO\ HYHQ \LHOG D SDUWLDO RUGHU 2I VRPH LQWHUHVW WKHQ DUH WKH FRQGLWLRQV UHTXLUHG IRU GHILQLWLRQ RI D SDUWLDO RUGHU VXFK WKDW QR H[WHQVLRQ WR D ZHDN WRWDO RUGHU LV JHQHUn DOO\ SRVVLEOH ,I ZH WDNH WKH QRWLRQ RI PLQLPDO FKDQJH IRU JUDQWHG WKHQ 3ROORFNnV DFFRXQW UHTXLUHV WKDW ZH VHOHFW IRU FRQVLGHUDWLRQ DOO WKRVH SZRUOGV WKDW UHVXOW IURP VRPH PLQLPDO FKDQJH LQ WKH DFWXDO ZRUOG WKDW PDNHV WKH DQWHFHn GHQW WUXH /RHZHU FODLPV > SS @ WKDW WKH VHPDQWLFV GHWHUPLQLQJ 3ROORFNnV ORJLF RI WKH VLPSOH VXEMXQFWLYH LV D VHOHFWLRQ IXQFWLRQ VHPDQn WLFV VDWLVI\LQJ FRQGLWLRQV Df WKURXJK Ff SOXV Hf DQG If :H VKDOO TXHVWLRQ WKLV FODLP LQ &+$37(5 3285 LQ RXU EURDGHU GLVFXVVLRQ RI FRQGLWLRQDO ORJLFV WKHUHf ,Q WHUPV RI WKLV VHOHFWLRQ IXQFWLRQ DQ RUGHU UHODWLRQ FDQ EH GHILQHG DV E\ /RHZHU > S @ E\ /HW 8 I EH DV LQ DQG DfFf Hf DQG If :H GHILQH 7 F 8 [ 8 [ 8 E\ Y7AZ LII IX^YZ`f ^Y` 7KH UHODWLRQ 7 DV GHILQHG GRHV QRW SHUPLW WLHV VLQFH LI Y7 Z DQG :7 Y ZH KDYH Y Z VR 7A LV DQWLV\PPHWULF :KLOH 7A LV WUDQVLWLYH H[WHQGLQJ 7X E\ DGGLQJ WKH FRQGLWLRQ RU IX^YZ`f ^YZ` UHVXOWV LQ D UHODWLRQ ZKLFK LV QRW JHQHUDOO\ WUDQVLWLYH > S QO@ $V IX^XZ`f ^X` X LV WKH 7 OHDVW HOHPHQW :KLOH I GRHV QRW SLFN RXW PHPEHUV RI DQ

PAGE 141

HTXLYDOHQFH FODVV RI ZRUOGV I GRHV SLFN RXW WKH VHW RI APLQLPDO PHPEHUV RI S 7KH GLVWLQFW PHPEHUV RI WKLV PLQLPDO VHW DUH 7 LQFRPSDUDEOH 7KXV ZKHWKHU RXU JRYHUQLQJ SULQFLSOH IRU FRXQWHUIDFWXDO GHOLEHUDn WLRQ LV FRPSDUDWLYH VLPLODULW\ RU PLQLPDO FKDQJH ZKLFK LV D YDULHW\ RI FRPSDUDWLYH GLIIHUHQFHf WKH UHVXOW LV DQ RUGHULQJ RI SRVVLEOH ZRUOGV LQ ZKLFK WKRVH DQWHFHGHQWZRUOGV PLQLPDO LQ WKH RUGHU DUH WKH FULWLFDO RQHV IRU HYDOXDWLQJ FRXQWHUIDFWXDOV ,Q &+$37(5 )285 ZH VKDOO FRQVLGHU D EURDGHU UDQJH RI FRQGLWLRQDO ORJLFV RQO\ VRPH RI ZKLFK UHVXOW LQ DQ RUGHULQJ RI SRVVLEOH ZRUOGV %XW ZH VKDOO DUJXH WKHUH DQG LQ &+$37(5 ),9( WKDW DQ\ FRQGLWLRQDO ORJLF DGHTXDWH WR UHSUHVHQW WKH FRXQWHUIDFWXDO FRQGLWLRQDO PXVW LQFRUSRUDWH VRPH RUGHULQJ RI SRVVLEOH ZRUOGV :H UHn SOLHG EULHIO\ LQ WKH SUHYLRXV VHFWLRQ WR /RHZHUnV FKDUJH WKDW VXFK DQ RUGHULQJ PXVW EH FRQVLGHUHG SULPLWLYH WR WKH VHPDQWLFV IRU WKH FRQGLWLRQDO DQG LQFDSDEOH RI VXSSRUWLQJ DQ DQDO\VLV 7KDW WKLV LV QRW WKH FDVH ZDOO EH DUJXHG DJDLQ LQ &+$37(5 ),9( 1RWHV 7KH VHPDQWLFV XVHG WR HVWDEOLVK WKH RUGHU DUH WKRVH VXJJHVWHG E\ /RHZHU >@ :H VKDOO DUJXH LQ 6HFWLRQ WKDW /RHZHUnV FRQGLWLRQV DFWXDOO\ GR QRW GHWHUPLQH 3ROORFNnV V\VWHP 66 FRQWUD /RHZHUnV FODLP > S QO@ +RZHYHU WKH FRQGLWLRQV GR GHWHUPLQH D SDUWLDO RUGHU DQG WKLV VXIILFHV IRU RXU SUHVHQW SXUSRVHV 7KH UHVXOWLQJ V\VWHP LV FORVH WR 3ROORFNnV KRZHYHU 6HH QRWH

PAGE 142

&+$37(5 3285 02'$/ $1' &21',7,21$/ /2*,&6 $ /E GDO&RQGLWLRQDO /DQJXDJH DQG 0RGDO 6\VWHPV ( 0 5 +HUHWRIRUH ZH KDYH PDGH IUHH XVH RI D IRUPDO ODQJXDJH WKDW ZH LQWURn GXFHG E\ SURYLGLQJ WUDQVODWLRQ GLUHFWLRQV IURP RUGLQDU\ ODQJXDJH LQWR LW ,Q ZKDW IROORZV ZH VKDOO UHLQWURGXFH WKLV ODQJXDJH PRUH IRUPDOO\ SURYLGLQJ WKH V\QWD[ RI D IRUPDO V\VWHP $ IRUPDO V\VWHP FRQVLVWV RI D ODQJXDJH D VHW RI IRUPXODV GHVLJQDWHG DV D[LRPV DQG D VHW RI UXOHV RI LQIHUHQFH WRJHWKHU ZLWK ZKDW FRQVWLWXWHV WKH QRWLRQ RI D GHULYDWLRQ GHGXFWLRQ SURRIf LQ WKDW V\VWHP :H ZLOO NHHS RXU ODQJXDJH VLPSOH LQWURGXFLQJ PRVW RI WKH RSHUDWRUV ZH KDYH EHHQ XVLQJ E\ ZD\ RI GHILQLWLRQ WKDW LV DV DEEUHYLDWLRQV RI PRUH FRPn SOH[ IRUPXODV LQ RXU ODQJXDJH 6LQFH ZH DUH LQWHUHVWHG LQ FRQGLWLRQDOV RXU ODQJXDJH LV D FRQGLWLRQDO ODQJXDJH DEEUHYLDWHG &/: 7KH IROORZLQJ WKUHH GHILQLWLRQV LQWURGXFH WKH SULPLWLYH DQG GHILQHG V\PEROV RI &+,nf 7KH SULPLWLYH V\PEROV RI &/: FRQVLVW RI Df 'HQXPHUDEO\ PDQ\ SURSRVLWLRQDO VHQWHQFHf OHWWHUV 3 T U Ef 7KH WUXWKIXQFWLRQDO RSHUDWRU & Ff 7KH FRQVWDQW IDOVH SURSRVLWLRQ R Gf 7KH PRGDO RSHUDWRU / IRU QHFHVVLW\ Hf 7KH FRXQWHUIDFWXDO RSHUDWRU : IRU FRXQWHUIDFWXDO FRQGLWLRQDOLW\

PAGE 143

3 7KH VHW RI IRUPXODV ZHOOIRUPHGIRUPXODV ZIIVf RI &/: LV WKH VHW FORVHG XQGHU WKH IROORZLQJ UXOHV RI FRPSRXQGLQJ RI WKH SULPLWLYH V\PEROV Df (YHU\ SURSRVLWLRQDO OHWWHU LV D IRUPXOD Ef R LV D IRUPXOD Ff ,I D DQG E DUH IRUPXODV WKHQ &DE LV D IRUPXOD Gf ,I D DQG E DUH IRUPXODV WKHQ :DE LV D IRUPXOD Hf ,I D LV D IRUPXOD WKHQ /D LV D IRUPXOD 3 7KH GHILQHG V\PEROV RI &/: DUH WKRVH IRU WKH FRQVWDQW WUXH SURn SRVLWLRQ f QHJDWLRQ 1f FRQMXQFWLRQ .f GLVMXQFWLRQ $f PDWHULDO HTXLYDOHQFH (f FRXQWHUIDFWXDO HTXLYDOHQFH )f DQG SRVVLELOLW\ 0f Df GI &RR Ef 1D GI &DR Ff .DE GI 1&D1E Gf $DE GI &1DE Hf (DE GI .&DE&ED If )DE GI .:DE:ED Jf 0D GI 1/1D 6RPH FODXVHV RI 3 GHILQH RQH GHILQHG V\PERO LQ WHUPV RI DQRWKHU 7KHVH FDQ REYLRXVO\ EH H[SDQGHG E\ RWKHU FODXVHV WR GHILQLWLRQV LQ WHUPV RI WKH SULPLWLYH V\PEROV DORQH )RU WKRVH ZLWK SHUVQLFNLW\ IRUPDO FRQn VFLHQFHV WKH UHIHUHQFHV WR FRQMXQFWLRQ DQG WKH OLNH PD\ EH WDNHQ DV WKH JLYLQJ RI QDPHV WR WKH RSHUDWRUV WKRXJK WKH\ GR UHIOHFW WKH LQWHQGHG LQWHUSUHWDWLRQ $V ZH JR RQ ZH ZLOO ZDQW WR DGG VHYHUDO FODXVHV WR HJ IRU WKH PLJKWFRQGLWLRQDO :H ZDOO OHW &/ GHVLJQDWH WKH VXEVHW RI &/: REWDLQHG E\ RPLWWLQJ 'Hf DQG VR 'Gf DQG 'If DQG &: WKH VXEVHW RI &/: REWDLQHG E\ HPLWWLQJ 'Gf DQG

PAGE 144

VR 'Hf DQG 'Jff $ PRGDO RSHUDWRU / FDQ EH LQWURGXFHG LQWR &: E\ GHILQLWLRQ WKRXJK WKH UHYHUVH LV QRW JHQHUDOO\ WKH FDVH )RU WKH IRUPHU SXUSRVH WKH IROORZLQJ GDWDVH PD\ EH DGGHG WR Kf /D :1DD 7KH IROORZLQJ GHILQLWLRQ IL[HV WKH QRWLRQV RI GHULYDWLRQ SURRI DQG WKHRUHPKRRG $ IRUPXOD D RI &/: LV GHULYDEOH GHGXFLEOHf LQ D GHVLJQDWHG V\VWHPf IURP VHW RI IRUPXODV 6 SURYLGHG WKHUH LV D ILQLWH VHTXHQFH RI IRUPXODV D D VXFK WKDW f f Q Df D LV D Q Ef IRU HDFK DA L Qf RQH RI WKH IROORZLQJ KROGV f D LV DQ D[LRP L f DA EHORQJV WR 6 f DA IROORZV IURP RQH RU PRUH SUHYLRXV PHPEHUV RI WKH VHTXHQFH E\ D UXOH RI LQIHUHQFH 7KH IRUPXOD D LV SURYDEOH DQG VR D WKHRUHP LI WKH DERYH KROGV ZKHUH 6 :H ZLOO XVH 6 _AD WR V\PEROL]H D LV GHULYDEOH IURP 6 LQ / DQG _AD WR V\PEROL]H D LV D WKHRUHP RI / GURSSLQJ WKH VXEVFULSW ZKHQ RXU DWn WHQWLRQ LV IRFXVHG RQ RQH ORJLF 7KH ORJLF IRU D V\VWHP LV LWV VHW RI WKHRUHPV $V LV ZHOONQRZQ WKH VDPH ORJLF LH VDPH WKHRUHPV LQ WKH VDPH ODQJXDJH PD\ EHORQJ WR GLIIHUHQW V\VWHPV LQ YLUWXH RI WKHLU GLIn IHULQJ D[LRPV RU UXOHV RI LQIHUHQFH :H KDYH EHHQ DQG ZLOO FRQWLQXH WR EH PRUH FRQFHUQHG ZLWK ORJLFV WKDQ ZLWK V\VWHPV ZLWK ZKLFK WKH\ PD\ EH D[LRPLWL]HG VR IRU RXU SXUSRVHV ZH PD\ XVH WKH WHUPV LQWHUFKDQJHDEO\ 7KLV ZRXOG EH D PLVWDNH LI ZH ZHUH LQWHUHVWHG LQ FRPSDULQJ GLIIHUHQW D[LRPDWL]DWLRQV IRU WKH VDPH ORJLFf

PAGE 145

/HW DQG GHQRWH WZR ORJLFV V\VWHPVf 7KHQ FRQWDLQV SURYLGHG HYHU\ WKHRUHP RI LV D WKHRUHP RI /A 8QGHU WKH VDPH FRQn GLWLRQV ZH DOVR VD\ LV DQ H[WHQVLRQ RI /A 7KH FRQWDLQPHQW LV SURSHU SURYLGHG WKHUH LV D WKHRUHP RI /A WKDW LV QRW D WKHRUHP RI (YHU\ V\VWHP ZKLFK ZH VKDOO FRQVLGHU LV DQ H[WHQVLRQ RI FODVVLFDO SURSRVLWLRQDO FDOFXOXV 3&f :H FDQ JXDUDQWHH WKLV E\ SURYLGLQJ RXU V\VWHPV DW D PLQLn PXP ZLWK WKH IROORZLQJ D[LRPV DQG UXOHV RI LQIHUHQFH '$ $[LRPV $O &S&TS $ &&S&TU&&ST&SU $ &&&SRRS 5XOHV RI LQIHUHQFH 03 0RGXV 3RQHQV )URP D DQG &DE LQIHU E 86 8QLIRUP 6XEVWLWXWLRQ )URP D LQIHU DEf ZKHUH DEf LV WKH UHVXOW RI XQLIRUPO\ VXEVWLWXWLQJ IRUPXOD E IRU HYHU\ RFFXUUHQFH RI DQ\ SURSRVLWLRQDO OHWWHU LQ D 7KLV LV RQH RI WKH VWDQGDUG D[LRPDWL]DWLRQV RI 3& 6HJHUEHUJ >@ =HPDQ >@f 2XU PLQLPDO ORJLF ZLOO EH WKH GHGXFWLYH FORVXUH RI &/: XQGHU WKH DERYH D[LRPV DQG UXOHV RI LQIHUHQFH DQG KHQFH ZLOO FRQWDLQ 3& ,Q YLUn WXH RI WKLV ZH ZLOO IUHHO\ PDNH XVH RI YDULRXV VWDQGDUG UHVXOWV IRU 3& $OO WUXWKIXQFWLRQDO WDXWRORJLHV DUH LQ RXU PLQLPDO ORJLF DQG WKURXJK 03 DQG 86 DOO VXEVWLWXWLRQ LQVWDQFHV RI WKHP LQ &/: 7KH SUHVHQFH RI / DQG : DUH REYLRXVO\ UDWKHU LQHVVHQWLDO VR ZH ZLOO XVXDOO\ UHIHU WR WKLV V\VWHP DV 3&

PAGE 146

7KH GHILQLWLRQ DQG UHVXOWV EHORZ DUH VWDQGDUG LQ DOO RI ZKDW IROORZV OHW / EH D V\VWHP FRQWDLQLQJ 3& Df / LV FRQVLVWHQW SURYLGHG WKH QHJDWLRQ RI D WDXWRORJ\ LV QRW DPRQJ LWV WKHRUHP RWKHUZLVH LQFRQVLVWHQW Ef $ VHW 6 RI IRUPXODV RI &/:f LV /FRQV LV WHQW SURYLGHG WKH QHJDWLRQ RI QR WDXWRORJ\ LV GHULYDEOH IURP 6 LQ / RWKHUZLVH /LQFRQVLVWHQW Ff $ VHW RI IRUPXODV 6 LV PD[LPDOO\ /FRQVLVWHQW SURYLGHG D IRUPXOD D LV LQ 6 LII 1D LV QRW LQ 6 $OWHUQDWHO\ SURYLGHG IRU DOO IRUPXODV D QRW LQ 6 6 8 Df LV /LQFRQVLVWHQWf 7 ,I D VHW 6 RI IRUPXODV LV /FRQVLVWHQW WKHQ WKHUH LV D PD[LPDOO\ /FRQVLVWHQW H[WHQVLRQ RI 6 6r /LQGHQEDXPnV /HQLQDf 6HH /HZLV > S @ IRU VLPSOH SURRIf 7 &DE LV GHULYDEOH LQ / IURP VHW RI IRUPXODV 6 LII E LV GHULYDEOH LQ / IURP 6 8 ^D` 'HGXFWLRQ 7KHRUHPf 7 (YHU\ PD[LPDOO\ /FRQVLVWHQW VHW 6 FRQWDLQV HYHU\ WKHRUHP RI / 7 6 _D LII HYHU\ PD[LPDOO\ /FRQVLVWHQW H[WHQVLRQ RI 6 FRQWDLQV D 7 $ PD[LPDOO\ /FRQVLVWHQW VHW 6 KDV HDFK RI WKH IROORZLQJ SURn SHUWLHV Df 1D H 6r LII D L 6r Ef .DE H 6 LII ERWK D H 6f DQG E H 6r Ff $DE H 6f LII HLWKHU D H 6f RU E H 6r Gf &DE H 6n LII LI D H 6a WKHQ E H 6r Hf (DE H 6f LII HLWKHU DE H 6f RU DE L 6 7KH D[LRPV DQG UXOHV VWDWHG LQ PD\ EH FDOOHG QRQQQGDO D[LRPV DQG UXOHV RI LQIHUHQFH )ROORZLQJ LV D OLVW RI PRGDO D[LRPV DQG

PAGE 147

UXOHV RI LQIHUHQFH LQ WHUPV RI ZKLFK FHUWDLQ ORJLFV DQG IDPLOLHV RI ORJLFV PD\ EH FODVVLILHG :H DVVXPH WKH ODQJXDJH LV &/ ( 0RGDO UXOHV RI LQIHUHQFH 5( )URP (DE LQIHU (/D/E (0 )URP &DE LQIHU &/DOE 55 )URP &.DEF LQIHU &./DOE/F 51 )UFP D LQIHU /D 5. ,QIHU &.. ./DA,A /DQ/ Q e IURP &.. .DQD D E Q &RQYHQWLRQDOO\ 5. IRU Q LV 51 DQG 5. IRU Q LV 50f ( 0RGDO D[LRPV 0 &/.ST./S/T 5 &./S/T/.ST &/&ST&/S/T 1 /, 4 /S 6 0S 7 &/SS 8 &/S//S % &0/SS ( 0/S/S $ EDVLV IRU D V\VWHP LV WKH VHW RI D[LRPV DQG UXOHV RI LQIHUHQFH IRU LW +HQFH $ WKURXJK $ SOXV 03 DQG 86 LV D EDVLV IRU 3& :H PD\ LQGLFDWH WKLV DV 3& $O $ $ 03 86 :KHUH / LV D ORJLF DQG /n DQ H[WHQVLRQ RI / ZH PD\ LQGLFDWH D EDVLV IRU /n E\ /n / 5 ZKHUH DGGLQJ 5 WR WKH EDVLV IRU / SURGXFHV D EDVLV IRU /n 7KH DPELJXLW\ ZKHUHn E\ 3& GHQRWHV ERWK V\VWHP DQG ORJLF ZLOO EH H[WHQGHG VR WKDW WKH H[SUHVVLRQ

PAGE 148

IRU WKH EDVLV IRU D V\VWHP ZLOO DOVR EH DQ H[SUHVVLRQ IRU WKH ORJLF RI WKDW V\VWHP 7KH IROORZLQJ V\VWHPV DUH FHQWUDO WR 6HJHUEHUJnV FODVVLILFDWLRQ RI PRGDO ORJLFV ( Df ( 3& 5( Ef 5 3& 5 (0 Ff 5 51 ,Q 6HJHUEHUJnV WHUPLQRORJ\ D ORJLF ZKLFK FRQWDLQV ( LV FDOOHG FODVVLFDO WKDW ZKLFK FRQWDLQV 5 UHJXODU DQG WKDW ZKLFK FRQWDLQV QRUPDO 7KH UXOH 5( LV VRPHWLPHV FDOOHG ,QWHUFKDQJH RU 6XEVWLWXWLRQ RI (TXLYDOHQWV (0 5HJXODULW\ LQ 6HJHUEHUJ >@ D PLVQRPHUf DQG 51 1HFHVVLWDWLRQ 7KDW 5 LV DQ H[WHQVLRQ RI ( IROORZV IURP WKH IDFW WKDW 5( LV GHULYDEOH LQ 5 )RU LI (DE EH DVVXPHG WKHQ E\ 3& ERWK &DE DQG &ED IROn ORZ ZKHQFH E\ 50 ERWK &/D/E DQG &/E/D IROORZ ZKHQFH E\ 3& (7Q7E IROn ORZV :KHUH ZH VD\ E\ 3& ZH PHDQ E\ VRPH UXOH RI LQIHUHQFH SULPLWLYH RU GHULYHG DOORZHG LQ 3&f 7KDW LV DQ H[WHQVLRQ RI 5 LV REYLRXV IURP WKH GHILQLWLRQ 7KH IROORZLQJ OHPPDV ZLOO DLG XV LQ SURYLGLQJ D PRUH XQLIRUP FKDUn DFWHUL]DWLRQ RI FODVVLFDO PRGDO ORJLFV / 5( LV GHULYDEOH LQ 3& 50 3URRI 7KLV LV SURYHG DERYH LQ VKRZLQJ 5 WR EH DQ H[WHQVLRQ RI ( / 55 LV GHULYDEOH LQ 5 3URRI $VVXPH &.DEF %\ 50 &/.DE/F IROORZV ZKLOH &./D/E/.DE LV DQ LQVWDQFH RI D[LRP 5 XQGHU 86 +HQFH E\ 3& WUDQVLWLYLW\ RI &f &./DOE/F IROORZV 4('

PAGE 149

/ 5 LV GHULYDEOH LQ 3& 55 3URRI &.ST.ST LV D WKHRUHP RI 3& +HQFH E\ 55 &./S/T/.ST IROORZV EXW WKDW LV 5 4(' ,Q YLHZ RI / ZH PD\ GHVLJQDWH IRXU V\VWHPV DV IROORZV ( Df ( 3& 5( Ef 0 3& 5( 50 ( 50 Ff 5 3& 5( 50 55 0 55 Gf 3& 5( 50 55 51 5 51 $ IXUWKHU OHPPD DOORZV XV WR VLPSOLI\ WKHVH EDVHV / (0 LV GHULYDEOH LQ 3& 55 3URRI $VVXPH &DE %\ 3& &.DDE IROORZV KHQFH E\ 55 ZH KDYH &./D/D/E %XW &/D./D/D LV D WKHRUHP RI 3& KHQFH E\ 3& ZH KDYH &/DOE 4(' ,Q YLHZ RI / DQG / ZH PD\ UHVWDWH (O DV IROORZV ( Df ( 3& 5( Ef 0 3& (0 Ff 5 3& 55 Gf 3& 55 51 8QIRUWXQDWHO\ 55 LV QRW GHULYDEOH LQ 3& 51 +RZHYHU 55 LV GHULYDEOH IURP 5. LQGHHG LV WKH FDVH ZKHUH Q ZKLOH 51 LV WKH FDVH ZKHUH Q KHQFH ZH PD\ UHSODFH Gf DERYH E\ Gf 3& 5. WKXV DFKLHYLQJ D FHUWDLQ QHDWQHVV $ ORJLF ZKLFK FRQWDLQV 0 LV FDOOHG PRQRWRQLF 7KLV WHUPLQRORJ\ LV LQWURGXFHG E\ &KHOODV DQG 0F.LQQH\ >@ 7KH\ GHVLJQDWH WKH VPDOOHVW PRQRWRQLF ORJLF E\ 5 DQG WKH VPDOOHVW UHJXODU ORJLF E\ &f ,W LV FOHDU WKDW FRQWDLQV 5 FRQWDLQV 0 FRQWDLQV (

PAGE 150

8VLQJ WKH IROORZLQJ OHQPDV ZH PD\ SURYLGH DOWHUQDWLYH V\VWHPV IRU 0 5 DQG XVLQJ RQO\ 3& 5( FHUWDLQ D[LRPV DV EDVHV / 50 LV GHULYDEOH LQ ( 0 3URRI $VVXPH &DE %\ 3& &D.DE IROORZV DQG &.DED LV D 3& WKHRUHP +HQFH E\ 3& (D.DE IROORZV 7KHQ E\ 5( ZH KDYH (/D/.DE DQG VR E\ 3& &/D/.DE 1RZ &/.DE./D/E LV DQ LQVWDQFH RI 0 DQG &./DOEOE LV D 3& WKHRUHP KHQFH E\ 3& ZH KDYH &/D/E WZR DSSOLFDWLRQV RI WUDQVLWLYLW\f 4(' / 51 LV GHULYDEOH LQ ( 1 3URRI $VVXQH D LV D WKHRUHP %\ 3& ZH KDYH (OD KHQFH E\ 5( (/O/D %XW /, LV 1 VR E\ 03 /D IROORZV 4(' 7KHVH OHQPDV DUH SURYHG LQ 6HJHUEHUJ > SS @ 7RJHWKHU ZLWK / ZH PD\ WKHQ FRQFOXGH ( Df 0 ( 0 Ef 5 ( 0 5 Ff (051 6WULFWO\ VSHDNLQJ ZH KDYH QRW SURYHG FRQWDLQPHQW ERWK ZD\V IRU (Df RU Ff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

PAGE 151

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

PAGE 152

ZRUOG VHPDQWLFV LQ RXU GLVFXVVLRQ RI PRGDO ORJLF LQ 6HFWLRQ ZH VKDOO KDYH RFFDVLRQ WR UHIHU WR LW DJDLQ EHORZ LQ GLVFXVVLQJ WKH FRQQHFWLRQ EHWZHHQ QHLJKERUKRRG DQG UHODWLRQDO VHPDQWLFV 7KH FHQWUDO QRWLRQV RI QHLJKERUKRRG VHPDQWLFV LQGHHG RI PRGHO WKHRUHWLF VHPDQWLFV DUH WKRVH RI D IUDPH D PRGHO RQ WKDW IUDPH DQG WUXWK RI D IRUPXOD ZLWK UHIHUHQFH WR WKDW PRGHO 7KH GHILQLWLRQV EHORZ DUH DGRSWHG IURP 6HJHUEHUJ > SS @ $ QHLJKERUKRRG IUDPH ) 81 LV DQ RUGHUHG SDLU VXFK WKDW Df 8 LV D VHW RI SRVVLEOH ZRUOGVf DQG Ef 18 rf 338ff LV D IXQFWLRQ IRU HDFK X H 8 LV D VHW RI VXEVHWV RI 8 FDOOHG WKH VHW RI QHLJKERUn KRRGV RI X $ PRGHO 0 819 RQ IUDPH ) 81 LV DQ RUGHUHG WULSOH VXFK WKDW 93 8 LV D IXQFWLRQ ZKHUH 3 LV WKH VHW RI SURSRVLWLRQDO OHWn WHUV ,I 3 KDV EHHQ RUGHUHG E\ WKH VHW RI QDWXUDO QXPEHUV DQG VR LV FRXQWDEOH WKHQ 91 r‘ 8 ZLOO GR DV ZHOOf 7KH WUXWK LQ 0 RI D IRUPXOD D DW ZRUOG X LQ 8 V\PEROL]HG DV D LV GHILQHG DV IROORZV Df )RU DOO S H 3 S LII X H 9Sf Ef 1RW R Ff &DE LII LI DWKHQ E Gf /D LII IRU VRPH $ H 1A $ ^ Y Y H 8 DQG D ` :H DUH SUHVHQWO\ UHVWULFWLQJ RXU DWWHQWLRQ WR ODQJXDJH &/ VR QR FODXVH LV UHTXLUHG IRU :DE DW WKLV WLQH 7KH VHW $ DERYH LV UHIHUUHG WR DV WKH VHW RI ZRUOGV ZKHUH IRUPXOD D LV WUXH DQG LV FRQYHQWLRQDOO\ V\PEROL]HG __D^>A $ IRUPXOD QRW WUXH E\ ZLOO EH VDLG WR EH IDOVH V\PEROL]HG M \ D :H ZLOO GURS WKH VXSHUVFULSWV ZKHQ VDIH WR GR VR

PAGE 153

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f ,I / LV D ORJLF DQG & D FODVV RI IUDPHV WKHQ / LV GHWHUPLQHG E\ & SURYLGHG / LV ERWK FRQVLVWHQW DQG FRPSOHWH ZLWK UHVSHFW WR & ZKHUH / LV FRQVLVWHQW ZLWK UHVSHFW WR & LI HYHU\ ) LQ & LV D IUDPH IRU / DQG / LV FRPSOHWH ZLWK UHVSHFW WR & LI HYHU\ IRUPXOD YDOLG LQ & LV D WKHRUHP RI / ,I & FRQWDLQV RQO\ ) WKHQ / LV GHWHUPLQHG E\ ) ,I HDFK PHPEHU RI & VDWLVILHV IL[HG FRQGLWLRQ F WKHQ / LV GHWHUPLQHG E\ F :H PD\ PDNH WKH LPPHGLDWH REVHUYDWLRQ WKDW HYHU\ QRQPRGDO D[LRP EHLQJ WUXWKIXQFWLRQDOO\ YDOLG LV YDOLG LQ WKH FODVV RI DOO QHLJKERUn KRRG IUDPHV ZKLOH 86 DQG 03 SUHVHUYH YDOLGLW\ VR QR FRQWUDGLFWLRQ FDQ EH GHULYHG +HQFH / 3& LV FRQVLVWHQW ZLWK UHVSHFW WR WKH FODVV RI DOO IUDPHV 7KXV LV ZLOO EH QHFHVVDU\ RQO\ WR FKHFN WKH PRGDO D[LRPV DQG UXOHV RI LQIHUHQFH WR VKRZ DQ\ PRGDO ORJLF FRQVLVWHQW ZLWK UHVSHFW WR VRPH FODVV RI IUDPHV 7KH FRQVLVWHQF\ RI ( LV WKXV QHDUO\ DV LLPHGLDWH / ( LV FRQVLVWHQW ZLWK UHVSHFW WR WKH FODVV RI DOO IUDPHV 3URRI :H VKRZ 5( SUHVHUYHV YDOLGLW\ 6XSSRVH (DE LV YDOLG LQ WKH FODVV RI DOO IUDPHVf 7KHQ IRU DQ\ PRGHO 0 RQ DQ\ IUDPH __D__ __E__ )RU LI QRW WKHQ WKHUH LV VRPH ZRUOG ZKHUH (DE GRHV QRW KROG +HQFH E\ 'Gf

PAGE 154

ZH KDYH HLWKHU ERWK /DDQG /ERU QHLWKHU IRU HDFK X LQ 8 +HQFH ZH KDYH (/D/E 4(' 7KH RWKHU ORJLFV ZH KDYH PHQWLRQHG DUH FRQVLVWHQW DQG DV ZH VKDOO VHH FRPSOHWH ZLWK UHVSHFW WR VXEFODVVHV RI WKH FODVV RI DOO IUDPHV 7R GLVWLQJXLVK WKHVH ZH PDNH QRWH RI WKH IROORZLQJ FRQGLWLRQV ZKLFK PD\ EH SODFHG RQ D IUDPH ( Pf $ $ % V 1 LPSOLHV $ H 1 DQG % H 1 X Y X X Uf $% H LPSOLHV $ $ % H 1 Qf 8 H 1X Tf 1X 38f Vf 1 X Wf 1A A LPSOLHV X H $1 5HIHUULQJ WR WKHVH FRQGLWLRQV ZH PD\ FODVVLI\ ZRUOGV DV IROORZV Df VLQJXODU ZRUOGV VDWLVI\ Vf Ef PRQRWRQLF ZRUOGV VDWLVI\ Pf Ff UHJXODU ZRUOGV VDWLVI\ ERWK Pf DQG Uf Gf QRUPDO ZRUOGV VDWLVI\ Pf Uf DQG Qf MRLQWO\ 1RWH WKDW PRQRWRQLF DQG UHJXODU ZRUOGV PD\ EH VLQJXODU LQ ZKLFK FDVH Pf RU Pf DQG Uf DUH YDFXRXVO\ VDWLVILHG $OVR WKH VDWLVIDFWLRQ RI Qf SUHFOXGHV Vf DQG FRQYHUVHO\ VR QRUPDO ZRUOGV DUH WKRVH IRU ZKLFK 1X L DQG $ $ % H LII $% H 1A RU LQ RWKHU ZRUOGV 1 LV D ILOWHU LQ WKH VXEVHW DOJHEUD RI 8 7KLV LV 6HJHUEHUJnV GHILQLWLRQ RI QRUPDO ZRUOGV > S @ 1RWH IXUWKHU WKDW Pf DQG QRW Vf WRJHWKHU LPSO\ Qf WKXV VKRZLQJ WKH HTXLYDOHQFH RI WKHVH WZR GHILQLWLRQV RI QRUPDO ZRUOGVf

PAGE 155

)UDQNV PD\ WKHQ EH FODVVLILHG EDVHG RQ WKH ZRUOGV LQ WKHP Df VLQJXODU IUDPHV FRQVLVW RI VLQJXODU ZRUOGV H[FOXVLYHO\ Ef PRQRWRQLF IUDPHV FRQVLVW RI PRQRWRQLF ZRUOGV H[FOXVLYHO\ Ff UHJXODU IUDPHV FRQVLVW RI UHJXODU ZRUOGV H[FOXVLYHO\ Gf QRUPDO IUDPHV FRQVLVW RI QRUPDO ZRUOGV H[FOXVLYHO\ 1RWH WKDW ERWK PRQRWRQLF DQG UHJXODU IUDPHV PD\ FRQWDLQ VLJXODU ZRUOGV WKRXJK QRW VR QRUPDO IUDPHV :H ZLOO GHQRWH E\ & WKH FODVV RI DOO IUDPHV DQG E\ &F WKH FODVV RI IUDPHV HDFK IUDPH RI ZKLFK VDWLVLIHV SRVVLEO\ PXOWLSOHf FRQGLWLRQ Ff 7KXV IRU H[DPSOH & LV WKH FODVV RI QRUPDO IUDPHV 7KH ILJXUH EHORZ UHSUHVHQWV WKH FRQWDLQPHQW UHODWLRQV RI VRPH RI WKHVH FODVVHV UHDGLQJ WKH DUURZ DV FRQWDLQV )LJXUH ,Q WKH QH[W VHULHV RI OHPPDV ZH SURYH WKH FRQVLVWHQF\ RI YDULRXV RI WKH ORJLFV ZH KDYH PHQWLRQHG ZLWK UHVSHFW WR DQ DSSURSULDWH FODVV RI IUDPHV $SSURSULDWH EHFDXVH LW ZLOO WXUQ RXW WR EH WKH ODUJHVW FODVV RI IUDPHV ZLWK UHVSHFW WR ZKLFK HDFK LV FRQVLVWHQWf )LUVW KRZHYHU ZH PDNH VRPH REVHUYDWLRQV DERXW GHILQHG FRQQHFWLYHV LQ HIIHFW H[SDQGLQJ XSRQ WKH WUXWK GHILQLWLRQ WKURXJK RXU GHILQLWLRQV RI WKH FRQQHFWLYHV DQG VHW WKHRUHWLF FRQVLGHUDWLRQV ( )RU DQ\ PRGHO Df __.DE__ __D__ $ __E__

PAGE 156

Ef __$DE__ ,, D ,, 8 __E__ Ff __1D__ 8 O_D__ Gf (DE LII D LII E fX fX nX Hf .DE LII ERWK D DQG E fX ‘X fX If $DE LII HLWKHU D RU E fX n8 fX Jf 1D nX LII @M D Kf 0D nX LII ,M  /1D / ( 6 LV FRQVLVWHQW ZUW ZLWK UHVSHFW WRf &f 2 3URRI ,Q YLHZ RI / LW LV VXIILFLHQW WR VKRZ 6 LV YDOLG LQ &f 6R E DVVXPH 0 LV DQ\ PRGHO RQ DQ\ IUDPH LQ & DQG X LV DQ\ ZRUOG %\ FRQGLWLRQ Vf KHQFH __1S__ O 1 6R E\ WKH WUXWK GHILQLWLRQ /1S KHQFH 03 4(' / 0 ( 0 LV FRQVLVWHQW ZUW & f§ f§ P 3URRI ,W LV VXIILFLHQW WR VKRZ 0 LV YDOLG LQ & /HW 0 EH DQ\ PRGHO RQ DQ\ IUDPH LQ DQG VXSSRVH /.ST 7KHQ E\ WUXWK GHILQLWLRQ __.ST_> H 1A KHQFH __S__ $ __T__ H 1 +HQFH E\ FRQGLWLRQ Pf ERWK __S__ __T__ H 1A 6R E\ WUXWK GHILQLWLRQ /S DQG /T DQG VR ./S/T 4(' / ( 5 LV FRQVLVWHQW ZUW & 3URRI ,W LV VXIILFLHQW WR VKRZ 5 LV YDOLG LQ & /HW 0 EH DQ\ PRGHO RQ DQ\ IUDPH LQ &A $VVXPH ./S/T 7KHQ E\ WKH WUXWK GHILQLWLRQ /S DQG n_ /T 6R E\ WKH WUXWK GHILQLWLRQ __S>_ H DQG __T__ H 1A +HQFH E\ FRQGLWLRQ Uf __S__ $ __T__ H 1A 6R __.ST__ H 1A KHQFH E\ WUXWK GHILQLn WLRQ /.ST 4(' / ( 4 LV FRQVLVWHQW ZUW & T 3URRI f /HWWLQJ 0 EH DQ\ PRGHO RQ DQ\ IUDPH LQ DQG X DQ\ ZRUOG E\ FRQGLWLRQ Tf 38f /HW S EH DQ\ SURSRVLWLRQ 7KHQ __S__ F 8 KHQFH E\ Tf @_S_M ( 1X! 7KHUHIRUH /S 4('

PAGE 157

/ ( 1 LV FRQVLVWHQW ZUW & 3URRI /HWWLQJ 0 EH DQ\ PRGHO RQ DQ\ IUDPH LQ & DQG X DQ\ ZRUOG LW IROORZV IURP FRQGLWLRQ Qf WKDW 8 H 1A 3& FRQVLGHUDWLRQV VKRZ ____ 8 KHQFH E\ WUXWK GHILQLWLRQ /O 4(' &RPELQDWLRQV RI WKH DERYH OHPPDV HVWDEOLVK WKH IROORZLQJ UHVXOWV IRU WKH ORJLFV ZH KDYH SULPDULO\ LGHQWLILHG /$ 5 ( 0 5LV FRQVLVWHQW ZUW & f§ f§ PU 3URRI )ROORZV IURP OHQUQDV / DQG / 4(' / ( 0 5 1 LV FRQVLVWHQW ZUW & f§ f§ PP 3URRI )ROORZV IURP OHQUQDV / DQG / 4(' :H PLJKW HVWDEOLVK VLPLODU UHVXOWV IRU RWKHU H[WHQVLRQV RI ( 2QH H[WHQVLRQ LQ SDUWLFXODU RI LQWHUHVW LV 7 WKH PRGDO ORJLF XVXDOO\ GHQRWHG E\ 7 LV DOVR GHQRWHG 7r =HPDQ >@f / 7 LV FRQVLVWHQW ZUW & LQGHHG ZUW &Mf f§ PPW W 3URRI ,W LV VXIILFLHQW WR VKRZ 7 LV YDOLG LQ & LQ YLHZ RI /f PPW $VVXPH /S 7KHQ __S__ H 1A VR 1 I 6R E\ FRQGLWLRQ Wf X H $1A KHQFH X H __S__ 7KHUHIRUH S 4(' 6\VWHP 7 DORQJ ZLWK V\VWHPV 6 7 8f DQG 6 7 8 (f DUH WKUHH RI WKH PRGDO ORJLFV WKDW KDYH GUDZQ WKH PRVW DWWHQWLRQ DV UHDVRQDEOH ORJLFV IRU RXU RUGLQDU\ QRWLRQV RI SRVVLELOLW\ DQG QHFHVVLW\ 5HFDOO ZH ,PSRVHG RQ WKH VSKHUH IXQFWLRQ FRQGLWLRQV VXIILFLHQW WR JXDUDQWHH WKH PRGDO ORJLF YDOLGDWHG WKHUHE\ ZDV 6f :H QRZ KDYH FRQVLVWHQF\ UHVXOWV IRU ( 0 5 DQG 7 7KH SURRI RI FRPSOHWHQHVV RI WKHVH ORJLFV ZLWK UHVSHFW WR WKH IUDPHV DOUHDG\ QRWHG IRU WKHP LV PRUH FRPSOLFDWHG 7KH QRWLRQ RI PD[LPDOO\ /FRQVLVWHQW VHWV SOD\V D PDMRU UROH ,W LV LQ WHUPV RI WKHVH WKDW VSHFLDO IUDPHV RI WKH UHTXLVLWH FODVVHV DUH FRQVWUXFWHG ,Q PRGHO WKHRUHWLF VHPDQWLFV VXFK

PAGE 158

IUDPHV ZKLFK ZH ZLOO GHILQH SUHFLVHO\ EHORZ DUH FDOOHG FDQRQLFDO IUDPHV DQG WKH PRGHOV FDQRQLFDO PRGHOV 7KH IROORZLQJ GHILQLWLRQ LV DGDSWHG IURP 6HJHUEHUJ > S @ /HW / EH D FODVVLFDO ORJLF DQG Df GHQRWH WKH VHW RI DOO PD[LPDOO\ /FRQVLVWHQW VHWV RI IRUPXODV Ef _D_A GHQRWH WKH PD[LPDOO\ /FRQVLVWHQW VHWV RI ZKLFK IRUPXOD D LV D PHPEHU VR 8f Ff 1A8 338ff EH D IXQFWLRQ VXFK WKDW /D H X LII _D_A H 1AG-f Gf )RU HDFK S H 3 9/Sf _S_/ 9A LV FDOOHG WKH QHLJKERUKRRG FDQRQLFDO PRGHO IRU / ,W LV HDV\ WR VKRZ WKDW LQ LV ZHOOGHILQHG DQG XQLTXH %HFDXVH / LV FODVVLFDO LV XQDPELJXRXV ZLWK UHVSHFW WR UHSUHVHQWDWLYHV RI D VHW RI PD[LPDOO\ /FRQVLVWHQW VHWV 3‘ LQ LV XQDPELJXRXV 7KDW LV LI _D_A _E>A WKHQ ODO H 1 LII ,E ,7 H 1 n/ X / X 3URRI +HUHLQ ZH GURS WKH VXSHUVFULSW DQG VXEVFULSW / DQG ZLOO GR VR LQ IXUWKHU SURRIV DV ZHOOf $VVXPH _D_ _E_ 7KHQ E\ WKH SURSHUWLHV RI PD[LPDOO\ /FRQVLVWHQW VHWV _f§ (DE +HQFH E\ 5( _f§ (/D/E 6R IRU DOO X /D H X LII /E H X 7KHUHIRUH E\ GHILQLWLRQ RI 1 _D_ H 1 LII _E_ H 1 X 6HJHUEHUJnV )XQGDPHQWDO 7KHRUHP IRU &ODVVLFDO /RJLFV > S @ LV DQ LLUPHGLDWH FRQVHTXHQFH 7 /HW EH D FDQRQLFDO QHLJKERUKRRG PRGHO IRU FODVVLFDO ORJLF / 7KHQ IRU DOO IRUPXODV D DQG DOO X H 8 7KHQ 0 0 \ f§ D LII D H X

PAGE 159

7KDW LV __D__A/ _D_A 3URRI 7KH GHILQLWLRQ RI 0 DQG FORVXUH RI PD[LPDOO\ /FRQVLVWHQW VHWV XQGHU 03 HVWDEOLVKHV WKH WKHRUHP IRU IRUPXOD D D SURSRVLWLRQDO OHWWHU R RU RI WKH IRUP &DE 6R DVVXPH DV DQ LQGXFWLYH K\SRWKHVLVf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nV /HQPD ZKLFK VWDWHV WKDW RQO\ WKHRUHPV EHRQJ WR HYHU\ PD[LPDOO\ /FRQVLVWHQW VHW 6HJHUEHUJ FDOOV VXFK ORJLFV QDWXUDO > S @ 2Q WKH RWKHU KDQG ZKHUH WKH FDQRQLFDO IUDPH LV QRW D PHPEHU RI WKH DSSURSULDWH FODVV RI IUDPHV WKH WHFKQLTXHV UHTXLUHG EHFRPH PRUH FRPSOH[ DQG HYHQ FXPEHUVRPH 7ZR DOWHUn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

PAGE 160

6HJHUEHUJ > SS @ GHILQHV WKH QRWLRQ RI WKH DXJPHQWDWLRQ RI D IUDPH VSHFLILFDOO\ WR JXDUDQWHH WKH VDWLVIDFWLRQ RI FRQGLWLRQV Pf DQG Uf DQG WKHQ SURYHV WKH OHPPD VWDWHG EHORZ /HW ) 81 EH D IUDPH 7KH DXJPHQWDWLRQ RI ) LV WKH IUDPH 81 VXFK WKDW X LI ? ^$ $ F 8 DQG $1 F $` LI 1 A f§ X f§ X / /HW / EH D UHJXODU ORJLF 0 WKH FDQRQLFDO PRGHO IRU / DQG fW LV DXJPHQWDWLRQ 7KHQ IRU DOO IRUPXODV D DQG DOO X H 8 0U 0O f§ D LII f§D nX rX :H VKRXOG UHPDUN WKDW WKH DXJPHQWDWLRQ RI D PRGHO LV WKH DXJPHQWDWLRQ RI LWV IUDPH DQG WKDW D PRGHO LGHQWLFDO ZLWK LWV DXJPHQWDWLRQ LV VDLG WR EH DXJPHQWHG ,W LV FOHDU WKDW WKH GHILQLWLRQ RI $ LQ WKH GHILQLWLRQ RI ,I X JXDUDQWHHV VXSHUVHW FORVXUH WKXV VDWLVILHV Pf DQG FORVXUH XQGHU LQWHUn VHFWLRQV WKXV VDWLVILHV Uf ,Q YLUWXH RI WKLV WKH GHILQLWLRQ RI 1 X SUHVHUYHV ERWK QRUPDO ZRUOGV DQG VLQJXODU ZRUOGV ,Q HIIHFW WKHQ IRU D UHJXODU ORJLF / IRUPV D EULGJH IURP WKH FDQRQLFDO PRGHO WR WKH DSSOLFDWLRQ RI 7 LQ FRPSOHWHQHVV SURRIV 7KRXJK DXJPHQWDWLRQ LV DSSURSULDWH IRU UHJXODU ORJLFV LW LV QRW IRU PRQRWRQLF ORJLFV WKDW DUH QRW UHJXODU 6HJHUEHUJnV HUURU > S @ LQ WKLV UHJDUG PRWLYDWHG &KHOODV DQG 0F.LQQH\ >@ :H PD\ H[SUHVV WKH GLIILFXOW\ LQ WKH IROORZLQJ OHPPD / ,I LV DQ DXJPHQWHG PRGHO IRU FODVVLFDO ORJLF / WKHQ ERWK D[LRPV 0 DQG 5 DUH WUXH

PAGE 161

3URRI )RU 0 DVVXPH /.ST 7KHQ __.ST__ H 1r VR __S__ $ _>T__ H +HQFH E\ Pf ERWK __S____T__ H 1M? 7KXV ERWK ,S DQG /T DQG VR ./S/T )RU 5 DVVXPH ./S/T 7KHQ ERWK __S____T__ H1VR E\ Uf __S__ $ __T_> H 1r? ,IHQFH __.ST > H 1A DQG VR ./ST 4(' 6R DXJPHQWDWLRQ RI ZRXOG FRQWUDU\ WR WKH H[SHFWHGf GLVWLQFWQHVV RI 0 DQG 5 IRUFH WKH VDWLVIDFWLRQ RI D[LRP 5 ,QGHHG XVLQJ DQ H[DPSOH RI &KHOODV DQG 0F.LQQH\ > S Q@ ZH FDQ SUHVHQW D FRXQWHUPRGHO IRU 5 WKDW VDWLVILHV Pf )LJXUH ,Q WKH DERYH GLDJUDP 8 ^XY` ZKLOH LV GHVLJQDWHG E\ WKH FLUFOHV FRQn QHFWHG WR X E\ VWUDLJKW OLQHV ,Q EUDFNHWV WKH DWRPLF IRUPXODV WUXH DW X UHVSHFWLYHO\ Yf DUH LQGLFDWHG :KLOH /S DQG /T DUH WUXH DW X __.ST__ LV QRW LQ 1A VR &./S/T/.ST IDLOV $VVXPLQJ LV WKH VDPH DV 1 WKH PRGHO VDWLVILHV Pf +HQFH WKH DERYH EHORQJV WR WKH FODVV RI IUDPHV

PAGE 162

VDWLVI\LQJ Pf EXW GRHV QRW YDOLGDWH 5 ,QFLGHQWDOO\ WKLV VKRZV WKH FRQWDLQPHQW RI 0 LQ 5 LV SURSHU 7KH DQDORJ\ RI / GRHV QRW KROG LI / LV D PRQRWRQLF ORJLF 7KH QRWLRQ RI DXJPHQWDWLRQ LV WRR VWURQJ LW PDNHV PRUH WUXH WKDQ UHTXLUHG 2Q WKH SDWWHUQ RI DXJPHQWDWLRQ &KHOODV DQG 0F.LQQH\ > S @ GHILQH D ZHDNHU QRWLRQ RI VXSS OHPHQW DW LRQ ZKHUHE\ WKH VDWLVIDFWLRQ RI Pf EXW QRW Uf DV ZHOO LV JXDUDQWHHG /HW ) 81 EH D IUDPH 7KH VXSSOHPHQWDWLRQ RI ) LV WKH IUDPH )r 81r VXFK WKDW 1r ^% % F 8 DQG $ F % IRU VRPH $ H 1 ` X f§ f§ X :H VKRXOG UHPDUN WKDW WKH VXSSOHPHQWDWLRQ RI D PRGHO LV WKH VXSSOHPHQWDn WLRQ RI LWV IUDPH DQG D IUDPH LGHQWLFDO ZLWK LWV VXSSOHPHQWDWLRQ LV VDLG WR EH VXSSOHPHQWHG 1RWH WKDW SUHVHUYHV VLQJXODU ZRUOGV DQG IRUFHV VXSHUVHW FORVXUH VR WKH VDWLVIDFWLRQ RI Pf +HQFH )r LV PRQRWRQLF &KHOODV DQG 0F.LQQH\ > S @ SURYH WKH DQDORJ RI / IRU VXSn SOHPHQWHG IUDPHV /$ /HW / EH D PRQRWRQLF ORJLF WKH FDQRQLFDO PRGHO IRU / DQG N LWV VXSSOHPHQWDWLRQ 7KHQ IRU DOO IRUPXODV D DQG IRU DOO ZRUOGV X 0W ,MLDLII f§D 3URRI ,Q WKH SURRI ZH GURS VXSHUVFULSWV DQG VXEVFULSW H[FHSW IRU r WR GLVWLQJXLVK WKH PRGHOVf 7KH SURRI LV E\ LQGXFWLRQ RQ WKH OHQJWK RI D RYHU DOO ZRUOGV DQG WKH RQO\ FDVH WKDW LV QRW LPPHGLDWH LV LI D LV RI WKH IRUP /E 6XSSRVH WKH WKHRUHP KROGV IRU IRUPXODV RI OHQJWK E DQG IRU HYHU\ X LQ 8A &RQVHTXHQWO\ __E__ __E__r $VVXPH /E 7KHQ __E__ H 1 DQG VR E\ GHILQLWLRQ RI 1r DV DQ\ VHW LV D VXEVHW RI LWVHOI __E__nn H 1r &RQVHTXHQWO\ r/E

PAGE 163

$VVW LPS r,E 7KHQ __E __ f H 1r %\ WKH LQGXFWLYH K\SRWKHVLV YH KDYH E LII nYEDQG E\ 7 ZH KDYH E LII E H X KHQFH __E__f _E_ 6R E\ WKH GHILQLWLRQ RI 1f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n ELQDWLRQ RI FRQGLWLRQV Uf Qf Vf Tf Wf WKHQ 3r GRHV DOVR 3URRI 6XSSRVH ) VDWLVILHV Uf 7R VKRZ )r VDWLVILHV Uf DVVXPH $% H ,7 7KHQ IRU VRPH & F $ DQG VRPH F % &' H 1 6R E\ Uf & $ H 1 EXW & $ F $ $ % KHQFH E\ GHILQLWLRQ RI 19 $ $ % H 1 X f§ X X DQG Uf LV VDWLVILHG 6XSSRVH ) VDWLVILHV Qf 7KHQ 8 H 1 EXW DV 8 F 8 LW IROORZV WKDW 8 H 1r X 6XSSRVH ) VDWLVILHV Vf :H KDYH DOUHDG\ QRWHG WKDW VXSSOHPHQWDWLRQ SUHVHUYHV VLQJXODU ZRUOGV VR )nn VDWLVILHV Vf 6XSSRVH ) VDWLVILHV Tf 7KHQ 1 38f KHQFH LV DOUHDG\ VXSSOHn PHQWHG VR LGHQWLFDO WR 1r X

PAGE 164

6XSSRVH ) VDWLVILHV Wf 7R VKRZ )r VDWLVILHV Wf DVVXPH LI 7KHQ 1 W VR X H $1 %XW $1 F $1 DQG VR X H $1 X f X X f§ X X &OHDUO\ LI ) VDWLVILHV VRPH SHUPLVVLEOH FRPELQDWLRQ RI FRQGLWLRQV WKHQ )r GRHV DV ZHOO 4(' 6XSSRVH ZH ZLVK WR SURYH WKDW PRQRWRQLF ORJLF / LV FRPSOHWH ZLWK UHVSHFW WR FODVV & ZKHUH F GHQRWHV VRPH FRPELQDWLRQ RI FRQGLWLRQV U FP 7KH JHQHUDO IRUPDW LV DV IROORZV ( f $VVXPH VRPH IRUPXOD D LV YDOLG LQ & FP f 6KRZ WKH IUDPH RI VDWLVILHV F f 7KHQ WKH IUDPH RI $Le VDWLVILHV FP E\ / f 6R D LV WUXH LQ 0r f +HQFH D LV WUXH LQ E\ / f +HQFH D LV D WKHRUHP RI / E\ 7 DQG WKH FRUROODU\ WR /LQGHQEDXQnV /HPPD f 7KHUHIRUH / LV FRPSOHWH ZLWK UHVSHFW WR :KHUH WKH ORJLF FRQFHUQHG LV QRW PRQRWRQLF EXW FODVVLFDO QHYHUWKHOHVV WKH VLPSOHU SODQ VXJJHVWHG HDUOLHU VXIILFHV LQ PRVW FDVHV ( 4 LV RQH H[FHSWLRQ > S @f ( f $VVXPH VRPH IRUPXOD D LV YDOLG LQ & F f 6KRZ WKH IUDPH RI 0A VDWLVILHV F f 7KHQ D LV WUXH LQ $IA f +HQFH D LV D WKHRUHP RI / E\ 7 DQG WKH FRUROODU\ WR /LQGHQEDXQnV /HPPD f +HQFH / LV FRPSOHWH ZLWK UHVSHFW WR &4

PAGE 165

6LQFH WKH UHVW LV URXWLQH RXU FRPSOHWHQHVV SURRIV ZLOO RQO\ KDYH WR HVWDEOLVK (f (ff DERYH LQ RUGHU WR EH VXIILFLHQW :H PD\ WKHUHIRUH VWDWH WKH IROORZLQJ FRPSOHWHQHVV OHUQQDV / ( LV FRPSOHWH ZUW & 3URRI &OHDUO\ WKH IUDPH RI $IJ LV LQ WKH FODVV RI DOO IUDPHV VR E\ ( WKH OHPPD IROORZV 0 ( 0 LV FRPSOHWH ZUW & r 3URRI 7KRXJK WKH IUDPH RI GRHV QRW VDWLVI\ Pf WKH IUDPH RI GRHV VR E\ ( WKH OHQPD IROORZV 5 ( 0 5LV FRPSOHWH ZUW & 3URRI :H QHHG RQO\ VKRZ WKH IUDPH RI VDWLVILHV Uf 7KHQ WKH OHPPD IROORZV DFFRUGLQJ WR ( $VVXPH $% H 1JXf 7KHQ E\ GHILQLWLRQ WKHUH DUH IRUPXODV D DQG E VXFK WKDW /D/E H X DQG $ _D_ DQG % _E_ 6LQFH /D /E H X E\ WKH SURSHUWLHV RI PD[LPDOO\ 5FRQVLVWHQW VHWV ./D/E H X %XW X H 8J VR LV FORVHG XQGHU DSSO\LQJ 03 WR LQVWDQFHV RI D[LRP 5 VXFK DV &./D/E/.DE H X ZLWK ./D/E H X KHQFH /.DE H X &RQVHTXHQWO\ E\ GHILQLWLRQ _.DE_ H 1JXf 6R E\ WKH SURSHUWLHV RI PD[LPDOO\ 5FRQVLVWHQW VHWV -D_ $ _E_ H 1JXf EXW WKDW LV $ $ % H 1JXf 4(' / ( 0 5 1LV FRPSOHWH ZUW & PP 3URRI ,Q YLHZ RI / LW LV VXIILFLHQW WR VKRZ WKH IUDPH RI 0Y VDWLVILHV Qf $VVXPH X H 8J 7KHQ /, H X DV D[LRP 1 6R E\ GHILQLWLRQ __ H 1J&Xf EXW 8 4(' 7 7 LV FRPSOHWH ZUW & f§ f§ PPW 3URRI ,Q YLHZ RI LW LV VXIILFLHQW WR VKRZ WKDW WKH IUDPH RI 0UM VDWLVILHV Wf

PAGE 166

/HW X H 8S DQG DVVXPH 1SXf :RUOG X FRQWDLQV DOO VXEVWLWXWLRQ LQVWDQFHV RI 7 &/SS VR IRU HDFK IRUPXOD D LI /D H X WKHQ D H X E\ FORVXUH XQGHU 03 6R IRU HDFK VXFK IRUPXOD D _D_ H 1SXf DQG X H _D> +HQFH X H $1S Xf 4(' ,Q YLHZ RI WKH DERYH OHPPDV DQG WKH HDUOLHU OHPPDV RQ FRQVLVWHQF\ ZH FDQ VWDWH WKH IROORZLQJ UHVXOWV FRQFHUQLQJ WKH GHWHUPLQDWLRQ RI ORJLFV E\ FODVVHV RI IUDPHV 7 7 7 7 7 ( LV GHWHUPLQHG E\ & 0 LV GHWHUPLQHG E\ & 5 LV GHWHUPLQHG E\ & PU LV GHWHUPLQHG E\ & PP 7 LV GHWHUPLQHG E\ & nPPI / /f / /f / /f / /f / /f &RPSOHWHQHVV UHVXOWV IRU PRGDO ORJLFV FRQWDLQLQJ 5 RU DUH PRUH HDVLO\ HVWDEOLVKHG XVLQJ WKH QRWLRQ RI UHODWLRQDO IUDPHV ZKLFK ZH EULHIO\ GLVFXVVHG LQ D SUHYLRXV VHFWLRQ ,W LV WKLV WDFWLF ZKLFK 6HJHUEHUJ > @ IROORZV KRZHYHU ZH VKDOO QRW JR VR IDU $ERYH ZH KDYH H[SORUHG UDWKHU EDVLF V\VWHPV H[SUHVVLEOH LQ &/ %HIRUH WXUQLQJ WR WKH V\VWHPV H[SUHVVLEOH LQ &: ZH VKDOO LQYHVWLJDWH WKH UHDOWLRQVKLS EHn WZHHQ UHODWLRQDO DQG QHLJKERUKRRG VHPDQWLFV DQG GHWHUPLQH ZKHWKHU RU QRW WKH VSKHUH IXQFWLRQ LV D QHLJKERUKRRG IXQFWLRQ RQ /HZLVn LQWHUSUHWDWLRQ 1HLJKERUKRRG DQG 5HODWLRQDO 6HPDQWLFV )ROORZLQJ 6HJHUEHUJ > S @ ZH PD\ GHILQH DQ DOWHUQDWLYH UHODWLRQ RQ DQ\ UHJXODU QHLJKERUKRRG IUDPH /HW ) 81 EH D UHJXODU IUDPH :H GHILQH DQ DOWHUQDWLYH UHODWLRQ 5 F 8 r 8 E\ X5Y LII X LV QRUPDO DQG Y H $1 X

PAGE 167

7KH IROORZLQJ OHPPD LV WKHQ LPPHGLDWH / )RU DQ\ UHJXODU IUDPH DQG PRGHO /DLII X LV QRUPDO DQG $1X F __ D__ IRU DOO IRUPXODV D 3URRI 6XSSRVH /D 7KHQ X FDQQRW EH VLQJXODU VR X LV QRUPDO DV WKH IUDPH LV UHJXODU DQG FRQVLVWV RQO\ RI QRUPDO DQG VLQJXODU ZRUOGV 6LQFH ,1, e 1X! F ,,D 6XSSRVH X LV QRUPDO DQG $1A F __D__ 6LQFH X LV QRUPDO LV D ILOWHU VR FORVHG XQGHU VXSHUVHWV KHQFH __D__ H 1A 7KHUHIRUH /D 4(' )RU DQ\ UHJXODU QHLJKERUKRRG IUDPH ZH PD\ GHILQH D FRUUHVSRQGLQJ UHODWLRQDO IUDPH /HW ) 81 EH D UHJXODU QHLJKERUKRRG IUDPH 7KHQ WKH UHODWLRQDO IUDPH FRUUHVSRQGLQJ WR ) ) LV JLYHQ E\ )A 854 VXFK WKDW Df 8 LV WKH VDPH VHW DV LQ ) Ef 5 LV WKH DOWHUQDWLYH UHODWLRQ GHILQHG RQ ) Ff 4 LV WKH VXEVHW RI 8 FRQVLVWLQJ RI VLQJXODU ZRUOGV %\ FKDQJLQJ FODXVH Gf RI WKH GHILQLWLRQ RI WUXWK LQ D PRGHO WR Gnf /D LII X L 4 DQG IRU DOO Y LI X5Y WKHQ D LW LV FOHDU WKDW ZH PD\ FDUU\ RXW RXU VHPDQWLFV IRU D ORJLF FRQWDLQLQJ 5 E\ FRQVLGHULQJ WKH FRUUHVSRQGLQJ UHODWLRQDO IUDPHV VHH .ULSNH > @f 2I LPSRUWDQFH LQ WKH DERYH LV WKH IDFW WKDW /D LII X LV QRUPDO DQG $1 BF __D__ &RQVLGHU WKH VSKHUH IXQFWLRQ LW LV FOHDU WKDW LW VDWLVILHV WKH IRUPDO FULWHULD IRU EHLQJ D QHLJKERUKRRG IXQFWLRQ VLQFH LWV UDQJH LV 338ff KRZHYHU LQ /HZLVn VHPDQWLFV LW GRHV QRW RSHUDWH DV D QHLJKERUKRRG IXQFWLRQ DV LQ QHLJKERUKRRG VHPDQWLFV )RU RQ /HZLVn VHPDQWLFV LW LV 8A ZKLFK FRUUHVSRQGV WR WKH VHW RI ZRUOGV WR ZKLFK X KDV DFFHVV LH ZH PXVW GHILQH WKH DOWHUQDWLYH RU DFFHVVLQJ UHODWLRQ

PAGE 168

LQ WHUPV RI 8X! UDWKHU WKDQ $A IRU WKH ORJLFDO PRGDOLWLHV *LYHQ WKH FRQGLWLRQV RQ A LW LV D PD[LPDO WRZHU RI WKH LPSURSHU ILOWHU 38f ,Q ZKLFK FDVH ZH ZRXOG KDYH /D IRU IRUPXOD D ZKHUH D LV D WUXWK IXQFWLRQDO FRQWUDGLFWLRQ LI ZH ZHUH WR FRQVLGHU D QHLJKERUKRRG IXQFWLRQ DQG DSSO\ WKH WUXWK GHILQLWLRQV RI QHLJKERUKRRG VHPDQWLFV VLQFH H 7KH LQHVFDSDEOH FRQFOXVLRQ LV WKDW LV QRW D QHLJKERUKRRG IXQFWLRQ DQG WKDW /HZLVn VHPDQWLFV LV QRW QHLJKERUKRRG VHPDQWLFV LQ GLVJXLVH /HZLVn V\VWHPRIVSKHUHV VHPDQWLFV PD\ EH ORRNHG XSRQ DV WKH JLYLQJ RI D FHUWDLQ LQWHUQDO VWUXFWXUH WR WKH DFFHVVLQJ UHODWLRQ EDVHG XSRQ FRQn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f 8 LV D VHW RI SRVVLEOH ZRUOGV Ef 5 LV D UHODWLRQ RQ 8 Ff 4 LV WKH VHW RI VLQJXODU ZRUOGV

PAGE 169

7KHQ WKH QHLJKERUKRRG IUDPH FRUUHVSRQGLQJ WR ) )n LV WKH RUGHUHG SDLU 81 VXFK WKDW 1 I LI X H 4 X >B^$ $F 8 DQG ^Y Y H 8 DQG X5Y` e $` LI X L 4 7KDW WKLV UHVXOWV LQ D UHJXODU IUDPH LV REYLRXV 7KDW H[DFWO\ WKH VDPH IUPDODV DUH YDOLG LQ DQ\ PRGHO RQ WKH IUDPHV LV DOVR FOHDU 2I FRXUVH RXU DVVHUWLRQ KHUH LV OLPLWHG WR &/ ZH KDYH QRW \HW GHILQHG PRGHOV IRU &: LQ WHUPV RI QHLJKERUKRRG IUDPHV ,W LV WR WKLV FRQGLWLRQDO ODQJXDJH DQG WKH QHLJKERUKRRG VHPDQWLFV IRU LW WKDW ZH WXUQ LQ WKH QH[W VHFWLRQ 7KHQ ZH FDQ DQVZHU WKH TXHVWLRQ DV WR ZKHWKHU HDFK V\VWHP RI VSKHUHV IUDPH RU PRGHOf FRUUHVSRQGV WR VRPH QHLJKERUKRRG IUDPH RU PRGHOf &RQGLWLRQDO /RJLF WKH 6\VWHPV &H &N &( &0 &5 &. ,Q 6HFWLRQ ZH LQWURGXFHG D ODQJXDJH &/: SHUPLWWLQJ WKH H[SUHVn VLRQ RI ERWK PRGDO DQG FRQGLWLRQDO VHQWHQFHV ,Q WKH UHPDLQGHU RI WKDW VHFWLRQ DQG LQ 6HFWLRQ ZH UHVWULFWHG RXU DWWHQWLRQ WR WKH PRGHO IUDJ PHQW RI WKDW ODQJXDJH &/ DQG VHYHUDO EDVLF ORJLFV H[SUHVVLEOH WKHUHLQ 7KH QHLJKERUKRRG VHPDQWLFV RI 6HJHUEHUJ >@ ZDV UHVWUXFWHG LQ D ZD\ ZKLFK FRQGXFHV WR WKH FRPSDULVRQ RI RXU GHYHORSPHQW RI &/ORJLFV WR WKH &Or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

PAGE 170

FRXQWHUIDFWXDO FRQGLWLRQDO DQG LQGHHG WKDW WKLV IDLOXUH ZDV YHU\ QHDUO\ D GHILQLQJ FKDUDFWHULVWLF RI FRXQWHUIDFWXDOV 7KH QRWLRQ RI WKH DQWHFHGHQW QHFHVVLWDWLQJ WKH FRQVHTXHQW ZDV QRW VLPSO\ WKH QRWLRQ RI D QHFHVVDU\ PDWHULDO FRQGLWLRQDO 1RZ RQH ZD\ RI H[SUHVVLQJ WKH QRWLRQ RI WKH DQWHFHGHQW QHFHVVLn WDWLQJ WKH FRQVHTXHQW ZKLFK ZH KDYH QRW KHUHWRIRUH FRQVLGHUHG LV WR DVVLJQ D QHFHVVLW\ RSHUDWRU WR HDFK VHQWHQFH '‘ /DE A E LV DQHFHVVDU\ 7KLV QRWLRQ RI D VHQWHQWLDOO\ LQGH[HG PRGDOLW\ LV PHQWLRQHG E\ /HZLV > S @ DQG LV FORVHO\ UHODWHG WR &KHOODVn >@ GHYHORSPHQW RI FRQGLWLRQDO ORJLF 7KH YLUWXH RI WKLV DV D VWDUWLQJ SRLQW LV WKDW LW ZLOO HQDEOH XV WR PDNH D UDWKHU HDV\ WUDQVLWLRQ IURP PRGDO ORJLFV WR FRQGLWLRQDO ORJLFV &RQVLGHU WKH FKDUDFWHULVWLF UXOH RI LQIHUHQFH IRU FODVVLFDO PRGDO ORJLFV 5( )UFWQ (DE LQIHU (/D/E 6XSSRVH WKH / LV D VHQWHQWLDOO\ LQGH[HG PRGDOLW\ 7KHQ 5( ZRXOG DSn SHDU DV 5(n )UDQ (DE LQIHU (/ D/ E F F 7KH IRUFH RI 5( LQ PRGDO ORJLFV LV WKDW LI WZR VHQWHQFHV H[SUHVV WKH VDPH SURSRVLWLRQ DQG VR DUH HTXLYDOHQWf WKHQ HLWKHU WKH\ DUH ERWK QHFHVn VDU\ RU QHLWKHU LV :LWK UHVSHFW WR VHQWHQWLDOO\ LQGH[HG PRGDOLWLHV WKH DQDORJRXV UHTXLUHPHQW LV H[SUHVVHG E\ 5(n +RZHYHU ZLWK VHQWHQWLDOO\ LQGH[HG PRGDOLWLHV D VHFRQG FRQVLGHUDn WLRQ DNLQ WR WKDW PHQWLRQHG DERYH DULVHV ,I D VHQWHQFH F LV DQHFHVVDU\ DQG D H[SUHVVHV WKH VDPH SURSRVLWLRQ DV E LV F DOVR EQHFHVVDU\" ,I VR

PAGE 171

WKHQ ZKDW ZH UHDOO\ KDYH DUH SURSRVLWLRQDOO\ LQGH[HG PRGDOLWLHV DQG WKLV UHTXLUHPHQW FRXOG EH H[SUHVVHG E\ WKH IROORZLQJ UXOH 5( )URP (DE LQIHU (/AF/AF :H ZRXOG WKHQ QRW GLVWLQJXLVK DPRQJ HTXLYDOHQW VHQWHQFHV LQ HLWKHU RSHUDWRU RU RSHUDQG /HW LRV GHVLJQDWH D ODQJXDJH RI VHQWHQWLDOO\ LQGH[HG PRGDOLWLHV DV &/, %HORZ DUH SDUDOOHO VHWV RI UXOHV RI LQIHUHQFH DQG D[LRPV DV WKH\ ZRXOG DSSHDU LQ D FRQGLWLRQDO ODQJXDJH &: DQG LQ &/, ( 5XOHV RI LQIHUHQFH &RQGLWLRQDO ,QGH[HG PRGDO 5&($ )URP (DE LQIHU (:DF:EF )URP (DE LQIHU (/AF/AF 5&(& )URP (DE LQIHU (:FD:FE )URP (DE LQIHU (/ D/ E F F 5&0 )URP &DE LQIHU &:FD:FE )URP &DE LQIHU &/ D/ E F F 5&5 )UFP &.DEF LQIHU &.:GD:GE:GF )URP &.DEF LQIHU &./AD/AE/AF 5&1 )URP E LQIHU :DE )URP E LQIHU / E D 5&. )URP &.. .DAA DAE )UFP &.. .DA DQE LQIHU &.. .:FDA:FD LQIHU &.. ./ D/ D F F :FD :HE IRU Q / D / E IRU Q F Q F ( $[LRPV &RQGLWLRQDO ,QGH[HG PRGDO &0 &:S.TU.:ST:SU &/ .TU./ T/ U 3 3 3 &5 &.:ST:SU:S.TU &./ T/ U/ .TU 3 3 3 &1 :SO / 3 &. &:S&TU&:ST:SU &/ &TU&/ T/ U S S0 S ,W LV FOHDU WKDW WKHUH LV D RQHWRRQH FRUUHVSRQGHQFH EHWZHHQ IRUPXODV RI &: DQG &/, GHILQHG E\ LGHQWLI\LQJ IRUPXODV RI WKH IRUP :DE ZLWK WKRVH RI WKH IRUP / E IRU DOO IRUPXODV D DQG E FO

PAGE 172

:KLOH RXU FKRLFH EHWZHHQ WKH WZR ODQJXDJHV LV RQH RI LQVFULSWLRQDO SUHIHUHQFH RXU LQWXLWLRQV DERXW FRQGLWLRQDOLW\ DQG LQGH[HG PRGDOLW\ PD\ GLIIHU )RU H[DPSOH LI WKH WUXWK RI D LV LUUHOHYDQW WR WKH WUXWK RI E VKRXOG LW IROORZ WKDW E LV ERWK DQHFHVVDU\ DQG 1DQHFHVVDU\" 7KH FRQn GLWLRQDO SDUDOOHO LV WKH W\SLFDO HYHQ LInFRQGLWLRQDO ZKHUH WKHUH LV LQWXLWLYHO\ QR QHFHVVLWDWLRQ FRQQHFWLQJ WKH DQWHFHGHQW WR WKH FRQVHTXHQW &RPSDULQJ WKH LQGH[HG PRGDO UXOHV DQG D[LRPV DERYH WR WKH PRGDO UXOHV DQG D[LRPV LQ ( DQG ( UHYHDOV WKH FOHDU SDUDOOHO EHWZHHQ WKH FRQVWUXFWLRQV ([FHSWLQJ 5&($ IRU WKH PRPHQW 5&(& FRUUHVSRQGV WR 5( 5&0 WR .0 HWF ZKHUH DQ DUELWUDU\ VHQWHQWLDOO\ LQGH[HG PRGDOLW\ KDV EHHQ VXEVWLWXWHG IRU WKH VLQJOH PRGDOLW\ H[SUHVVHG E\ / LQ WKH ODWn WHU VHULHV 7KRXJK LQ ZKDW IROORZV ZH ZLOO EH FRQVLGHULQJ RXU EDVLF ODQn JXDJH WR EH &: LQ YLHZ RI WKH FORVH FRUUHVSRQGHQFH EHWZHHQ LQGH[HG PRGDO ODQJXDJHV DQG FRQGLWLRQDO ODQJXDJHV WKH WHUPLQRORJ\ DQG PDQ\ RI WKH WHFKQLTXHV GHYHORSHG LQ 6HFWLRQV DQG ZLOO DSSO\ WR RXU SUHn VHQW HQGHDYRUV :H KDYH DOUHDG\ REVHUYHG LQIRUPDOO\ WKDW WKH SUHVHQFH RI 5&($ UHQGHUV VHQWHQWLDOO\ LQGH[HG PRGDOLWLHV HTXLYDOHQW WR SURSRVLWLRQDOO\ LQGH[HG RQHV LQ WKH VHQVH WKDW DQ HTXLYDOHQFH FODVV RI VHQWHQWLDOO\ LQn GH[HG PRGDOLWLHV FRUUHVSRQGV WR D SURSRVLWLRQDOO\ LQGH[HG PRGDOLW\f 7KH IRUPDO VHPDQWLF GLVWLQFWLRQ ZLOO EH GHYHORSHG LQ 6HFWLRQ 2QH PLJKW ZRQGHU ZK\ LI 5( GLYLGHV LQWR WZR FRQGLWLRQDO UXOHV WKH VDPH SUDFWLFH LV QRW IROORZHG IRU .0 DQG WKH RWKHUV )RU H[DPSOH ZKDW DERXW 5&0$ )URP &DE LQIHU &:DF:EF RU H[SUHVVHG LQ WHUPV RI LQGH[HG PRGDOLWLHV )URP &DE LQIHU &/ 7KH DQVZHU LV WKDW WKH\ FRXOG KDYH EHHQ LQFOXGHG DQG ZRXOG KDYH EHHQ EXW

PAGE 173

IRU WKH IDFW WKDW QR FRQGLWLRQDO RU LQGH[HG PRGDO ORJLF ZLWK DQ\ SUHn WHQVLRQV WR UHOHYDQFH WR DQ\ RI RXU QDLYH QRWLRQV ZRXOG FRQWDLQ 5&0$ 7KH IROORZLQJ FRQVLGHUDWLRQV PDNH WKLV HYLGHQW 6XSSRVH &DE LV D WKHRUHP DQG IXUWKHU WKDW VDQH SURSRVLWLRQ F LV DQHFHVVDU\ LH / F KROGV 6LQFH &DE LV D WKHRUHP D UHDVRQDEOH VRQDQWLFV PLJKW ZHOO FO KDYH WKH VHW RI DZRUOGV D VXEVHW RI WKH VHW RI EZRUOGV LH ,ODOO e __E__ ,W LV HYLGHQW WKDW LI ZH WDNH DQHFHVVDU\ WR PHDQ TXLWH UHDVRQDEO\ WUXH DW HYHU\ DZRUOG WKDW LW QHHG QRW IROORZ WKDW F EH WUXH DW HYHU\ EZRUOG DV ZHOO 1RZ WKH DERYH ZRXOG EH D UDWKHU VWURQJ WUXWK FRQGLWLRQ IRU DQHFHVVLW\ VR WKH LQIHUHQFH ZRXOG FHUWDLQO\ IDLO IRU DQ\ ZHDNHU FRQGLWLRQ 2Q WKH RWKHU KDQG VLPLODU UHDVRQLQJ PLJKW ZHOO SHUVXDGH RQH WKDW WKH RUGHUUHYHUVLQJ 5&0$nn )UDQ &DE LQIHU &/AF/AF RXJKW WR KROG VLQFH LI F LV EQHFHVVDU\ DQG HYHU\ DZRUOG LV D EZRUOG WKHQ F LV DQHFHVVDU\ DV ZHOO +RZHYHU HYHQ KHUH ZH VKRXOG SDXVH DQG UHFDOO WKDW E\ WKH GHGXFWLRQ WKHRUHP 7f 5&0$ FDQ EH H[SUHVVHG DV )URP &DE DQG /AF LQIHU /DF RU LQ FRQGLWLRQDO QRWDWLRQ )URP &DE DQG :EF LQIHU :DF %XW ZH UHMHFWHG WKLV LQIHUHQFH SDWWHUQ IRU FRXQWHUIDFWXDOV LQ 6HFWLRQ VHH (f )ROORZLQJ &KHOODV > SS @ DQG 1XWH > S @ FRQGLn WLRQDO ORJLFV PD\ EH FODVVLILHG RQ WKH EDVLV RI WKH IROORZLQJ V\VWHPV FRQSDUH (f

PAGE 174

( &RQGLWLRQDO V\VWHPV Df &H 3& 5&(& KDOIFODVVLFDO Ef &N 3& 5&. KDOIQRUPDO Ff &( 3& 5&($ 5&(& FODVVLFDO Gf &0 3& 5&($ 5&0 PRQRWRQLF Hf &5 3& 5&($ 5&5 UHJXODU If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f / 5&0 LV GHULYDEOH LQ 3& 5&5 &RPSDUH /f / 5&. LV GHULYDEOH LQ 3& 5&5 5&1 DQG FRQYHUVHO\ 3URRI 5&1 LV 5&. IRU Q DQG 5&5 LV 5&. IRU Q &RQYHUVHO\ 5&1 JLYHV XV 5&. IRU Q DQG 5&5 JLYHV XV 5&. IRU Q 7KH DVVRFLDWLYLW\ RI FRQMXQFWLRQ DQG DQ LQGXFWLRQ JLYH XV 5&. IRU Q 4('

PAGE 175

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f / &0 LV GHULYDEOH LQ 3& 5&0 3URRI %RWK &.TUT DQG &.TUU DUH 3& WKHRUHPV 6R E\ 5&0 ZH KDYH ERWK &:S.TU:ST DQG &:S.TU:SU DQG KHQFH E\ 3& &:S.TU.:ST:SU ZKLFK LV &0 4(' &RQSDUH /f / 5&5 LV GHULYDEOH LQ &H &0 &5 &RPSDUH /f / &5 LV GHULYDEOH LQ 3& 5&5 &RQSDUH /f / 5&1 LV GHULYDEOH LQ *H &1 &RPSDUH /f / &1 LV GHULYDEOH LQ &H 5&1 &RQSDUH /f ,Q YLHZ RI / WKH IROORZLQJ DUH DOWHUQDWLYH EDVHV IRU WKH FODVVLFDO FRQGLWLRQDO ORJLFV QRWHG DERYH

PAGE 176

( Df &0 &( &0 Ef &5 &( &0 &5 Ff &. &( &0 &5 &1 &RQSDUH WKLV WR WKH VLPLODU EDVHV IRU PRGDO ORJLFV LQ ( ,Q 6HFWLRQ ZH LQGLFDWHG DQ DOWHUQDWLYH EDVLV IRU WKH VPDOOHVW QRUPDO ORJLF LQ WHUPV RI DGGLQJ D[LRP DQG UXOH 51 WR 3& $[LRP &. LV WKH FRQGLWLRQDO DQDORJ RI D[LRP :H SURYH WKH FRQGLWLRQDO DQDORJV RI / EHORZ 7KH DQDORJV RI 6HJHUEHUJn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

PAGE 177

&. 3& 5&($ &. 5&1 $V ZLWK WKH SDUDOOHO EDVLV IRU WKLV LV D SDUWLFXODUO\ HFRQRPLFDO EDVLV WKRXJK LW FRQFHDOV WKH IXQGDPHQWDO QDWXUH RI D[LDQV &0 DQG &5 LQ FRPSDULVRQ WR WKH VHPDQWLFV DSSURSULDWH WR FODVVLFDO FRQGLWLRQDO ORJLFV 7KH VHPDQWLFV WR EH GHYHORSHG LQ 6HFWLRQ ZLOO DOORZ XV WR VKRZ WKH FRQWDLQPHQWV QRWHG DIWHU / DUH DOO SURSHU $V ZH VKDOO VHH WKHVH VHPDQWLFV DUH DQ DGDSWDWLRQ RI WKH QHLJKERUKRRG VHPDQWLFV SUHVHQWHG LQ 6HFWLRQ WR FRQGLWLRQDO RU LQGH[HG PRGDOf ORJLF 1HLJKERUKRRG 6HPDQWLFV IRU &RQGLWLRQDO /RJLF $ YHUVLRQ RI QHLJKERUKRRG VHPDQWLFV IRU FRQGLWLRQDO ORJLF LV GHYHOn RSHG E\ &KHOODV XQGHU WKH UXEULF RI PLQLPDO IUDPHV DQG PRGHOV > SS @ 7KLV VHPDQWLFV LV DGHTXDWH WR FODVVLFDO FRQGLWLRQDO ORJLFV ,Q D QRWH > S QO@ &KHOODV LQGLFDWHV WKDW WKHUH LV D VHQWHQWLDO YDULDQW WR KLV VWDQGDUG VHPDQWLFV ZKLFK LV DSSURSULDWH WR KDOIQRUPDO ORJLFV 1XWH > S @ GHYHORSV WKH YHUVLRQ RI QHLJKERUKRRG VHPDQWLFV DSSURSULDWH WR KDOIFODVVLFDO ORJLFV 5HFDOO WKDW LQ QHLJKERUKRRG VHPDQWLFV IRU PRGDO ORJLF WKH WUXWK FRQ GLWLRQ IRU D IRUPXOD /D LV JLYHQ E\ /D LII __D H 1X ZKHUH LV D VHW RI VXEVHWV RI WKH VHW RI SRVVLEOH ZRUOGV 8 ,I ZH UHn SODFH / DERYH E\ D VHQWHQWLDOO\ RU SURSRVLWLRQDOO\ LQGH[HG PRGDOLW\ WKHQ HLWKHU RXU WUXWK FRQGLWLRQ RU RXU GHILQLWLRQ RI 1 PXVW WDNH DFFRXQW X RI WKH LQGH[ 7KH ODWWHU DOWHUQDWLYH LV WKH FDVH LQ WKH WZR NLQGV RI IUDPHV WR EH GHILQHG EHORZ

PAGE 178

:H WDNH DV RXU ODQJXDJH &: DQG SHQGLQJ WKH GHILQLWLRQ RI PRGHO DQG WUXWK EHORZ GHILQH YDOLGLW\ FRQVLVWHQF\ DQG FRPSOHWHQHVV LQ WKH VWDQGDUG ZD\ RI 6HFWLRQ $ VHQWHQWLDO QHLJKERUKRRG IUDPH ) 81 LV DQ RUGHUHG SDLU VXFK WKDW Df 8 LV D VHW RI SRVVLEOH ZRUOGV DQG Ef 1 8 [ &: r‘ 338ff LV D IXQFWLRQ $ SURSRVLWLRQDO QHLJKERUKRRG IUDPH ) 81 LV DQ RUGHUHG SDLU VXFK WKDW Df 8 LV D VHW RI SRVVLEOH ZRUOGV DQG Ef 1 8 [ S8f r 338ff LV D IXQFWLRQ 7KH VDPH GHILQLWLRQ RI PRGHO ZLOO GR IRU ERWK W\SHV RI IUDPHV $ PRGHO 0 819 LV DQ RUGHUHG WULSOH VXFK WKDW ) 81 LV D IUDPH DQG 93 r 38f LV D IXQFWLRQ ZKHUH 3 LV WKH VHW RI SURSRVLWLRQDO OHWWHUV 0 LV FDOOHG D PRGHO RQ 3 7KH GHILQLWLRQ RI WUXWK IRU QRQFRQGLWLRQDO VHQWHQFHV LV DV LQ VR RQO\ WKH FODXVH IRU FRQGLWLRQDO IRUPXODV LV VWDWHG EHORZ 7KH WUXWK LQ VHQWHQWLDO QHLJKERUKRRGf PRGHO 0 RI IRUPXOD :DE LV GHILQHG DV :DE LII __E__0 H 1XDf 7KH WUXWK LQ SURSRVLWLRQDO QHLJKERUKRRGf PRGHO 0 RI IRUPXOD :DE LV GHILQHG DV :DE LII __E__A H 1X __D__9 :H ZLOO GURS VXSHUVFULSWV LQ D DQG >_D__: ZKHQHYHU SRVVLEOH WR GR VR ZLWKRXW FRQIXVLRQ 1RWH WKDW LQ ZH WDFLWO\ XVH &: WR VWDQG IRU WKH ZHOOIRUPHG IRUPXODV RI &:

PAGE 179

7KH IROORZLQJ OHPPD LV LPPHGLDWH / 3& LV FRQVLVWHQW ZUW WKH FODVV RI DOO VHQWHQWLDO SURSRVLWLRQDOf IUDPHV ,W ZRXOG QRUPDOO\ EH RXU LQWHQWLRQ WR FODVVLI\ FRQGLWLRQDO ORJLFV LQ WHUPV RI FODVVHV RI IUDPHV DV ZH GLG ZLWK PRGDO ORJLFV 7KLV ZLOO QRW JHQHUDOO\ EH SRVVLEOH LQ WKH FDVH RI KDOIFODVVLFDO ORJLFV ODUJHU WKDQ &HA VHH &KDOODV > S QO@f :H VKDOO VHH EHORZ ZK\ WKLV LV WKH FDVH 3DUWO\ IRU WKLV UHDVRQ DQG SDUWO\ IRU UHDVRQV RI XQLIRUPLW\ 1XWH >@ FODVVLILHV FRQGLWLRQDO ORJLFV LQ WHUPV RI FODVVHV RI PRGHOV SUHIHUULQJ WR LQFRUSRUDWH WKH YDOXDWLRQ IXQFWLRQ 9 GLUHFWO\ LQWR WKH GHILQLWLRQ RI WKH VHW RI SRVn VLEOH ZRUOGV > SS @ :H SUHIHU WR UHWDLQ WKH PRUH DOJHEUDLF FODVVLILFDWLRQ LQ WHUPV RI IUDPHV ZKHQHYHU SRVVLEOH $V PRVW FRQGLWLRQDO ORJLFV LQWHQGHG WR UHSUHVHQW WKH FRXQWHUIDFWXDO FRQGLWLRQDO DUH FODVVLFDO LQFOXGLQJ /HZLVn WKLV ZLOO QRW EH RYHUO\ UHVWULFWLYH $V LQ 6HFWLRQ ZH ZLOO VKRZ ILUVW FRQVLVWHQF\ UHVXOWV DQG WKHQ FRPSOHWHQHVV UHVXOWV ,Q YLHZ RI / LW ZLOO EH QHFHVVDU\ WR FKHFN RQO\ WKH FRQGLWLRQDO D[LRPV DQG UXOHV RI LQIHUHQFH IRU YDOLGLW\ LQ D SDUn WLFXODU FODVV RI IUDPHV RU UDUHO\ PRGHOVf / &H LV FRQVLVWHQW ZUW WKH FODVV RI DOO VHQWHQWLDO IUDPHV RU PRGHOVf 3URRI :H QHHG RQO\ VKRZ 5&(& SUHVHUYHV YDOLGLW\ 6XSSRVH (DE LV YDOLG 7KHQ IRU DQ\ VHQWLHQWLDO PRGHO __D__ __E__ 6R IRU HDFK X H 8 ZH KDYH HLWKHU ERWK :FD DQG :HE RU QHLWKHU E\ +HQFH ZH KDYH (:FD:FE 4(' 1RWH WKDW WKLV SURRI JRHV WKURXJK EHFDXVH WKH VHW RI ZRUOGV ZKHUH D RU Ef LV WUXH DSSHDUV LQ WKH WUXWK GHILQLWLRQ DV D PHPEHU RI 1XFf

PAGE 180

/ &( LV FRQVLVWHQW ZUW WKH FODVV RI DOO VHQWHQWLDO PRGHOV VDWLVI\n LQJ WKH FRQGLWLRQ 1XDf 1XEf LII __D__ __E__ 3URRI :H QHHG RQO\ FKHFN WKDW 5&($ SUHVHUYHV YDOLGLW\ 6XSSRVH (DE LV YDOLG WUXH LQ HYHU\ GHVLJQDWHG PRGHOf 7KHQ __D__ __E__ &RQVHTXHQWO\ E\ WKH VWDWHG FRQGLWLRQ 1XDf 1XEf 7KHQ __F__ H 1XDf LII __F__ H 1XEf DQG VR ZH KDYH (:DF:EF 4(' 7KH FRQGLWLRQ ZKLFK DOORZV XV WR UHVWRUH 5&($ WR D KDOIFODVVLFDO ORJLF PXVW EH VWDWHG LQ WHUPV QRW VROHO\ RI IUDPHV EXW RI PRGHOV +RZn HYHU DQ DOWHUQDWLYH LV DYDLODEOH IRU &( / &( LV FRQVLVWHQW ZUW WKH FODVV RI DOO SURSRVLWLRQDO IUDPHV 3URRI :H QHHG RQO\ FKHFN WKDW 5&(& DQG 5&($ SUHVHUYH YDOLGLW\ 6XSSRVH (DE LV YDOLG 7KHQ LQ DQ\ PRGHO __D__ __E__ &RQVHTXHQWO\ ZH KDYH ERWK 1X __Df 1X __Ef DQG __D__ H1X__F@_f LII __E__ H 1X __F__f 7KHUHn IRUH ZH KDYH ERWK (:DF:EF DQG (:FD:FE 4(' )RU WKH FRQVLVWHQF\ DQG ODWHU FRPSOHWHQHVVf RI &0 &5 DQG &. ZH ZLOO UHTXLUH WKH FRQGLWLRQV WR EH VWDWHG EHORZ EH LPSRVHG DV LQGLFDWHG RQ SURSRVLWLRQDO IUDPHV $V LQ 6HFWLRQ ZH ZLOO OHW & GHQRWH WKH FODVV RI DOO SURSRVLWLRQDO IUDPHV DQG &GHQRWH WKH FODVV RI IUDPHV VDWLVI\LQJ D SDUWLFXODU FRQGLWLRQ RU FRPELQDWLRQ RI FRQGLWLRQV )RU VLPSOLFLW\ ZH ZLOO GURS WKH LQLWLDO F LQ WKH GHVLJQDWLRQ RI WKH FRQGLWLRQV EHORZ ZKHQ XVLQJ WKHP DV VXEVFULSWVf ( FPf $ $ % H 1X;f LPSOLHV $ H 1X;f DQG % H 1X;f FUf $% H 1X;f LPSOLHV $ $ % H 1X;f FQf 8 H 1X;f

PAGE 181

FTf 1X;f 38f FVf 1X;f %HFDXVH 1 LV D IXQFWLRQ ZKRVH GRPDLQ LV 8 [ S8f ZH FDQQRW GLUHFWO\ FODVVLI\ ZRUOGV DV VD\ VLQJXODU DV LQ 6HFWLRQ EXW ZLOO ILUVW KDYH WR FODVVLI\ ZRUOGSURSRVLWLRQ SDLUV WKHQ ZRUOGV WKHQ IUDPHV Df $ SURSRVLWLRQ ; LV VLQJXODU DW X LII FVf LV VDWLVILHG Ef $ SURSRVLWLRQ ; LV PRQRWRQLF DW X LII FPf LV VDWLVILHG Ff $ SURSRVLWLRQ ; LV UHJXODU DW X LII FPf DQG FUf DUH VDWLVILHG Gf $ SURSRVLWLRQ ; LV QRUPDO DW X LII FPf FUf FQf DUH MRLQWO\ VDWLVILHG $ ZRUOG LV VLQJXODU PRQRWRQLF HWFf LII HYHU\ SURSRVLWLRQ LV VLQJXODU PRQRWRQLF HWFf DW WKDW ZRUOG $ IUDPH LV VLQJXODU PRQRWRQLF HWFf LII HYHU\ ZRUOG LQ WKDW IUDPH LV VLQJXODU PRQRWRQLF HWFf 1RWH WKDW WKH MRLQW VDWLVIDFWLRQ RI FPf FUf DQG FQf DPRXQWV WR UHn TXLULQJ WKDW 1X;f EH D ILOWHU )RU DQ\ PRGHO FODXVHV Df WKURXJK Jf RI ( KROG VR WKDW ZH PD\ ZRUN ZLWK WKH VDPH H[WHQVLRQV RI RXU WUXWK GHILQLWLRQV WR GHILQHG FRQn QHFWLYHV ,Q DGGLWLRQ ZH PD\ GHILQH WKH IROORZLQJ FRQQHFWLYH LQWHQGHG WR UHSUHVHQW WKH PLJKWFRQGLWLRQDO DQG D FRUUHVSRQGLQJ WUXWK FRQGLWLRQ Df 9DE A 1:D1E Ef 9DE LII :D1E :H DOVR FDOO DWWHQWLRQ WR WKH IROORZLQJ FRQGLWLRQDO D[LRPV

PAGE 182

( &4 :ST &6 9ST ,Q WKH IROORZLQJ VHULHV RI OHPPDV ZH VWDWH WKH FRQVLVWHQF\ RI YDULRXV FRQGLWLRQDO ORJLFV ZLWK UHVSHFW WR FHUWDLQ FODVVHV RI SURSRVLn WLRQDO IUDPHV +HUHDIWHU XQOHVV RWKHUZLVH QRWHG IUDPHV ZLOO EH SUR SRVLWLRQDO QHLJKERUKRRG IUDPHV :H SURYH RQO\ D VDPSOH RI WKH IROORZLQJ OHPPDV VLQFH WKH SURRIV IROORZ WKRVH RI FRPSDUDEOH OHPPDV LQ 6HFWLRQ VR FORVHO\ ,Q YLHZ RI / ZH QHHG RQO\ FKHFN WKH YDOLGLW\ RI WKH FRQGLWLRQDO D[LRPV LQ WKH GHVLJQDWHG FODVV RI IUDPHV / &( &6 LV FRQVLVWHQW ZUW & f§ V 3URRI $VVXPH 0 LV DQ\ PRGHO RQ DQ\ IUDPH LQ & DQG X DQ\ ZRUOG %\ V FRQGLWLRQ FVf 1X __S__f KHQFH __1T__ O 1XSf 6R E\ WKH WUXWK GHILQLWLRQ ZH KDYH :S1T DQG VR E\ ZH KDYH 9ST 4(' &RPSDUH /f / &( 24 LV FRQVLVWHQW ZUW &A &RPSDUH /f / &0 &( &0 LV FRQVLVWHQW ZUW & &RPSDUH /f 3URRI ,W LV VXIILFLHQW WR VKRZ WKDW &0 LV YDOLG LQ & /HW 0 EH DQ\ P PRGHO RQ DQ\ IUDPH LQ &A DQG VXSSRVH :S.TU 7KHQ E\ WKH WUXWK GHILQLn WLRQ __.TU H 1XSf %XW WKHQ __T__ $ __U__ H 1XSf DQG VR E\ FPf ,, T },, U H 1XSf 6R E\ WKH WUXWK GHILQLWLRQ ERWK :ST DQG :SU DQG KHQFH .:ST:SU 4(' / &5 LV FRQVLVWHQW ZUW &n &RPSDUH /f / &1 LV FRQVLVWHQW ZUW & &RPSDUH /f / p IIL &0 &5LV FRQVLVWHQW ZUW & &RPSDUH /f

PAGE 183

/$ &. &( &0&5 &1LV FRQVLVWHQW ZUW & &RQSDUH /$ f f§ f§ PP %RWK &6 DQG &( 24 DUH RI VRPH SDWKRORJLFDO LQWHUHVW ,Q WKH IRUPHU 9ST LV DQ D[LRP DQG LQ WKH ODWWHU :ST LV :RUOGV ZKLFK VDWLVI\ HLWKHU FVf RU FTf DUH XQSUHGLFWDEOH ZRUOGV WKRXJK LQ GLIIHULQJ VHQVHV LQ FVZRUOGV DQ\WKLQJ PLJKW KDSSHQ DQG LQ FTZRUOGV HYHU\WKLQJ ZLOO KDSSHQ 1RWH WKDW 9ST DOVR KDV D SURSRVLWLRQDOO\ LQGH[HG PRGDOLW\ FRUUHVSRQGLQJ WR LW ,I ZH GHILQH 0 T DV 1/A1T WKHQ DV 1/A1T FRUUHVSRQGV WR 1:S1T E\ 9ST FRUUHVSRQGV WR 0AT :H DOVR UHPDUN XSRQ WKH IROORZLQJ ZLWKRXW SURRI / &N LV FRQVLVWHQW ZLWK UHVSHFW WR WKH FODVV RI VHQWHQWLDO PRGHOV VDWLVI\LQJ WKH VHQWHQWLDO DQDORJV RI FPf FUf DQG FQf 7KH SURRIV RI FRPSOHWHQHVV IRU WKH ORJLFV UHPDUNHG DERYH DUH DJDLQ FRPSOLFDWHG E\ WKH IDFW WKDW WKH PRVW QDWXUDO FDQRQLFDO PRGHOV DUH LQVXIn ILFLHQW LQ WKH FDVH RI D PRQRWRQLF FRQGLWLRQDO ORJLF )XUWKHUPRUH WKH IDFW WKDW 1 KDV DV D VHFRQG DUJXPHQW D VXEVHW RI 8 PHDQV WKDW FDQRQLFDO IUDPHV DUH LQ JHQHUDO QRW XQLTXH IRU JLYHQ ORJLF / &KHOODV > SS @ SURYLGHV WKH UHTXLVLWH WHFKQLTXHV DQG ZH UHO\ XSRQ KLV PHWKRGV DQG WHUPLQRORJ\ LQ ZKDW IROORZV /HW / EH D FODVVLFDO FRQGLWLRQDO ORJLF DQG Df GHQRWH WKH VHW RI PD[LPDOO\ /FRQVLVWHQW IRUPXODV Ef _D_A GHQRWH WKH PD[LPDOO\ /FRQVLVWHQW VHWV RI ZKLFK D LV D PHPEHU VR __ f Ff 1 [ S8Af ‘r 338Aff LV DQ\ IXQFWLRQ VXFK WKDW :DE H X LII _E_A H 1X _D_Af Gf )RU HDFK S H 3 9ASf _Sc/

PAGE 184

7KHQ ) LV D FDQRQLFDO SURSRVLWLRQDO QHLJKERUKRRGf IUDPH IRU / DQG 0 AAA LV D FDQRQLFDO SURSRVLWLRQDO QHLJKERUKRRGf PRGHO IRU / 7KDW WKH GHILQLWLRQ RI LV XQDPELJXRXV ZLWK UHVSHFW WR UHSUHn VHQWDWLYHV RI VHWV RI PD[LPDOO\ /FRQVLVWHQW VHWV IROORZV IURP WKH IDFW WKDW / LV FODVVLFDO )URP WKLV WKH IXQGDPHQWDO WKHRUHP IRU FODVVLFDO FRQGLWLRQDO ORJLFV IROORZV UHDGLO\ &RPSDUH &KHOODV > SS QO@f _E_/ G_/ WKHQ _E_/ H 1X_D_/f LII _G_/ H 1X_F_/f 3URRI $V EHIRUH ZH GURS ZKDWHYHU VXEVFULSWV ZH FDQf $VVXPH _D_ _F_ DQG _E_ _G_ 7KHQ E\ WKH SURSHUWLHV RI PD[LPDOO\ / FRQVLVWHQW VHWV I (DF DQG I (EG 6R E\ 5&($ ZH KDYH I (:DE:FE DQG E\ 5&(& ZH KDYH f§ (:FE:FG 6R E\ FORVXUH XQGHU 03 ZH KDYH I (:DE:FG 7KHQ E\ WKH SURSHUWLHV RI PD[LPDOO\ /FRQVLVWHQW VHWV ZH KDYH :DE H X LII :HG H X IRU DOO X H 8 7KHQ E\ WKH GHILQLWLRQ RI 1 ZH KDYH _E_ H 1X _D_f LII _G_ H 1X_F_f 4(' 7 /HW EH D FDQRQLFDO SURSRVLWLRQDO QHLJKERUKRRG PRGHO IRU FODVVLn FDO FRQGLWLRQDO ORJLF / 7KHQ IRU DOO IRUPXODV D DQG DOO ZRUOGV X H 8 7KDW LV __D__A/ _D_A IRU DOO IRUPXODV D 3URRI :H GR WKH LQGXFWLRQ RQO\ IRU IRUPXOD D RI WKH IRUP :EF 6R DVVXPH WKH WKHRUHP KROGV IRU DOO ZRUOGV DQG IRUPXODV E DQG F %\ WKH WUXWK GHILQLn WLRQ :EF LII __E__ H 1X __D__f 6LQFH E\ K\SRWKHVLV __E__ _E_ DQG __D _D_ ZH KDYH :EF LII _E_ H 1X _F_f +HQFH E\ WKH GHILQLWLRQ RI 1 DQG 3 ZH KDYH :EF LII :EF H X 4(' 3 LQ LV XQDPELJXRXV 7KDW LV LI _D_

PAGE 185

:H QRWHG DERYH WKDW FDQRQLFDO IUDPHV DUH JHQHUDOO\ QRW XQLTXHO\ VSHFLILHG E\ FRQGLWLRQ 'Ff 6XSSRVH ; F 38Af VXFK WKDW ; I _D_A IRU DOO IRUPXODV D :KDW HOHPHQW RI 338Aff GR ZH DVVLJQ WR WKH SDLU X;f" 'HSHQGLQJ XSRQ RXU FKRLFH ZH ZLOO JHW GLIIHUHQW FDQRQLFDO IUDPHV $ SDUWLFXODU IDPLO\ RI FDQRQLFDO IUDPHV FDOOHG SURSHU E\ &KHOODV > S @ LV LGHQWLILHG E\ WKH IROORZLQJ VSHFLILFDWLRQ RI 1A ( 1X _D_/f ^ME_/ _E_/ F DQG :DE H X` 7KDW WKLV GHWHUPLQHV D VHW RI FDQRQLFDO IUDPHV LV HYLGHQW E\ FDQSDULQJ & WR 'Ff )RU VXEVHWV RI 38Af VXFK DV ; DERYH D VPDOOHVW SURSHU FDQRQLFDO IUDPH LV GHWHUPLQHG E\ VHWWLQJ 1AX;f IRU DOO VXFK ; DQG DOO ZRUOGV X D ODUJHVW SURSHU FDQRQLFDO IUDPH LV GHWHUPLQHG E\ VHWWLQJ 1AX;f 38Af 6HH > S @ EXW QRWH WKH DGDSWDWLRQ WR QHLJKERUKRRG PRGHOV LV XVHG DERYHf &RPSOHWHQHVV SURRIV IRU FRQGLWLRQDO ORJLFV DUH VLPLODU WR WKRVH IRU PRGDO ORJLFV ,Q DOO FDVHV IRU D FODVVLFDO FRQGLWLRQDO ORJLF / ZH PD\ SURFHHG DFFRUGLQJ WR WKH IROORZLQJ SODQ ( f $VVXPH VRPH IRUPXOD D LV YDOLG LQ FODVV RI IUDPHV & F f 6KRZ WKDW WKH IUDPH RI VRPH SURSHUf FDQRQLFDO PRGHO 0A VDWLVILHV Ff f 7KHQ D LV WUXH LQ 0a f +HQFH D LV D WKHRUHP RI / E\ 7 DQG WKH FRUROODU\ WR /LQGHQEDXPnV /HQLQD f 7KHUHIRUH / LV FRPSOHWH ZLWK UHVSHFW WR & ,Q WKH VLPSOHVW FDVHV ZH ZLOO EH DEOH WR ILQG D SURSHU FDQRQLFDO PRGHO ZKLFK VDWLVILHV WKH VWDWHG FRQGLWLRQV RQ WKH IUDPHV +RZHYHU DQDORJRXV

PAGE 186

WR PRGDO ORJLFV QR SURSHU FDQRQLFDO IUDPH IRU D PRQRWRQLF FRQGLWLRQDO ORJLF VDWLVILHV FPf $ FDQRQLFDO IUDPH WKDW GRHV VDWLVI\ FPf FDQ EH IRXQG XVLQJ WKH QRWLRQ RI VXSS OHPHQW DW LRQ GHILQHG E\ &KHOODV > S @ ,Q WKH FDVH RI PRQRWRQLF PRGDO ORJLFV WKH VXSSOHPHQWHG IUDPH ZDV QRW LWVHOI FDQRQLFDO 7KH FDVH LV RWKHUZLVH ZLWK FRQGLWLRQDO ORJLFV EHFDXVH WKH FDQRQLFDO IUDPHV DUH QRW XQLTXH 7KH GHILQLWLRQ DQG OHQPD EHORZ DGDSWHG IURP &KHOODV > SS @ HVWDEOLVK WKH UHTXLVLWH WHFKQLTXHV /HW ) 81 EH D FDQRQLFDO SURSRVLWLRQDO QHLJKERUKRRG IUDPH IRU PRQRWRQLF FRQGLWLRQDO ORJLF / 7KHQ WKH VXSSOHPHQWDWLRQ RI ) LV )r 81r VXFK WKDW 1rX;f ^< < F 8 DQG = F < IRU VRPH = H 1X;f` $ IUDPH LGHQWLFDO ZLWK LWV RZQ VXSSOHPHQWDWLRQ LV VDLG WR EH VXSSOHPHQWHG / ,I ) 81 LV D SURSHU FDQRQLFDO IUDPH IRU PRQR WRQLF FRQGLn WLRQDO ORJLF / WKHQ )r 81f WKH VXSSOHPHQWDWLRQ RI ) LV D FDQRQLFDO IUDPH IRU / 3URRI $VVXPH ) LV DV VWDWHG DQG VKRZ :DE H X LII _E_ H 1rX _D_f 6XSSRVH :DE H 8 7KHQ _E_ H 1X _D_f VR E\ GHILQLWLRQ _E_ H 1rX _D_f 6XSSRVH _E_ H 1rX _D_f 7KHQ IRU VRPH = H 1X _D_f =F _E_ 6LQFH = H 1X _D_f E\ GHILQLWLRQ RI 1 IRU VRPH F :DF H X DQG = _F_ +HQFH OFO eB LE_ VR f§ &FE +HQFH E\ 5&0 _f§ &:DF:DE 7KHQ DV PD[LPDOO\ / FRQVLVWHQW VHWV DUH FORVHG XQGHU 03 :DE H X 4(' )RU PRQR WRQLF FRQGLWLRQDO ORJLFV ZH ZLOO KDYH WR VKRZ (f Df 7KH IUDPH RI VRPH SURSHU FDQRQLFDO PRGHO 0 VDWLVILHV FRQGL -M WLRQV Ff OHVV FPf

PAGE 187

Ef 7KH VXSSOHPHQWDWLRQ RI WKDW IUDPH VWLOO VDWLVILHV Ff LQFOXGLQJ FPf E\ / EHORZ Ff +HQFH WKH VXSSOHPHQWHG IUDPH LV FDQRQLFDO IRU / E\ /$ 7KH IROORZLQJ OHPPD LV XVHIXO LQ WKLV FRQQHFWLRQ / ,I ) 81 LV D SURSHU FDQRQLFDO IUDPH VDWLVI\LQJ VRPH SHUn PLVVLEOH FRPELQDWLRQ RI FRQGLWLRQV FUf FQf FTf DQG FVf WKHQ f -/ ) 81 VDWLVILHV WKDW FRPELQDWLRQ RI FRQGLWLRQV DQG FPf 3URRI 7KDW )r VDWLVILHV FPf LV REYLRXV 7KH SURRIV IRU HDFK VHSDUDWH FRQGLWLRQ DQG WKH FRPELQDWLRQV DUH SUHFLVHO\ VLPLODU WR WKRVH RI /$ 4(' ,Q YLHZ RI ($ $ DV VXSSOHPHQWHG DERYH ZH ZLOO RQO\ KDYH WR VKRZ ($f KROGV LQ RXU FRPSOHWHQHVV SURRIV 6HYHUDO LPPHGLDWHO\ IROORZ /$ &( LV FRPSOHWH ZUW & 3URRI &OHDUO\ WKH IUDPH RI LV LQ WKH FODVV RI DOO IUDPHV & VR WKH OHPPD IROORZV E\ ( 4(' /$ &( &6 LV FRPSOHWH ZUW & 3URRI /HW EH D SURSHU FDQRQLFDO PRGHO IRU &( &6 VDWLVI\LQJ FVf ZKHQHYHU ; _D_ IRU HYHU\ IRUPXOD D WKDW LV WKH VPDOOHVW SURSHU FDQRQLFDO PRGHO 7KDW VDWLVILHV FVf ZKHQ ; _D_ IRU VRPH IRUPXOD D IROORZV 9DE H X IRU DQ\ ZRUOG X DQG DQ\ IRUPXODV D DQG E VLQFH &( &6 LV WKH GHGXFWLYH FORVXUH RI WKH V\VWHP FRQWDLQLQJ &6 &RQVHTXHQWO\ 1:FG H X IRU DOO X DQG DOO IRUPXODV F DQG G VR :HG O X 6R E\ GHILQLWLRQ RI 1T 1JX;f IRU DOO X DQG ; &RQVHTXHQWO\ FVf LV VDWLVILHG %\ ( WKH OHPPD IROORZV 4('

PAGE 188

/$ &0 &( 2, LV FRPSOHWH ZUW &P 3URRI 7KRXJK QR SURSHU FDQRQLFDO PRGHO VDWLVILHV FPf HYHU\ VXSn SOHPHQWHG PRGHO GRHV +HQFH WKH VXSSOHPHQWDWLRQ RI DQ\ SURSHU FDQRQLFDO PRGHO VD\ WKH ODUJHVW VDWLVILHV FPf VR E\ ( WKH OHPPD IROORZV 4(' / &5 &( 2, &5 LV FRPSOHWH ZUW & f§ f§ PU 3URRI :H QHHG RQO\ VKRZ VRPH SURSHU FDQRQLFDO PRGHO IRU &5 VDWLVILHV FUf /HW EH WKH ODUJHVW SURSHU FDQRQLFDO PRGHO IRU &5 7KHQ 05 VDWLVILHV FUf ZKHQHYHU ; A _D_ IRU HYHU\ IRUPXOD D VLQFH LQ WKRVH FDVHV ZH KDYH VHW 1X;f 38f 6XSSRVH ; _D_ IRU VRPH IRUPXOD D /HW _E_ F_ e 1X _D_f 7KHQ :DE:DF H X DQG VR .:DE:DF H X E\ WKH SURSHUWLHV RI PD[LPDOO\ /FRQVLVWHQW VHWV %\ FORVXUH XQGHU 03 DQG XVLQJ WKH LQVWDQFH RI D[LRP &5 LQ X RI WKH IRUP &.:DE:DF:D.EF :D.EF H X &RQVHTXHQWO\ _.EF_ H 1X_D_f 6R _E_ $ _F_ H1X_D_f 4(' / &. &( &0 &5 &1LV FRPSOHWH ZUW & 3URRI ,Q YLHZ RI WKH SURRI RI / ZH QHHG RQO\ VKRZ WKH ODUJHVW SURSHU FDQRQLFDO PRGHO IRU &. VDWLVILHV FQf ,W REYLRXVO\ GRHV LI ; _DM IRU HYHU\ IRUPXOD D VR DVVXPH ; _D_ IRU VRPH IRUPXOD D 6LQFH &1 KROGV IRU HYHU\ X :DO H X +HQFH __ H 1X _D_f EXW __ 8 VR 8 H 1X_D_f 4(' 7KH FRPSOHWHQHVV RI &( &5 DQG &( &1 DUH REYLRXV LQ YLHZ RI WKH SURRIV RI WKH DERYH 7KH FRPSOHWHQHVV RI &( 24 RIIHUV DQ LQWHUHVWLQJ WZLVW DV QR SURSHU FDQRQLFDO PRGHO VDWLVILHV FTf / &( 24 LV FRPSOHWH ZUW & 3URRI /HW EH WKH VXSSOHPHQWDWLRQ RI WKH ODUJHVW SURSHU FDQRQLFDO PRGHO IRU 4( 24 7KHQ IRU ; L DIRU HYHU\ IRUPXOD D 1X;f 38f VR FTf KROGV $VVXPH ; _D_ IRU VRPH IRUPXOD D 6LQFH 24 KROGV LW

PAGE 189

IROORZV WKDW IRU DOO X :DR H X &RQVHTXHQWO\ _R_ H 1X _D_f %XW >R_ VR E\ VXSHUVHW FORVXUH 1X_D_f 38f 4(' 7KH DERYH OHPPDV DQG WKH HDUOLHU RQHV RQ FRQVLVWHQF\ RI FODVVLFDO FRQGLWLRQDO ORJLFV LPPHGLDWHO\ JLYH XV WKH IROORZLQJ UHVXOWV FRQFHUQLQJ WKH GHWHUPLQDWLRQ RI ORJLFV E\ FODVVHV RI IUDPHV 7 &( LV GHWHUPLQHG E\ & WKH FODVV RI DOO SURSRVLWLRQDO QHLJKERUn KRRG IUDPHV 7 &0 LV GHWHUPLQHG E\ & 7 &5 LV GHWHUPLQHG E\ & 7 &. LV GHWHUPLQHG E\ J WKH FODVV RI DOO SURSRVLWLRQDO QHLJKn ERUKRRG IUDPHV ZKHUH 1X;f LV D ILOWHU 7 &( LV GHWHUPLQHG E\ & 7 &( &6 LV GHWHUPLQHG E\ & &RPSOHWHQHVV SURRIV IRU WKH KDOIFODVVLFDO FRQGLWLRQDO ORJLF &H DQG WKH KDOIQRUPDO FRQGLWLRQDO ORJLF &N FDQ EH SURGXFHG E\ VOLJKW PRGLn ILFDWLRQ RI WKH DERYH WHFKQLTXHV &RPSOHWHQHVV DQG FRQVLVWHQF\ SURRIV IRU H[WHQVLRQV RI &. ZLOO EH FRQVLGHUHG LQ 6HFWLRQ DIWHU VHYHUDO H[n WHQVLRQV DUH SUHVHQWHG LQ 6HFWLRQ %HIRUH WXUQLQJ WR H[WHQVLRQV RI &. ZH ZLOO GLVFXVV DOWHUQDWLYH VHPDQWLFV IRU &. DQG WKH UHODWLRQVKLS EHWZHHQ QHLJKERUKRRG IUDPHV DQG VSKHUH IXQFWLRQ IUDPHV SDUWLDOO\ DQVZHUn LQJ WKH TXHVWLRQ RI 6HFWLRQ $OWHUQDWLYH 6HPDQWLFV IRU &RQGLWLRQDO /RJLFV ,Q 6HFWLRQ ZH VKRZHG WKDW UHODWLRQDO VHPDQWLFV IRU PRGDO ORJLFV FRXOG EH GHILQHG DV D VSHFLDO FDVH RI QHLJKERUKRRG VHPDQWLFV IRU UHJXODU PRGDO ORJLFV 7KH DQDORJRXV VLWXDWLRQ REWDLQV IRU FRQGLWLRQDO ORJLFV

PAGE 190

IRU DQ\ UHJXODU SURSRVLWLRQDO QHLJKERUKRRG IUDPH IRU FRQGLWLRQDO ORJLF ZH FDQ FRQVWUXFW D SURSRVLWLRQDO DFFHVVLELOLW\ IXQFWLRQ DV ZHOO DV D SURSRVLWLRQDO DOWHUQDWLYH UHODWLRQ /HW ) 81 EH D UHJXODU SURSRVLWLRQDO QHLJKERUKRRG IUDPH /HW ^X 1X;f ` 7KHQ 4 LV WKH VHW RI DOO ZRUOGV VLQJXODU IRU SURSRVLWLRQ ; /HW 4 WKH VHW RI DOO VXFK 4A ^X;f X H 4A` LV WKH VHW RI VLQJXODULWLHV RI 6 :H GHILQH D SURSRVLWLRQDO DFFHVVLELOLW\ IXQFWLRQ 6 8 [ 38f 38f E\ 6X;f $1X;f SURYLGHG 1X;f I /HW ) 81 EH D UHJXODU SURSRVLWLRQDO QHLJKERUKRRG IUDPH DQG 6 GHILQHG DV LQ :H GHILQH D SURSRVLWLRQDO DFFHVVLELOLW\ RU DOWHUQDWLYHf UHODWLRQ 5 F 8 [ S8f [ 8 VXFK WKDW X5AY LII X L 4[ DQG Y H 6X;f 6LPLODU GHILQLWLRQV FRXOG EH SURGXFHG IRU UHJXODU VHQWHQWLDO QHLJKERUn KRRG IUDPHV ,W LV FOHDU IURP WKH DERYH GHILQLWLRQV WKDW WR HDFK UHJXODU SUR SRVLWLRQDO QHLJKERUKRRG IUDPH WKHUH FRUUHVSRQGV D SURSRVLWLRQDO DFFHVn VLELOLW\ IXQFWLRQ IUDPH )r 864 DQG D SURSRVLWLRQDO UHODWLRQDO IUDPH ) 854 7KH WUXWK GHILQLWLRQV IRU FRQGLWLRQDOV LQ SURSRVLn WLRQDO DFFHVVLELOLW\ IXQFWLRQ DQG SURSRVLWLRQDO UHODWLRQDO PRGHOV DUH JLYHQ E\ OHW 0 864 EH D SURSRVLWLRQDO DFFHVVLELOLW\ IXQFWLRQ PRGHO 7KHQ ZH GHILQH WUXWK LQ 0 DW ZRUOG X IRU FRQGLWLRQDO IRUPXODV E\ :DE LII X L 4__DMM DQG 6X __D__f F __E__

PAGE 191

' /HW 0 854 EH D SUHSRVLWLRQDO UHODWLRQDO PRGHO 7KHQ ZH GHILQH WUXWK LQ 0 DW ZRUOG X IRU FRQGLWLRQDO IRUPXODV E\ :DE LII X L 4__D__ DQG Y X5__D__9` F __E__ &KHOODV > SS @ GLVFXVVHV SURSRVLWLRQDO DFFHVVLELOLW\ IXQFn WLRQ IUDPHV IRU WKH FDVHV ZKHUH 4 WKDW LV QR VLQJXODULWLHV DUH SUHn VHQW DV VWDQGDUG IUDPHV DQG PHQWLRQV WKH FRUUHVSRQGLQJ UHODWLRQDO IUDPHV 1XWH > FKDSWHU @ FRQSDUHV YDULRXV VHPDQWLFV IRU FRQGLWLRQDO ORJLFV DQG GLVFXVVHV ERWK WKH VHQWHQWLDO DQG SURSRVLWLRQDO YHUVLRQV RI QRUPDO DFFHVVLELOLW\ IXQFWLRQ DQG UHODWLRQDO IUDPHV ZKHUH QRUPDO PHDQV 4 f $OO RI 1XWHn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f r‘ 338ff GHILQHG E\ > LI X H H 4 1X;f ? O ^< < F 8 DQG ^Y Y H 8 DQG Y H 6X;f` F <` RWKHUZLVH ,W LV FOHDU WKDW ZH FDQ GR WKH VDPH IRU SURSRVLWLRQDO UHODWLRQDO IUDPHV DV ZHOO DQG LQ RQHWRRQH FRUUHVSRQGHQFH ZLWK WKH DERYH UHSODFH Y H 6X;f E\ X5AY LQ WKH GHILQLWLRQ RI 1X;f ,Q YLHZ RI RXU HDUOLHU FRPSOHWHQHVV UHVXOWV DQG WKH DERYH GHILQLn WLRQV ZH PD\ VWDWH WKH IROORZLQJ WKHRUHPV

PAGE 192

7 &5 LV GHWHUPLQHG E\ WKH FODVV RI DOO SURSRVLWLRQDO DFFHVVLn ELOLW\ IXQFWLRQ IUDPHV DQG &. LV GHWHUPLQHG E\ WKH FODVV RI DOO VXFK IUDPHV IRU ZKLFK 4 7 &5 LV GHWHUPLQHG E\ WKH FODVV RI DOO SURSRVLWLRQDO UHODWLRQDO IUDPHV DQG &. LV GHWHUPLQHG E\ WKH FODVV RI DOO VXFK IUDPHV IRU ZKLFK 4 6LPLODU UHVXOWV KROG IRU WKH VHQWHQWLDO YHUVLRQV WKRXJK LQ WHUPV RI FODVVHV RI PRGHOV ,W LV LPSRUWDQW WR QRWH WKDW 1XWH GHQRWHV E\ SURSRVLWLRQDO UHODWLRQDO PRGHO RQO\ WKRVH PRGHOV ZKRVH IUDPH LV VXFK WKDW QR VLQJXODULWLHV DUH SUHVHQW > S @ WKDW LV QRUPDO SURSRVLn WLRQDO UHODWLRQDO IUDPHV ,Q LQYHVWLJDWLQJ WKH UHODWLRQVKLS EHWZHHQ VSKHUH IXQFWLRQ IUDPHV DQG DOWHUQDWLYH VHPDQWLFV IRU FRQGLWLRQDO ORJLFV LW ZLOO EH XVHIXO WR KDYH D IRUPDO GHILQLWLRQ RI D VSKHUH IXQFWLRQ IUDPH DQG DVVRFLDWHG PRGHOV DQG WUXWK GHILQLWLRQV 6HYHUDO GHILQLWLRQV WRZDUG WKLV HQG DUH VWDWHG EHORZ $ VSKHUH IXQFWLRQ IUDPH ) 8 LV DQ RUGHUHG SDLU VXFK WKDW Df 8 LV D VHW RI SRVVLEOH ZRUOGV DQG Ef 8 ‘r 338ff LV D IXQFWLRQ VXFK WKDW ;< H X LPSOLHV ;F
PAGE 193

' /HW ) EH D VSKHUH IXQFWLRQ IUDPH 7KHQ ) LV Df OLPLWHG LII ;F X LPSOLHV $; H ; Ef QRUPDO LII 8X I ,Q LW ZLOO EH QRWHG WKDW WKH FORVXUH DQG FHQWHULQJ FRQGLWLRQV DUH QRW SUHVHQW 7KH IRUPHU DUH QRW UHTXLUHG LQ RUGHU WR GHWHUPLQH WKH VHW RI YDOLG IRUPXODV VHH /HZLV > S Q@f DQG WKH ODWWHU LV QRW UHn TXLUHG WR GHWHUPLQH WKH ZHDNHVW ORJLF /HZLV UHFRJQL]HV 9 $V ZH VKDOO VHH ODWHU ZH VKDOO DGG FRQGLWLRQV WR JHW WKH FODVV RI IUDPHV WKDW GHWHUn PLQHV 9& DQG 9&8 ,Q YLHZ RI 'Df D IUDPH LV OLPLWHG MXVW LQ FDVH LW VDWLVILHV WKH OLPLW DVVXPSWLRQ /$ 7KLV ZLOO QRW DIIHFW YDOLGLW\ HLWKHU /HZLVn GHILQLWLRQ RI QRUPDO > S @ KDV EHHQ JLYHQ KHUH DV QRUPDOnn VLQFH KLV GHILQLWLRQ LV DW YDULDQFH ZLWK WKH GHILQLWLRQV ZH KDYH EHHQ XVLQJ (YHU\ OLPLWHG VSKHUH IXQFWLRQ IUDPH LV QRUPDO LQ RXU VHQVH VLQFH WKH DFFHVVLELOLW\ IXQFWLRQ IUDPH FRUUHVSRQGLQJ WR LW KDV 4 /HZLV HVVHQWLDOO\ SURYHV WKLV KLPVHOI > S @ DQG ZH SURYH D YHUVLRQ EHORZ /HZLV FDOOV VXFK IXQFWLRQV VHW VHOHFWLRQ IXQFWLRQVf 7 (DFK OLPLWHG VSKHUH IXQFWLRQ IUDPH LV HTXLYDOHQW WR VRPH QRUPDO DFFHVVLELOLW\ IXQFWLRQ IUDPH $OWHUQDWHO\ FODVV VHOHFWLRQ IXQFWLRQ IUDPHf 3URRI /HW ) EH D OLPLWHG VSKHUH IXQFWLRQ IUDPH DQG 0 DQ\ PRGHO RQ WKDW IUDPH :H GHILQH D FRUUHVSRQGLQJ DFFHVVLELOLW\ IXQFWLRQ 6 8 [ S8f 38f E\ 0 ,W LV VXIILFLHQW WR VKRZ :DE LII 6X __D__f F __E__ 1RWH ZH KDYH VDLG

PAGE 194

QRWKLQJ DERXW 4 DQG DVVXPH LW LV HPSW\ WKXV ) 86 LV QRUPDO 6XSSRVH :DE 7KHQ HLWKHU __D__ $ 8X RU QRW ,I VR WKHQ 6X @MDf F __E__ ,I QRW WKHQ E\ WKH WUXWK GHILQLWLRQ IRU VRPH ; H I __D__ $;f F __E__ %XW E\ GHILQLWLRQ 6 X __ D_ _f BF __ D $;f VR 6X __D__f F __E__ 6XSSRVH 6 X __ D f F __E__ ,I 6X __Df WKHQ E\ GHILQLWLRQ __DM_ $ 8X VR :DE 6R DVVXPH RWKHUZLVH /HW ; EH WKH VPDOOHVW VSKHUH VXFK WKDW __D__ $ ; F __E>_ VR 6 X __ D f __ D __ $; :H PD\ GR WKLV E\ GHILQLWLRQ RI 6 DQG EHFDXVH ) LV OLPLWHG 6LQFH 6X __D__f F __E__ LW IROORZV WKDW I __D__ $ ;f e __E>_ IRU VRPH ; H KHQFH :DE 4(' 7KH UHTXLUHPHQW WKDW ) EH OLPLWHG FDQQRW EH UHPRYHG IURP 7 HOVH 6 ZLOO QRW SURGXFH FRUUHFW UHVXOWV IRU WUXWK RI QRQYDOLG FRQGLWLRQDO IRUPXODV ,I ZH KDG GHILQHG 6X;f DV WKH VHW RI SZRUOGV LQ WKH VPDOOHVW SSHUPLWWLQJ VSKHUH WKHQ IRU QRQOLPLWHG IUDPHV 6 ZRXOG QRW HYHQ EH ZHOO GHILQHG 7KH UHTXLUHPHQW WKDW ) EH OLPLWHG QHHG QRW DSSHDU LQ WKH IROn ORZLQJ WKHRUHP DGDSWHG IURP 1XWH > S @ ,W VKRXOG EH QRWHG WKDW 1XWH DVVXPHV OLPLWHG VSKHUH IXQFWLRQ IUDPHV DUH DOO WKDW RQH QHHG FRQVLGHU :KLOH WKLV LV WUXH IRU TXHVWLRQV RI YDOLGLW\ RI WKH WKHRUHPV RI D ORJLF LW LV LQVXIILFLHQW IRU WUXWK LQ JHQHUDO VR ZH SUHIHU WKH JUHDWHU JHQHUn DOLW\ REWDLQHG E\ DVVXPLQJ WKH VSKHUH IXQFWLRQ IUDPH LV QRW QHFHVVDULO\ OLPLWHG 7 (DFK VSKHUH IXQFWLRQ IUDPH LV HTXLYDOHQW WR VRPH QRUPDO QHLJKERUn KRRG IUDPH 3URRI /HW ) 8 EH D VSKHUH IXQFWLRQ IUDPH /HW 0 EH DQ\ PRGHO RQ ) :H GHILQH D FRUUHVSRQGLQJ QHLJKERUKRRG IXQFWLRQ 1 8 [ S8f 338ff E\

PAGE 195

I 38f LI 8 $ ; 1X;f ^< < F 8 DQG IRU VRPH = H A = $ ;f F <` RWKHUZLVH ,W LV VXIILFLHQW WR VKRZ :DE LII __E>_ H 1X __D_>f IRU HTXLYDOHQFH WR VRPH QHLJKERUKRRG PRGHO )RU QRUPDOLW\ FPf DQG FQf REYLRXVO\ KROG VR ZH RQO\ QHHG WR VKRZ FUf KROGV 0 6XSSRVH :DE 7KHQ E\ WKH WUXWK GHILQLWLRQ HLWKHU __D $ 8X RU IRU VRPH = H I = $ __D__f F __E__ ,I WKH IRUPHU WKHQ __Ec, H 38f 1X__D__f 6XSSRVH _@E__ H 1X_>DM_f ,I _MD__ $ 8X WKHQ E\ WKH WUXWK GHILQLWLRQ :DE WULYLDOO\ 6R DVVXPH RWKHUZLVH 7KHQ IRU VRPH = H X! = $ __Df F __E__ 6R E\ WKH WUXWK GHILQLWLRQ :DE :H VKRZ QRUPDOLW\ E\ DVVXPLQJ $% H 1X;f ,W LV VXIILFLHQW WR FRQVLGHU WKH QRQWULYLDO FDVH VR ZH KDYH ERWK IRU VRPH = H I = $ ;f F $ DQG IRU VRPH : H : $ ;f F % 1RZ E\ FRQGLWLRQ 'Ef RQ D VSKHUH IXQFWLRQ IUDPH HLWKHU = F : RU : F = KHQFH HLWKHU = $ ;f F : $ ;f RU : $ ;f F = $ ;f :LWKRXW ORVV RI JHQHUDOLW\ DVn VXPH WKH IRUPHU 7KHQ ERWK $ DQG % FRQWDLQ = $ ; KHQFH I = $ ;f e $ $ %f +HQFH $ $ % H 1X;f 4(' 7KXV WKH TXHVWLRQ RI 6HFWLRQ DV WR ZKHWKHU HDFK V\VWHP RI VSKHUHV IUDPH VSKHUH IXQFWLRQ IUDPHf KDV D FRUUHVSRQGLQJ QHLJKERUKRRG IUDPH KDV DQ DIILUPDWLYH DQVZHU ,W LV HYLGHQW WKDW ZH PD\ UHJDUG QHLJKERUKRRG VHPDQWLFV IRU FRQGLWLRQDO ORJLF DV PRUH JHQHUDO WKDQ VSKHUH IXQFWLRQ VHPDQWLFV LI RQO\ EHFDXVH QHLJKERUKRRG VHPDQWLFV LV DGHTXDWH WR QRQQRUPDO FRQGLWLRQDO ORJLFV ,Q 6HFWLRQ ZH VKDOO LQWURGXFH VRPH DGGLWLRQDO FRQGLWLRQDO D[LRPV DQG WKH ORJLFV REWDLQDEOH WKHUHIURP DQG LQ 6HFWLRQ ZH VKDOO GHYHORS DSSURSULDWH VHPDQWLFV IRU VXFK ORJLFV

PAGE 196

([WHQVLRQV RI &. 7KH DQDO\VHV RI FRXQWHUIDFWXDO FRQGLWLRQDOV E\ /HZLV 6WDOQDNHU DQG 3ROORFN DOO UHVXOW LQ ORJLFV IRU WKH FRQGLWLRQDO WKDW DUH H[n WHQVLRQV RI &. /RHZHU >@ VKRZV WKDW RQH SODXVLEOH UHDGLQJ RI *RRGPDQnV IDLOHG DQDO\VLV DOVR \LHOGV D ORJLF IRU WKH FRXQWHUIDFWXDO ZKLFK LV DQ H[WHQVLRQ RI &. ,Q >@ 1XWH GLVFXVVHV LQ VRPH GHWDLO D QXPEHU RI H[WHQVLRQV RI &. DQG RXU GLVFXVVLRQ LQ WKH QH[W WZR VHFWLRQV LV ODUJHO\ EDVHG XSRQ KLV +RZHYHU 1XWHnV SUHIHUUHG DQDO\VLV DFFHSWV 6'$ DQG UHMHFWV VXEVWLWXWLRQ RI HTXLYDOHQWV VR LV DQ H[WHQVLRQ RI &N WKH VPDOOHVW KDOIQRUPDO FRQGLWLRQDO ORJLF :H VKDOO FRQILQH RXU DWn WHQWLRQ WR QRUPDO FRQGLWLRQDO ORJLFV ,Q RXU YLHZ WKH RUGHULQJ UHODWLRQV LI DQ\ LPSRVHG XSRQ WKH VHW RI SRVVLEOH ZRUOGV E\ D JLYHQ DQDO\VLV RI FRXQWHUIDFWXDOV FRQVWLWXWH WKH VLQJOH PRVW LPSRUWDQW GLVWLQJXLVKLQJ FKDUDFWHULVWLF RI YDULRXV DQDO\VHV 7KH WKHVHV VWDWHG LQ WKLV VHFWLRQ DQG WKH VHPDQWLFV RI 6HFWLRQ DUH FKRVHQ VR DV WR PDNH WKLV RUGHU PRUH WUDQVSDUHQW 7KH IROORZLQJ FRQGLn WLRQDO WKHVHV PD\ EH DGGHG WR &. WR \LHOG YDULRXV H[WHQVLRQV ZH FRQWLQXH WR DVVXPH RXU ODQJXDJH LV &:f ( &RQGLWLRQDO 7KHVHV ,' :SS 03 :ST&ST && &.ST:ST 0' &:1SS:TS &.:ST:TS(:SU:TU &$ &.:SI:TU:$STU &% &:$STU$:SU:TU

PAGE 197

&9 &.:ST0-S1U:.SUT &(0 $:ST:S1T 7KH 7KHVHV RI ( PD\ DOO EH IRXQG LQ RWKHU VRXUFHV SULQFLSDOO\ 1XWH >@ WKRXJK RXU QDPLQJ FRQYHQWLRQV GLIIHU VRPHZKDW ,Q RUGHU WR VLPSOLI\ WKH VWDWHPHQW RI WZR RI WKHVH WKHVHV ZH LQWURGXFH DV LQ &+$37(5 21(f WKH IROORZLQJ GHILQHG V\PEROV )RU DQ\ IRUPXODV DE Df 9DE GI 1:D1E Ef )DE .:DE:ED 7KXV ZH PD\ UHVWDWH WZR RI RXU WKHVHV DV ( &2 &)ST(:SU:TU &9 &.:ST9SU:.SUT ([WHQVLRQV RI &. ZKLFK ZH VKDOO GLVFXVV LQ WKLV DQG WKH VXFFHHGLQJ VHFWLRQ LQFOXGH ( ([WHQVLRQV RI &. &. ,' &. 03 % &. ,' 03 % && &. ,' 0' &2 03 &* && &3 &$ 9 &9 &$ &2 &$ 9: 9 03

PAGE 198

66 &$ && 9& 9: && & 66 &(0 )ROORZLQJ WKH WHUPLQRORJ\ RI 1XWH > SS @ DV ZH GR WKURXJKRXW WKLV VHFWLRQf ZH FDOO DQ\ QRUPDO ORJLF FRQWDLQLQJ ,' GHSHQGDEOH DQG D QRUPDO ORJLF FRQWDLQLQJ 03 ZHDNO\ PDWHULDO $ GHSHQGDEOH ZHDNO\ PDWHULDO ORJLF FRQWDLQLQJ && LV FDOOHG PDWHULDO 7KXV LV WKH VPDOOHVW PDWHULDO ORJLF LV GLVFXVVHG XQGHU WKDW QDPH E\ /RHZHU DV WKH ORJLF FRUUHVSRQGLQJ WR *RRGPDQnV DFFRXQW RI WKH FRXQWHUIDFWXDO FRQGLWLRQDO > S @ % &. ,' DQG &. 03 DUH GLVFXVVHG E\ &KHOODV >@ % FDQ SODXVLEO\ EH FRQVLGHUHG WKH ORJLF RI WKH YHUVLRQ RI *RRGPDQnV DFFRXQW DGn YRFDWHG E\ %HQQHWW >@ 6HH /RHZHU > S @f 03 SURYLGHV DQ DQDORJ RI WKH UXOH RI LQIHUHQFH RI PRGXV SRQHQV RU GHWDFKPHQW IRU WKH FRQGLWLRQDO ZKLOH ,' JXDUGV DJDLQVW WKH SRVVLELOLW\ WKDW LI S ZHUH WUXH S PLJKW QRW EH WUXH :LWKRXW WKHVH LW LV GLIILFXOW WR VHH KRZ D ORJLF FRXOG EH FRQn VLGHUHG WR UHSUHVHQW WKH FRXQWHUIDFWXDO FRQGLWLRQDO VR LQ VRPH VHQVH % LV WKH DEVROXWHO\ PLQLPDO QRUPDO ORJLF IRU WKH FRXQWHUIDFWXDO FRQGLWLRQDO $ ORJLF FRQWDLQLQJ 0' ZLOO EH FDOOHG PRGDO 7KDW WKLV DSSHOODWLRQ LV DSSURSULDWH ZLOO EH VKRZQ LQ WKH QH[W VHFWLRQ :H QRWH DW WKLV SRLQW WKDW LI 0' LV D WKHVLV RI D ORJLF ZH PD\ GHILQH D PRGDO RSHUDWRU DV IROn ORZV /D :1DD 7KDW / FRUUHVSRQGV WR WKH PRGDO RSHUDWRU RI VRPH QRUPDO PRGDO ORJLF ZLOO EH VKRZQ LQ 6HFWLRQ $ GHSHQGDEOH ORJLF FRQWDLQLQJ ERWK ,' DQG ZLOO EH FDOOHG RUGHUHG 7KXV LV WKH VPDOOHVW RUGHUHG ORJLF 1XWH FDOOV ORJLFV FRQWDLQLQJ RUGHUHG EHFDXVH RI WKH SURSHUWLHV RI WKH DOJHEUDV ZKLFK PRGHO WKHP >

PAGE 199

SS @ )URP RXU SRLQW RI YLHZ WKLV LV D PLVQRPHU DV LWVHOI GRHV QRW LPSRVH DQ RUGHU UHODWLRQ RQ WKH VHW RI SRVVLEOH ZRUOGV DV ZLOO EH VKRZQ LQ WKH QH[W VHFWLRQ :H VKDOO VKRZ WKDW DQ RUGHU UHODWLRQ 5 F 8 [ 8 [ 8 PD\ EH GHILQHG IRU ORJLFV FRQWDLQLQJ &3 LQ 6HFWLRQ 5HODWLYH WR D JLYHQ ZRUOG X WKH UHODWLRQ ZLOO EH VKRZQ WR EH D SDUWLDO RUGHU ,I 03 LV SUHVHQW X ZLOO EH 5APLQLPDO DQG LI && DQG 03 DUH SUHn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n VHQW X LV 5 OHDVW $OO RI /HZLVn 9ORJLFV DUH YDULDEO\ VWULFW DQG 9: DQG 9& DUH DPRQJ WKHP 9: EHLQJ ZHDNO\ PDWHULDO DQG 9& EHLQJ PDWHULDO 66 WKH VPDOOHVW DGGLWLYH PDWHULDO ORJLF LV VWXGLHG XQGHU WKDW QDPH E\ 3ROORFN >@ %\ DGGLQJ &(0 WR 66 RU 9& ZH JHW WKH ORJLF & WKH FRQGLWLRQDO ORJLF RI 6WDOQDNHUnV DQDO\VLV & LV WKH VPDOOHVW VLQJXODU ORJLF LQ 1XWH V WHUPLQRORJ\ ZKLFK VKRXOG QRW EH FRQIXVHG ZLWK WKH WHUP VLQJXODU DSSOLHG WR ZRUOGV RI QRQQRUPDO ORJLFV LQ 6HFWLRQ DQG ZKLFK VDWLVI\ &6 9ST )LJXUH GLDJUDPV WKH FRQWDLQPHQW UHODWLRQV RI WKH ORJLFV ZH KDYH GLVFXVVHG DERYH :H VKDOO VKRZ LQ WKH QH[W VHFWLRQ WKDW WKHUH LV D ORJLF EHWZHHQ &3 DQG 9 DQG FRQVHTXHQWO\ ORJLFV EHWZHHQ &$ DQG 9: DQG EHWZHHQ 66 DQG 9& ZKLFK KDYH QRW SUHYLRXVO\ EHHQ QRWHG

PAGE 200

& &* 66 9& &. 03 ‘ &$ 9: &. &. ,' &3 9 )LJXUH $ VLPLODU GLDJUDP LV IRXQG LQ 1XWH > S @ :H QRWH WKDW WKH IRXUWK DQG ILIWK FROXPQV FRUUHVSRQG WR ZKDW ZH PD\ FDOO IDPLOLHV RI SDUWLDOO\ RUGHULQJ ORJLFV DQG WRWDOO\RUGHULQJ ORJLFV UHVSHFWLYHO\ ,Q IDFW RQH PD\ UHJDUG WKH KRUL]RQWDO GLPHQVLRQ DV RQH RI LQFUHDVLQJ VWUHQJWK RI WKH FRPSDULVRQ RI SRVVLEOH ZRUOGV UHTXLUHG E\ WKH ORJLF ZKLOH WKH YHUWLFDO GLPHQVLRQ UHSUHVHQWV LQFUHDVLQJ PDWHULDOLW\ RI WKH FRXQWHUIDFWXDO FRQGLn WLRQDO 2I WKHVH ORJLFV WKH ZHDNHVW WKDW FDQ SODXVLEO\ EH FRQVLGHUHG D ORJLF RI WKH FRXQWHU IDFWXDO FRQGLWLRQDO LV &$ :H VKDOO VXSSRUW WKLV FODLP E\ DUJXLQJ LQ &+$37(5 ),9( WKDW D ORJLF IRU WKH FRXQWHUIDFWXDO FRQn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

PAGE 201

VHOHFWLRQ IXQFWLRQV VLQFH IRU D SURSRVLWLRQ DQG D ZRUOG WKH\ VHOHFW WKRVH ZRUOGV ZKLFK DUH RI FRQFHUQ LQ HYDOXDWLQJ D FRXQWHUIDFWXDO :H FRXOG XVH SURSRVLWLRQDO QHLJKERUKRRG IXQFWLRQV DQG ZRXOG KDYH WR LI ZHUH ZHUH JRLQJ WR FRQVLGHU QRPRUPDO FRQGLWLRQDO ORJLFV 6LQFH ZH VKDOO OLPLW RXU DWn WHQWLRQ WR H[WHQVLRQV RI &. VXFK JHQHUDOLW\ PXVW JLYH ZD\ WR WKH UHODWLYH VLPSOLFLW\ RI VHOHFWLRQ IXQFWLRQV )UDQ WKH SRLQW RI YLHZ RI DQ DQDO\VLV RI FRXQWHUIDFWXDOV WKHVH FKRLFHV DOO KDYH RQH VLJQLILFDQW GUDZEDFN WKH VHOHFWLRQ RI ZRUOGV IRUPDOO\ GHSHQGV XSRQ ERWK D ZRUOG DQG D SURSRVLWLRQ 2QH QRWDEOH DGn YDQWDJH RI /HZLVn DFFRXQW LV WKDW WKH DUUDQJHPHQW RI SRVVLEOH ZRUOGV IRU WKH SXUSRVH RI FRXQWHUIDFWXDO GHOLEHUDWLRQ LV VWDEOH DPRQJ GLIIHUHQW DQWHFHGHQWV 7KLV LV D FRQVHTXHQFH RI VLPLODULW\ RUGHULQJ 7KH DQWHFHn GHQW VKRXOG QRW FRQGLWLRQ RXU MXGJPHQW DV WR ZKLFK ZRUOGV DUH PRUH VLPLn ODU WR WKH DFWXDO ZRUOG )RU VXIILFLHQWO\ VWURQJ H[WHQVLRQV RI &. ZH ZLOO EH DEOH WR UHJDLQ SDUW RI WKH DGYDQWDJH RI /HZLVn DQWHFHGHQWLQGHSHQGHQW DFFRXQW RI WKH DUn UDQJHPHQW RI SRVVLEOH ZRUOGV :H ZLOO EH DEOH WR GHILQH DQ RUGHULQJ UHn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

PAGE 202

QRQFRQGLWLRQDO IRUPXODV DQG ZKDWHYHU HOVH LV UHTXLUHG DUH DV EHIRUH 7KH WUXWK FRQGLWLRQ IRU 9DE LV GHULYHG IURP WKDW RI :DE DQG 'f 6HOHFWLRQ IXQFWLRQ VHPDQWLFV /HW 8 EH D VHW RI SRVVLEOH ZRUOGV Df $ VHOHFWLRQ IXQFWLRQ LV DQ\ IXQFWLRQ I8 [ S8f 38f Ef $ VHOHFWLRQ IXQFWLRQ IUDPH ) 8I LV DQ RUGHUHG SDLU ZKHUH 8 LV D VHW RI SRVVLEOH ZRUOGV DQG I D VHOHFWLRQ IXQFWLRQ Ff $ VHOHFWLRQ IXQFWLRQ PRGHO 0 8I 9 LV DQ RUGHUHG WULSOH ZKHUH 8I LV D VHOHFWLRQ IXQFWLRQ IUDPH DQG 93 r‘ 38f 3 LV WKH VHW RI DWRPLF VHQWHQFHV RI &:f LV D YDOXDWLRQ Gf :DE LII IX__DM_0f F __E__0 Hf _J 9DE LII IX__D__0f $ __E__: &ODVVHV RI IUDPHV ZLOO EH VSHFLILHG E\ VWDWLQJ WKH FRQGLWLRQV LI DQ\ ZKLFK DSSO\ WR WKH VHOHFWLRQ IXQFWLRQ 7KH IROORZLQJ OLVW FRPSULVHV WKH FRQGLWLRQV ZH VKDOO EH XVLQJ & )RU DOO X H 8 DQG DOO ;<= H 38f LGf IX;f F ; PSf ,I X H ; WKHQ X H I X;f FFf ,I X H ; WKHQ IX;f ^X` PGf ,I I X;f WKHQ IX
PAGE 203

0RVW RI WKH DERYH FRQGLWLRQV DUH IRXQG LQ 1XWH > S @ +H GHYHORSV DOJHEUDLF VHPDQWLFV IRU H[WHQVLRQV RI 2. DQG XVHV WKH VHOHFWLRQ IXQFWLRQ FRQGLWLRQV WKURXJK WKH HTXLYDOHQFH RI WKH WZR W\SHV RI VHPDQWLFV 7KH FRPSOHWHQHVV RI &' &$ 66 9 9: 9& DQG & ZLWK UHVSHFW WR WKH DSSURn SULDWH FRQGLWLRQV LV FODLPHG E\ KLP &RQGLWLRQV f DQG f DUH XVHG E\ /HZLV > S @ LQ GHILQLQJ KLV VHWVHOHFWLRQ IXQFWLRQV 7KH\ DUH LQFOXGHG VR WKDW ZH PD\ VKRZ WKH UHODWLRQ RI WKHP WR WKH RWKHU FRQGLWLRQV RI & )ROORZLQJ 1XWH ZH H[WHQG WKH WHUPLQRORJ\ DSSOLHG WR WKH H[WHQVLRQV RI &. PHQWLRQHG LQ 6HFWLRQ WR WKH FRUUHVSRQGLQJ VHOHFWLRQ IXQFWLRQV DQG IUDPHV $ IUDPH ) 8I DQG VHOHFWLRQ IXQFWLRQ I LV Df GHSHQGDEOH LI LGf LV VDWLVILHG Ef ZHDNO\ PDWHULDO LI PSf LV VDWLVILHG Ff PDWHULDO LI LGf PSf DQG FFf DUH VDWLVILHG Gf PRGDO LI PGf LV VDWLVILHG Hf RUGHUHG LI LGf PGf DQG FRf DUH VDWLVILHG If YDULDEO\ VWULFW LI LGf PGf FRf DQG FYf DUH VDWLVILHG Jf DGGLWLYH LI LGf PSf PGf FRf DQG FDf DUH VDWLVILHG Kf VLQJXODU LI LGf PSf FFf PGf FDf DQG FHPf DUH VDWLVILHG 3URRIV RI WKH FRQVLVWHQF\ DQG FRPSOHWHQHVV RI H[WHQVLRQV RI &. ZLWK UHVSHFW WR WKH DSSURSULDWH FODVVHV RI IUDPHV ZLOO IROORZ WKH PHWKRGV RI &KHOODV >@ &KHOODV FDOOV VHOHFWLRQ IXQFWLRQ IUDPHV VWDQGDUG IUDPHV DQG WUHDWV 2. ,' &. 03 DQG ,6 DV H[DPSOHV ,Q ZKDW IROORZV EHFDXVH

PAGE 204

RI WKH ODUJH QXPEHU RI FRQGLWLRQV ZH ZLOO JHQHUDOO\ UHIHU WR FODVVHV RI IUDPHV E\ WKH WHUPV JLYHQ LQ )LUVW ZH VKRZ WKH FRQVLVWHQF\ RI YDULRXV H[WHQVLRQV RI &. ZLWK UHVSHFW WR DQ DSSURSULDWH FODVV RI IUDPHV SDXVH WR VKRZ WKDW QRW DOO RI WKH FRQGLWLRQV DUH LQGHSHQGHQW WKHQ VKRZ WKH FRPSOHWHQHVV RI YDULRXV H[WHQVLRQV RI &. ZLWK UHVSHFW WR WKH DSSURSULDWH FODVV RI IUDPHV &RQn VLGHUDWLRQ RI WLPH DQG VSDFH UHTXLUH WKDW FRPSOHWH SURRIV QRW EH SURn YLGHG LQ VRPH FDVHV )RU FRQVLVWHQF\ ZH VKRZ WKH YDOLGLW\ RI WKH FKDUn DFWHULVWLF D[LRPVf LQ WKH FODVV RI IUDPHV LQGLFDWHG / &. ,' LV FRQVLVWHQW ZUW WKH FODVV RI GHSHQGDEOH IUDPHV 3URRI )RU DQ\ GHSHQGDEOH PRGHO DQG DQ\ ZRUOG X H 8 LGf UHTXLUHV WKDW I X __S__f F __S__ +HQFH E\ WKH WUXWK GHILQLWLRQ :SS 4(' / &. 03 LV FRQVLVWHQW ZUW WKH FODVV RI ZHDNO\ PDWHULDO IUDPHV 3URRI )RU DQ\ ZHDNO\ PDWHULDO PRGHO DQG DQ\ ZRUOG X DVVXPH :ST ,I X L O_S WKHQ &ST WULYLDOO\ ,I X H __S__ WKHQ E\ PSf X H I X __S f $V I X __S __f F __T __ E\ DVVXPSWLRQ X H __T__ +HQFH &ST 4(' / % &. ,'03LV FRQVLVWHQW ZUW WKH FODVV RI GHSHQGDEOH ZHDNO\ PDWHULDO IUDPHV / DQG /f / % && LV FRQVLVWHQW ZUW WKH FODVV RI PDWHULDO IUDPHV 3URRI ,Q YLHZ RI / ZH VKRZ WKH YDOLGLW\ RI && )RU DQ\ PDWHULDO PRGHO DQG DQ\ ZRUOG X DVVXPH .ST 7KHQ ERWK S DQG T VR X H __S__ +HQFH E\ FFf IX __S__f ^X` $V X H __T__ IX __S__f F __T__ VR :ST 4(' / &. ,'0' 2LV FRQVLVWHQW ZUW WKH FODVV RI RUGHUHG IUDPHV

PAGE 205

3URRI ,Q YLHZ RI / ZH VKRZ WKH YDOLGLW\ RI 0' DQG &2 /HW DQ RUGHUHG PRGHO EH JLYHQ DQG X DQ\ ZRUOG LQ 8 )RU 0' DVVXPH :1SS 7KHQ I X __ 1S Mf F __S__ %\ LGf IX __1S__f F __1S__ 1RZ __S__ 8 __1S__ VR IX __1S__f +HQFH E\ PGf IX __T__f $ __1S__ 7KXV IX __T__f F __SOc +HQFH :TS )RU DVVXPH )ST 7KDW LV ERWK :ST DQG :TS 6R IX __S__f F __T__ DQG IX __T__f F __S__ 6R E\ FRf IX __S__f IX __T__f +HQFH IX__S__f F __U__ LII IX __T__f F __U__ 7KHUHIRUH :SU LII :TU DQG VR (:SU:TU 4(' / 03 LV FRQVLVWHQW ZUW WKH FODVV RI RUGHUHG ZHDNO\ PDWHULDO IUDPHV / DQG/f / &* && LV FRQVLVWHQW ZUW WKH FODVV RI RUGHUHG PDWHULDO IUDPHV / DQG /$f / &3 &$ LV FRQVLVWHQW ZUW WKH FODVV RI RUGHUHG IUDPHV VDWLVI\LQJ FDf 3URRI ,Q YLHZ RI / ZH VKRZ WKH YDOLGLW\ RI &$ )RU DQ\ PRGHO RQ VXFK D IUDPH DQG DQ\ ZRUOG X LQ 8 DVVXPH .:SU:TU 7KHQ ZH KDYH ERWK :SU DQG :TU DQG VR IX __S__f F __U__ DQG IX __T__f F __U__ +HQFH I X __S __f 8IX __T__f F __ U 1RZ __$ST__ __S__ 8 __T__ VR E\ FDf DV I X __$STf F IX __S__f 8 IX __T__f ZH KDYH IX __$ST__f F __U__ 7KXV :$STU 4(' / 9 &9 LV FRQVLVWHQW ZUW WKH FODVV RI YDULDEO\ VWULFW IUDPHV 3URRI ,Q YLHZ RI / ZH VKRZ WKH YDOLGLW\ RI &9 /HW D YDULDEO\ VWULFW PRGHO EH JLYHQ DQG X DQ\ ZRUOG LQ 8 $VVXPH .:ST9SU 7KHQ ZH KDYH :ST DQG 9SU DQG VR IX __S__f F __T__ DQG IX __S__f $ __U__ L 6R I X __S__f L __1U 8 __U +HQFH E\ FYf I X __S__ $ __U__f F __T__ %XW __S__ D __ U __.SU__ 7KHUHIRUH :.SUT 4('

PAGE 206

/ &$ &3 03 LV FRQVLVWHQW ZUW WKH FODVV RI DGGLWLYH IUDPHV / DQG /f / 9: 9 03 LV FRQVLVWHQW ZUW WKH FODVV RI YDULDEO\ VWULFW PDWHULDO IUDPHV / DQG /f / 66 &$ && LV FRQVLVWHQW ZUW WKH FODVV RI DGGLWLYH PDWHULDO IUDPHV / DQG /f / 9& 9: && LV FRQVLVWHQW ZUW WKH FODVV RI YDULDEO\ VWULFW PDWHULDO IUDPHV / DQG /f /$ & 66 &(0 LV FRQVLVWHQW ZUW WKH FODVV RI VLQJXODU IUDPHV 3URRI ,Q YLHZ RI / ZH VKRZ WKH YDOLGLW\ RI &(0 /HW DQ\ VLQJXODU PRGHO EH JLYHQ DQG X DQ\ ZRUOG LQ 8 %\ FHPf IX __S__f LV HLWKHU D VLQJOHWRQ RU ,I WKH ODWWHU WKHQ ZH KDYH ERWK IX __S__f F __T__ DQG I X __S f F __1T VR KDYH :ST DQG :S1T DQG VR FHUWDLQO\ $:ST:S1T ,I WKH IRUPHU VXSSRVH IX __S__f ^Z` $V Z H __T__ RU Z H __1T 8 M@T__ ZH KDYH HLWKHU IX __Sc_f F __T__ RU IX __S__f F __1T +HQFH HLWKHU :ST RU :S1T DQG VR $:ST:S1T 4(' 7KH FRQGLWLRQV RI & DUH QRW FRPSOHWHO\ LQGHSHQGHQW VR EHIRUH SURFHHGLQJ WR FRPSOHWHQHVV WKHRUHPV ZH VWDWH VHYHUDO RI WKH UHODWLRQV DPRQJ WKH FRQGLWLRQV / ,Q DQ\ VHOHFWLRQ IXQFWLRQ IUDPH FFf LPSOLHV PSf / ,Q DQ\ VHOHFWLRQ IXQFWLRQ IUDPH LGf DQG PGf LPSO\ f / ,Q DQ\ VHOHFWLRQ IXQFWLRQ IUDPH LGf PGf FRf DQG FYf LPSO\ f DQG f / ,Q DQ\ VHOHFWLRQ IXQFWLRQ IUDPH LGf PGf FRf DQG FYf LPSO\ FDf DQG FEf / ,Q DQ\ VHOHFWLRQ IXQFWLRQ IUDPH LGf f DQG f LPSO\ PGf FRf DQG FYf

PAGE 207

/ ,Q DQ\ VHOHFWLRQ IXQFWLRQ IUDPH LGf PGf FRf FDf DQG FHPf LPSO\ FYf / ,Q DQ\ VHOHFWLRQ IXQFWLRQ IUDPH LGf f FDf DQG FEf LPSO\ PGf DQG FRf 7KH SURRIV RI WKH FRPSOHWHQHVV RI WKH YDULRXV H[WHQVLRQV RI &. ZLOO DJDLQ UHTXLUH WKH QRWLRQV RI FDQRQLFDO IUDPH DQG PRGHO $V LV WKH FDVH LQ RXU WUHDWPHQW RI QHLJKERUKRRG VHPDQWLFV IRU FODVVLFDO FRQn GLWLRQDO ORJLFV WKH IUDPHV GHILQHG LQ DUH QRW XQLTXH VR IROn ORZLQJ &KHOODV ZH ZLOO GHVLJQDWH D FHUWDLQ VXEFODVV RI FDQRQLFDO IUDPHV DV SURSHU DQG PDNH RXU VHOHFWLRQ RI VSHFLILF FDQRQLFDO IUDPHV IURP DPRQJ WKRVH /HW / EH D QRUPDO FRQGLWLRQDO ORJLF DQG 8 WKH VHW RI PD[LPDOO\ /FRQVLVWHQW H[WHQVLRQV RI / /HW I 8 [ S8f 38f EH DQ\ IXQFWLRQ VXFK WKDW IRU DOO VHQWHQFHV DE DQG DOO X H 8 :DE e X LII IX _D_Af F _E/ )XUWKHUPRUH LI 93 38f LV D YDOXDWLRQ VXFK WKDW 93f _S_A WKHQ ) 8I DQG 0 8I 9 DUH FDQRQLFDO VHOHFWLRQ IXQFWLRQ IUDPH DQG PRGHO IRU / UHVSHFWLYHO\ 7KH IROORZLQJ WKHRUHPV DUH DQDORJV WR WKRVH IRU FDQRQLFDO QHLJKERUn KRRG IUDPHV DQG PRGHOV IRU FRQGLWLRQDO ORJLFV VR QR SURRIV DUH SUHVHQWHG 6HH &KHOODV > SS @ DQG 6HFWLRQ f 7 /HW 0 EH D FDQRQLFDO VHOHFWLRQ IXQFWLRQf PRGHO IRU / D QRUPDO FRQGLWLRQDO ORJLF 7KHQ IRU DOO VHQWHQFHV D DQG DOO ZRUOGV X LQ 8 D LII D H X 7KXV __D__: _D_/ 7-/ :LWK 0/ DV LQ 7 IRU HYHU\ VHQWHQFH D D LII _M D

PAGE 208

7 ,I D VHOHFWLRQ IXQFWLRQ I VDWLVILHV WKH FRQGLWLRQ IX -D_Af Y H 8 ^E H &: :DE H X` F Y` WKHQ I LV D FDQRQLFDO VHOHFWLRQ IXQFWLRQ FDOOHG D SURSHU FDQRQLFDO VHOHFn WLRQ IXQFWLRQ )XUWKHUPRUH 8I DQG 8I9 DUH SURSHU FDQRQLFDO IUDPH DQG PRGHO UHVSHFWLYHO\ SURYLGHG 9Sf _S_A $V 7 GRHV QRW SODFH DQ\ FRQGLWLRQ XSRQ WKRVH ; F 8 IRU ZKLFK ; I _D_ IRU HYHU\ VHQWHQFH D WKHUH LV QR XQLTXHO\ GHWHUPLQHG SURSHU FDQRQLFDO IUDPH EXW UDWKHU D UDQJH EHWZHHQ D ODUJHVW IRU ZKLFK IX;f 8 ZKHQHYHU ; I _D_ IRU HYHU\ VHQWHQFH D DQG D VPDOOHVW IRU ZKLFK I X;f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f 3URRI /HW 3 8I EH D SURSHU FDQRQLFDO IUDPH IRU &. ,' WKDW VDWLVn ILHV LGf ZKHQHYHU ; I -D_ IRU HYHU\ VHQWHQFH D VD\ WKH VPDOOHVW VXFK IUDPH VR IX;f :H PXVW VKRZ ) VDWLVILHV LGf ZKHQ ; _D_ IRU VRPH VHQWHQFH D 6XSSRVH Y H IX _D_f DQG VKRZ Y H _D_ %\ GHILQLWLRQ RI I E :DE H X` F Y %\ ,' :DD H X VR D H Y +HQFH Y H _D_ 4(' 7A &. 03 LV GHWHUPLQHG E\ WKH FODVV RI ZHDNO\ PDWHULDO IUDPHV /f 3URRI OHW 3 a 8I EH D SURSHU FDQRQLFDO IUDPH IRU &. 03 WKDW VDWn LVILHV PSf ZKHQHYHU ; A @D@ IRU HYHU\ VHQWHQFH D VD\ WKH ODUJHVW VXFK

PAGE 209

IUDPH VR IX;f 8 :H VKRZ ) VDWLVILHV PSf ZKHQ ; _D_ IRU VRPH VHQWHQFH D 6XSSRVH X H _D_ DQG VKRZ X H IX _D_f 6R ZH PXVW VKRZ ^E :DE H X` F X 6XSSRVH E LV VXFK WKDW :DE H X %\ 03 &DE H X DQG DV D H X ZH KDYH E H X E\ GHGXFWLYH FORVXUH 4(' &KHOODV SRLQWV RXW WKDW QHLWKHU WKH ODUJHVW QRU WKH VPDOOHVW SURSHU FDQRQLFDO IUDPH VDWLVILHV ERWK LGf DQG PSf 7KLV GLFWDWHV D GLIIHUHQW FKRLFH RI I LQ WKH IROORZLQJ > S @ 7 % &. ,'03LV GHWHUPLQHG E\ WKH FODVV RI GHSHQGDEOH ZHDNO\ PDWHULDO IUDPHV /$f 3URRI /HW ) 8I !EH D SURSHU FDQRQLFDO IUDPH IRU % WKDW VDWLVILHV LGf DQG PSf ZKHQHYHU ; I _D_ IRU HYHU\ VHQWHQFH D )RU LQVWDQFH OHW IX;f ; 7KH SURRIV RI 7 DQG 7 VXIILFH WR VKRZ LGf DQG PSf DUH VDWLVILHG ZKHQHYHU ; _D_ IRU VRPH VHQWHQFH D 4(' 7 % && LV GHWHUPLQHG E\ WKH FODVV RI PDWHULDO IUDPHV /f 3URRI /HW ) 8I EH D SURSHU FDQRQLFDO IUDPH IRU WKDW VDWLVILHV LGf PSf DQG FFf ZKHQHYHU ; I _D_ IRU HYHU\ VHQWHQFH D ,QGHHG IRU ; I _D_ IRU HYHU\ VHQWHQFH D GHILQH I E\ ^X` LI X H ; I X;f ; RWKHUZLVH 7KDW I VDWLVILHV LGf DQG PSf ZKHQHYHU ; _D_ IRU VRPH VHQWHQFH D LV REYLRXV IURP 7 :H VKRZ I VDWLVILHV FFf ZKHQHYHU ; _D_ IRU VRPH VHQWHQFH D /HW X H _D_ 6KRZ IX _D_f ^X` 7KDW LV ZH PXVW VKRZ Y ^E :DE H X` FA Y` ^X` 6XSSRVH Y LV VXFK WKDW E H Y DQG :DE H X :H FODLP Y X 6XSSRVH F H X $V D H X .DF H X VR E\ && :DF H X

PAGE 210

+HQFH F H Y )RU WKH FRQYHUVH VXSSRVH F L X VR 1F H X E\ PD[LPDOOLW\ 7KHQ .D1F H X DQG E\ && :D1F H X +HQFH 1F H Y 6R E\ FRQVLVWHQF\ FLY 4(' 7 &. ,'0' &'LV GHWHUPLQHG E\ WKH FODVV RI RUGHUHG IUDPHV /f 3URRI /HW ) 8I EH D SURSHU FDQRQLFDO IUDPH IRU WKDW VDWLVILHV LGf PGf DQG FRf ZKHQHYHU ; I _D_ IRU HYHU\ VHQWHQFH D 7KLV FDQ EH DFFRPSOLVKHG E\ VHWWLQJ I X;f ; ZKHQHYHU ; I _D_ IRU HYHU\ VHQWHQFH D LGf LV FOHDUO\ VDWLVILHG DQG DV IX;f RQO\ LI ; PGf LV VDWLVILHG ,W LV FOHDU WKDW LGf LV VDWLVILHG IRU DOO < F 8 IURP 7 VR IRU DQ\ < IX S @ LW LV IRU WKRVH ZH FRQVLGHU KHUH

PAGE 211

7 2 03 LV GHWHUPLQHG E\ WKH FODVV RI RUGHUHG ZHDNO\ PDWHULDO IUDPHV /f 3URRI $ SURRI FDQ EH FRQVWUXFWHG E\ LQVSHFWLQJ WKH SURRIV RI 7 DQG 7 7 &* &2 && LV GHWHUPLQHG E\ WKH FODVV RI RUGHUHG PDWHULDO IUDPHV /f 3URRI 6HH 7 DQG 7 7 &3 &$ LV GHWHUPLQHG E\ WKH FODVV RI RUGHUHG IUDPHV VDWLVI\LQJ FDf /f 3URRI /HW ) 8I EH D SURSHU FDQRQLFDO IUDPH IRU &3 WKDW VDWLVILHV LGf PGf FRf DQG FDf ZKHQHYHU ; I _DM IRU HYHU\ VHQWHQFH D ,Q YLHZ RI 7 ZH ZLOO VKRZ FDf LV VDWLVILHG ZKHQHYHU ; _D_ DQG < _E_ IRU VRPH VHQWHQFHV D DQG E :H PXVW VKRZ IX _D_ 8 _E_f e I X M D _f 8 IX_EMf %\ &$ DQG GHGXFWLYH FORVXUH LI :DF H X DQG :EF H X WKHQ :$DEF H X +HQFH ^F :DF H X` $ ^F :EF H X` F ^F :$DEF H X` 7KHQ E\ D WKHRUHP RI PD[LPDOO\ /FRQVLVWHQW H[WHQVLRQV ^Y ^F :$DEF H X` e Y` F Y F :DF H X` e Y` 8 ^Y ^F :EF H X` F Y` 7KHUHIRUH IX _$DE_f e I X D _f 8 I X E Mf 4(' 7 9 &9 LV GHWHUPLQHG E\ WKH FODVV RI YDULDEO\ VWULFW IUDPHV /f 3URRI /HW ) 8I EH D SURSHU FDQRQLFDO IUDPH IRU 9 WKDW VDWLVILHV LGf PGf FRf DQG FYf ZKHQHYHU ; A _D_ IRU HYHU\ VHQWHQFH D ,Q YLHZ RI 7 ZH VKRZ WKDW FYf LV VDWLVILHG ZKHQHYHU ; _D_ < _E_ DQG = _F_ IRU VRPH VHQWHQFHV D E DQG F 6XSSRVH IX_D_f F _E_ DQG IXMD_f D >F_ I 6KRZ IX_.DF_f e _E_ 7KLV LV EHFDXVH _.DF_ D $ F _f )URP RXU DVVXPSWLRQV ZH KDYH :DE H X DQG :D1F L X 6R E\

PAGE 212

PD[LPDOOLW\ 1:D1F H X 7KDW LV 9DF H X DQG VR .:DE9DF H X +HQFH E\ &9 :.DFE H X 7KHQ IX _.DF_f F _E_ 4(' 7 &$ &3 03 LV GHWHUPLQHG E\ WKH FODVV RI DGGLWLYH IUDPHV /f 3URRI 6HH 7 DQG 7 7 9: 9 03 LV GHWHUPLQHG E\ WKH FODVV RI YDULDEO\ VWULFW ZHDNO\ PDWHULDO IUDPHV /f 3URRI 6HH 7 DQG 7 7 66 &$ && LV GHWHUPLQHG E\ WKH FODVV RI DGGLWLYH PDWHULDO IUDPHV /f 3URRI 6HH 7 DQG 7 7 9& 9: && LV GHWHUPLQHG E\ WKH FODVV RI YDULDEO\ VWULFW PDWHULDO IUDPHV /f 3URRI 6HH 7 DQG 7 7 & 66 &(0 LV GHWHUPLQHG E\ WKH FODVV RI VLQJXODU IUDPHV f 3URRI 7KH SURRI UHTXLUHV VKRZLQJ WKDW &(0 OHDGV WR FHPf EHLQJ VDWLVILHG 6XFK D SURRI FDQ EH FRQVWUXFWHG $OWHUQDWLYH D[LRPDWL]DWLRQV RI WKH H[WHQVLRQV RI &. FDQ EH IRXQG 7KH HTXLYDOHQFH RI FHUWDLQ VHWV RI FRQGLWLRQV \LHOGHG E\ / / UHYHDOV VHYHUDO 3ROORFN > S @ KDV DQ DOWHUQDWLYH D[LRPDWL]DWLRQ RI 66 QRWDEOH IRU WKH SUHVHQFH RI WKH D[LRP 3ROORFNnV $f &3 &.:ST:SU:.SUT ZKLFK FRUUHVSRQGV WR WKH FRQGLWLRQ FSf IX;f F < DQG IX;f F = LPSOLHV IX; $ =f F < 7KLV D[LRP LV DOPRVW EXW QRW TXLWH &9 WKH FRQGLWLRQ :SU UDWKHU WKDQ 9SU LV UHTXLUHG IRU FRQMRLQLQJ S DQG U LQ WKH DQWHFHGHQW

PAGE 213

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f LV VDWLVILHG :H VKDOO JR RQ WR HVWDEOLVK FRPSDUDWLYH RUGHULQJ VHPDQWLFV D JHQHUDOL]DWLRQ RI /HZLVn FRPSDUDWLYH VLPLODULW\ VHPDQWLFV IRU WKRVH H[WHQVLRQV RI &. ZKLFK FRQWDLQ &3 7R VKRZ WKDW GRHV QRW FRQWDLQ &$ ZH UHTXLUH D VHW RI SRVVLEOH ZRUOGV 8 DQG D VHOHFWLRQ IXQFWLRQ I WKDW VDWLVILHV LGf PGf DQG FRf EXW GRHV QRW VDWLVI\ FDf ,W ZLOO EH LQVWUXFWLYH LI ZH VR FRQVWUXFW 8I WKDW ZLWK D UHODWLRQ Y5AZ GHILQHG E\ IX^YZ`f ^Y` 5 ZLOO QRW EH WUDQVLWLYH /HW 8 ^XYZ]` DQG GHILQH I VR WKDW I VDWLVILHV LGf PGf FRf DQG HYHQ FDf IRU ZRUOGV YZ] )RU ZRUOG X DQG DQ\ ; F 8 ZH GHILQH IX;f E\ IX;f ^X` LI X H ; DQG RWKHUZLVH E\ WKH IROORZLQJ WDEOH ; I X;f ^Y` ^YZ` ^9` ^Z` ^Z ]` ^Z` ^]` ^Y]` ^]` ^9Z]` ^Y:]`

PAGE 214

1RWH WKDW IX^YZ`f ^Y` DQG IX^Z]`f ^Z` ZKLOH IX^Y]`f ^]` +HQFH Y5 Z DQG Z5 ] EXW Y ] VR 5 LV QRW WUDQVLWLYH 6LQFH IX;f F ; LGf LV VDWLVILHG 6LQFH IX;f I IRU DQ\ ; H[FHSW UDGf LV VDWLVILHG &KHFNLQJ WKH VDWLVIDFWLRQ RI FRf LV PRUH WHGLRXV ,Q PRVW FDVHV ZKHUH I X;f F < LV VDWLVILHG IX SS @ ,Q RUGHU WR VKRZ WKDW D SDUWLDO RUGHULQJ RI SRVVLEOH ZRUOGV LPSOLHV WKDW FDf LV VDWLVILHG ZH VKDOO UHTXLUH WKH GHILQLWLRQ RI FRPSDUDWLYH RUGHU UHODWLRQV IUDPHV PRGHOV DQG WUXWK IRU FRQGLWLRQDOV :DE DQG GHULYHG IURP 'f 9DE

PAGE 215

' &RPSDUDWLYH RUGHU VHPDQWLFV OHW 8 EH D VHW RI SRVVLEOH ZRUOGV Df $ FRPSDUDWLYH RUGHU UHODWLRQ 5 F 8 [ 8 [ 8 LV DQ\ UHODWLRQ VXFK WKDW UHODWLYH WR HDFK ZRUOG X LQ 8 Lf5 SDUWLDOO\ RUGHUV ZHDNO\ RUGHUV WRWDOO\ RUGHUVf WKDW VXEVHW RI 8 LQ WKH GRPDLQ RI 5 GHVLJQDWHG 'RUQ 5 f LLf(YHU\ QRQHPSW\ VXEVHW RI 8 WKDW PHHWV 'RP 5 KDV DW OHDVW RQH 5 PLQLPDO HOHPHQW X Ef $ FRPSDUDWLYH RUGHU IUDPH ) 85 LV DQ RUGHUHG SDLU VXFK WKDW 8 LV D VHW RI SRVVLEOH ZRUOGV DQG 5 LV D FRPSDUDWLYH RUGHU UHODWLRQ Ff ,I 93 38f LV D YDOXDWLRQ WKHQ WKH RUGHUHG WULSOH 0 859 LV D FRPSDUDWLYH RUGHU PRGHO Gf )RU ; F 8 ZH GHVLJQDWH WKH VHW RI 5 PLQLPDO HOHPHQWV RI ; E\ 5 ; )XUWKHUPRUH ZN Y U QRW Z5 Yf X rX GI X Lf ,I 5A LV D SDUWLDO RUGHU WKHQ 5 ; Y H ; $ 'RP 5 IRU DOO Z H ; ZLWK Z I Y ZL Y` X X f X LLf ,I 5X LV D ZHDN WRWDO RUGHU WKHQ 5 ; Y H ; $ 'RP 5 IRU DOO Z H ; Y5 Z X X X LLLf,I 5 LV D ZHOORUGHU WKHQ 5 ; LV WKH 5 OHDVW X X X HOHPHQW LI DQ\ RI ; $ 'RP 5A DQG RWKHUZLVH 5X; Hf ,MMZ£E ,II 5X__D__F __E__ If _J 9DE LII 5A+DOO $ c_E_>0 A %\ FRQGLWLRQ DLLf ZH KDYH DVVXPHG WKDW 5 VDWLVILHV WKH OLPLW DVVXPSWLRQ 7KLV VLPSOLILHV WKH IROORZLQJ SURRIV :H FRXOG GHILQH FRPSDUDWLYH RUGHULQJV WKDW YLRODWH WKH OLPLW DVVXPSWLRQ DQG FKDQJH RXU GHILQLWLRQV DFFRUGLQJO\ ,Q ZKDW IROORZV ZH VKDOO DVVXPH WKDW ZKHQ D VXEVHW ; RI 8 LV PHQWLRQHG ZH

PAGE 216

PHDQ ; $ 'DP 5 :LWK WKLV LQ PLQG ZH FDQ FKDUDFWHUL]H 5 PLQLPDO X X HOHPHQWV RI ; /$ 7KH 5APLQLPDO HOHPHQWV RI ; F 8 DUH JLYHQ DV IROORZV Df ,I 5X LV D SDUWLDO RUGHU WKHQ LI ; 5 ; X ^Y` LI Y5AZ IRU DOO Z H ; ^YZ H ; 9 Z DQG : Y` RWKHUZLVH X X Ef ,I 5 LV D ZHDN WRWDO RUGHU WKHQ X LI ; 9[ L ^YZ H ; Y5 Z DQG Z5 Y` RWKHUZLVH X X 3URRI 7KH SURRI LV REYLRXV IURP WKH GHILQLWLRQ DQG WKH SURSHUWLHV RI SDUWLDO DQG ZHDN RUGHUV UHVSHFWLYHO\ :H QRWH WKDW ZKHQ 5 LV D ZHDN WRWDO RUGHU 5X; FRQVWLWXWHV DQ HTXLYDOHQFH FODVV XQGHU WKH FRQGLWLRQ JLYHQ LQ Ef 4(' 7KH IROORZLQJ WKHRUHP HVWDEOLVKHV WKH IDFW WKDW QR ORJLF VPDOOHU WKDQ &3 TXDOLILHV DV D ORJLF ZKLFK SDUWLDOO\ RUGHUV WKH VHW RI SRVVLEOH ZRUOGV UHODWLYH WR HDFK ZRUOG 7$ 7KH ORJLF GHWHUPLQHG E\ WKH FODVV RI DOO SDUWLDOO\ RUGHUHG FRPn SDUDWLYH RUGHU IUDPHV FRQWDLQV &3 3URRI :H VKDOO VKRZ WKDW JLYHQ D SDUWLDOO\ RUGHUHG FRPSDUDWLYH RUGHU IUDPH ZH FDQ GHILQH DQ HTXLYDOHQW VHOHFWLRQ IXQFWLRQ IUDPH ZKLFK VDWLVILHV LGf PGf FRf DQG FDf 'HILQH I8 [ S8f r 38f E\ I X;f 5X; ,W VKRXOG EH REYLRXV WKDW ) 85 DQG )n 8I DUH HTXLYDOHQW )RU LGf E\ GHILQLWLRQ 5X; F ; VR IX;f F ; DQG LGf LV VDWLVn ILHG

PAGE 217

)RU PGf ZH PXVW VKRZ 5A; LPSOLHV 5A SS @ GHILQHV D SDUWLDO RUGHU IRU HDFK ZRUOG X RI D IUDPH VDWLVI\LQJ LGf FFf f FDf DQG FEf ,Q OLJKW RI WKLV WKH IROORZLQJ LV DQ LQWHUHVWLQJ UHVXOW 7 7KHUH LV D SDUWLDOO\ RUGHUHG FRPSDUDWLYH RUGHULQJ IUDPH WKDW GRHV QRW VDWLVI\ FEf 3URRI /HW 8 ^XYZ\]` )RU HDFK [ H 8 OHW 5 EH D SDUWLDO RUGHULQJ RI 8 DQG VSHFLILFDOO\ IRU ZRUOG X OHW 5A EH JLYHQ E\ )LJXUH

PAGE 218

)LJXUH ,Q YLHZ RI 7 ZH NQRZ WKDW I GHILQHG E\ IX;f 5A; VDWLVILHV LGf PGf FRf DQG FDf IRU DOO ZRUOGV LQ 8 /HW ; ^\Y` DQG < ^Z ]` 7KHQ ; 8 < ^YZ]\` 1RWH WKDW 5A; ^\Y` DQG 5X< ^Z]` DQG 58; 8 S QO@ FODLPV WKDW KLV V\WHP *r GHWHUPLQHG E\ D VHOHFWLRQ IXQFWLRQ VHPDQWLFV VDWLVI\LQJ WKH FRQGLWLRQV QRWHG DERYH LQFOXGLQJ FEff LV HTXLYDOHQW WR 3ROORFNnV V\VWHP 66 (YLGHQWO\ WKLV LV QRW WKH FDVH 7KH IDFW WKDW WKH IUDPH RI 7 VDWLVILHV WKH D[LRPV DQG UXOHV RI LQIHUHQFH RI 3ROORFNnV D[LRPDWL]DWLRQ RI 66 > SS @ FDQ EH GLUHFWO\ YHULILHG 7KH IROORZLQJ WKHRUHPV VKRZ WKDW IXUWKHU FRQWDLQPHQWV DUH SURSHU DQG WKDW FEf LV VDWLVILHG ZKHQ WKH RUGHULQJ LV D ZHDN WRWDO RUGHU 7 (YHU\ ZHDNWRWDOO\ RUGHUHG FRPSDUDWLYH RUGHULQJ IUDPH VDWLVILHV FEf 3URRI /HW ) 85 EH D ZHDNWRWDOO\ RUGHUHG FRPSDUDWLYH RUGHU IUDPH DQG GHILQH I E\ IX;f 5A; $V EHIRUH WKH VHOHFWLRQ IXQFWLRQ IUDPH

PAGE 219

8I LV HTXLYDOHQW 7R VKRZ FEf KROGV ZH PRVW VKRZ WKDW HLWKHU 5 ; F 5 ; 8 SS @

PAGE 220

:H SUHIHU WKH DERYH SURRI VKRZLQJ &% LV QRW FRQWDLQHG LQ 66 WR 1XWHnV SURRI VKRZLQJ &9 LV QRW FRQWDLQHG LQ 66 )LUVW RXU SURRI LV VLPSOHU DQG VHFRQG LW UHYHDOV DQRWKHU IDPLO\ RI ORJLFV ZKLFK DUH SDUWLDOO\ EXW QRW WRWDOO\ RUGHULQJ 7KLV IROORZV IURP WKH IDFW WKDW ZH FDQ DGG &% WR &3 ZLWKRXW SURGXFLQJ WKH ZHDN WRWDO RUGHU WKDW FKDUDFWHUL]HV 9 )LUVW ZH REVHUYH WKDW WKH FRQGLWLRQV RQ D VHOHFWLRQ IXQFWLRQ IUDPH GHWHUPLQLQJ &3 VXIILFH WR GHILQH DQ HTXLYDOHQW FRPSDUDWLYH RUGHU IUDPH ZLWK Y5Z LII IX^YZ`f ^Y` DQG WKDW WKH FRQGLWLRQV RQ D VHOHFWLRQ IXQFWLRQ IUDPH GHWHUPLQLQJ 9 VXIILFH WR GHILQH DQ HTXLYDOHQW FRPSDUDWLYH RUGHU IUDPH ZLWK Y5 Z LII HLWKHU IX^YZ`f ^Y` RU IX^YZ`f ^YZ` %DVLFDOO\ WKH SURRIV RI 7 DQG 7 DUH UHYHUVHG WR VKRZ WKLV 7 $ VHOHFWLRQ IXQFWLRQ IUDPH VDWLVI\LQJ LGf PGf FRf DQG FDf FDQ EH SDUWLDOO\ RUGHUHG 7KDW LV WKHUH LV DQ HTXLYDOHQW SDUWLDOO\ RUn GHULQJ FRPSDUDWLYH RUGHU IUDPH 3URRI /HW ) 8I EH D VHOHFWLRQ IXQFWLRQ IUDPH VDWLVI\LQJ LGf PGf FRf DQG FDf 'HILQH 5 F 8 [ 8 [ 8 E\ Y5AZ LII I X ^YZ`f ^Y` 7KDW WKH UHVXOWLQJ IUDPH 85 LV HTXLYDOHQW JLYHQ ZKDW ZH VKRZ EHORZ LV QRW GLIILFXOW WR VKRZ :H QRWH WKDW LV DQWLV\QPHWULF E\ GHILQLWLRQ %\ LGf HLWKHU IX^Y`f RU IX^Y`f Yf ,I WKH IRUPHU WKHQ E\ PGf IX
PAGE 221

^Y` 6XSSRVH IX^YZ]`f 7KHQ E\ PGf IX^YZ`f $ ^YZ]` D FRQWUDGLFWLRQ 6R IX^YZ]`f ^Y` :H WKHQ KDYH IX^YZ]`f F ^Y]` DQG IX^Y]`f F ^YZ]` WKH ODWWHU E\ LGf 6R E\ FRf IX^YZ]`f IX^Y]`f +HQFH IX^Y]`f ^Y` 4(' 7 $ VHOHFWLRQ IXQFWLRQ IUDPH VDWLVI\LQJ LGf PGf FRf DQG FYf FDQ EH ZHDNWRWDOO\ RUGHUHG 3URRI $ SURRI VLPLODU WR WKDW RI 7 FDQ EH GHYLVHG &RQGLWLRQ FYf LPSOLHV ERWK WUDQVLWLYLW\ DQG FRQQHFWHGQHVV LQ WKH SUHVHQFH RI WKH RWKHU FRQGLWLRQV 4(' /RHZHU > S QO@ FRQWDLQV D SURRI RI WKH WUDQVLWLYLW\ RI D UHODWLRQ GHILQHG DV LQ 7 EDVHG XSRQ D VHW RI FRQGLWLRQV LQFOXGLQJ FDf DQG FEf DV SUHYLRXVO\ PHQWLRQHG 7KHUH DUH D QXPEHU RI W\SRJUDSKn LFDO HUURUV LQ WKH SURRI DQG LW XVHV FRQGLWLRQ FEf ZKLFK WKH DERYH SURRI GRHV QRW ,W ZDV /RHZHUnV RWKHUZLVH ZHOOH[HFXWHG DUWLFOH ZKLFK LPSHOOHG PH WR VWXG\ RUGHULQJ UHODWLRQV PRUH FORVHO\ ,Q YLHZ RI 7 DQG 7 ZH PD\ LPPHGLDWHO\ FRQFOXGH 7 &3 LV GHWHUPLQHG E\ WKH FODVV RI SDUWLDOO\ RUGHUHG FRPSDUDWLYH RUGHU IUDPHV 7 9 LV GHWHUPLQHG E\ WKH FODVV RI ZHDNWRWDOO\ RUGHUHG FRPSDUDWLYH RUGHU IUDPHV 6HFRQG FRQVLGHU WKH IUDPH ) 85 ZLWK 8 ^XYZ\]` 'HILQH 5[ DV D WRWDO RUGHU IRU DOO [ H 8 VXFK WKDW [ X /HW 5 EH JLYHQ E\ )LJXUH

PAGE 222

Z X )LJXUH $V SUHYLRXVO\ GHILQH IX;f 5A; IRU DOO ZRUOGV X DQG ; F 8 7KH WRWDO RUGHU IRU YZ\] LPSOLHV WKH VDWLVIDFWLRQ RI LGf PGf FRf FYf FDf DQG FEf KRZHYHU LV QRW D WRWDO RUGHU 7KDW LV D SDUWLDO RUGHU LPSOLHV WKH VDWLVIDFWLRQ RI DOO EXW FRQGLWLRQV FEf DQG FYf ,W FDQ EH YHULILHG WKDW 5A DV GHSLFWHG LQ )LJXUH VDWLVILHV FEf +RZHYHU FYf LV QRW VDWLVILHG E\ 5 DV ZH PD\ REVHUYH E\ VHWWLQJ ; ^YZ\]! < ^\]` DQG = ^YZ\` 7KHQ WKH IROORZLQJ KROG 5A; < DQG 5A;f $ = I EXW 5A&; $ =f  < +HQFH FYf GRHV QRW KROG LQ WKLV IUDPH 4(' 1RWH WKDW ZH FKRVH 5 VR WKDW XSf DQG FFf ZHUH DOVR VDWLVILHG :H FRXOG GR WKH VDPH IRU WKH RWKHU ZRUOGV LQ 8 7KXV WKH IROORZLQJ WKHRUHPV DUH LPPHGLDWH 7 &3 &% &$ &% 66 &% GR QRW FRQWDLQ &9 7 &3 &% &$ &% 66 &% DUH SURSHUO\ FRQWDLQHG LQ 9 9: DQG 9& UHVSHFWLYHO\ ,W LV HDV\ WR VKRZ WKDW 03 LV QRW FRQWDLQHG LQ &. ,' DQG && LV QRW FRQWDLQHG LQ % 6R WKH RWKHU FRQWDLQPHQWV RI )LJXUH DUH SURSHU )LJXUH H[SDQGV RXU IDPLOLHV RI ORJLFV E\ WKH PHPEHUV QRWHG LQ 7

PAGE 223

& 66 66 &% 9& &$ &$ &% 9: &3 &3 &% 9 )LJXUH :H FORVH WKLV VHFWLRQ ZLWK VDQH REVHUYDWLRQV DERXW FRQGLWLRQV XSf FFf DQG PGf )URP RXU H[DPSOHV LW KDV EHFRPH HYLGHQW WKDW PSf LV HTXLYDOHQW WR WKH FRQGLWLRQ WKDW LI X H ; WKHQ X H 5A; ZKLOH FFf LV HTXLYDOHQW WR WKH FRQGLWLRQ WKDW LI X H ; WKHQ 5X; ^X` 7KXV X LV 5APLQLPDO LQ D FRPSDUDWLYH RUGHU IUDPH VDWLVI\LQJ PSf DQG X LV 5AOHDVW LQ D FRPSDUDWLYH RUGHU IUDPH VDWLVI\LQJ FFf 7KLV KDV FHUWDLQ LPSOLFDn WLRQV IRU DQ DQDO\VLV RI FRXQWHUIDFWXDOV ZKLFK ZLOO EH GHYHORSHG LQ &+$37(5 ),9( ,Q YLHZ RI WKHVH REVHUYDWLRQV ZH PD\ VWDWH 7 &$ 66 DUH GHWHUPLQHG E\ WKH FODVV RI SDUWLDOO\ RUGHUHG FRPSDUDn WLYH RUGHU IUDPHV ZLWK X 5 PLQLPDO IRU HDFK X H 8 DQG X 5 OHDVW IRU X X HDFK X H 8 UHVSHFWLYHO\ 7 9: 9& DUH GHWHUPLQHG E\ WKH FODVV RI ZHDNWRWDOO\ RUGHUHG FRPn SDUDWLYH RUGHU IUDPHV ZLWK X 5 PLQLPDO IRU HDFK X H 8 DQG X 5 OHDVW IRU X X HDFK X H 8 UHVSHFWLYHO\ :H PD\ DOVR VWDWH WKRXJK ZLWKRXW IXOO SURRI WKH IROORZLQJ 7 &3 &% &$ &% 66 &% DUH GHWHUPLQHG E\ WKH FODVV RI SDUWLDOO\ RUGHUHG FRPSDUDWLYH RUGHU IUDPHV VDWLVI\LQJ FEf FEf PSf DQG FEf PSf FFf UHVSHFWLYHO\

PAGE 224

,Q WKH SUHYLRXV VHFWLRQ ZH QRWHG WKDW D ORJLF FRQWDLQLQJ 0' DQG GHWHUPLQHG E\ D VHPDQWLFV VDWLVI\LQJ PGf FRXOG DSSURSULDWHO\ EH FRQn VLGHUHG D PRGDO ORJLF 7KH IROORZLQJ WKHRUHP HVWDEOLVKHV WKLV 7 ,Q D FRQGLWLRQDO ORJLF FRQWDLQLQJ &. 0' /D GHILQHG E\ /D :1DD LV D QRUPDO PRGDO RSHUDWRU 3URRI /HW 8I EH D VHOHFWLRQ IXQFWLRQ IUDPH VDWLVI\LQJ PGf 'HILQH )8 !‘ 38f E\ )Xf 4 I X;f :H FODLP ) LV DQ DFFHVVLELOLW\ IXQFWLRQ DQG VR GHWHUPLQHV WKH ORJLF RI WKH PRGDO IUDJPHQW RI WKH ORJLF FRQWDLQLQJ &. 0' 7KH VWDQGDUG WUXWK FRQGLWLRQ IRU VXFK IUDPHV LV /D LII )Xf F +D_> SURYLGHG WKDW WKHUH DUH QR VLQJXODU ZRUOGV 7KDW WKHUH DUH QR VLQJXODU ZRUOGV IROORZV IURP WKH QRUPDOLW\ RI &. 0' +HQFH LW VXIILFHV WR VKRZ WKDW :1DD LII )Xf F __D__ 6XSSRVH :1DD 7KHQ E\ 0' :ED IRU DOO VHQWHQFHV E +HQFH IX __Ef F __D IRU DOO VHWV __E__ ,I ; I __E__ IRU HYHU\ IRUPXOD E DGGLQJ WKHVH ;nV WR WKH LQWHUVHFWLRQ FDQQRW HQODUJH LW 7KHUHIRUH )Xf F __ D )RU WKH FRQYHUVH ZH PD\ HTXLYDOHQWO\ VKRZ WKDW )Xf $ __D__ I LQSOLHV 9DD 6R VXSSRVH )Xf $ __D>_ A 7KHQ LQ SDUWLFXODU IRU ; __D I X __Df $ __D A 6R ZH KDYH 9DD 4(' 1RWHV 3URRI RI WKHVH WKHRUHPV PD\ EH IRXQG LQ %HWK >@ RU PRVW DQ\ ZRUN LQ WKH IRXQGDWLRQV RI ORJLF

PAGE 225

0XFK RI WKH WHUPLQRORJ\ DQG UHVXOWV RI 6HFWLRQV DQG DUH WR EH IRXQG LQ 6HJHUEHUJ >@ DQG &KHOODV DQG 0F.LQQH\ >@ :H KDYH DOWHUHG VRPH WHUPLQRORJ\ LQ WKH LQWHUHVWV RI JUHDWHU XQLIRUPLW\ DQG WR EULQJ RXU PRGDO WHUPLQRORJ\ LQWR OLQH ZLWK WKH WHUPLQRORJ\ WR EH XVHG ZLWK FRQGLWLRQDO ORJLFV LQ VXEVHTXHQW VHFWLRQV :H SURYLGH WKH IRO IORZLQJ OLVW RI FRUUHVSRQGHQFHV ZKHUH ZH GLIIHU ZLWK WKRVH RI 6HJHUEHUJ RU &KHOODV DQG 0F.LQQH\ 7+,6 (66$< 6(*(5%(5* 50 55 55 5. 0 (5 5 & 0 5 &/.DE/D 5 . .n 8 Pf Uf $ $ % H 1 LPSOLHV Uf $ F % DQG $ H 1A LPSOLHV $ H 1 % H 1 X X Uf Nf Nf :H GLVFXVV D QXPEHU RI ORJLFV QRW VSHFLILFDOO\ GLVFXVVHG LQ 1XWH >@ WKRXJK ZKHUH ZH KDYH GLVFXVVHG WKH VDPH ORJLF ZH KDYH UHWDLQHG KLV RU KLVWRULFDOO\ HDUOLHU WHUPLQRORJ\ +RZHYHU ZH KDYH DOWHUHG VRPH QDPHV RI VSHFLILF WKHVHV 03 && 0' DQG &2 DUH &DOOHG 03n &6 DQG *2 LQ “XWH &% LV f RI /RHZHU >@ /RJLFV % DQG DUH GLVFXVVHG E\ /RHZHU DV LV&. ,'0' && &$ &% ZKLFK KH FDOOV *r DQG RU VR ZH EHOLHYH PLVWDNHQO\ LGHQWLILHV ZLWK 66 &* DQG &3 DUH LPSOLFLW LQ 1XWH WKRXJK ILUVW H[SOLFLWO\ GLVFXVVHG KHUH 7KH H[WHQVLRQV RI &3 &$ DQG 66 IRUPHG E\ DGGLQJ &% DV DQ D[LRP DUH WR P\ NQRZOHGJH ILUVW PHQn WLRQHG LQ WKH SUHVHQW HVVD\ )RU VHWV RI VHQWHQFHV ;<= OHW ;r@ IRU SURRI DQG .UDEEH >@ IRU VOLJKW FRUUHFWLRQ WR SURRI )RU ODFN RI PD[LPDO GHSWK RI ERWK VHOHFWLRQ IXQFWLRQ DQG QHLJKERUn KRRG VHPDQWLFV VHH *HUVRQ >@ DQG 1XWH > @ 7KHUH LV DQ H[WHQVLRQ RI 6 QRW FRPSOHWH ZUW DQ\ FODVV RI QHLJKERUKRRG IUDPHV

PAGE 226

&+$37(5 ),9( &281,(5)$&78$/6 $1' 7+( &203$5,621 2) :25/'6 $Q $GHTXDWH &RXQWHUIDFWXDO /RJLF ,Q WKLV VHFWLRQ ZH ZLOO FRQVLGHU VHYHUDO FDQGLGDWHV IRU D PLQLPDOO\ DGHTXDWH FRXQWHUIDFWXDO ORJLF DOO FRQWDLQLQJ &$ $Q\ RI WKHP LQSRVHV DW OHDVW D SDUWLDO RUGHU RQ WKH VHW RI SRVVLEOH ZRUOGV UHODWLYH WR HDFK EDVH ZRUOG ZLWK WKH EDVH ZRUOG PLQLPDO LQ WKH RUGHU 2XU DUJXPHQW IRU WKLV LV EDVHG XSRQ WKH SODXVLELOLW\ RI FHUWDLQ FRQGLWLRQDO WKHVHV 3UHSDUDWRU\ WR WKLV ZH PDNH VHYHUDO REVHUYDWLRQV DERXW FRPSDUDWLYH RUGHU VHPDQWLFV DQG WKH FRQGLWLRQDO ORJLFV GLVFXVVHG LQ 6HFWLRQ )LJXUH GLDJUDPV WKH FRQWDLQPHQW UHODWLRQV RI DOO WKH ORJLFV GLVn FXVVHG LQ WKDW VHFWLRQ ,QFOXGHG DUH GHVFULSWLYH QDPHV IRU HDFK IDPLO\ RI ORJLFV RI LQFUHDVLQJ PDWHULDOLW\ ZKHUH HDFK IDPLO\ LV RI LQFUHDVLQJ VWUHQJWK LQ WHUPV RI WKH FRPSDULVRQ RI SRVVLEOH ZRUOGV UHTXLUHG E\ WKH DSSURSULDWH VHPDQWLFV IRU WKDW IDPLO\ )RU HDVH RI UHIHUHQFH ZH UHVWDWH WKH VHPDQWLF FRQGLWLRQV RULJLn QDOO\ SODFHG RQ VHOHFWLRQ IXQFWLRQV LQ WHUPV RI D FRPSDUDWLYH RUGHU UHODWLRQ 5 F 8 [ 8 [ 8 ZKHUH 5 LV WKH RUGHU UHODWLYH WR ZRUOG X 5H FDOO WKDW 5 ; UHSUHVHQWV WKH 5 PLQLPDO HOHPHQWV RI ; $ 'RP 5 X U X X & )RU DOO ;<= F 8 DQG DOO X H 8 LGf 5X; F ; PSf ,I X H ; WKHQ X H 5 ; X FFf ,I X H ; WKHQ ^X` 5A;

PAGE 227

1RUPDO /DZIXO 0RGDO 3DUWLDO 6HPLFRQQHFWHG :HDNf 6XEVWLWXWLRQDO 2UGHU 3DUWLDO 7RWDO 2UGHU 2UGHU & &* 66 &. 03 % &2 &$ &. &. ,' &3 0DWHULDOLW\ &RPSDULVRQ r (;7(16,216 2) &. )LJXUH 66 &% &$ &% &3 &% 9& : .! 1 2 9

PAGE 228

PGf ,I 5X; WKHQ 5
PAGE 229

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f DQG VR UHTXLUH D VXEVHW RI ; $ 'RP 5 DSSHDU DV LQ )LJXUH Df 6HPLFRQQHFWHGQHVV IRUFHV WKH RFFXUUHQFH RI )LJXUH Ef RU Ff LQ VXFK FDVHV Y Y Y \ Df Ef Z \ Ff Z \ Z ] ] ] )LJXUH :H FDQ WKHQ VKRZ WKH IROORZLQJ 7 $ SDUWLDOO\ RUGHUHG FRPSDUDWLYH RUGHU IUDPH VDWLVILHV FEf LII LW LV VHPLFRQQHFWHG 7 &3 &% &$ &% 66 &% DUH GHWHUPLQHG E\ WKH FODVV RI VHPLFRQn QHFWHG SDUWLDOO\ RUGHUHG FRPSDUDWLYH RUGHU IUDPHV SHU VH ZLWK X 5 PLQLPDO IRU HDFK X H 8 DQG ZLWK X 5 OHDVW IRU HDFK X H 8 UHVSHFWLYHO\ &RQGLWLRQ FSf DQG VR WKHVLV &3f LV LQFOXGHG EHFDXVH LW DOVR LV FKDUDFWHULVWLF RI FRQGLWLRQDO ORJLFV WKDW UHTXLUH D SDUWLDO RUGHU RI SRVn VLEOH ZRUOGV :H UHFDOO WKDW LQ 6HFWLRQ LW ZDV VKRZQ WKDW &3 LV GHWHUPLQHG E\ WKH FODVV RI DOO SDUWLDOO\ RUGHUHG FRPSDUDWLYH RUGHU IUDPHV

PAGE 230

7 DQG 7f 7KLV ZDV EDVHG XSRQ WKH IDFW WKDW DQ\ SDUWLDOO\ RUGHUHG FRPSDUDWLYH RUGHU IUDPH VDWLVILHV FRQGLWLRQV LGf PGf FRf DQG FDf 7f DQG DQ\ IUDPH VDWLVI\LQJ WKHVH FRQGLWLRQV FDQ EH SDUWLDOO\ RUGHUHG 7f 7KH IROORZLQJ WKHRUHPV UHODWH FSf WR WKHVH REVHUYDWLRQV 7 (YHU\ SDUWLDOO\ RUGHUHG FRPSDUDWLYH RUGHU IUDPH VDWLVILHV FSf 3URRI :H NQRZ LGf PGf DQG FDf DUH VDWLVILHG E\ 7 6XSSRVH WKDW 5X; F < DQG 5X;F = %\ ZD\ RI FRQWUDGLFWLRQ VXSSRVH Y H 5A ; $ =f DQG Y L < +HQFH Y O 5A; %HFDXVH Y e ; $ = E\ LGf DQG VLQFH Y O (O ; LW IROORZV WKDW WKHUH LV VRPH Z H 5 ; VXFK WKDW Z Y DQG Y Z 7KHQ E\ X X LGf DQG RXU DVVXPSWLRQV ZH;$<$= ,I Z5 Y DQG Y H ; WKHQ Y H 5A; +HQFH Y H < D FRQWUDGLFWLRQ 6R Z5 Y %XW Z5 Y DQG Z H ; $ = LPSOLHV X X U Y L 5X; $ =f D FRQWUDGLFWLRQ 4(' 7 $Q\ VHOHFWLRQ IXQFWLRQ IUDPH VDWLVI\LQJ LGf PGf FDf DQG FSf FDQ EH SDUWLDOO\ RUGHUHG 3URRI /HW ) 8I EH VXFK D VHOHFWLRQ IXQFWLRQ IUDPH 'HILQH D UHODWLRQ 5 F 8 [ 8 [ 8 E\ Y5AZ LII IX^YZ`f ^Y` ,Q YLHZ RI 7 ZH VKDOO VKRZ WKDW 5 LV WUDQVLWLYH 5HFDOO WKDW DQWLn V\PPHWU\ IROORZV E\ GHILQLWLRQ DQG LGf SOXV PGf \LHOG UHIOH[LYLW\ 6XSSRVH Y5AZ DQG Z5A] DQG VKRZ Y5A] :H PD\ WKHQ DVVXPH WKDW IX^YZ`f ^Y` DQG WKDW IX^Z]`f ^Z` )UFP WKH SURRI RI 7 ZH NQRZ WKDW FDf LPSOLHV IX^YZ]`f F ^Y` $JDLQ PGf LPSOLHV IX^YZ]`f ^Y` 7KHQ LQ DFFRUGDQFH ZLWK FSf ZH KDYH IX^YZ]`f F ^Y` DQG IX^YZ]`f F Y]f 6R ZH KDYH IX^YZ]` $ ^Y]`f F ^Y` +HQFH IX^Y]`f F ^Y` )URP PGf ZH LQIHU IX^Y]`f I 7KHUHIRUH IX^Y]`f ^Y` 4('

PAGE 231

:H FRQFOXGH WKDW DV FSf DQG FRf VWDQG LQ YDOLGLW\SUHVHUYLQJ HTXLYDOHQFH LQ WKH SUHVHQFH RI LGf PGf DQG FDf &3 DQG DUH GHGXFWLYHO\ HTXLYDOHQW LQ &. ,' 0' &$ 7 &3 ,'0' &$ ,'0' &3 &$ ,Q ZKDW IROORZV ZH VKDOO DUJXH WKDW FHUWDLQ WKHVHV DUH UHTXLVLWH IRU DQ\ DGHTXDWH ORJLF RI FRXQWHUIDFWXDOV 7KHVH PLQLPDO WKHVHV DUH OLVWHG EHORZ ( 0LQLPDO FRXQWHUIDFWXDO WKHVHV /,' &/&ST:ST 03 &:ST&ST ,0' &/S:TS &3 &.:ST:SU:.SUT &$ &.:SU:TU:$STU $OO EXW /,' DQG ,0' DUH UHFRJQL]DEOH IURP RXU SUHYLRXV OLVWV :H REVHUYH WKDW /,' DQG ,0' DUH WKH PRGDO HTXLYDOHQWV RI ,' DQG 0' UHVSHFWLYHO\ SURn YLGHG WKH ORJLF SHUPLWV WKH GHILQLWLRQ RI D PRGDO RSHUDWRU 7R DFFHSW /,' DQG ,0' LV WR DFFHSW ,' DQG 0' ,Q DGGLWLRQ WR WKH DERYH PLQLPDO WKHVHV WKH IROORZLQJ DOVR GHVHUYH VHPH FRQVLGHUDWLRQ ( 2SWLRQDO FRXQWHUIDFWXDO WKHVHV && &.ST:SU &% &:$STU$:SU:TU &9 &.:ST9SU:.SUT :H KDYH GLVFXVVHG WKHVH SUHYLRXVO\ LQ D IRUPDO FRQWH[W DV ZHOO :H DVVXPH ZLWKRXW IXUWKHU DUJXPHQW WKDW DQ\ DGHTXDWH FRXQWHUIDFWXDO ORJLF PXVW EH QRUPDO WKDW LV FRQWDLQ &. :H DUJXHG LQ 6HFWLRQ WKDW WKH VWULFW FRQGLWLRQDO ZDV VWURQJHU WKDQ WKH FRXQWHUIDFWXDO FRQGLWLRQDO

PAGE 232

DQG LQ VRPH VHQVH ZDV WKH RXWHU OLPLW RI WKH FRXQWHU IDFWXDO ,I SURn SRVLWLRQ S VWULFWO\ LPSOLHV SURSRVLWLRQ T WKHQ VXUHO\ T ZRXOG EH WUXH LI S ZHUH 6LPLODUO\ LI SURSRVLWLRQ S LV QHFHVVDU\ WKHQ LW ZRXOG EH WUXH XQGHU DQ\ FRQGLWLRQ &RQVHTXHQWO\ ZH VKRXOG DFFHSW ERWK /,' DQG /0' DQG VR ,' DQG 0' 6XSSRVH ZH DJUHH WKDW LI S ZHUH WUXH T ZRXOG EH WUXH ,I ZH VXEn VHTXHQWO\ GLVFRYHU WKDW S LV WUXH EXW T LV IDOVH WKLV HIIHFWLYHO\ UHn EXWV RXU RULJLQDO FRXQWHUIDFWXDO WKDW LV &.S1T1:ST 7KH FRQWUDSRVLWLYH RI WKLV LV 03 5HJDUGLQJ :ST DV D FRQGLWLRQDO SUHGLFWLRQ OHDGV WR WKH VDPH FRQFOXVLRQ 5HFDOO WKH RLO\ HQJLQH RI 6HFWLRQ f $ FRQGLWLRQDO SUHGLFWLRQ ZRXOG KDUGO\ EH LQIRUPDWLYH LI D UXOH RI GHWDFKPHQW ZHUH QRW RSHUDWLYH IRU LW 6R ZH DUH ERXQG WR DFFHSW 03 :H KDYH SUHYLRXVO\ REVHUYHG WKDW FRXQWHUIDFWXDOV GR QRW JHQHUDOO\ SHUPLW VWUHQJWKHQLQJ WKH DQWHFHGHQW +RZHYHU LQ FHUWDLQ FDVHV WKH DQWHn FHGHQW FDQ EH VWUHQJWKHQHG 2QH FDVH LQ ZKLFK LW VHHPV HYLGHQW WKDW ZH FDQ FRQMRLQ D SURSRVLWLRQ WR WKH DQWHFHGHQW RI D FRXQWHUIDFWXDO LV ZKHQ WKH FRQMXQFW LWVHOI LV D FRXQWHUIDFWXDO FRQVHTXHQW RI WKH RULJLQDO DQWHFHn GHQW 6XSSRVH NDQJDURRV ZRXOG WRSSOH RYHU LI WKH\ KDG QR WDLOV DQG NDQJDn URRV ZRXOG ORVH WKHLU EDODQFH LI WKH\ KDG QR WDLOV ,W VHHPV WR IROORZ WKDW LI NDQJDURRV KDG QR WDLOV DQG ORVW WKHLU EDODQFH WKHQ WKH\ ZRXOG WRSn SOH RYHU $V 3ROORFN DVVHUWA LI U ZRXOG EH WUXH LI S ZHUH WKHQ LQ VRPH VHQVH .SU EHLQJ WUXH LV QRW D GLIIHUHQW FLUFXPVWDQFH IURP S EHLQJ WUXH > S @ 6R ZH DUH FRPPLWWHG WR DFFHSWLQJ &3 6XSSRVH WKDW U ZRXOG EH WUXH LI S ZHUH DQG DOVR WKDW U ZRXOG EH WUXH LI T ZHUH :H WKHQ DJUHH WKDW ZKLFKHYHU RI S nDQG T LV WUXH U ZLOO EH WUXH 7KXV LW ZRXOG VHDQ ZH DJUHH WR WKH YDOLGLW\ RI &$ 7KH FDVHV ZKHUH RQH PLJKW GRXEW WKH YDOLGLW\ RI &$ DUH WKRVH ZKHUH $ST VRPHKRZ

PAGE 233

HQFRPSDVVHV PRUH FDVHV WKDQ WKH FODVVLFDO %RROHDQf VXP RI S DQG T $V $ LV D FODVVLFDO RSHUDWRU LQ DOO WKH ORJLFV ZH KDYH FRQVLGHUHG WKLV FDVH GRHV QRW DULVH 6HH KRZHYHU +DUGHJUHH >@ DQG =HPDQ >@@ IRU QRQFODVVLFDO GLVMXQFWLRQf :H KDYH VXJJHVWHG WKDW D PLQLPDO FRXQWHUIDFWXDO ORJLF LV &$ &.,'030' &3 &$ :H KDYH DJUHHG LQ WKH SUHFHGLQJ SDUDn JUDSKV WR HDFK RI WKH FKDUDFWHULVWLF D[LRPV RI &$ &$ LV GHWHUPLQHG E\ WKH FODVV RI SDUWLDOO\ RUGHUHG FRPSDUDWLYH RUGHU IUDPHV ZLWK ZRUOG X 5 PLQLPDO 7KXV HYHQ ZLWKRXW DQ\ RI WKH WKHVHV RI ( ZH DUH FRPn PLWWHG WR D ORJLF ZLWK D SDUWLDOO\ RUGHUHG VHPDQWLFV DV D ORJLF RI FRXQWHU IDFWXDOV 2I WKH WKUHH WKHVHV RI ( && DQG &9 KDYH FRPH XQGHU WKH JUHDWHVW DWWDFN 1XWH > @ KDV DUJXHG DJDLQVW ERWK && DQG &9 %HQQHWW >@ UHn JDUGHG WKH SUHVHQFH RI && DV RQH RI WKH PRVW FRXQWHULQWXLWLYH DVSHFWV RI /HZLVn DQDO\VLV DV KDV %LJHORZ >@ 3ROORFN >@ DUJXHV IRU && EXW GHQLHV &9 2WKHUV FRXOG EH PHQWLRQHG 7R P\ NQRZOHGJH RQO\ /RHZHU >@ KDV VXJn JHVWHG WKDW &% LV RI DQ\ VLJQLILFDQFH &% LV WKXV QRW VR PXFK SUREOHPDWLF DV LJQRUHG &% EHDUV VRPH UHODWLRQ WR 6'$ &:$STU.:SU:TU 7KH GLVWLQFWLRQ LV LQ WKH FRQVHTXHQWnV EHLQJ D GLVMXQFWLRQ LQ WKH IRUPHU DQG D FRQMXQFWLRQ LQ WKH ODWWHU +RZHYHU &% XQOLNH 6'$ GRHV QRW UHVXOW LQ WKH HTXLYDOHQFH RI WKH FRXQWHUIDFWXDO DQG WKH VWULFW FRQGLWLRQDO LQ D FODVVLFDO FRQGLWLRQDO ORJLF )RU WKH PRVW SDUW ZH KDYH VLGHVWHSSHG WKH GHEDWH RYHU 6'$ E\ OLPLWLQJ RXU FRQVLGHUDWLRQ WR QRUPDO FRQGLWLRQDO ORJLFVf &% DQG 6'$ DUH DOLNH LQ KDYLQJ D GLVMXQFWLYH DQWHFHGHQW DQG LW LV WKLV FKDUDFWHULVWLF WKDW UHQGHUV WKHP WURXEOHVRPH ,Q FRQYHUVDWLRQ ZKHQ ZH DVVHUW D FRXQWHU IDFWXDO ZLWK D GLVMXQFWLYH DQWHFHGHQW ZH DUH JHQHUDOO\ SUHSDUHG WR GHIHQG

PAGE 234

HLWKHU VLPSOLILHG FRXQWHUIDFWXDO WKXV ZH REVHUYHG LQ 6HFWLRQ WKDW 6'$ ZDV SUDJPDWLFDOO\ YDOLG 7KDW LV ERWK GLVMXQFWV DUH XVXDOO\ HQWHU WDLQDEOH ZLWK UHVSHFW WR WKH VDPH VRUWV RI SRVVLEOH ZRUOGV ,I VR WKHQ 6'$ ZRUNV LI QRW WKHQ LW GRHV QRW 7KH VLWXDWLRQ LV QRW WKH VDPH ZLWK &% (YHQ LI WKH WZR GLVMXQFWV DUH WUXH LQ TXLWH GLVVLPLODU SRVVLEOH ZRUOGV WKHQ DW OHDVW RQH RI WKH WZR VLPSOLILHG FRXQWHUI DFWXDOV ZRXOG VHHP WR KROG +RZHYHU ZH DUH DOn UHDG\ GLVFXVVLQJ DQ DQDO\VLV SUHVXPLQJ VRPH VRUW RI GHIHQVLEOH FRPSDULVRQ RI SRVVLEOH ZRUOGV VR SHUKDSV RXU IXUWKHU GLVFXVVLRQ RI &% VKRXOG EH OHIW WR WKH QH[W VHFWLRQ FDQ WKLQN RI QR FRXQWHUH[DPSOH WR &% DQG VR DP LQFOLQHG WR DFFHSW LW &RXQWHUH[DPSOHV WR && VXFK DV WKRVH VXJJHVWHG E\ %HQQHWW >@ DQG %LJHORZ >@ DUH GUDZQ IURP WKH YDVW SKLORVRSKLFDO VWRUHKRXVH RI XQLQWXLn WLYH LQGLFDWLYH FRQGLWLRQDOV ZKHUH WKH DQWHFHGHQW DQG FRQVHTXHQW DUH LUn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

PAGE 235

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f WRWDO RUGHU ,I ZH FRQSDUH &3 DQG &9 ZH VHH WKDW &3 SHUPLWV WKH VWUHQJWKHQLQJ RI DQ DQWHn FHGHQW ZLWK DQ\ FRXQWHUIDFWXDO FRQVHTXHQW RI WKDW DQWHFHGHQW 2Q WKH RWKHU KDQG &9 SHUPLWV VWUHQJWKHQLQJ WKH DQWHFHGHQW ZLWK DQ\ SURSRVLWLRQ FRWHQDEOH ZLWK WKDW DQWHFHGHQW 7KH XVH RI FRWHQDEOH LV DSSURSULDWH LQ WKLV FRQQHFWLRQ VLQFH *RRGPDQ > S @ GHILQHV S LV FRWHQDEOH ZLWK T DV 1:S1T ZKLFK LV MXVW 9ST LQ RXU V\PERORJ\ ,Q IDFW /RHZHU XVHV &RWSTf SUHFLVHO\ ZKHUH ZH XVH 9ST > S @ $Q DOOHJHG FRXQWHUH[DPSOH WR &9 KDV EHHQ SUHVHQWHG E\ 3ROORFN DQG ZH UHSURGXFH LW EHORZ LQ RXU V\PERORJ\ > SS @ 6XSSRVH S T DQG U DUH WKUHH XQUHODWHG IDOVH VWDWHPHQWV 0\ FDU LV SDLQWHG EODFN 0\ JDUEDJH FDQ EOHZ RYHU DQG 0\ PDSOH WUHH GLHG $ VXEVWLWXWLRQ LQVWDQFH RI &9 LV ( &.:$ST1U9$ST$UT:.$ST$UT1U )URP ( ZH PD\ GHULYH ( &.1:$STS1:$.SUTT9$STU

PAGE 236

3ROORFN WKHQ DUJXHV WKDW WKH DQWHFHGHQW RI ( LV WUXH EXW WKH FRQVHTXHQW LV IDOVH 3ROORFN VXJJHVWV WKDW LI S DQG T DUH XQUHODWHG WKHQ WKHLU GLVMXQFn WLRQ FDQQRW EULQJ DERXW WKH WUXWK RI RQH RI WKH GLVMXQFWV VR :$STS LV IDOVH DQG 1:$STS LV WKXV WUXH 6LPLODUO\ IRU WKH VHFRQG FRQMXQFW RI WKH DQWHFHGHQW RI ( LI .SU DQG T DUH XQUHODWHG WKHQ WKHLU GLVMXQFn WLRQ DJDLQ FDQQRW EULQJ DERXW RQH GLVMXQFW VR 1:$.SUTT LV WUXH 7KXV WKH DQWHFHGHQW RI ( LV WUXH 1RZ FRQVLGHU WKH FRQVHTXHQW 9$STU 6LQFH S DQG T DUH XQUHODWHG DQG LUUHOHYDQW WR U LW IROORZV WKDW HYHQ LI HLWKHU S RU T ZHUH WXUH 1U ZRXOG VWLOO EH WUXH +HQFH :$ST1U LV WUXH VR 9$STU LV IDOVH 7KXV JRHV 3ROORFNnV DUJXPHQW /RHZHU > S ,OO@ ZULWHV ( HQWLUHO\ LQ WHUPV RI WKH HTXLYDn OHQW FRWHQDELOLW\ VWDWHPHQWV RU LQ RXU V\PERORJ\ DV PLJKWFRQGLWLRQDOV ( &.9$ST1S9$.SUT1T9$STU 7KLV LV UHYHDOLQJ RQ VHYHUDO FRXQWV 2QH FDQ DUJXH IRU WKH WUXWK RI ERWK FRQMXQFWV RI WKH DQWHFHGHQW RI ( ZLWKRXW HPSOR\LQJ WKH SULQFLSOH WKDW D GLVMXQFWLRQ RI XQUHODWHG SURSRVLWLRQV FDQQRW EULQJ DERXW RQH RI WKH GLVn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

PAGE 237

WKH UHDVRQDEOHVQHVV GHSHQGV XSRQ KRZ OLNHO\ S T DQG U DUH ,I WKH\ DUH DOO DERXW HTXDOO\ OLNHO\ WKHQ WKH DQWHFHGHQW RI ( LV WUXH %XW WKHQ E\ WKH VDPH UHDVRQLQJ VR LV WKH FRQVHTXHQW LI HLWKHU S RU T ZHUH WUXH WKHQ U PLJKW DOVR EH WUXH *LYHQ WKUHH HTXDOO\ OLNHO\ XQn UHODWHG SURSRVLWLRQV DP LQFOLQHG WR WKLQN WKDW LI HLWKHU RI DQ\ WZR ZHUH WUXH WKH RWKHU PLJKW VWLOO EH IDOVH RU PLJKW DOVR EH WUXH 7KXV 3ROORFNnV DOOHJHG FRXQWHUH[DPSOH LV RI GXELRXV LPSRUW ,I WKH XQUHODWHG SURSRVLWLRQV DUH DERXW HTXDOO\ OLNHO\ DV WKH\ DUH LQ 3ROORFNnV H[DPSOH WKHQ WKH FRQVHTXHQW LV WUXH DORQJ ZLWK WKH DQWHFHGHQW RI ( 2Q WKH RWKHU KDQG LI WKH XQUHODWHG SURSRVLWLRQV DUH QRW HTXDOO\ OLNHO\ WKHQ RQH RU WKH RWKHU FRQMXQFW RI WKH DQWHFHGHQW LV QRW WUXH ,Q HLWKHU FDVH ZH KDYH QR FRXQWHUH[DPSOH (YHQ LI 3ROORFNnV DUJXPHQW LV WKXV ZHDNHQHG FDQ WKLQN RI QR GHFLVLYH DUJXPHQW IRU &9 DQG VR DP ZLOOLQJ WR UHJDUG LW DV SUREOHPDWLF DORQJ ZLWK && DQG SHUKDSV &% 1HYHUWKHOHVV ZH DUH OHIW ZLWK D PLQLPDO FRXQWHUIDFWXDO ORJLF ZKLFK UHTXLUHV D SDUWLDO RUGHU RI SRVVLEOH ZRUOGV ,W LV QRW FOHDU WKDW HYHQ VR D SDUWLDO RUGHU PXVW IRUP D SDUW RI DQ DQDO\VLV RI FRXQWHUIDFWXDOV 7KLV LV WKH SRLQW PDGH E\ /RHZHU DOWKRXJK RXU UHDVRQLQJ ZLWK FRXQWHUIDFWXDOV GRHV LQn YROYH D VLPLODULW\ RUGHULQJ RI ZRUOGV WKH FRQFHSW RI VLPn LODULW\ LV SULPLWLYH DQG GRHV QRW VXSSRUW DQ DQDO\VLV RI FRXQWHUIDFWXDOV > SS @ :H VKDOO DUJXH LQ WKH QH[W VHFWLRQ WKDW RQ WKH FRQWUDU\ WKH SDUWLDO RUGHULQJ UHTXLUHG E\ DQ\ DGHTXDWH FRXQWHUIDFWXDO ORJLF FDQ EH UHJDUGHG DV HPHUJLQJ IURP DQ DQDO\VLV RI FRXQWHUIDFWXDOV &RPSDUDWLYH 2UGHU $QDO\VLV :H KDYH UHWXUQHG UHSHDWHGO\ LQ WKLV HVVD\ WR WKH GLVWLQFWLRQ EHWZHHQ WKH ORJLF RI D FRQFHSW DQG DQ DQDO\VLV RI D FRQFHSW 7KH LVVXH LV UDLVHG

PAGE 238

RQFH PRUH E\ WKH GLOHPPD /RHZHU >@ SRVHV IRU WKRVH DQDO\VHV RI WKH FRXQWHUIDFWXDO FRQGLWLRQDO WKDW XWLOL]H D UHODWLRQ RI FRPSDUDWLYH VLPLn ODULW\ ( Df ,I VLPLODULW\ RUGHULQJ RI SRVVLEOH ZRUOGV LV SULPLWLYH WR D VHPDQWLFV IRU WKH FRQGLWLRQDO WKHQ LW FDQQRW VXSSRUW DQ DQDO\VLV Ef ,I VLPLODULW\ RUGHULQJ LV QRW DQWHFHGHQWO\ ZHOOHQRXJK XQGHUVWRRG WKHQ LW FDQQRW VXSSRUW DQ DQDO\VLV Ff 6LPLODULW\ RUGHULQJ LV HLWKHU SULPLWLYH RU QRW ZHOO XQGHUVWRRG Gf 6LPLODULW\ RUGHULQJ FDQQRW VXSSRUW DQ DQDO\VLV 8OWLPDWHO\ ZH VKDOO LQGLFDWH KRZ RQH PD\ JR EHWZHHQ WKH KRUQV RI WKLV GLOHPPD DV VRPH RI RXU SULRU FRPPHQWV KDYH VXJJHVWHG 7KH GLOHPPD VKRXOG EH WDNHQ VHULRXVO\ KRZHYHU 7KH GLVWLQFWLRQ EHn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n I\LQJ FHUWDLQ FRQWLQJHQW VHQWHQFHV DV WUXH DQG RWKHUV DV IDOVH :H GR DFn FHSW FHUWDLQ VHQWHQFHV DQG UHMHFW RWKHUV DQG WR WKH H[WHQW WKDW ZH DUH V\VWHPDWLF LQ GRLQJ VR DQ DQDO\VLV PXVW SURYLGH D WKHRUHWLFDO EDVLV :H KDYH DUJXHG SUHYLRXVO\ WKDW ZKDW LV UHTXLUHG LV DQ H[SODQDWRU\ DQDO\VLV

PAGE 239

:KLOH JHQHUDOO\ ZH H[SHFW DQ DQDO\VLV WR FODULI\ VHPH RI WKH SX]]OLQJ FDVHV DQG RWKHUZLVH WR FRQIRUP WR RXU PRUH ILUPO\ KHOG LQWXLWLRQV DERXW WKH FRQFHSW XQGHUJRLQJ DQDO\VLV D ZHOOFRQVWUXFWHG DQDO\VLV FDQ VRPH WLUUHV SHUVXDGH XV WR UHYLVH RXU LQWXLWLRQV IRU WKH VDNH RI JUHDWHU FODULW\ HOVHZKHUH 7KH DEDQGRQPHQW RI H[LVWHQWLDO LPSRUW IRU XQLYHUVDO JHQHUDOLn ]DWLRQV LV D FDVH LQ SRLQW :H VDZ LQ WKH SUHYLRXV VHFWLRQ WKDW D PLQLPDOO\ DGHTXDWH ORJLF IRU WKH FRXQWHU IDFWXDO FRQGLWLRQDO PXVW LQFOXGH &$ DQG WKXV FRQIRUPV WR D VHPDQWLFV ZLWK D SDUWLDO RUGHULQJ RI SRVVLEOH ZRUOGV UHODWLYH WR HDFK EDVH ZRUOG +RZHYHU FRPSDUDWLYH RUGHU IUDPHV FRQVWLWXWH D IRUPDO VHPDQn WLFV :LWKRXW VRPHWKLQJ H[WHUQDO WR WKLV IRUPDO VWUXFWXUH XSRQ ZKLFK WR EDVH WKH FRPSDUDWLYH RUGHU WKDW RUGHU UHPDLQV PHUHO\ SULPLWLYH WR WKH VHPDQWLFV DQG LQFDSDEOH RI VXSSRUWLQJ DQ DQDO\VLV 5HVWULFWLQJ RXU DWn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

PAGE 240

7KH GLOHPD IRU DOO RI WKHVH DQDO\VHV LV WKDW WKH RUGHULQJ SULQFLSOH LV QRW VXIILFLHQWO\ ZHOO XQGHUVWRRG WR DYRLG GLVDJUHHPHQW DERXW KRZ WR DSn SO\ LW LQ SDUWLFXODU FDVHV $QG WKH OHVV VDLG DERXW WKH SULQFLSOH WKH PRUH URRP WKHUH LV IRU GLVDJUHHPHQW 6R DV /HZLV VD\V OLWWOH DERXW FRPSDUDWLYH VLPLODULW\ WKHUH LV FRQVLGHUDEOH ODWLWXGH IRU DOOHJHG FRXQWHUn H[DPSOHV $V 3ROORFN VD\V PXFK DERXW WKH QRWLRQ RI PLQLPDO FKDQJH SURn YLGLQJ DQ DQDO\VLV RI LW DV ZHOO WKH ODWLWXGH LV JUHDWO\ UHGXFHG EXW XSRQ WKH SHQDOW\ RI KDYLQJ WR EULQJ LQ D QXPEHU RI VXEVLGLDU\ QRWLRQV RI TXHVWLRQDEOH SHGLJUHH VXFK DV WKH FRQFHSW RI D VLPSOH SURSRVLWLRQ > SS @ 7KH IDFW WKDW WKHUH LV LQIRUPHG GLVDJUHHPHQW QRW RQO\ DERXW KRZ WR DSSO\ DQ\ JLYHQ RUGHULQJ SULQFLSOH EXW DOVR DERXW ZKDW LV WKH RSHUDWLYH RUGHULQJ SULQFLSOH VHHPV RQO\ WR VWUHQJWKHQ /RHZHUnV GLOHPn PD ZLWK UHIHUHQFH WR DOO RI WKHVH DQDO\VHV &RXQWHUIDFWXDOV LW KDV RIWHQ EHHQ UHPDUNHG DUH QRWRULRXVO\ YDJXH DP LQ DJUHHPHQW ZLWK /HZLV WKDW LW LV QHLWKHU WR EH H[SHFWHG QRU LV LW GHVLUDEOH WKDW DQ DQDO\VLV VKRXOG DWWHPSW WR UHSODFH RXU YDJXH FRQFHSW RI FRQGLWLRQDOLW\ ZLWK D VKDUSO\ GHOLPLWHG RQH )RU YHU\ OLNHO\ LI ZH ZHUH WR GR VR ZH FRXOG QRW FRQVLVWHQWO\ DFFRUUPRGDWH DOO RI RXU LQWXLWLRQV DERXW FRQGLWLRQDOV +HQFH /HZLV DUJXHV WKDW D FRUUHVSRQGLQJO\ YDJXH QRn WLRQ WKDW RI FRPSDUDWLYH VLPLODULW\ LV MXVW ZKDW LV UHTXLUHG 2QH GLIILFXOW\ ZLWK FRPSDUDWLYH RYHUDOO VLPLODULW\ LV WKDW ZH DUH ERXQG WR DFFHSW D ZHDN WRWDO RUGHU DV ZH DUJXHG LQ 6HFWLRQ :KDWHYHU WKH PHULWV RI 3ROORFNnV FRXQWHUH[DPSOH WR &9 LW VHHPV WR PH WKDW LQ FRXQWHUIDFWXDO GHOLEHUDWLRQ ZH URXWLQHO\ FRQVLGHU DV UHDVRQDEOH VLWXDn WLRQV ZKLFK GR QRW OLH DW D VLQJOH OHYHO RI FRPSDUDWLYH VLPLODULW\ WR WKH DFWXDO ZRUOG WKDW LV DUH QRW LQ D VLQJOH HTXLYDOHQFH FODVV RI SRVVLEOH ZRUOGV JLYHQ D IDLUO\ QDUURZ VHQVH RI FRPSDUDWLYH VLPLODULW\

PAGE 241

$ FRPSOHPHQWDU\ GLIILFXOW\ LQIHFWV WKH QRWLRQ RI PLQLPDO FKDQJH &RQVLGHU WKH IROORZLQJ H[DPSOH ZKLFK LV EDVHG XSRQ D UHFHQW DFWXDO HYHQW DW D 1HZ
PAGE 242

U ERWK SODFH ^Z]` V WKH *DOORSHU ZLQV ^\]` W WKH )O\HU ZLQV ^YZ` GR QRW NQRZ DQG GRXEW LI DQ\RQH GRHV KRZ WR GHWHUPLQH WKH VLPSOH SURSRVLWLRQV ZKLFK HQDEOH RQH WR UDQN WKH ZRUOGV LQ WHUPV RI FKDQJH IUDQ WKH DFWXDO ZDU ,G +RZHYHU EHOLHYH WKH UDQNLQJ RI )LJXUH Df LV SUHIHUDEOH WR WKDW RI Ef Df Ef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f ERWK RI WKH IROORZLQJ LQVWDQFHV RI &% DUH WUXH QRQWULYLDOO\ ( &:$STU$:SU:TU &:$VWU$:VU:WU

PAGE 243

7KH LQVWDQFHV RI ( DUH WUXH RQ WKH VHFRQG UHDGLQJ DOVR WKRXJK WULYLDOO\ VLQFH DOO WKH FRXQWHUIDFWXDOV LQ ( DUH IDOVH +RZHYHU FRQVLGHU WKH IROORZLQJ SURSRVLWLRQV D ^Y]` E ^Z\` DQG F ^Y\` )LJXUH Ef LV D FRXQWHUPRGHO WR WKH LQVWDQFH RI &% ( &:$DEF$:DF:EF 7KDW LV WKHUH LV VRPH GLVMXQFWLYH FRQGLWLRQ $DE ^YZ\]`f ZKLFK UHVXOWV LQ H[DFWO\ RQH KRUVH JHWWLQJ DURXQG WKH SLOHXS EXW QHLWKHU GLVn MXQFW LV FRXQWHUIDFWXDOO\ VXIILFLHQW IRU WKLV $V VWDWHG HDUOLHU FDQQRW WKLQN RI D FRXQWHUH[DPSOH WR &% DQG VR GR QRW WKLQN WKLV H[DPSOH LV RQH 7KH VLWXDWLRQ RI ( VHHPV KLJKO\ LPSODXVLEOH WR PH LI VRPH GLVMXQFWLYH FRQGLWLRQ KDV WKH UHVXOW LQGLFDWHG WKHQ VRPH RQH RI WKH GLVMXQFWV GRHV ,I RQH LV ORRNLQJ IRU D ZD\ WR H[n SUHVV SURSRVLWLRQV D DQG E SHUKDSV WKH )O\HU*DOORSHU MXVW EDUHO\ ILQLVKHV ZLOO GR DV WKLV OHDYHV RSHQ ZKHWKHU WKH RWKHU KRUVH JHWV DURXQG WKH SLOHXS PRUH HDVLO\ RU QRW DW DOO 7R DFFRPQRGDWH 3ROORFNnV DFFRXQW WR WKLV YLHZ UHTXLUHV HLWKHU WKH FKDQJH WKDW UHVXOWV LQ Z RU ] WR FRQWDLQ WKH FKDQJHV WKDW UHVXOW LQ Y DQG \ WKXV UHVWRULQJ WKH PLVVLQJ OLQNV RI )LJXUH Ef RU WR GHQ\ WKDW DQ\ RI WKH FKDQJHV FRQWDLQ DQ\ RI WKH RWKHUV DV LQ )LJXUH Df Df Ef )LJXUH ,W ZDV DUJXHG LQ 6HFWLRQ DJDLQVW D VLPLODU H[DPSOH RI 1XWHnV (f WKDW RQFH ZH KDYH UHDFKHG WKH VLWXDWLRQ GHSLFWHG LQ )LJXUH

PAGE 244

Df ZH PLJKW DV ZHOO FROODSVH LW WR )LJXUH Ef ZKHUH WKH ZRUOGV XQGHU FRQVLGHUDWLRQ DOO IDOO LQWR DQ HTXLYDOHQFH FODVV RI ZRUOGV PRUH RU OHVV HTXDOO\ VLPLODU WR WKH EDVH ZRUOG JLYHQ D ORRVH HQRXJK VHQVH RI FRPSDUDWLYH VLPLODULW\ 1XWHnV SRLQW ZDV WKDW ZH ZRXOG QRUPDOO\ FRQVLGHU HDFK RI WKH ZRUOGV Y Z \ DQG ] DV D UHDVRQDEOH VLWXDWLRQ IURP WKH SRLQW RI YLHZ RI WKH DFWXDO ZRUOG :LWK WKDW DVVHVVPHQW DP LQ DJUHHPHQW :KDW GLVDJUHH ZLWK LV WKDW LW LV SDUW RI /HZLVn DQDO\VLV WKDW ZH JR RQ WR UDQN WKHVH ZRUOGV LQ WHUPV RI FRPSDUDWLYH RYHUDOO VLPLODULW\ GLVVLPLODUO\ &RPSDUDWLYH VLPLODULW\ LV QRW QHFHVVDULO\ WKDW ILQH D QRWLRQ QRU VKRXOG LW EH 3ROORFNnV DWWHPSW WR HVFDSH WKH GLOHPPD SRVHG E\ /RHZHU E\ LQ HIn IHFW JUDVSLQJ WKH VHFRQG KRUQ VHHPV PLVJXLGHG WR PH %\ DQDO\]LQJ WKH QRWLRQ RI PLQLPDO FKDQJH DV DQ RUGHULQJ SULQFLSOH DQG LQ WKH SURFHVV PDNLQJ LW PRUH SUHFLVH WKH UHVXOW PD\ EH VRPHWKLQJ WKDW LV DEOH WR VXSn SRUW DQ DQDO\VLV EHFDXVH ZHOOHQRXJK H[SODLQHG WR EH DQWHFHGHQWO\ XQGHUn VWDQGDEOH EXW WKDW JLYHV WKH ZURQJ WUXWK YDOXHV IRU LPSUHFLVH FRXQWHU IDFWXDOV VR QRW EH D FRUUHFW DQDO\VLV ,I ZH FRQVLGHU WKH FRXQWHUIDFWXDOV ZH KDYH UHIHUUHG WR DERYH VXFK DV :$STU :SU :TU HWF ZH VHH WKDW LQ DFWXDOLW\ DOO DUH IDOVH *LYHQ WKDW WKH *DOORSHU SODFHV WKH )O\HU PLJKW SODFH RU PLJKW QRW WKH )O\HU PLJKW QRW HYHQ ILQLVK RU LI VR PLJKW HYHQ ZLQ 7KXV RQO\ )LJXUHV Df RU Ef ZLOO JLYH WKH FRUUHFW WUXWK YDOXHV IRU WKH FRXQWHUIDFWXDOV 7R FODLP KRZHYHU WKDW DQ RUGHULQJ SULQFLSOH EDVHG RQ PLQLPDO FKDQJH ZRXOG \LHOG )LJXUH Df UDWKHU WKDQ HLWKHU Df Ef DV LV RU Ef ZLWK WKH PLVVLQJ OLQNV UHVWRUHG VHHPV VOLJKWO\ DG KRF 7KHUH DUH UHDVRQV IRU EHOLHYLQJ D FRPSDUDWLYH RUGHU EDVHG RQ FKDQJH IURP WKH DFWXDO ZRUOG ZLOO \LHOG VRPHWKLQJ OLNH Df RU Ef (LWKHU WKH

PAGE 245

FKDQJH ZKLFK DOORZV RQH KRUVH WR ILQLVK LV FRQWDLQHG LQ D FKDQJH ZKLFK DOORZV WZR WR ILQLVK RU WKH RWKHU ZD\ DURXQG :KLOH DP QRW VXUH ZKLFK LW LV GRXEW LI LW LV QHLWKHU 6R LW DSSHDUV WKDW 3ROORFNnV DFFRXQW UHTXLUHV HYHQ JUHDWHU VNLOOV DW UDQNLQJ ZRUOGV WKDQ /HZLVn DQG 1XWHn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n QRWLRQ RI FRPSDUDWLYH RYHUDOO VLPLODULW\ DV ERWK SULPLWLYH DQG DQWHFHGHQWO\ XQGHUn VWRRG E\ VHHLQJ LW DV DQDORJRXV WR WKH H[WHQVLRQ RI D SK\VLFDO FRQFHSW VXFK DV PRWLRQ WR D QHZ GRPDLQ ,W LV LQ WKLV ZD\ WKDW ZH VHH WKHUH LV DQRWKHU DOWHUQDWLYH QRW H[SUHVVHG LQ WKH GLVMXQFWLYH SUHPLVH RI /RHZHUnV GLOHPPD ,W LV QRW WKH FDVH WKDW VLPLODULW\ RUGHULQJ LV PHUHO\ SULPLWLYH QRU LV LW SRRUO\ XQGHUVWRRG ,W LV UHDVRQDEO\ ZHOOXQGHUVWRRG LQ RUGLQDU\ FRQWH[WV EXW WKH DSSOLFDWLRQ WR SRVVLEOH ZRUOGV LQWURGXFHV IDFWRUV LQWR WKH FRPSDULVRQ WKDW DUH QRW RSHUDWLYH RU QRW VLJQLILFDQW LQ PRUH RUGLQDU\

PAGE 246

FRQWH[WV 7KXV LQ JLYLQJ D FRXQWHUIDFWXDO DQDO\VLV RI FDXVDWLRQ /HZLV LQWURGXFHV FRPSDUDWLYH VLPLODULW\ DQG ZDUQV KDYH QRW VDLG MXVW KRZ WR EDODQFH WKH UHVSHFWV RI FRPSDULVRQ DJDLQVW HDFK RWKHU VR KDYH QRW VDLG MXVW ZKDW RXU UHODWLRQ RI FRPSDUDWLYH VLPLODULW\ LV WR EH 1RW IRU QRWKLQJ GLG FDOO LW SULPLWLYH %XW KDYH VDLG ZKDW VRUW RI UHODWLRQ LW LV DQG ZH DUH IDPLOLDU ZLWK UHODWLRQV RI WKDW VRUW > S @ &RPSDUDWLYH VLPLODULW\ LV QRW PHUHO\ SULPLWLYH %XW DOVR ZKDW LV RI LPn SRUWDQFH LQ FRPSDULQJ ZRUOGV IRU WKH SXUSRVH RI FRXQWHUIDFWXDO GHOLEHUDWLRQ LV QRW VHOIHYLGHQWO\ SDUW RI RXU RUGLQDU\ QRWLRQ RI RYHUDOO VLPLODULW\ 6RPH IDFWRUV DUH PDQLIHVW RQO\ LQ DSSO\LQJ WKH FRQFHSW WR SRVVLEOH ZRUOGV 7KXV LW LV SURSHU WR FDOO /HZLVn DFFRXQW D WKHRU\ RI FRXQWHUIDFWXDOV &RPSDUDWLYH RYHUDOO VLPLODULW\ LV QRW WKH IXOO DQVZHU WR WKH FRPSDUDn WLYH RUGHU RI SRVVLEOH ZRUOGV WKDW RFFXUV LQ FRXQWHUIDFWXDO UHDVRQLQJ /HZLV UHVWULFWV KLV DFFRXQW WR GHWHUPLQLVP EXW LW VHHPV FOHDU WKDW LQn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n DELOLW\ VXQ LV QHDUO\ XQLW\ DQG D VOLJKW SUREDELOLW\ WKDW QR GHFD\ ZLOO RFFXU /DEHO WKHVH (A (A DQG (A ,I HLWKHU (A RU RFFXUV WKH SULVRQHU ZLOO GLH 7KH GHDWK ZLOO EH SURORQJHG DQG SDLQIXO LI (A RFFXUV VZLIW DQG SDLQOHVV LI ( RFFXUV ,I (A QR GHFD\ RFFXUV WKH SULVRQHU ZLOO EH UHOHDVHG

PAGE 247

6XSSRVH LQ WKH DFWXDO ZRUOG D FHUWDLQ SULVRQHU LV UHOHDVHG EHFDXVH RFFXUUHG (QWHUWDLQ WKH FRXQWHUIDFWXDO VXSSRVLWLRQ WKDW WKH SULVRQHU ZDV H[HFXWHG &HUWDLQO\ WKH ZRUOGV LQ ZKLFK WKH SULVRQHU GLHV D ORQJ DQG SDLQIXO GHDWK DUH VLJQLILFDQWO\ GLVVLPLODU SDUWLFXODUO\ IRU WKH SULVRQHUf WR WKRVH LQ ZKLFK WKH GHDWK LV SDLQOHVV DQG VZLIW
PAGE 248

IURP WKH SRLQW RI YLHZ RI WJ ZKDW ZLOO KDSSHQ VKRXOG D SDUWLFXODU EUDQFK EH UHDOL]HG 6RPHWKLQJ RI WKLV VRUW LV DSSDUHQWO\ SDUW RI WKH EDFNJURXQG WR 7KRPDVRQ DQG *XSWDn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nV ZKLFK DVVLJQV D KLJK SULRULW\ WR SUHVHUYLQJ ODZV RI QDWXUH ZRXOG QRW GR MXVWLFH WR WDLOOHVV NDQJDURRV 7KH SDWWHUQ IRU FRXQWHUIDFWXDO GHOLEHUDWLRQ WKDW VXJJHVW JHQHUDOO\ FRQIRUPV WR /HZLVn DFFRXQW ,Q FRQVLGHULQJ D JLYHQ FRXQWHUIDFWXDO RXU RUGHULQJ RI SRVVLEOH ZRUOGV DW WKH FRDUVHVW OHYHO LV D ZHDN WRWDO RUGHU 2IWHQ WKLV ZLOO VXIILFH $V ZH KDYH D PRUH SUHFLVHO\ GHWHUPLQHG DQWHFHGHQW WKLV RUGHU FDQ EH UHILQHG WR D ZHDN SDUWLDO RUGHU RU HYHQ D SDUWLDO RUGHU 7KH GHJUHH WR ZKLFK ZH UHILQH WKH RUGHU ZH LPSRVH ZLOO GHSHQG XSRQ WKH GHn JUHH WR ZKLFK ZH DUH FDSDEOH RI PDNLQJ WKH FRQGLWLRQDO SUHFLVH

PAGE 249

,Q FRQGLWLRQDO SUHGLFWLRQV WKDW IRUP SDUW RI D VFLHQWLILF H[SHULn PHQW D KLJK GHJUHH RI SUHFLVLRQ LV GHVLUDEOH ,Q VXSSRVLWLRQV DERXW WDLOOHVV NDQJDURRV RU IXWXUH 3UHVLGHQWV RU SDVW RQHV ZKR PLJKW QRW KDYH EHHQf D SUHFLVH RUGHU LV QHLWKHU GHVLUDEOH QRU SRVVLEOH &RPSDUDWLYH VLPLODULW\ FDQ EH UHOD[HG VXIILFLHQWO\ WR OHDYH URRP IRU ERWK NLQGV RI VLWXDWLRQV $Q\WKLQJ PRUH LV OLNHO\ D 3URFUXVWHDQ EHG IRU FRQGLWLRQDOV :KDW ZH KDYH LV OHVV EXW QRW IDLUO\ FRQVWUXHG DV PHUHO\ SULPLWLYH

PAGE 250

5()(5(1&(6 >@ ƒTYLVW /HQQDUW 0RGDO /RJLF ZLWK 6XEMXQFWLYH &RQGLWLRQDOV DQG 'LVSRVLWLRQDO 3UHGLFDWHV -RXUQDO RI 3KLORVRSKLFDO /RJLF f SS >@ %DUNHU -RKQ $ &DXVDWLRQ DQG &RXQWHUIDFWXDOV SUHSULQW SDSHU SUHVHQWHG DW WKH $3$ :HVWHUQ 'LYLVLRQ LQ $SULO f >@ %HQQHWW ) &RXQWHUIDFWXDOV DQG 3RVVLEOH :RUOGV &DQDGLDQ -RXUQDO RI 3KLORVRSK\ f SS >@ %HWK ( : 7KH )RXQGDWLRQV RI 0DWKHPDWLFV 1RUWK+ROODQG $PVWHUGDP >@ %LJHORZ -RKQ & ,I7KHQ 0HHWV WKH 3RVVLEOH :RUOGV 3KLORVRSKLD f SS >@ %RDV 0DULH 7KH (VWDEOLVKPHQW RI WKH 0HFKDQLFDO 3KLORVRSK\ 2VLULV f SS >@ %RGH -DPHV 5 7KH 3RVVLELOLW\ RI D &RQGLWLRQDO /RJLF 1RWUH 'DPH -RXUQDO RI )RUPDO /RJLF f SS >@ %UDGOH\ 5D\PRQG DQG 6ZDUW] 1RUPDQ 3RVVLEOH :RUOGV +DFNHWW ,QGLDQDSROLV >@ %XWFKHU 'DYLG 6XEMXQFWLYH &RQGLWLRQDO 0RGDO /RJLF 3K' WKHVLV 6WDQIRUG 8QLYHUVLW\ >@ &DUQDS 5XGROI 0HDQLQJ DQG 1HFHVVLW\ 8QLYHUVLW\ RI &KLFDJR 3UHVV &KLFDJR >@ &KHOODV %ULDQ ) %DVLF &RQGLWLRQDO /RJLF -RXUQDO RI 3KLORVRSKLFDO /RJLF f SS >@ &KHOODV %ULDQ ) DQG 0F.LQQH\ $XGUH\ 7KH &RPSOHWHQHVV RI 0RQRWRQLF 0RGDO /RJLFV =HLWVFKULIW IXU 0DWKHPDWLVFKH /RJLF XQG *UXQGODJHQ GHU 0DWKHPDWLF f SS >@ &KLVKROP 5RGHULFN 7KH &RQWUDU\WRIDFW &RQGLWLRQDO 0LQG f SS >@ /DZ 6WDWHPHQWV DQG &RXQWHUIDFWXDO ,QIHUHQFH $QDO\VLV f SS 5HSULQWHG LQ &DXVDWLRQ DQG &RQGLWLRQDOV >@

PAGE 251

>@ 7KHRU\ RI .QRZOHGJH QG HGLWLRQ 3UHQWLFH+DOO (QJOHZRRG &OLIIV 1>@ 'DYLV : $ ,QGLFDWLYH DQG 6XEMXQFWLYH &RQGLWLRQDOV 3KLORVRSKLFDO 5HYLHZ f SS >@ 'H:LWW %U\FH DQG *UDKDP 1HLOO HGV 7KH 0DQ\:RUOGV ,QWHUSUHWD WLRQ RI 4XDQWXP 0HFKDQLFV 3ULQFHWRQ 8QLYHUVLW\ 3UHVV 3ULQFHWRQ >@ (OOLV %ULDQ (SLVWHPLF )RXQGDWLRQV RI /RJLF -RXUQDO RI 3KLORVRSKLFDO /RJLF f SS >@ $ 8QLILHG 7KHRU\ RI &RQGLWLRQV -RXUQDO RI 3KLORVRSKLFDO /RJLF f SS >@ (OOLV %ULDQ -DFNVRQ )UDQN DQG 3DUJHWWHU 5REHUW $Q 2EMHFWLRQ WR 3RVVLEOH:RUOGV 6HPDQWLFV IRU &RXQWHUIDFWXDO /RJLFV -RXUQDO RI 3KLORVRSKLFDO /RJLF f SS >@ )LQH .LW 0RGHO 7KHRU\ IRU 0RGDO /RJLF 3DUW WKH 'H 5H'H 'LFWR 'LVWLQFWLRQ -RXUQDO RI 3KLORVRSKLFDO /RJLF f SS >@ )RXOLV DQG 5DQGDOO & + 2SHUDWLRQDO 6WDWLVWLFV -RXUQDO RI 0DWKHPDWLFDO 3K\VLFV f SS >@ 2SHUDWLRQDO 6WDWLVWLFV ,, -RXUQDO RI 0DWKHPDWLFDO 3K\VLFV f SS >@ )XPHUWRQ 5 $ 6XEMXQFWLYH &RQGLWLRQDOV 3KLORVRSK\ RI 6FLHQFH f SS >@ *HUVRQ 0DUWLQ 7KH ,QDGHTXDF\ RI 1HLJKERUKRRG 6HPDQWLFV IRU 0RGDO /RJLF -RXUQDO RI 6\PEROLF /RJLF f SS >@ *RRGPDQ 1HOVRQ 7KH 3UREOHP RI &RXQWHUIDFWXDO &RQGLWLRQDOV -RXUQDO RI 3KLORVRSK\ f SS 5HSULQWHG LQ )DFW )LFWLRQ DQG )RUHFDVW >@ >@ )DFW )LFWLRQ DQG )RUHFDVW UG HGLWLRQ +DFNHWW ,QGLDQDSROLV >@ *RRVHQV : &DXVDO &KDLQV DQG &RXQWHUIDFWXDOV -RXUQDO RI 3KLORVRSK\ f SS HUUDWXP >@ +DDFN 6XVDQ 5HFHQW 3XEOLFDWLRQV LQ /RJLF 3KLORVRSK\ f SS A >@ +DFNLQJ ,DQ :KDW LV 6WULFW ,PSOLFDWLRQ" -RXUQDO RI 6\PEROLF /RJLF f SS

PAGE 252

>@ :KDW LV /RJLF" -RXUQDO RI 3KLORVRSK\ f SS >@ +DUGHJUHH *DU\ 0 6WDOQDNHU &RQGLWLRQDOV DQG 4XDQWXP /RJLF -RXUQDO RI 3KLORVRSKLFDO /RJLF f SS >@ +D]HQ $OOHQ DQG 6ORWH 0LFKDHO (YHQ LI $QDO\VLV f SS >@ +HU]EHUJHU +DQV &RXQWHUIDFWXDOV DQG &RQVLVWHQF\ -RXUQDO RI 3KLORVRSK\ f SS >@ +RQGHULFN 7HG &DXVHV DQG &DXVDO &LUFXPVWDQFHV DV 1HFHVVLWDWLQJ 3URFHHGLQJV RI WKH $ULVWRWOHDQ 6RFLHW\ f SS >@ -DFNVRQ )UDQN $ &DXVDO 7KHRU\ RI &RXQWHUI DFWXDOV $XVWUDODVLDQ -RXUQDO RI 3KLORVRSK\ f SS >@ .LP -DHJZRQ &DXVHV DQG &RXQWHUIDFWXDOV -RXUQDO RI 3KLORVRSK\ f SS >@ .QHDOH :LOOLDP 1DWXUDO /DZV DQG &RQWUDU\WRIDFW &RQGLWLRQDOV $QDO\VLV f SS >@ .UDEEH (ULN 1RWH RQ D &RPSOHWHQHVV 7KHRUHP LQ WKH 7KHRU\ RI &RXQWHUIDFWXDOV -RXUQDO RI 3KLORVRSKLFDO /RJLF f SS >@ .ULSNH 6DXO 6HPDQWLFDO $QDO\VLV RI 0RGDO /RJLF 1RUPDO 3URSRVLWLRQDO &DOFXOL =HLWVFKULIW IXU 0DWKHPDWLVFKH /RJLF XQG *UXQGODJHQ GHU 0DWKHPDWLF f SS >@ 6HPDQWLFDO $QDO\VLV RI 0RGDO /RJLF ,, 1RQQRUPDO 3URSRVLWLRQDO &DOFXOL LQ : $GGLVRQ / +HQNLQ DQG $ 7DUVNL HGLWRUV 7KH 7KHRU\ RI 0RGHOV 1RUWK+ROODQG $PVWHUGDP SS >@ .XKQ 7KRPDV 6 7KH 6WUXFWXUH RI 6FLHQWLILF 5HYROXWLRQV QG HGLWLRQ 8QLYHUVLW\ RI &KLFDJR 3UHVV &KLFDJR >@ /DXPHU .HLWK :RUOGV RI WKH ,PSHULXP %HUNHOH\ 1HZ @ OHWPDQQ 6FRWW $ *HQHUDO 3URSRVLWLRQDO /RJLF RI &RQGLWLRQDOV 1RWUH 'DPH -RXUQDO RI )RUPDO /RJLF f SS >@ /HQQRQ ( $OJHEUDLF 6HPDQWLFV IRU 0RGDO /RJLFV -RXUQDO RI 6\PEROLF /RJLF f SS >@ $OJHEUDLF 6HPDQWLFV IRU 0RGDO /RJLFV ,, -RXUQDO RI 6\PEROLF /RJLF f SS >@ /HZLV & 7KH 0RGHV RI 0HDQLQJ 3KLORVRSKLFDO DQG 3KHQRPHQR ORJLFDO 5HVHDUFK f SS

PAGE 253

>@ /HZLV & DQG /DQJIRUG & + 6\PEROLF /RJLF QG HGLWLRQ 'RYHU 1HZ @ /HZLV 'DYLG *HQHUDO 6HPDQWLFV 6\QWKHVH f SS >@ &RXQWHUIDFWXDOV DQG &RPSDUDWLYH 3RVVLELOLW\ -RXUQDO RI 3KLORVRSKLFDO /RJLF f SS >@ &RXQWHUIDFWXDOV +DUYDUG &DPEULGJH >@ &DXVDWLRQ -RXUQDO RI 3KLORVRSK\ f SS 5HSULQWHG LQ &DXVDWLRQ DQG &RQGLWLRQDOV >@ >@ 3RVVLEOH:RUOGV 6HPDQWLFV IRU &RXQWHUIDFWXDO /RJLFV $ 5HMRLQGHU -RXUQDO RI 3KLORVRSKLFDO /RJLF f SS >@ &RQYHUVDWLRQDO 6FRUH -RXUQDO RI 3KLORVRSKLFDO /RJLF WR DSSHDU >@ /RHE /RXLV ( &DXVDO 2YHUGHWHUPLQLVP DQG &RXQWHUIDFWXDOV 5HYLVLWHG 3KLORVRSKLFDO 6WXGLHV f SS >@ /RHZHU %DUU\ &RXQWHUIDFWXDOV ZLWK 'LVMXQFWLYH $QWHFHGHQWV -RXUQDO RI 3KLORVRSK\ f SS >@ &RWHQDELOLW\ DQG &RXQWHUIDFWXDO /RJLF -RXUQDO RI 3KLORVRSKLFDO /RJLF f SS >@ /\RQ $UGRRQ 7KH ,QPXWDEOH /DZV RI 1DWXUH 3URFHHGLQJV RI WKH $ULVWRWOHDQ 6RFLHW\ f SS >@ 0DFNHQ]LH 1ROODLJ $QDO\]LQJ ZLWK 6XEMXQFWLYHV 'LDORJXH &DQDGDf f SS >@ 0DFNLH -RKQ / &RXQWHUIDFWXDOV DQG &DXVDO /DZV LQ 5 %XWOHU $QDO\WLFDO 3KLORVRSK\ %ODFNZHOO 2[IRUG SS >@ &DLDVHV DQG &RQGLWLRQV $PHULFDQ 3KLORVRSKLFDO 4XDUWHUO\ f SS 5HSULQWHG LQ &DXVDWLRQ DQG &RQGLWLRQDOV >@ >@ 7KH &HPHQW RI WKH 8QLYHUVH 2[IRUG 8QLYHUVLW\ 3UHVV 2[IRUG >@ 0F.D\ 7KRPDV DQG 9DQ ,QZDJHQ 3HWHU &RXQWHUIDFWXDOV ZLWK 'LVn MXQFWLYH $QWHFHGHQWV 3KLORVRSKLFDO 6WXGLHV f SS >@ 0HUULOO + )RUPDOL]DWLRQ 3RVVLEOH:RUOGV DQG WKH )RXQGDWLRQV RI 0RGDO ORJLF (UNHQQWQLV f SS >@ 0RQWDJXH 5LFKDUG 3UDJPDWLVP LQ 5 .OLEDUVN\ HGLWRU &RQWHPSRUDU\ 3KLORVRSK\ /RJLF DQG )RXQGDWLRQV RI 0DWKHPDWLFV /D 1XRYD ,WDOLH )ORUHQFH

PAGE 254

>@ 1DJHO (UQVW 7KH 6WUXFWXUH RI 6FLHQFH +DUFRXUW %UDFH DQG :RUOG 1HZ @ 1XWH 'RQDOG &RXQWHUIDFWXDOV 1RWUH 'DPH -RXUQDO RI )RUPDO /RJLF f SS >@ &RXQWHUIDFWXDOV DQG WKH 6LPLODULW\ RI :RUOGV -RXUQDO RI 3KLORVRSK\ f SS >@ &RQGLWLRQDO /RJLF 3RWSRXUUL 5HOHYDQFH /RJLF 1HZV f SS >@ 7KH /RJLF RI &DXVDO &RQGLWLRQDOV RI 8QLYHUVDO 6WUHQJWK 5HOHYDQFH /RJLF 1HZV f SS >@ 'DYLG /HZLV DQG WKH $QDO\VLV RI &RXQW HUI DFWXDOV 1RXV f SS >@ $Q ,QFRPSOHWHQHVV 7KHRUHP IRU &RQGLWLRQDO /RJLF 1RWUH 'DPH -RXUQDO RI )RUPDO /RJLF f SS >@ 6LPSOLILFDWLRQ DQG 6XEVWLWXWLRQ RI &RXQWHUIDFWXDO $QWHn FHGHQWV 3KLORVRJKLD ,VUDHOf f SS >@ 6FLHQWLILF /DZ DQG 1RPRORJLFDO &RQGLWLRQDOV 16) 7HFKQLFDO 5HSRUW 62& f >@ &RQYHUVDWLRQDO 6FRUH DQG &RQGLWLRQDOV -RXUQDO RI 3KLORVRSKLFDO /RJLF WR DSSHDU >@ 7RSLFV LQ &RQGLWLRQDOV 5HLGHO %RVWRQ >@ 3LSHU + %HDP /RUG .DOYDQ RI 2WKHUZKHQ $FH 1HZ @ 3ROORFN -RKQ / .QRZOHGJH DQG -XVWLILFDWLRQ 3ULQFHWRQ 8QLYHUVLW\ 3UHVV 3ULQFHWRQ >@ 7KH n3RVVLEOH:RUOGVn $QDO\VLV RI &RXQWHUIDFWXDOV 3KLORVRSKLFDO 6WXGLHV f SS >@ 6XEMXQFWLYH 5HDVRQLQJ 5HLGHO %RVWRQ >@ 3ULRU $ 1 7LPH DQG 0RGDOLW\ &ODUHQGRQ 2[IRUG >@ 3DVW 3UHVHQW DQG )XWXUH &ODUHQGRQ 2[IRUG >@ 5HVFKHU 1LFKRODV %HOLHI&RQWUDYHQLQJ 6XSSRVLWLRQV DQG WKH 3UREOHP RI &RQWUDU\WRIDFW &RQGLWLRQDOV 3KLORVRSKLFDO 5HYLHZ f SS 5HSULQWHG LQ &DXVDWLRQ DQG &RQGLWLRQDOV >@ >@ 5REHUWV 'RQ 7KH ([LVWHQWLDO *UDSKV RI & 6 3HLUFH $SSURDFKHV WR 6HPLRWLFV 0RXWRQ 7KH +DJXH

PAGE 255

>@ >@ 6DEHUKDJHQ )UHG 0DVN RI WKH 6XQ $FH 1HZ @ 6FKQHLGHU (PD 5HFHQW 'LVFXVVLRQ RI 6XEMXQFWLYH &RQGLWLRQDOV 5HYLHZ RI 0HWDSK\VLFV f SS >@ 6FRWW 'DQD $GYLFH RQ 0RGDO /RJLF LQ /DPEHUW 3KLORVRSKLFDO 3UREOHPV LQ /RJLF 5HLGHO 'RUGUHFKW >@ 6HJHUEHUJ .ULVWHU 'HFLGDELOLW\ RI 6 7KHRULD f SS >@ 'HFLGDELOLW\ RI )RXU 0RGDO /RJLFV 7KHRULD f SS >@ $Q (VVD\ LQ &ODVVLFDO 0RGDO /RJLF 8QLYHUVLW\ RI 8SSVDOD 8SSVDOD 6ZHGHQ >@ 6KRUWHU 0 &DXVDOLW\ DQG D 0HWKRG RI $QDOD\VLV LQ 5 %XWOHU HGLWRU $QDO\WLFDO 3KLORVRSK\ QG VHULHV %DVLO %ODFNZHOO 2[IRUG >@ 6N\UPV %ULDQ 3RVVLEOH :RUOGV 3K\VLFV DQG 0HWDSK\VLFV 3KLORVRSKLFDO 6WXGLHV f SS >@ 6ORWH 0LFKDHO $ 7LPH LQ &RXQWHUIDFWXDOV 3KLORVRSKLFDO 5HYLHZ f SS >@ 6RVD (UQHVW HG &DXVDWLRQ DQG &RQGLWLRQDOV 8QLYHUVLW\ 3UHVV 2[IRUG >@ 6WDOQDNHU 5REHUW & $ 7KHRU\ RI &RQGLWLRQDOV IURP 6WXGLHV LQ /RJLFDO 7KHRU\ $PHULFDQ 3KLORVRSKLFDO 4XDUWHUO\ 0RQRJUDSK 6HULHV HGLWRU 1 5HVFKHU %ODFNZHOO 2[IRUG SS 5HSULQWHG LQ &DXVDWLRQ DQG &RQGLWLRQDOV >@ >@ 6WDOQDNHU 5REHUW & DQG 7KRPDVRQ 5LFKPRQG + $ 6HPDQWLF $QDO\VLV RI &RQGLWLRQDO /RJLF 7KHRULD f SS >@ 6ZDLQ 0LFKDHO $ &RXQWHUIDFWXDO $QDO\VLV RI (YHQW &DXVDWLRQ 3KLORVRSKLFDO 6WXGLHV f SS >@ 7HPSOH 'HQQLV 1RPLF 1HFHVVLW\ DQG &RXQWHUIDFWXDO )RUFH $PHULFDQ 3KLORVRSKLFDO 4XDUWHUO\ f SS >@7KRPDVRQ 5 DQG *XSWD $ $ 7KHRU\ RI &RQGLWLRQDOV LQ WKH &RQWH[W RI %UDQFKLQJ 7LPH 3KLORVRSKLFDO 5HYLHZ f SS >@9HQGOHU =HQR &DXVDO 5HODWLRQV -RXUQDO RI 3KLORVRSK\ f SS

PAGE 256

>@ :DVVHUPDQ +RZDUG & $Q $QDO\VLV RI WKH &EXQWHUIDFWXDO &RQGLn WLRQDO 1RWUH 'DPH -RXUQDO RI )RUPDO /RJLF f SS >@ @ =HPDQ -D\ 0RGDO /RJLF &ODUHQGRQ 2[IRUG >@ 3LHUFHnV /RJLFDO *UDSKV 6HPLWLFD f SS >@ 2UWKRPRGXODU /RJLF FKDSWHUV DQG SUHSULQW >@ *HQHUDOL]HG 1RUPDO /RJLF -RXUQDO RI 3KLORVRSKLFDO /RJLF f SS >@ 1RUPDO ,PSOLFDWLRQV %RXQGHG 3RVHWV DQG WKH ([LVWHQFH RI 0HHWV 1RWUH 'DPH -RXUQDO RI )RUPDO /RJLF f SS >@ 7ZR %DVLF 3XUH,PSOLFDWLRQDO 6\VWHPV 1RWUH 'DPH -RXUQDO RI )RUPDO /RJLF f SS >@ 1RUPDO 6DVDNL DQG &ODVVLFDO ,PSOLFDWLRQV -RXUQDO RI 3KLORVRSKLFDO /RJLF f SS

PAGE 257

%,2*5$3+,&$/ 6.(7&+ -RKQ &O\GH 0D\HU ERP 0D\ LQ .LQJVWRQ 3HQQV\OYDQLD UHFHLYHG KLV %$ GHJUHH LQ IURP 5DQGROSK0DFRQ &ROOHJH $VKODQG 9LUJLQLD $IWHU RQH \HDU RI JUDGXDWH ZRUN LQ SKLORVRSK\ DW
PAGE 258

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t 3URIHVVRU RI &RPSWWHU DQG ,QIRUPDWLRQ 6FLHQFHV DQG 0DWKHPDWLFV 7KLV GLVVHUWDWLRQ ZDV VXEPLWWHG WR WKH *UDGXDWH )DFXOW\ RI WKH 'HSDUWPHQW RI 3KLORVRSK\ LQ WKH &ROOHJH RI /LEHUDO $UWV DQG 6FLHQFHV DQG WR WKH *UDGXDWH &RXQFLO DQG ZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHn PHQWV IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ $XJXVW 'HDQ *UDGXDWH 6FKRRO


120
analogy or no del. That the motion of molecules has properties that the
motion of billiard balls does not have is learned in the context of the
new theory after the bridging analogy has been made.
Loewer [57, p. 113] charges that either comparative similarity is
not well enough understood to support an analysis of counterfactuals or
else it appears in Lewis' account as a primitive technical concept and
so cannot be antecedently understood. He charges that Lewis inconsis
tently seems to want to have it both ways: comparative similarity is
antecedently understood and also is a new technical concept, primitive
to Lewis' theory. The grounds for this charge are that on the one hand,
Lewis appeals to our antecedent understanding of comparative similarity
in ordinary situations to argue, for example, for centering on the grounds
that nothing nothing is more similar to a given thing than the given thing
is to itself, while on the other hand, he argues for the special principle
that similarity before the event counts for more than similarity after,
which surely has no application to cities or faces, and thus appears only
in this particular technical application of comparative similarity. If
comparative similarity is primitive, Lewis can simply stipulate the lat
ter, but then what he offers is not an analysis. However, there would ap
pear to be no ordinary applications of comparative similarity that would
suggest the need for the special principles involved in applying it to
possible worlds, thus these special principles cannot be antecedently
understood.
On the contrary, I am suggesting that Lewis can have it both ways
without inconsistency. The concept of comparative overall similarity
can be introduced through analogy to its antecedently understood appli
cations. To that extent it is not simply a prirmitive technical concept.


41
mask computes the possibilities based upon a comprehensive access to
facts about the present. This is reminiscent of Stalnaker's claim
that "one can sometimes have evidence about non-actual situations"
[96, p. 166]. Such evidence is acquired from the actual situation
in non-mysterious ways [96, pp. 178-179].
Both the actual and the speculative history of possible worlds
have something to offer us. They give us a variety of analogies
with which to test our grasp of the concept of possible worlds.
2.2 Possible Worlds: Modality and the Strict Conditional
The naive concept of a possible world seems natural and obvious:
we all understand what is meant by saying "things could be otherwise."
If the actual world is the way things are, then a possible world is
another way things could have been. We can think of possible worlds
as variants on the actual world. A critic might suggest that intro
ducing possible worlds, when we have enough difficulty determining
what the actual world is, is to compound our problems to no purpose.
Our problems, however, are already compounded: the difficulty in
determining what the actual world is lies in the fact that the extent
of our knowledge and belief (or true belief) seriously underdeter
mines it.
The critic would express this under determination by saying we
do not or cannot know everything about the actual world, while I would
express it by saying we do not or cannot know which world is actual,
except within certain limits. While the critic can only say that our
knowledge underdetermines the world that is, I can make sense of a
positive assertion as to what it does determine: the set of worlds
that, for all we know, any one of which might be the actual world. If


REFERENCES
[1] qvist, Lennart, "Modal Logic with Subjunctive Conditionals and
Dispositional Predicates," Journal of Philosophical Logic 2 (1973),
pp. 1-76.
[2] Barker, John A., "Causation and Counterfactuals," preprint (paper
presented at the APA Western Division in April, 1980).
[3] Bennett, J. F., "Counterfactuals and Possible Worlds," Canadian
Journal of Philosophy 5 (1974), pp. 381-402.
[4] Beth, E. W., The Foundations of Mathematics, North-Holland,
Amsterdam, 1965.
[5] Bigelow, John C., "If-Then Meets the Possible Worlds," Philosophia
6 (1976), pp. 215-235.
[6] Boas, Marie, "The Establishment of the Mechanical Philosophy,"
Osiris 10 (1952), pp. 412-541.
[7] Bode, James R., "The Possibility of a Conditional Logic," Notre Dame
Journal of Formal Logic 20 (1979), pp. 147-154.
[8] Bradley, Raymond and Swartz, Norman, Possible Worlds, Hackett,
Indianapolis, 1979.
[9] Butcher, David 0., "Subjunctive Conditional Modal Logic," Ph.D.
thesis, Stanford University, 1979.
[10] Carnap, Rudolf, Meaning and Necessity, University of Chicago Press,
Chicago, 1947.
[11] Chellas, Brian F., "Basic Conditional Logic," Journal of
Philosophical Logic 4 (1975), pp. 133-153.
[12] Chellas, Brian F. and McKinney, Audrey, "The Completeness of
Monotonic Modal Logics," Zeitschrift fur Mathematische Logic und
Grundlagen der Mathematic 21 (1975), pp. 379-383.
[13] Chisholm, Roderick, "The Contrary-to-fact Conditional," Mind 55
(1946), pp. 289-307.
[14] "Law Statements and Counterfactual Inference," Analysis
15 (1955), pp. 97-105. Reprinted in Causation and Conditionals [67].
243


97
However, VCP is not even generalizable to the n = 2 level, as Figure
2.6.2 is a countermodel to the following argument, involving a set of
just two "might"-consequences.
E2.6.7: Vpq
Vpr
LCKqrs
. Vps
Figure 2.6.2
Whether a generalized consequence principle should be valid or not
depends upon the type of conditional in question. It has yet to be shown
that the counterfactual conditional, i.e., the "would"-conditional is of
such a type. Clearly the "might"-conditional is not. This seems to be
connected to the fact that the "might"-conditional is closely associated
with the notion of possibility, while the ''would''-conditional is usually
seen as a case of necessity. While everything that might be the case
should such-and-such be true is not necessarily, or even likely consistent,
just as the combination of everything that is possible is not consistent,
we may feel that the combination of everthing that is necessary being


75
E2.5.6:
Wpq If the U. S. were to threaten to cut off wheat sales
to OPEC, then OPEC would not embargo oil sales to
the U. S.
. WNqNp
. If OPEC were to embargo oil sales to the U. S.,
then the U. S. wDuld not threaten to cut off
wheat sales to OPEC.
One could argue that the threat to cut off wheat sales becomes even more
likely if an oil embargo goes through, so the premise could be true,
while the conclusion is false. Figure 2.5.3 is a countermodel to this
inference pattern. Once should note that though contraposition fails,
modus tollens does not, as Wpq entails Cpq and modus toliens is valid
for the latter.
Figure 2.5.3
The question arises as to why hypothetical syllogism should ever
have been thought to be valid for ordinary language conditionals? The
reason is simple: it often is. And it often is under conditions which
are weaker than those of SCE, but stronger than those of RET. The


130
account for situations satisfying the limit assumption. While it seems
that Nute is informally arguing for a more liberal sense of similarity
than he supposes Lewis to have, his arguments do contain an insight which
he never quite directly expresses: considering counterfactual situations
from the point of view of similarity reverses the description of counter-
factual deliberation in our informal model.
In our informal model of counterfactual deliberation we were led
to consider hypothetical situations that result from changes made in the
actual situation to accommodate the counterfactual supposition. Stalnaker
suggested that there was always some single minimal change that could be
made, and therefore there was a world in which the antecedent was true
least different from (so most similar to) the actual world. Lewis sug
gested that perhaps no changes were minimal, but even if there were, more
than one change might be minimal, but these would all result in worlds of
equal similarity to the actual world. The issue raised by Nute is that
the making of different changes may result in worlds which differ in
their similarity to the actual world, so that a minimal change order may
not be the same as a comparative similarity order.
This notion of different, but in some sense equally small, changes
is central to Pollock's account in [80] of counterfactuals. In fact
Pollock specifically claims that the order based upon minimal change is
a partial, rather than a total, order [80, p. 22], Two different changes
in the actual world may be such that neither contains a smaller change,
yet neither contains the other. Such an order is not connected.
Pollock's argument for the limit assumption, discussed in Section
2.6, is ultimately aimed at discrediting comparative similarity in favor
of minimal change as the operative principle in counterfactual evaluation.


177
Then F = < > is a canonical (propositional neighborhood) frame
for L, and M = < 1^,1^,^ > is a canonical (propositional neighborhood)
model for L.
That the definition of is unambiguous with respect to repre
sentatives of sets of maximally L-consistent sets follows from the fact
that L is classical. From this the fundamental theorem for classical
conditional logics follows readily. (Compare Chellas [11, pp. 139-140,
149nl2].)
|b|L = 1d|L then |b|L e N(u,|a|L) iff |d|L e N(u,|c|L).
Proof: (As before, we drop whatever subscripts we can.)
Assume |a| = |c| and |b| = |d|. Then by the properties of maximally L-
consistent sets, f- Eac and f- Ebd. So by RCEA we have f- EWabWcb and by
RCEC we have ( EWcbWcd. So by closure under MP we have f- EWabWcd. Then
by the properties of maximally L-consistent sets we have Wab e u iff Wed e
u for all u e U. Then by the definition of N we have |b| e N(u, |a|) iff
|d| e N(u,|c|). QED
T4.5.1: Let be a canonical propositional neighborhood model for classi
cal conditional logic L. Then for all formulas a and all worlds u e U:
That is, ||a||^L = |a|^ for all formulas a.
Proof: We do the induction only for formula a of the form Wbc. So assume
the theorem holds for all worlds and formulas b and c. By the truth defini
tion, |= Wbc iff ||b|| e N(u, ||a||). Since by hypothesis ||b|| = |b| and
||a11 = |a|, we have |= Wbc iff |b| e N(u, |c|). Hence by the definition
of N and P4.5.1 we have |= Wbc iff Wbc e u. QED
P4.5.1: in D4.5.10 is unambiguous. That is, if |a|


69
We may divide our concerns into those inference patterns that ought
to be valid for the counterfactual conditional and those which ought not
to be valid. Among the first will be those patterns we would expect any
conditional to adhere to. Many authors (including Zeman [104], Hardegree
[32]) consider two requirements absolutely minimal for a conditional
function:
C2.5.1: If p[= q then Wpq is true.
C2.5.2: If Wpq and p are true, then q is true.
The first is a simplified semantic version of the deduction theorem, and
the second is modus ponens or detachment. Both are satisfied by the
material conditional as well as the strict conditional (as we have defined
it). But they also hold for the counterfactual conditional. From p|= q
we have observed that LCpq follows. But from LCpq it follows that Wpq,
as we noted in Section 2.4. Hence C2.5.1 holds for the counterfactual.
Since Cpq follows from Wpq, also noted in Section 2.4, and modus ponens
holds for the material conditional, C2.5.2 holds as well for the counter-
factual .
Lewis draws particular attention to three inference patterns valid
for both the material and the strict conditional which are not valid for
the counterfactual conditionals on his semantics [51, pp. 31-35]; and
demonstrably ought not to be valid for conditionals in general. These
are strengthening the antecedent, hypothetical syllogism (transitivity
of implication), and contraposition. The first two are immediate corol
laries to a stronger principle identified by J. Jay Zeman, that of semi-
substiutivity of implication (respectively, strict implication) which is
valid for the material (respectively, strict) conditional [104, pp. 11,


138
Let and denote two logics (systems). Then contains
provided every theorem of is a theorem of L-^. Under the same con
ditions we also say is an extension of L^. The containment is proper
provided there is a theorem of L-^ that is not a theorem of Every
system which we shall consider is an extension of classical propositional
calculus (PC). We can guarantee this by providing our systems at a mini
mum with the following axioms and rules of inference:
DA.1.5:
Axioms:
Al: CpCqp
A2: CCpCqrCCpqCpr
A3: CCCpoop
Rules of inference:
MP: Modus Ponens: From a and Cab infer b.
US: Uniform Substitution: From a infer a(b), where a(b) is the
result of uniformly substituting formula b for every occurrence
of any propositional letter in a.
This is one of the standard axiomatizations of PC (Segerberg [91],
Zeman [104]).
Our minimal logic will be the deductive closure of CLW under the
above axioms and rules of inference, and hence will contain PC. In vir
tue of this we will freely make use of various standard results for PC.
All truth-functional tautologies are in our minimal logic, and through
MP and US all substitution instances of them in CLW. The presence of
L and W are obviously rather inessential, so we will usually refer to
this system as PC.


82
As Loewer point out [56, p. 535], the conditions under which a
counterfactual with a disjunctive antecedent is uttered are usually such
that we would be prepared to defend either SDA conjunct; our intention
is to make a more inclusive statement than either counter factual with
single antecedent alone. The following examples and accompanying diagrams
will illustrate my point.
E2.5.11:
(a) WApqr If Prof. X were to work less or were under
less pressure to publish, she would be
. less tense.
. KWprWpq
. If Prof. X were to work less, she would
be less tense, and, if she were under
less pressure to publish, she would
be less tense.
(b) WApsr If Prof. X were to work less or to die, she
would be less tense.
. KWprWsr
. If Prof. X were to work less, she would
be less tense, and, if she were to
die, she would be less tense.
Figure 2.5.7


79
E2.5.8:
Wqr
If Prof. X were to work less, she would be less tense.
Wpq
If Prof. X were to die, she would work less.
. Wpr
If Prof. X were to die, she would be less tense.
If these conditionals were to be uttered in the order given, it WDuld be
to invite the same surprised response, and for the same reason: the
basis for evaluation has been unexpectedly altered. In such a case I
think we would be correct in charging that the premises offer no support
for the conclusion, though the conclusion happens to be true anyway.
An inference is semantically valid if it is impossible for the
premises to be true and the conclusion false. Correspondingly, an in
ference is pragmatically valid if it is impossible for the premises to
be true with respect to the same sphere, and the conclusion false.
E2.5.7(a), unlike either E2.5.7(b) or E2.5.8, is pragmatically valid.
Adopting such a view of hypothetical syllogism, and hence of transitivity
of counterfactual implication, allows us to make sense both of those
instances where it seems to hold, and those where it certainly fails.
A similar situation arises in the case of the inference pattern
simplification of disjunctive antecedents (SDA). A considerable litera
ture has arisen in recent papers on the topic of the counterfactual con
ditional with respect to this inference alone [20, 53, 56, 63, 68, 73,
74, 76]. The inference pattern, an example, and the appropriate diagram
follow (adapted from Nute [68]):


88
we could hardly defend the supposition. In fact, our best defense is
that both p and Nq are possible simultaneously. So, of course p is pos
sible, and hence not impossible. The following chain of inferences sum
marizes the above argument:
E2.5.13: NWpq
. VpNq
. MKpq
. NLNp
E2.5.13 is semantically valid on the analysis we have given, as I believe
it should be, but it is also pragmatically valid, intuitively, and in
terms of our definition. But we cannot hold it semantically valid with
out holding E2.5.12(d) valid as well, since it is the contrapositive of
the above argument.
There are two apparent oddities about pragmatic validity as we have
defined it: first, semantic validity does not guarantee pragmatic valid
ity, as witness E2.5.12(d); second, the contrapositive argument to a
pragmatically valid one may not itself be pragmatically valid (same ex
ample). These situations depend upon the presence of clause D2.5.2(3),
so may be avoided if that clause is dropped. I am reluctant to call any
argument "pragmatically valid" when it contains vacuously true counter-
factuals, so am willing to put up with these oddities. I am influenced
perhaps by my feeling that though the inference from "All unicorns are
furry" to "It is not the case that some unicorns are bald" is semantically
valid, it is decidedly odd, in view of the nonexistence of unicorns.


136
P4.1.2: The set of formulas (well-formed-formulas, wffs) of CLW is the
set closed under the following rules of compounding of the primitive
symbols:
(a) Every propositional letter is a formula.
(b) o is a formula.
(c) If a and b are formulas, then Cab is a formula.
(d) If a and b are formulas, then Wab is a formula.
(e) If a is a formula, then La is a formula.
P4.1.3: The defined symbols of CLW are those for the constant true pro
position (1), negation (N), conjunction (K), disjunction (A), material
equivalence (E), counterfactual equivalence (F), and possibility (M):
(a)
1
= df
Coo
(b)
Na
= df
Cao
(c)
Kab
= df
NCaNb
(d)
Aab
= df
CNab
(e)
Eab
= df
KCabCba
(f)
Fab
= df
KWabWba
(g)
Ma
= df
NLNa
Some clauses of P4.1.3 define one defined symbol in terms of another.
These can obviously be expanded by other clauses to definitions in terms
of the primitive symbols alone. For those with persnickity formal con
sciences, the references to "conjunction" and the like may be taken as
the giving of names to the operators, though they do reflect the intended
interpretation. As we go on we will want to add several clauses to
D4.1.3, e.g., for the "might"-conditional. We wall let CL designate
the subset of CLW obtained by omitting D4.1.1(e) (and so D4.1.2(d) and
D4.1.3(f) and CW the subset of CLW obtained by emitting D4.1.1(d) (and


169
E4.4.4:
(a) CM = CE + CM
(b) CR = CE + CM + CR
(c) CK = CE + CM + CR + CN
Conpare this to the similar bases for modal logics in E4.1.6.
In Section 4.1 we indicated an alternative basis for the smallest
normal logic, K, in terms of adding axiom K and rule RN to PC. Axiom CK
is the conditional analog of axiom K. We prove the conditional analogs
of L4.1.9-11 below. The analogs of Segerberg's results about the non-
derivability of K in E + M and E + R and of M and R in E + K will require
the development of the semantics for conditional logics in Section 4.5.
LA.4.10: CK is derivable in Ce + CM + CR.
Proof: Assume WpCqr and Wpq. We will show that Wpr follows, hence
CWpCqrCWpqWpr will be a theorem. Conjoining our assumptions we have
KWpCqrWpq. An instance of CR is CKWpCqrWpqWpKCqrq, so by MP we have
WpKCqrq. Now CKCqrqr is a PC theorem, and so by RCM, which is derivable
in CE + CM, CWpCKqrqWpr follows. Hence by MP we have Wpr. QED
L4.4.11: RCM is derivable in PC + CK + RCN.
Proof: Assume Cab. By RCN, WcCab follows. An instance of CK is
CWcCabWcaWcb, hence by MP we have CWcaWcb. QED
L4.4.12: CR is derivable in PC + CK + RCN.
Proof: Assume KWpqWpr. GqCrKqr is a PC theorem. By L4.4.11, RCM balds,
so we have CWpqWpCrKqr. Hence by simplification of our assumption and
MP, we have WpCrKqr. But CWpCqKqrCWpqWpKqr is an instance of CK, so by
MP we have CWpqWpKqr. Thus by simplification and MP, we have WpKqr. QED
L4.4.10-12 permit us to state another basis for CK, noted in
Chellas [11, p. 149n7]:


These families are classified in terms of two dimensions: one of in
creasing materiality of the conditional connective, and the other of
increasing strength in the comparison of possible worlds implicit in
any semantics for the logics. Mach of this work is a continuation of
that of Brian Chellas and Donald Nute.
Comparative order semantics, a generalization of Lewis' comparative
similarity semantics, is developed. In comparative order semantics, pos
sible worlds are ordered relative to each world as a basis for comparison.
The smallest logic, CP, in which the order relative to each world is a
partial order, is identified. A number of logics of the counterfactual
conditional that have been suggested contain CP, though in some cases the
results presented in this essay are surprising, and at variance with what
has been claimed elsewhere. In particular, a family of logics is identified
which lies between previously identified logics whose comparative order
semantics are partially ordered (such as system SS of John Pollock), and
certain of the V-logics of Lewis (whose comparative order semantics are
a weak total order, relative to each base world). The smallest member
of this intervening family is characterized by a semiconnected partial
order.
The view that some version of comparative similarity ordering is
capable of supporting an analysis of the counterfactual conditional is
defended.
vii


205
maximallity, NWaNc e u. That is, Vac e u, and so KWabVac e u. Hence by
CV, WKacb e u. Then f(u, |Kac|) c |b|. QED
T4.8.13: CA = CP + MP is determined by the class of additive frames.
(L4.8.10.)
Proof: See T4.8.11 and T4.8.5.
T4.8.14: VW = V + MP is determined by the class of variably strict weakly
material frames. (L4.8.11.)
Proof: See T4.8.12 and T4.8.5.
T4.8.15: SS = CA + CC is determined by the class of additive material
frames. (L4.8.12.)
Proof: See T4.8.13 and T4.8.7.
T4.8.16: VC = VW + CC is determined by the class of variably strict
material frames. (L4.8.13.)
Proof: See T4.8.14 and T4.8.7.
T4.8.17: C2 = SS + CEM is determined by the class of singular frames.
(14.8.14.)
Proof: The proof requires showing that CEM leads to (cem) being satisfied.
Such a proof can be constructed.
Alternative axiomatizations of the extensions of CK can be found.
The equivalence of certain sets of conditions yielded by L4.8.15 L4.8.21
reveals several. Pollock [80, p. 42] has an alternative axiomatization of
SS notable for the presence of the axiom (Pollock's A4):
CP: CKWpqWprWKprq
which corresponds to the condition:
(cp) f(u,X) c Y and f(u,X) c Z implies f(u,X A Z) c Y.
This axiom is almost, but not quite CV; the condition Wpr rather than Vpr
is required for conjoining p and r in the antecedent.


10
have a set of facts and laws, and if it is a consequence of this set
that kangaroos topple over, then the conditional in question is ac
cepted. The only difference in the two cases is the specific temporal
order in the former not in the latter. Presumably this difference is
incorporated in large part in the laws applicable to the differing
situations. Otherwise it seems I can handle them quite similarly.
In applying a similar procedure to El. 1.8 I find that the changes
I make in the known facts and laws in order to accommodate the assump
tion that kangaroos have no tails leave unchanged that fact that kan
garoos are vegetarians. Hence this fact appears in the set of facts
and laws, so as a consequence of it. Hence the "even if-conditional
is accepted.
Note also that an asequential conditional (see Jackson [36]) is
still not that far removed from a sequential conditional. If El. 1.2
is acceptable, and we were to somehow bring it about that kangaroos
became tailless, we would expect them to topple over. Hence we can
also make the prediction that "If kangaroos are de-tailed, then they
will topple over."
The similarity in the informally sketched methods above suggests
that both sequential and asequential conditionals may be accessible to
the same analysis in terms of truth conditions. The reverse or back
wards sequential (see Jackson [36]) of El. 1.5 can also be seen as
similar. Where with a forwards sequential we consider whether the
antecedent is sufficient for the consequent in terms of the laws, for
the backwards sequential we consider whether the consequent is neces
sary for the antecedent to be subsequently realized in terms of the
laws involved.


128
In [68] ute argues that it is not sufficient to consider just
the world(s) most similar to the actual world in which the antecedent
is true, nor worlds more similar to the actual world in which antecedent
and consequent are true than any worlds in which the antecedent and conse
quent are false, but rather that in evaluating a counterfactual we must
consider "all worlds in which the antecedent is true that are similar
enough to our world for consideration" [68, p. 776]. We discussed the
example Nute uses to illustrate his meaning in Section 2.5, and repeat
it here:
E3.2.1: If we were to have good weather this summer, or if the sun were
to grow cold before the end of the summer, we would have a bumper crop.
Symbolize this as WApqr.
Nute champions the principle of simplification of disjunctive ante
cedents
SDA: CWApqrKWprWqr
also discussed in Section 2.5. If E3.2.1 is evaluated as true, as would
seem reasonable on Lewn.s' semantics, then SDA licenses the inference of:
E3.2.2: If the sun were to grow cold before the end of the summer, we
would have a bumper crop.
But this is certainly false. To avoid this inference one must either re
ject SDA or bar the evaluation of E3.2.1 as true.
We have indicated that our preference is to reject SDA, since ac
cepting it requires we give up substitution of equivalents. Nute pays
this price, and offers a semantics in which generally we must consider
worlds at which each disjunct is true in evaluating counterfactuals with
disjunctive antecedents [68, p. 776]. However, the worlds we then
sider may differ in similarity to the actual world. Though Nute's
con-


155
satisfying (m), but does not validate R. Incidentally, this shows the
containment of M in R is proper.
The analogy of L4.2.11 does not hold if L is a monotonic logic.
The notion of augmentation is too strong; it makes more true than required.
On the pattern of augmentation Chellas and McKinney [12, p. 382] define
a weaker notion of supp lement at ion, whereby the satisfaction of (m), but
not (r) as well, is guaranteed.
D4.2.8: Let F = < U,N > be a frame. The supplementation of F is the frame
F* = < U,N* > such that
N* = {B : B c U and A c B for some A e N }
u u
We should remark that the supplementation of a model is the supplementa
tion of its frame, and a frame identical with its supplementation is said
to be supplemented. Note that preserves singular worlds and forces
superset closure, so the satisfaction of (m). Hence F* is monotonic.
Chellas and McKinney [12, p. 382] prove the analog of L4.2.11 for sup
plemented frames:
LA.2.13: Let L be a monotonic logic, the canonical model for L, and
"k
its supplementation. Then for all formulas a, and for all worlds u,
Mt
Ijiaiff |=a.
Proof: (In the proof we drop superscripts and subscript except for *
to distinguish the models.) The proof is by induction on the length of
a over all worlds, and the only case that is not immediate is if a is
of the form Lb. Suppose the theorem holds for formulas of length b and
for every u in U^. Consequently, ||b|| = ||b||*.
Assume |=Lb. Then ||b|| e N and so by definition of N*, as any
set is a subset of itself, ||b||'' e N*. Consequently, |=*Lb.


226
encompasses more cases than the classical (Boolean) sum of p and q. As
"A" is a classical operator in all the logics we have considered, this
case does not arise. (See, however, Hardegree [32] and Zeman [105]] for
nonclassical disjunction.)
We have suggested that a minimal counterfactual logic is
CA=CK+ID+MP+MD + CP + CA. We have agreed in the preceding para
graphs to each of the characteristic axioms of CA. CA is determined by
the class of partially ordered comparative order frames with world u
R -minimal. Thus even without any of the theses of E5.1.3, we are com
mitted to a logic with a partially ordered semantics as a logic of counter-
factuals.
Of the three theses of E5.1.3, CC and CV have come under the greatest
attack. Nute [68, 74] has argued against both CC and CV. Bennett [3] re
garded the presence of CC as one of the most counterintuitive aspects of
Lewis' analysis, as has Bigelow [5]. Pollock [80] argues for CC, but denies
CV. Others could be mentioned. To my knowledge, only Loewer [57] has sug
gested that CB is of any significance. CB is thus not so much problematic
as ignored.
CB bears some relation to SDA: CWApqrKWprWqr. The distinction is in
the consequent's being a disjunction in the former, and a conjunction in
the latter. However, CB, unlike SDA, does not result in the equivalence
of the counterfactual and the strict conditional in a classical conditional
logic. (For the most part we have side-stepped the debate over SDA by
limiting our consideration to normal conditional logics.) CB and SDA are
alike in having a disjunctive antecedent, and it is this characteristic
that renders them troublesome. In conversation, when we assert a counter-
factual with a disjunctive antecedent, we are generally prepared to defend


64
However, Lewis' analysis allows us to avoid having to toss these into
the "irredeemably ambiguous" bin. Both E2.3.4 and E2.3.5 are false,
hence not rebutting, while the corresponding "might"-conditionals, as
already noted, are true. Similar considerations may be applicable to
E2.3.6 and E2.3.7, however} the so-called ambiguity may be even more
simply resolved. E2.3.6 is false and E2.3.7 is true because satisfying
the antecedent does not require we go so far as to change the physical
location of political subdivisions. If we had such a possibility in
mind then we would utter instead:
E2.3.8: If New York City were in Georgia, then this city would lie
south of the Mason-Dixon line.
As Goodman pointed out, the direction of the expressions in the ante
cedents of E2.3.6 and E2.3.7 assume importance which allows us to view
them as unambiguous [27, p. 15]. It is assigning too much to contextual
ambiguity not to take note of this fact.
With respect to E2.3.4 and E2.3.5 however, it seems the only
alternative to consigning these to the fog of ambiguity is to give
serio\as consideration to "might"-conditionals. That Lewis' analysis
provides for this is a point in its favor.
2.4 Modal logic and the System of Spheres
In the preceding sections we indicated that Lewis' analysis of
the counterfactual conditional is an extension or application of possible
worlds semantics developed for modal logic. In the version of Lewis'
analysis that we have given, the logical modalities, including the strict
conditional, can be expressed.


209
mean X A Dam R With this in mind, we can characterize R -minimal
u u
elements of X:
LA. 8.22: The R^-minimal elements of X c U are given as follows:
(a) If Ru is a partial order, then
0, if X = 0.
R /X = <
u
{v}, if vR^w for all w e X.
{v,w e X : V0. w and W0. v}, otherwise,
u u
(b) If R is a weak total order, then
u
0, if X = 0.
Vx -
i {v,w e X : vR w and wR v}, otherwise,
u u
Proof: The proof is obvious from the definition and the properties of
partial and weak orders, respectively. We note that when R is a weak
total order, Ru/X constitutes an equivalence class under the condition
given in (b). QED
The following theorem establishes the fact that no logic smaller
than CP qualifies as a logic which partially orders the set of possible
worlds relative to each world:
TA.8.20: The logic determined by the class of all partially ordered com
parative order frames contains CP.
Proof: We shall show that given a partially ordered comparative order
frame we can define an equivalent selection function frame which satisfies
(id), (md), (co), and (ca). Define f:U x p(U) -* P(U) by:
f (u,X) = Ru/X.
It should be obvious that F = < U,R > and F' = < U,f > are equivalent.
For (id), by definition, Ru/X c X, so f(u,X) c X and (id) is satis
fied.


76
following pair of examples was suggested by remarks of Donald Nute
[75]3 (the order of the premises has been reversed for a reason to
be explained):
E2.5.7:
(a) Wqr
Wpq
. Wpr
(b) Wqr
Wsq
If Prof. X were to work less, she would be less
tense.
If Prof. X were to delegate her authority, she
would work less.
. If Prof. X were to delegate her authority, she
would be less tense.
If Prof. X were to work less, she would be less
tense.
If Prof. X were to be canned, she would work less
. Wsr
. If Prof. X were to be canned, she would be
less tense.
If your intuition is to accept (a) and reject (b) then I would agree.
What sense or system can we make of this on Lewis' analysis? Figure
2.5.4 diagrams a situation where the premises of the two arguments
above are true, (a) succeeds, and (b) fails (i.e., Wpr is true and Wsr
is false). It is instructive to compare that figure with 2.5.2(b) and
(d). Fpq requires that there be some p-permitting sphere where p-worlds
exactly coincide with q-worlds, as diagrammed in 2.5.2(b). But 2.5.4 is
weaker since there the p-worlds need merely be a subset of the q-worlds
in some p-permitting sphere. On the other hand, in 2.5.2(d), while the
set of worlds where r is true and the set of worlds where q is true must
intersect in such a way as to contain all the p-worlds in some p-permitting
sphere, it is not necessarily the case that this intersection exhausts the
q-worlds in that sphere, as it must to make Wqr true. Those cases where a


156
Asst imp. |=*Ib. Then ||b || e N*. By the inductive hypothesis ve
have |=b iff |='vband by T4.2.1 we have |=b iff b e u, hence ||b|| = |b|.
So by the definition of N there is seme formula c such that |c| c |b|
for some formula Lc e u. But then by the properties of maximally L-
consistent sets, Ccb is a theorem of L. Then by rule KM, so is CLcLb,
which is therefore in u. As we already have Lc in u, it follows by
closure under MP, that Lb e u, so by T4.2.1, | = Lb. QED
To utilize L4.2.13 in completeness proofs for monotonic logics it
will be necessary to show that not only does the canonical model for a
logic satisfy a certain condition defining a class of frames, but that
after supplementation the supplemented canonical model still satisfies
those conditions. We can state this in the form of a general lemma
where "permissible" means that the combination of conditions is not a
set-theoretic contradiction.
L4.2.14: Let F be a frame. Then if F satisfies some permissible com
bination of conditions (r), (n), (s), (q), (t), then P* does also.
Proof: Suppose F satisfies (r). To show F* satisfies (r) assume
A,B e IT. Then for some C c A and some D c B, C,D e N So by (r)
C A D e N but C A D c A A B, hence by definition of N/V, A A B e N ,
u J u u
and (r) is satisfied.
Suppose F satisfies (n). Then U e N but as U c U, it follows
that U e N*.
u
Suppose F satisfies (s). We have already noted that supplementation
preserves singular worlds, so F'' satisfies (s).
Suppose F satisfies (q). Then N = P(U), hence is already supple
mented, so identical to N*.
u


122
3.2 Varieties of Order
As developed by Lewis, comparative overall similarity is a family
of ordering relations on the set of possible worlds. We indicated in
Section 2.2 how Lewis' sphere function produced for each world a system
of spheres about that world which carried information about comparative
similarity. Suppose A and B are two of the spheres in $u with Ac B,
Further suppose worlds v,w are such that v e A and v i B while w e B and
w i A. Then we said that v was more similar to u than w. On the other
hand, if every sphere that contains world w also contains world v, then
v is at least as similar to u as w. Either of the underlined phrases
above constitutes an ordering relation with respect to u, one strict, the
other weak.
In his chapter on reformulations Lewis shows that one can construct
a semantics equivalent to his sphere function semantics directly in terms
of a three-place comparative similarity relation [51, pp. 48-50]. We may
define a relation RonUxUxUas follows:
D3.2.1: Let U be a set of possible worlds and $ a sphere function. We
define R £ U x U x U by:
vR^w iff for all A e $^, if w e A, then v e A.
For u,v,w e U, vR^w is to be read as v is at least as similar to u as w.
We will continue to assume that U$u = U, so that we do not need a specifi
cation of some accessibility function or relation for each u e U in addi
tion to the above comparative similarity relation.
Let us fix our attention on one u e U, for convenience, the actual
world. R then inposes a weak total order on the worlds in U, with u
the strictly R^-minimal element. That is, R satisfies the following
conditions:


11
While tie thrust of the above remarks is to broaden the scope of
the conditionals with which we will be concerned, and to indicate that
"counterfactual" or "subjunctive" is not a necessary mark of such con
ditionals, nevertheless most of the conditionals we are concerned with
can be expressed as subjunctive conditionals with propositional con
stituents. For example, we can paraphrase "If kangaroos had no tails,
then they would topple over" as "If it were the case that kangaroos had
no tails, then it would be the case that kangaroos topple over." With
this in mind we offer the following symbolization of such a conditional
Wpq = j r "If it were the case that p, then it would be the
case that q."
Without implying the need for separate truth conditions, but for
ease of reference, we shall also adopt the following symbolizations:
Tpq = ^ "Even if it were the case that p, it would (still)
be the case that q."
Upq = ,r "If it were the case that p, then it could not be
^ false that q."
The former is intended to symbolize those conditionals where there is
an absence of a connection between antecedent and consequent ("even if"
conditionals) and the latter those conditionals where there is a con
nection between antecedent and consequent, i.e., where the antecedent
"brings about" the consequent (what Pollock [80, p. 27] calls "necessi-
tation"-conditionals). We shall also adopt the standard Polish, or
prefix, notation for the usual logical operations of material condi
tionality, material bioconditionality, negation, conjunction, and (in
clusive) disjunction. These and the symbolization for strict condi
tionality, necessity, and possibility in modal logics are listed below:


208
D4.8.3: Comparative order semantics: let U be a set of possible worlds:
(a) A comparative order relation R c U x U x U is any
relation such that, relative to each world u in U:
(i)R partially orders (weakly orders, totally orders)
that subset of U in the domain of R (designated Dorn R ).
(ii)Every nonempty subset of U that meets Dom R has at
least one R -minimal element,
u
(b) A comparative order frame F = < U,R > is an ordered pair
such that U is a set of possible worlds and R is a comparative
order relation.
(c) If V:P -> P(U) is a valuation, then the ordered triple
M = < U,R,V > is a comparative order model.
(d) For X c U we designate the set of R -minimal elements of
X by R /X. Furthermore (wk v = ,r not wR v):
J u *u df u
(i) If R^ is a partial order, then
R /X = (v e X A Dom R : for all w e X with w f v, wi( v}.
u u u
(ii) If Ru is a weak total order, then
R /X = (v e X A Dom R : for all w e X, vR w .
u u u
(iii)If R is a well-order, then R /X is the R -least
u u u
element, if any, of X A Dom R^, and otherwise Ru/X = 0.
(e) Ijjwb Iff Ru/||a||"c ||b||".
(f) |g Vab iff R^/Hall" A ¡|b|[M ^ 0.
By condition (aii) we have assumed that R satisfies the limit assumption.
This simplifies the following proofs. We could define comparative orderings
that violate the limit assumption, and change our definitions accordingly.
In what follows we shall assume that when a subset X of U is mentioned, we


102
so LCsr (equivalently, s|=r) is true. But by CP, Wps and LCsr being
true requires Wpr be true.
That the validity of GCP implies LA is shown as follows:
Suppose GCP is valid, and (by way of contradiction) IA does not hold
for entertainable antecedent p. Then A0p = 0 since LA does not hold.
But 0p is a set such that Wpq is true for each q e 0p. Since A0p = 0,
and 0 c ||Np|| i.e., the empty set is a subset of the set of worlds
where Np is true, by definition, 0p| = Np. Hence by GCP, WpNp is true.
Then LNp is true. But if so, p is not entertainable, contrary to hy
pothesis .
Thus GCP, the limit assumption, and counterfactual consistency
are all equivalent, on Lewis' semantics. To reject one is to reject
all, other things being equal. So rather than debate the intuitive
merits of GCP versus the limit assumption, we can consider the merits
of counterfactual inconsistency versus IA.
Herzberger, claiming that counterfactual consistency is eminently
desirable for counterfactual languages, concludes "one can now argue
rather assume that there must be a closest [antecedent]-world" [34,
p. 88], Though he seems here to have slipped from the limit assumption
to the uniqueness assumption, his point is clear: if counterfactual
inconsistency is undesirable, and equivalent to rejecting the limit as
sumption, then the limit assumption should not be rejected.
The question, then, is how undesirable is counterfactual in
consistency? Or, how desirable counterfactual consistency? Should
the story of how things would be be a consistent story? Pollock notes
that Lewis considered GCP and rejected it because of the conflict with


188
f P(U), if U$ A X = 0
N(u,X) =
{Y : Y c U and for some Z e $ 0 ^ (Z A X) c Y}, otherwise
It is sufficient to show |= Wab iff ||b[| e N(u, ||a|[) for equivalence to
some neighborhood model. For normality (cm) and (cn) obviously hold,
so we only need to show (cr) holds.
M
Suppose |= Wab. Then by the truth definition D4.6.7, either
||a11 A U$u = 0 or for some Z e $ 0 f (Z A ||a||) c ||b||. If the former,
then ||b¡I e P(U) =N(u,||a||).
Suppose |]b|| e N(u,|[aj|). If |ja|| A U$u = 0, then by the truth
definition, |= Wab trivially. So assume otherwise. Then for some
Z e $u> 0 = (Z A ||a11) c ||b||. So by the truth definition, |= Wab.
We show normality by assuming A,B e N(u,X). It is sufficient to
consider the nontrivial case, so we have both for some Z e $ ,
0 f (Z A X) c A, and for some W e $ 0 + (W A X) c B. Now by condition
D4.6.6(b) on a sphere function frame, either Z c W or W c Z, hence either
(Z A X) c (W A X) or (W A X) c (Z A X). Without loss of generality, as
sume the former. Then both A and B contain Z A X, hence 0 f (Z A X) £
(A A B). Hence A A B e N(u,X). QED
Thus the question of Section 4.3 as to whether each system of
spheres frame (sphere function frame) has a corresponding neighborhood
frame has an affirmative answer. It is evident that we may regard
neighborhood semantics for conditional logic as more general than sphere
function semantics, if only because neighborhood semantics is adequate
to nonnormal conditional logics. In Section 4.7 we shall introduce some
additional conditional axioms and the logics obtainable therefrom, and
in Section 4.8 we shall develop appropriate semantics for such logics.


84
One should note that these figures are only slight variations of
Figures 2.5.5 and 2.5.7(b), where we agreed the inference should not
be considered pragmatically valid. But both of the above meet the con
dition that all statements be evaluated with respect to the same sphere
(the third non-trivial one). For the SDA example, both disjuncts occur
in the same larger sphere. However, in both cases there is a smaller
sphere that would make some premise true. It would therefore seem that
we should define pragmatic validity as follows:
D2.5.1: An inference is pragmatically valid iff it is impossible for
the premises to be true and the conclusion false under the conditions
that:
(1) all premises are evaluated as true with respect to the
same sphere, and
(2) no premise is true with respect to any smaller sphere.
This has the effect of requiring that all variably strict conditionals
in the inference be of the same degree of strictness, and squares with
our intuition that we need consider no worlds more dissimilar to the
actual world than to some fixed degree. This would eliminate the situ
ations of Figure 2.5.8, but it has two consequences which may be unwelcome.
First, it imposes on the system of spheres the limit assumption:
C2.5.1: If 0 / X c $u> then AX e X.
That is, there is some smallest sphere in the set of spheres making
the premises true. Lewis, for reasons we shall discuss in Section 2.6,
wants to reject the limit assumption.
Second, it makes the following inference appear pragmatically in
valid:


CHAPTER POUR
MODAL AND CONDITIONAL LOGICS
4.1 A Lb dal/Conditional Language and Modal Systems E, M, R, K
Heretofore we have made free use of a formal language that we intro
duced by providing "translation directions" from ordinary language into
it. In what follows we shall reintroduce this language more formally,
providing the syntax of a formal system.
A formal system consists of a language, a set of formulas designated
as axioms, and a set of rules of inference, together with what constitutes
the notion of a derivation (deduction, proof) in that system. We will
keep our language simple, introducing most of the operators we have
been using by way of definition, that is, as abbreviations of more com
plex formulas in our language. Since we are interested in conditionals,
our language is a conditional language, abbreviated CLW. The following
three definitions introduce the primitive and defined symbols of CHI'):
D4.1.1: The primitive symbols of CLW consist of
(a) Denumerably many propositional (sentence) letters:
P, q, r . .
(b) The truth-functional operator C
(c) The constant false proposition o
(d) The modal operator L for necessity
(e) The counterfactual operator W for counterfactual conditionality.
135


77
hypothetical syllogism "works," but for which SCE is too strong, may
be like that diagramed in Figure 2.5.4, while those where SCE fails
must be. (Except that there could be some Krs-worlds in the s-permitting
sphere, but then also some KrNs-worlds, else Wsr is true contrary to the
assumption that the hypothetical syllogism failed.)
Figure 2.5.5 also represents a case in which hypothetical syllogism works,
but I shall argue that this is less usual.
Figure 2.5.5


173
L4.5.3: CE is consistent wrt the class of all sentential models satisfy
ing the condition:
N(u,a) = N(u,b) iff ||a|| = ||b||.
Proof: We need only check that RCEA preserves validity. Suppose Eab is
valid (true in every designated model). Then ||a|| = ||b||. Consequently,
by the stated condition, N(u,a) = N(u,b). Then ||c|| e N(u,a) iff ||c|| e
N(u,b), and so we have EWacWbc. QED
The condition which allows us to restore RCEA to a half-classical
logic must be stated in terms not solely of frames, but of models. How
ever, an alternative is available for CE.
L4.5.4: CE is consistent wrt the class of all propositional frames.
Proof: We need only check that RCEC and RCEA preserve validity. Suppose
Eab is valid. Then in any model ||a|| = ||b||. Consequently, we have both
N(u, ||a11) = N(u, ||b11), and ||a|| eN(u,||c]|) iff ||b|| e N(u, ||c||). There
fore, we have both EWacWbc and EWcaWcb. QED
For the consistency (and later completeness) of CM, CR, and CK we
will require the conditions to be stated below be imposed as indicated on
propositional frames. As in Section 4.2, we will let C denote the class
of all propositional frames and Cdenote the class of frames satisfying
a particular condition or combination of conditions. (For simplicity we
will drop the initial "c" in the designation of the conditions below when
using them as subscripts.)
E4.5.1:
(cm) A A B e N(u,X) implies A e N(u,X) and B e N(u,X).
(cr) A,B e N(u,X) implies A A B e N(u,X).
(cn) U e N(u,X).


94
to be a finite conjunction, the situation described above is, granting
Pollock's assumptions, a countermodel to GCP on Lewis' semantics. This
reflects the rejection of the limit assumption: the intersection of
all the p-permitting spheres is not itself p-permitting.
By the reasoning above, Pollock concludes that Lewis' semantics
is inadequate, as GCP should be a valid inference principle for counter-
factuals on the grounds that it is as intuitively valid as CP [80, p. 20].
There are two problems with Pollock's suggestion: first, it is not clear
to me that GCP is as intuitively valid as CP; second, the assumption we
granted about the ordering of worlds is stronger than that which Lewis
makes in first offering the example, and the additional strength is un
justifiable. (Herzberger points this out, crediting Isaac Levi, [34,
p. 85n].)
My intuitions have been shocked sufficiently often in the imovement
from the finite to the infinite that I am habitually suspicious of in
finite sets. What is true of their finite counterparts is often not
true of the infinite. So that while a finite intersection of towered
open intervals of the number line is nonempty, it is easy to construct
infinite towers of open intervals whose intersection is empty. In fact,
Lewis' line example suggests one. The assumption that if every finite
subset of a certain infinite set has a certain property, then every
subset does, is itself a powerful assumption in mathematics, and not
always justified. Hence my intuitions concerning GCP are at best no
less trustworthy than my intuitions about similarity orderings of worlds.
Why should not our notions of similarity result in open (noncompact)
rather than closed (compact) sets? GCP and the rejection of LA are on


70
162]; the strongest version of SSI(SSS) holds only for the material
conditional and for strict conditionals at least as strong as that
of S30.
We have already considered the inference of strengthening the
antecedent; the counterexample and corresponding inference pattern
are repeated below:
E2.5.1:
Wpq If this match were struck, it would light.

. WKprq . If this match were soaked in water and struck,
it would light.
The conjoining of r with p removes us to more remote worlds where the
consequent is no longer true, as Figure 2.5.1(a) illustrates.
Figure 2.5.1
The same figure also serves as a countermodel to transitivity, a
counterexample to which is given on the following page:


198
Proof: In view of L4.8.1 we show the validity of MD and CO. Let an
ordered model be given and u any world in U.
For MD, assume |=WNpp. Then f (u, || Np | j) c ||p||. By (id),
f(u, ||Np||) c ||Np||. Now ||p|| = U ||Np||, so f(u, ||Np||) = 0. Hence
by (md), f(u, ||q||) A ||Np|| = 0. Thus f(u, ||q||) c ||pl¡. Hence |=Wqp.
For 00, assume |= Fpq. That is, both |= Wpq and |= Wqp. So
f(u, ||p||) c ||q|| and f(u, ||q||) c ||p|| So by (co), f(u, ||p||) = f(u, ||q||).
Hence f(u,||p||) c ||r|| iff f(u, ||q||) c ||r||. Therefore |= Wpr iff |= Wqr,
and so |= EWprWqr. QED
L4.8.6: 00 = 0 + MP is consistent wrt the class of ordered weakly
material frames. (L4.8.2 andL4.8.5.)
L4.8.7: CG = 00 + CC is consistent wrt the class of ordered material
frames. (L4.8.6 and LA.8.4.)
L4.8.8: CP = 0 + CA is consistent wrt the class of ordered frames
satisfying (ca).
Proof: In view of L4.8.5 we show the validity of CA.. For any model on
such a frame and any world u in U assume |= KWprWqr. Then we have both
|= Wpr and |= Wqr, and so f(u, ||p||) c ||r|| and f(u, ||q||) c ||r||. Hence
f (u, ||p ||) Uf(u, ||q||) c || r 11. Now ||Apq|| = ||p|| U ||q||, so by (ca),
as f (u, ||Apq11) c f(u, ||p||) U f(u, ||q||), we have f(u, ||Apq||) c ||r||.
Thus |= WApqr. QED
L4.8.9: V = 0 + CV is consistent wrt the class of variably strict frames.
Proof: In view of L4.8.5 we show the validity of CV. Let a variably
strict model be given and u any world in U. Assume |= KWpqVpr. Then we
have |= Wpq and |= Vpr, and so f(u, ||p||) c ||q|| and f(u, ||p||) A ||r|| i 0.
So f (u, ||p||) i ||Nr 11 = U ||r 11. Hence by (cv), f (u, ||p|| A ||r||) c ||q||.
But ||p|| a || r 11 = ||Kpr||. Therefore, |= WKprq. QED


134
equivalence class of worlds, f does pick out the set of ^-minimal
members of p. The distinct members of this minimal set are T -
incomparable.
Thus, whether our governing principle for counterfactual delibera
tion is comparative similarity or minimal change (which is a variety of
comparative difference), the result is an ordering of possible worlds
in which those antecedent-worlds minimal in the order are the critical
ones for evaluating counterfactuals. In CHAPTER FOUR we shall consider
a broader range of conditional logics, only some of which result in an
ordering of possible worlds. But we shall argue, there and in CHAPTER
FIVE, that any conditional logic adequate to represent the counterfactual
conditional, must incorporate some ordering of possible worlds. We re
plied briefly in the previous section to Loewer's charge that such an
ordering must be considered primitive to the semantics for the conditional
and incapable of supporting an analysis. That this is not the case wall
be argued again in CHAPTER FIVE.
3.3 Notes
1The semantics used to establish the order are those suggested by
Loewer [57]. We shall argue in Section 4.8 that Loewer's conditions
actually do not determine Pollock's system SS, contra Loewer's claim
[57, p. 115nl6]. However, the conditions do determine a partial order,
and this suffices for our present purposes. The resulting system is
close to Pollock's, however.
2See note 1.


4
second subjunctive, the subjunctive mood is not essential to such con
ditionals. Clearly the actual falsity of antecedent and consequent is
not essential either, though from what has been said so far one might
claim that the conditional is only assertable when the antecedent and
consequent are not known to be true. Thus the conditional would still
be "potentially" counterfactual. It is certainly not correct to say
that such a conditional is only assertable on the assunption that the
antecedent and consequent are false, since the third party above makes
no such assunption.
So a conditional of the sort we are interested in need not be
subjunctive, nor need it be either actually counterfactual or assumed
to be counterfactual in order to be asserted appropriately, as Goodman
[27] recognized. Nor apparently does the actual truth of the antece
dent and consequent count against the truth of the conditional. (In
deed, Lewis' account [51] incorporates the contrary view.)
The third party's choice of a past indicative or subjunctive
sentence to express himself does not, I believe, reflect a choice be
tween two different propositions which he could alternatively express,
biat rather a choice between two ways of expressing the same proposition.
So he "says" nothing different by expressing himself one way rather than
the other. However, he may show something different about his attitude
toward the situation described to him by his choice of words. The use
of the subjunctive mood in the second case strongly suggests that he
regards the situation on which he has been asked to pass judgment as
hypothetical. From this it is an easy step to counterfactual, and the
association of subjunctive conditionals with counterfactual ones.


161
in terms of U$u> rather than A$^, for the logical modalities. Given the
conditions on $^, it is a maximal tower of the improper filter, P(U).
In which case we would have La for formula a where a is a truth-
functional contradiction if we were to consider $ a neighborhood function
and apply the truth definitions of neighborhood semantics, since 0 e $ .
The inescapable conclusion is that $ is not a neighborhood function,
and that Lewis' semantics is not neighborhood semantics in disguise.
Lewis' system-of-spheres semantics may be looked upon as the giving
of a certain internal structure to the accessing relation based upon con
siderations of overall comparative similarity: to each sphere S e $u
there corresponds a relation R such that uR v iff v e S. The most
s s
comprehensive of these relations is R..^ and it is in terms of this
relation that the semantics for the logical modalities must be defined.
On the other hand, neighborhood semantics may be looked upon as furnishing
an external structure in terms of which the accessing relation may be
defined. There does not, however, correspond a relation for each
neighborhood, since it is the intersection of the neighborhoods that
determines the largest relation.
Now each relational frame F = < U,R,Q > corresponds to sane
neighborhood frame. The following definition will serve as a basis for
this assertion:
DA.3.3: Let F = < U,R,Q > be a relational frame where
(a) U is a set of possible worlds.
(b) R is a relation on U.
(c) Q is the set of singular worlds.


201
T4.8.3: If a selection function f satisfies the condition:
f(u, Ja|^) = (v e U : {b e CW : Wab e u} c v}
then f is a canonical selection function called a proper canonical selec
tion function. Furthermore, < U,f > and < U,f,V > are proper canonical
frame and model respectively, provided V(p) = |p|^. As T4.8.3 does not
place any condition upon those X c U for which X f |a| for every sentence
a, there is no uniquely determined proper canonical frame, but rather a
range between a largest, for which f(u,X) = U whenever X f |a| for every
sentence a, and a smallest, for which f (u,X) = 0 whenever X f |a| for
every sentence a.
That CK is determined by the class of all selection function frames
was proved in Section 4.6. For the extensions of CK we consider it will
be sufficient to show that seme proper canonical frame for the extension
satisfies the appropriate conditions. Completeness will then follow by
T4.8.2. In the following we state that a frame determines a logic and
note which lemma preceding is required for the consistency half. The
proofs for CK + ID, CK + MP, and B are from Chellas [11, pp. 141-143].
T4.8.4: CK + ID is determined by the class of dependable frames. (L4.8.1.)
Proof: Let P = < U,f > be a proper canonical frame for CK + ID that satis
fies (id) whenever X f Ja| for every sentence a, say, the smallest such
frame, so f(u,X) =0. We must show F satisfies (id) when X = |a| for some
sentence a. Suppose v e f(u, |a|) and show v e |a|. By definition of
f, (b : Wab e u} c v. By ID, Waa e u, so a e v. Hence v e |a| QED
T^-8.5: CK + MP is determined by the class of weakly material frames.
(L4.8.2.)
Proof: let P ~ < U,f > be a proper canonical frame for CK + MP that sat
isfies (mp) whenever X ^ ]a] for every sentence a, say, the largest such


238
change which allows one horse to finish is contained in a change which
allows two to finish or the other way around. While I am not sure which
it is, I doubt if it is neither. So it appears that Pollock's account
requires even greater skills at ranking worlds than Lewis', and Nute's
attack is better aimed at the former than the latter.
I do not propose to settle even the formal issue as to whether an
adequate counterfactual logic ought to result in a total order or only
a partial order of the set of possible worlds. I suspect that one could
construct a series of counterfactuals with disjunctive antecedents where
at one limit, worlds that satisfied one disjunct would be roughly as
similar to the base world as worlds that satisfied the other, or both,
disjuncts. At the other limit, worlds satisfying one disjunct would be
considerably more like the base world than worlds that satisfied the other.
At some point in between we would be lumping together for consideration
worlds that differed noticeably in comparative overall similarity to the
base world, but still had to be considered reasonable situations. If so,
then perhaps a weak partial order accords best with our actual deliberations.
In Section 3.2, we discussed the need for viewing Lewis' notion of
comparative overall similarity as both primitive and antecedently under
stood, by seeing it as analogous to the extension of a physical concept,
such as motion, to a new domain. It is in this way that we see there is
another alternative not expressed in the disjunctive premise of Loewer's
dilemma. It is not the case that similarity ordering is merely primitive,
nor is it poorly understood. It is reasonably well-understood in ordinary
contexts, but the application to possible worlds introduces factors into
the comparison that are not operative, or not significant, in more ordinary


37
brings out the significant parallels between Peirce's "garrma" system
and possible world semantics. Evidently, having developed graphical
systems for propositional and quantificational calculus (the "alpha"
and "beta" systems), Peirce experimented with a third system (or frag
ments of several systems) in which he endeavored to make possible the
representation of universes of discourse other than the actual:
. . these would be "worlds of possibility." . .
He proposed that instead of considering just one SA
... we think of ourselves as working with a book
of such sheets, with each sheet in the book repre
senting a possible world much as Kripkean semantics
correlates a semantic tableau with each possible
world. [105, p. 252]
In the above quotation from Zeman, SA refers to the Sheet of Assertion
upon which graphical signs are written as assertions about the universe
of discourse.
Peirce did not quite reach conceptual closure on this idea, due
to the fact that he did not have an adequate way to represent the ac
cessibility relation. Though even here, Zeman notes [105, p. 253], he
came close. Peirce did hit upon a predicate which bears interpretation
as an accessibility relation, but did not develop it.
In addition to relational possible worlds semantics of the Kripkean
variety, Scott [88] and Montague [65] introduced a variant approach:
neighborhood semantics. A comprehensive treatment of modal logics in
terms of neighborhood semantics is found in Segerberg [91] which forms
an important basis for our presentation in Sections 4.1-4.3. Most of
the results therein are first brought together by Segerberg. The ap
plication of neighborhood semantics to conditionals is developed by
Chellas [11], and a systematic comparison of the varieties of relational


132
C3.2.2: For all u e U and propositions p,q e P(U) (again we identify a
proposition with a set of possible worlds):
(a) f (u,p) c p.
(b) if p @ u, then f(u,p) = {u}.
(c) if p c q and f(u,p) ^ 0, then f(u,q) ^ 0.
(d) if p c q and p A f(u,q) ^ 0,
then f(u,p) = p A f(u,q).
(e) f (u,p) c f(u,Apq) or f(u,q) c f(u,Apq).
(f) f(u,Apq) c f(u,p) U f (u,q).
Lewis shows that a selection function satisfying conditions (a) through
(d), which he calls a centered set-selection function, is equivalent to
some limited sphere function [51, pp. 58-59]. The truth conditions for
the "would"-conditional and the "might"-conditional can then be stated as:
D3.2.10: Wpq @ u iff for all w e f(u,p), q @ w.
D3.2.11: Vpq @ u iff for some w e f (u,p), q @ w.
For a given sphere function $ satisfying the limit assumption, the
equivalent set-selection function picks out the p-worlds in the inter
section of all p-permitting spheres, that is, the set of closest p-worlds.
Comparing this to our definition of the order relation in D3.2.1, and
the subsequent induced relation R on equivalence classes of worlds, we
see that f picks out the p-worlds in the R -least p-permitting equivalence
class. Consequently, we may define the order relation for Lewis' account
by:
D3.2.12: Let U,f be w in D3.2.11 and C3.2.2(a)-(d). We define
R c U x U x U by:
vRuw iff either f(u,{v,w}) = {v} or f(u,{v,w}) = {v,w}.


39
composed of many worlds" [17, p. v]. For the difficulties involved
in drawing any significant positive philosophical conclusions frcm
this theory one should see Skyrms' [93] criticism of "realistic" pos
sible worlds views.
Making no pretense to realism are the speculative excursions by
many contemporary science fiction authors into the realm of possible
worlds. On the one hand are the many "what if" themes which concentrate
on alternate histories of the actual world. Of more interest are those
speculations which postulate the simultaneous existence of a variety
of "parallel" worlds, usually with some means of enabling access from
one to another. In this connection the Lord Kalvan of Otherwhen
stories of H. Beam Piper [77] are typical, being adventure stories with
little conceptual meat.
A more highly developed parallel worlds theory is found in Worlds
of the Imperiun, by Keith Laumer [43] which is suggestive of Lewis'
employment of comparative overall similarity as a way of ordering pos
sible worlds for the purpose of determining the truth value of counter-
factuals. As we shall discover in Section 2.3, it is not enough to
analyze counterfactuals in terms of a single accessing relation on a
set of possible worlds. In addition to the concept of possible worlds
themselves, we must also have a concept of "distance" of them from our
actual world, which ones are in our immediate neighborhood, and which
father away? Lewis suggests that the concept needed here is the quite
ordinary one of comparative overall similarity [51, p. 1]. That is, we
can conpare possible worlds, much as we compare other things, in respect
of their overall similarity to a given, possibly the actual, world. If
we imagine an instant of time in our world as a point, then in Laumer's


FIVE OOUNIERFACIUALS AND COMPARISON OF WORLDS 219
5.1 An Adequate Counterfactual Logic 219
5.2 Comparative Order Analysis 230
REFERENCES 243
BIOGRAPHICAL SKETCH 250
v


72
the second premise is true. However, if I had started at 5 a.m. I
would have been very tired, and so would have forgotten to take the
shortcut that I actually did take, thus lengthening my trip by over
an hour. Then the conclusion fails.
There are several patterns related to transitivity that are
valid for the counterfactual conditional: (a) half of substitution
under strict equivalence (SSE), (b) substitution under counterfactual
equivalence (SCE), (c) the other half of SSE, the consequence princi
ple (CP), and (d) a strengthened version of transitivity (RRI). The
patterns, and diagrams corresponding, which suggest how one might argue
for their validity on Lewis' semantics, are represented in E2.5.4 and
in Figure 2.5.2.
E2.5.4:
LEpq
or
Wqr
LEpq
Wpr
Wpr
(a) SSE
Wqr
Fpq
Fpq
or
Wqr
Wpr
. Wpr . Wqr
(b) SCE
Wpq
LCqr
. Wpr
(c) CP
Wpq
WKpqr

. Wpr
(d) RRT


91
D2.6.1: Wpq @ u iff every p-world in the closest p-permitting sphere
in $ is a q-world,
u n
which would be nonvacuously true in case there was a p-permitting sphere,
and vacuously true otherwise. The spheres (though not the worlds they
contain) would be well-ordered with respect to subset containment. That
is, every subset of $u would have a least element.
But Lewis questions whether this would in general be a suitable
restriction on $. He argues:
Suppose we entertain the counterfactual supposition that
at this point
there appears a line more than an inch long. (Actually
it is just under an inch.) There are worlds with a line
2" long; worlds presumably closer to ours with a line
1% long; worlds presumably still closer to ours with a
line iy long; worlds presumably still closer
But how long is the line in the closest worlds with a
line more than an inch long? If it is 1+x" for any x
however small, why are there not other worlds still
closer to ours in which it is l+J^x", a length still
closer to its actual length? . Just as there is no
shortest possible length above 1", so there is no
closest world to ours among the worlds with lines more
than an inch long. . [51, p. 21]
On the basis of this example, Lewis rejects the limit assumption. Before
considering the consequences of this rejection, we note that Lewis recog
nizes an even stronger version of LA: the uniqueness assumption, LA.
(Called Stalnaker's assumption by Lewis because it is associated with
Stalnaker's semantics for conditionals. We use Nute's [74, p. 100] more
descriptive term.).
We will discuss Stalnaker' s semantics in CHAPTER THREE as part of
a general discussion of similarity. For now it suffices to indicate that
the uniqueness assumption requires that the closest p-permitting sphere
for any proposition p contains but one p-world. This means the sphere


183
for any regular propositional neighborhood frame for conditional logic
we can construct a propositional accessibility function, as well as a
propositional alternative relation:
D4.6.1: Let F = < U,N > be a regular propositional neighborhood frame.
Let = {u : N(u,X) = 0}. Then Q is the set of all worlds singular
for proposition X. Let Q = the set of all such Q^. {(u,X) : u e Q^}
is the set of singularities of S. We define a propositional accessibility
function S: U x P(U) -> P(U) by
S(u,X) = AN(u,X)
provided N(u,X) f 0.
D4.6.2: Let F = < U,N > be a regular propositional neighborhood frame
and S defined as in D4.6.1. We define a propositional accessibility
(or alternative) relation R c U x p(U) x U such that
uR^v iff u i Qx and v e S(u,X).
Similar definitions could be produced for regular sentential neighbor
hood frames.
It is clear from the above definitions that to each regular pro-
positional neighborhood frame there corresponds a propositional acces
sibility function frame F = < U,S,Q > and a propositional relational
frame F$ = < U,R,Q >. The truth definitions for conditionals in proposi
tional accessibility function and propositional relational models are
given by:
D4.6.3: let M = < U,S,Q > be a propositional accessibility function
model. Then we define truth in M at world u for conditional formulas by:
|= Wab iff u i Q||ajj and S(u, ||a||) c ||b||.


14
Wpq
. WKprq
is not valid. Speaking of a certain dry match in favorable conditions
(enough oxygen, etc.) I may say,"If this match were struck, it would
light." But it does not follow fran this that "If this match were
soaked in water and struck, it would light." The failure of Strength
ening the Antecedent is one of the striking peculiarities of counter-
factual s, and the single strongest argument against the counterfactual
conditional being a strict conditional.
1.2 The Metalinguistic Analysis
I referred earlier to a certain procedure whereby a counterfac-
tual could be evaluated as the starting point of a number of analyses
of counterfactuals. This is what has been called the "linguistic" or
"metalinguistic" account (by Pollock [80] and Lewis [51], respectively).
Because most hold that the consequent of a counterfactual is a logical
consequence of the antecedent conjoined with other statements, they are
also called "consequence theories." According to such accounts the
truth of a counterfactual conditional is largely based upon the rela
tions among certain linguistic entities, such as sentences or, in sane
cases, beliefs. (Such accounts have been offered by Goodman [26],
Chisholm [13], Mackie [59], Rescher [83], Jackson [36], Ellis [19], and
others.)1 I shall sketch a general outline of such an account which
does not do full justice to any of those that have actually been of
fered, but is sufficient to form a starting point for criticism.
Consider the well-worked-over example concerning a certain pres
ently unlit match:


237
5.2.2(a) we might as well collapse it to Figure 5.2.2(b), where the worlds
under consideration all fall into an equivalence class of worlds more or
less equally similar to the base world, given a loose enough sense of
comparative similarity. Nute's point was that we would normally consider
each of the worlds v, w, y, and z as a reasonable situation from the point
of view of the actual world. With that assessment, I am in agreement.
What I disagree with is that it is part of Lewis' analysis that we go on
to rank these worlds in terms of comparative overall similarity dissimilarly.
Comparative similarity is not necessarily that fine a notion, nor should it
be.
Pollock's attempt to escape the dilemma posed by Loewer by, in ef
fect, grasping the second horn, seems misguided to me. By analyzing the
notion of minimal change as an ordering principle, and in the process
making it more precise, the result may be something that is able to sup
port an analysis, because well-enough explained to be antecedently under
standable, but that gives the wrong truth values for imprecise counter-
factuals, so not be a correct analysis. If we consider the counterfactuals
we have referred to above, such as WApqr, Wpr, Wqr, etc., we see that in
actuality all are false. Given that the Galloper places, the Flyer might
place, or might not; the Flyer might not even finish, or if so, might
even win. Thus only Figures 5.2.2(a) or (b) will give the correct truth
values for the counterfactuals.
To claim, however, that an ordering principle based on minimal
change would yield Figure 5.2.2(a) rather than either 5.2.1(a), 5.2.1(b)
as is, or 5.2.1(b) with the missing links restored, seems slightly ad hoc.
There are reasons for believing a comparative order based on change from
the actual world will yield something like 5.2.1(a) or (b). Either the


170
CK = PC + RCEA + CK + RCN.
As with the parallel basis for K, this is a particularly economical
basis, though it conceals the fundamental nature of axians CM and CR
in comparison to the semantics appropriate to classical conditional
logics.
The semantics to be developed in Section 4.5 will allow us to
show the containments noted after L4.4.3 are all proper. As we shall
see, these semantics are an adaptation of the neighborhood semantics
presented in Section 4.2 to conditional (or indexed modal) logic.
4.5 Neighborhood Semantics for Conditional Logic
A version of neighborhood semantics for conditional logic is devel
oped by Chellas under the rubric of "minimal frames and models" [11,
pp. 144-147]. This semantics is adequate to classical conditional logics
In a note [11, p. 149nl4] Chellas indicates that there is a sentential
variant to his "standard" semantics which is appropriate to half-normal
logics. Nute [74, p. 65] develops the version of neighborhood semantics
appropriate to half-classical logics.
Recall that in neighborhood semantics for modal logic the truth con
dition for a formula La is given by
|= La iff ||a11 e Nu
where is a set of subsets of the set of possible worlds, U. If we re
place "L" above by a sententially or propositionally indexed modality,
then either our truth condition or our definition of N must take account
u
of the index. The latter alternative is the case in the two kinds of
frames to be defined below.


215
w
u
Figure 4.8.3
As previously, define f(u,X) = R^/X for all worlds u and X c U. The
total order for v,w,y,z implies the satisfaction of (id), (md), (co),
(cv), (ca), and (cb); however is not a total order. That is a
partial order implies the satisfaction of all but conditions (cb) and
(cv). It can be verified that R^ as depicted in Figure 4.8.3 satisfies
(cb). However, (cv) is not satisfied by R as we may observe by setting
X = {v,w,y,z>, Y = {y,z}, and Z = {v,w,y}. Then the following hold:
R^/X <: Y and (R^/X) A Z f 0, but R^/CX A Z) Y. Hence (cv) does not
hold in this frame. QED
Note that we chose R so that (up) and (cc) were also satisfied.
We could do the same for the other worlds in U. Thus the following
theorems are immediate:
T4.8.30: CP + CB, CA + CB, SS + CB do not contain CV.
T4.8.31: CP + CB, CA + CB, SS + CB are properly contained in V, VW, and
VC, respectively. 5
It is easy to show that MP is not contained in CK + ID and CC is
not contained in B. So the other containments of Figure 4.7.1 are proper.
Figure 4.8.4 expands our families of logics by the members noted in
T4.8.31.


222
D5.1.1: Let F = < U,R > be a comparative order frame which is partially
ordered. If for each u e U, satisfies the condition stated below,
then we say R^, R, and F are semiconnected.
Condition: For all X,Y c U such that each of X and Y is a pair of distinct
elements of U standing in relation and X f Y, some element of X stands
in relation R to some element of Y.
u
Semiconnectedness is hardly an obvious concept to apply to partial
orders. It is included here to characterize the subclass of partially
ordered comparative order frames which determine logics containing CB.
Countermodels to CB must fail to satisfy condition (cb), and so require
a subset of X A Dom R appear as in Figure 5.1.2(a). Semiconnectedness
forces the occurrence of Figure 5.1.2(b) or (c) in such cases.
v v
v y
(a)
(b) w
y
(c) w
y
w z z z
Figure 5.1.2
We can then show the following:
T5.1.1: A partially ordered comparative order frame satisfies (cb) iff
it is semiconnected.
T5.1.2: CP + CB, CA + CB, SS + CB are determined by the class of semicon
nected partially ordered comparative order frames per se, with u R -minimal
for each u e U, and with u R -least for each u e U, respectively.
Condition (cp) (and so thesis CP) is included because it also is
characteristic of conditional logics that require a partial order of pos
sible worlds. We recall that in Section 4.8 it was shown that CP is
determined by the class of all partially ordered comparative order frames


32
conditions, and is thus an exercise in confirmation theory. The con
firmation theory is found in [78], while [80] takes the claim that
laws can be analyzed in terms of their justification conditions for
granted and proceeds to analyze counterfactuals in terms of laws and
cotenability. Then cotenability is given an explanatory analysis in
terms of possible worlds. Thus the analysis Pollock offers is only
partly an analysis in terms of truth conditions, resting as it does
upon an analysis of laws in terms of justification conditions.
In partial contrast, Goodman [27] also provides an analysis of
laws in terms of confirmation theory and so in terms of justification
conditions. However, he seems to regard the solution to the coten
ability problem as fall-out from the analysis of laws [27, p. 122],
An account such as Jackson's [36] or Barker's [2] in terms of causal
laws, while having virtues and defects of their own (most counter legis
become irredeemably ambiguous), certainly are predicated upon an anal
ysis of causal laws if they are to have any explanatory force. The
same can be said of Rescher's [83] analysis of nomological counterfac
tuals (all others are irredeemably ambiguous) in terms of laws.
With the exception of Pollock's reliance on possible worlds to
analyze cotenability, all of these accounts share the assumption that
an analysis of law is prior to an analysis of counterfactuals and that
this analysis occurs in the context of confirmation theory and is an
analysis in terms of justification conditions. The breakout from the
circle thus comes in the analysis of law. Pollock included, these
variants are all direct inheritors of the metalinguistic approach.
The second approach is to attack counterfactuals directly by pro
viding an explanatory truth condition account of them. It is here that


22
concept, the claim certainly is that once one has grasped the truth
conditions one has grasped the meaning of the concept.
But here we must be careful. Not just any set of truth con
ditions will do (as Pollock points out in a different connection in
[78, p. 8]). Stalnaker indicates this when he says above that the
truth conditions must "explain" something. Judging by his analysis
Lewis has something similar in mind. The question to be answered is:
When should we be satisfied with a purported analysis? Goodman rightly
rejects his own analysis as circular, but Ellis offers an analysis con
taining formal elements with precisely the same characteristics.
Wasserman's analysis provides a logic for the conditional, but would we
be justified in claiming on that basis to have grasped its meaning?
Stalnaker, who shares with Ellis the conviction that the mood or
factual status of a conditional is a secondary consideration, distin
guishes two problems involved in analyzing counterfactuals. The first
he calls "the logical problem of conditionals" which is "the task of
describing the formal properties of the conditional function ..."
[96, p. 165]. The second is the "pragmatic problem of counterfactuals"
which concerns the fact that
. . the formal properties of the conditional function,
together with all of the facts, may not be sufficient
for determining the truth value of a counterfactual; that
is, different truth valuations of conditional statements
may be consistent with a single valuation of all non
conditional statements. [96, pp. 165-166]
The development of a semantic theory for counterfactuals Stalnaker re
gards as part of the logical problem. The semantic theory that he
does develop sheds light on the second problem as well, in his view,


104
E2.6.8: If the line were more than an inch long, the printer would
have made a mistake.
Let us symbolize the consequent of E2.6.8 by r, retaining p for the
antecedent and for the critical consequents. Any reasonable sphere
function wherein p failed the limit assumption would nevertheless make
E2.6.8 true: that is, there is seme p-permitting sphere, and every
p-world in that sphere is an r-world. Designate a p-permitting sphere
satisfying this truth condition for Wpr as S^. Let Q represent the sub
set of the set of critical consequents such that each critical conse
quent (considered as a set of worlds) is a subset of S^. It is easy to
see that Q is still inconsistent.
However, for each qy e Q, r is true at each world in q^. If it
were not, then the truth condition for Wpr would not be satisfied, as
was assumed. (Recall each world in q^ is a p-world.) So, for each
q^ e Q, qw £ ||r || A Sp. We may express this by saying that there are
many different possible ways the printer could have made a mistake.
What would truly be surprising is that the totality of such ways should
turn out to be consistent.
The way in which the printer made a mistake is irrelevant to the
truth of E2.6.8. There is no full story because there are too many
possible stories. Fortunately there is considerable overlap among the
otherwise inconsistent possible stories. This overlap is of a "family
resemblance" rather than "coirmon condition" type. This is because
AQ = 0, and so does not contain a world at which r is true.
One can still argue for the limit assumption, but not quite as
definitively as Herzberger suggests. Such an argument might proceed
by drawing attention to the fact that in the above example, since the


CHAPTER FIVE
COUNIERFACTUALS AND THE COMPARISON OF WORLDS
5.1 An Adequate Counterfactual Logic
In this section we will consider several candidates for a minimally
adequate counterfactual logic, all containing CA. Any of them inposes at
least a partial order on the set of possible worlds, relative to each
base world, with the base world minimal in the order. Our argument for
this is based upon the plausibility of certain conditional theses.
Preparatory to this we make several observations about comparative
order semantics and the conditional logics discussed in Section 4.8.
Figure 5.1.1 diagrams the containment relations of all the logics dis
cussed in that section. Included are descriptive names for each family
of logics of increasing materiality, where each family is of increasing
strength in terms of the comparison of possible worlds required by the
appropriate semantics for that family.
For ease of reference, we restate the semantic conditions, origi
nally placed on selection functions, in terms of a comparative order
relation R c U x U x U, where R is the order relative to world u. Re-
call that R /X represents the R -minimal elements of X A Dom R .
u r u u
C5.1.1: For all X,Y,Z c U, and all u e U:
(id) Ru/X c X.
(mp) If u e X, then u e R /X.
u
(cc) If u e X, then {u} = R^/X.
219


15
El.2.1: If that match were struck, then it would light.
We will symbolize El.2.1 as Wpq. The linguistic account attempts to
formalize our earlier procedure:
Dl.2.1: Wpq is true just in case there is a set of true factual
statements F and a set of laws L such that the conjunction of F, L
and p logically implies q.
As Goodman [27] pointed out, determining just what should go into F and
L is no easy task. Certainly such facts as that the match is well-made,
there is enough oxygen present, the match is dry, etc., should belong to
F, while certain chemical and physical laws belong to L. In fact, if we
let L consist of the single physical law "Matches satisfying conditions
C light when struck," where C incorporates the circumstances referred
to as facts above, then this law together with F and p logically imply
q, since the truths in F guarantee the satisfaction of conditions C.
We can just check each condition in C and see if it is satisfied by the
circumstances surrounding this particular match.
This approach would require that for each counterfactual we have
a highly specific covering law, the law itself specifying what must gp
into F. This shifts the problem of determining the truth of the counter-
factual to a problem of determining whether a certain highly specific law
is true, perhaps on the basis of other less specific, more general laws.
In either case, we somehow have to identify the relevant conditions F.
Shifting the problem to specifying a particular law of limited
generality does not solve it, since the problem of detennining the spe
cific facts F is now transformed into the problem of determining the spe
cific conditions C under which the law holds. Furthermore, this approach
would not work for "even if'-conditionals where there is no covering law


90
do so when the strictness of the variably strict conditionals involved
coincide. That the analysis here presented does help to explain this,
is, I think, a strong point in its favor.
In the next two sections and in CHAPTER THREE we shall consider
aspects of Lewis' analysis that are more problematic: his rejection of
the limit assumption, his possible worlds realism and the notion of
comparative overall similarity itself.
2.6 The Limit Assumption
In Section 2.3 we observed that closure under unions and inter
sections imposes a certain kind of bounding condition on $u and subsets
thereof. That is, there is a smallest and largest sphere in $ A$u
and U$u, respectively. Also any nonempty subset X of $u is bounded both
above and below by spheres UX and AX respectively. However, the closure
conditions are not as strong as they might be; it is not the case that
for all X c $ either UX e X or that AX e X. We restate here what Lewis
calls the Limit Assumption, LA (actually it is a "lower-limit" assumption)
C2.5.1: If 0 + X c $ then AX e X.
If we replaced the closure under (nonempty) intersections condition
(C2.3.4) on $ by this, we would have a sphere function which satisfied
the limit assumption.
A sphere function which satisfied LA would allow us to speak of a
"closest" sphere satisfying any given condition, since it would be the
intersection of all spheres satisfying that condition. For instance, for
proposition p we would be assured of a closest p-permitting sphere. We
could then define truth for the "would"-conditional more simply:


63
If for some reason it is desirable to preserve all of the relations
of the traditional square of opposition, then we must exclude vacuous
truth for the conditional Wpq. That is, if the antecedent of the con
ditional is impossible, we require it to be false. For this purpose the
following two definitions suffice:
D2.3.4: Wpq @ u iff there is some p-permitting sphere in $ and Cpq
@ every world in that sphere.
D2.3.5: Vpq @ u iff every p-permitting sphere in $u contains at least
one Kpq-world.
These have the virtue of still preserving the interdefinability of the
"would"- and "might"-conditionals.
Lewis' definition (either one) of the "might"-conditional is of
considerable importance, offering as it does a way of resolving problems
which had hertofore been labeled irredeemable. Rescher gives the fol
lowing examples of "purely hypothetical counterfactuals," i.e., those
not thoroughly grounded in laws [83, p. 162]:
E2.3.4: If Bizet and Verdi were compatriots, Bizet would be an Italian.
E2.3.5: If Bizet and Verdi were compatriots, Verdi vrould be a Frenchman.
E2.3.6: If Georgia included New York City, this city would lie south of
the Mason-Dixon line.
E2.3.7If Georgia included New York City, this state would extend
north of the Mason-Dixon line.
His view is that "these opposed results cannot be avoided" because
"The contextual ambiguity of the antecedent gives us no way of choosing
among the various mutually rebutting counterfactuals" [83, p. 162].
It is clear that these are rebutting only if one holds that con
ditional excluded middle is valid for the counterf actual conditional.


234
A complementary difficulty infects the notion of minimal change.
Consider the following example, which is based upon a recent actual
event at a New York racetrack in which a 99 to 1 longshot was the only
horse to finish. In a horserace there are two longshots, the Gainesville
Galloper and the Florida Flyer. In every race they have run, the Galloper
and/or the Flyer are trailing behind the pack at the finish. The only
way either could win is if something untoward were to happen simultaneous
ly to the other horses in the race.
In a particular race the Galloper and the Flyer are both entered
and, as usual, are trailing near the finish. Suddenly the leader falls,
and the other horses, bunched together, do not have enough time to get
around, so pile up with the leader. Except, the two slowpokes are well
behind, and have a chance to get out of the way. In the actual world,
neither horse is quite quick enough, so no horse finishes the race. We
consider four alternative worlds:
E5.2.2:
u: the actual world where no horse finishes
v: a world in which only the Flyer gets around the pile-up
y: a world in which only the Galloper gets around the pile-up
w: a world in which both get around, the Flyer cones in first,
and the Galloper second
z: a world in which both get around, the Galloper comes in first,
and the Flyer second
The following propositions will figure in what we have to say:
E5.2.3:
p: the Galloper places = {y,w,z}
q: the Flyer places = {v,w,z}


25
employment of the concept works if only because it will fail to dis
tinguish between correct and incorrect applications of it.
Ellis' analysis fails for just the reason we supposed: the
concepts in terms of which the truth conditions for the conditional
are stated require in part that we already understand the concept of
the conditional. We cannot construct the belief system unless we
antecedently understand under what conditions statements of the form
Wpr are held true in system B.
This is not to say that Ellis' semantics is in a formal sense
ill-defined, but rather that there are non-formal criteria that truth
conditions must meet in order to qualify as an analysis. Thus an anal
ysis of the conditional must be in terms of non-conditional notions
just because we are regarding the conditional notions as problematic.
Ellis' definitions amount to a recasting of cotenability in terms of
belief systems. Until we have an analysis of cotenability independent
of the concept of the conditional our analysis will fall short.
A general shortcoming of both linguistic accounts and related
belief-based accounts is that they attempt to model our informal pro
cedure for evaluating conditionals rather than explain it. Of course,
we do take as our assumed basis of reasoning on the assuiption that p
is true what we believe would still be the case on that assumption,
but this is just to say what it is we do, not to explain how or why it
works.
Returning once again to Goodman's analysis, suppose we could
satisfactorily settle the problem of what to include in the set of
specific facts F. Let us turn our attention to the set of laws L.
Three problems immediately arise:


65
With more than one sphere of accessibility assigned to each world
u in U, the concepts of possibility and necessity in their widest sense
need to be correlated with the "largest" sphere. The conditions we have
placed on the neighborhood function $ require that the largest sphere be
U$ for each u in U, and furthermore that U$ = U for all u in U. What
u u
Lewis calls the "outer modalities" [51, p. 22] are defined as follows:
D2.4.1: Lp @ u iff every world in U$u is a p-world.
D2.4.2: Mp @ u iff some world in U$u is a p-world.
In view of the fact that U$u = U for all u in U, these outer modalities
correspond to the logical modalities of S5.
Hence the requirement that p be entertainable can be expressed as
the requirement that Lp be true. Given the definitions D2.3.2 and D2.3.3
of Wpq and Vpq, it then follows that Wpq does not while Vpq does entail
that Lp is true.
We may also, given the above definition of necessity, define a
strict conditional which will be the strict conditional of S5:
D2,4.3: LCpq @ u iff every world in U$u is a Cpq-world.
Referring back to Figure 2.3.4(a), that diagrams a situation in which
LCpq is true. It is then readily seen that LCpq entails Wpq on Lewis'
analysis, since if Cpq is true at all worlds in U$u, it must then be true
at every world in some p-permitting sphere. The converse, however, does
not hold, as Figures 2.3.3 and 2.3.4(d) illustrate: Wpq may be true,
though there are NCpq-worlds (that is, KpNq-worlds).
One may easily confirm from the definition of Wpq that it entails
Cpq, hence we have a hierarchy of conditionals: Lcpq entials Wpq entails
Cpq. However, in no case does the converse entailment hold.


47
need be necessary in themselves, relative to those worlds where the
antecedent is true, the consequent is necessary, i.e., true in all of
them. There are variations on this, of course, but many of them are
amenable to rephrasing the claim in terms of accessibility. Returning
to our future possibles example, a certain conditional presently false
(since in those futures where the antecedent is true, it is not in all
the case that the consequent is true), may be true from the point of
view of one of those futures (since by then certain possibilities may
no longer be accessible, perhaps including those in which the antece
dent was true and the consequent false). What is not inevitable today
may become so by tomorrow, as we often find out to our regret. Hence
the definition, or truth conditions, for the strict conditional should
permit at least the flexibility of D2.2.4.
With this in mind, the following suggests itself:
D2.2.11: Cpq @ u iff for all w e S if p @ w, then q @ w.
Given the usual interpretation of Cpq and in view of D2.2.9 the above
reduces to:
D2.2.12: Cpq @ u iff LCpq @ u.
Henceforth we will use LCpq to denote the strict conditional, unless we
have reason to materially alter our definitions.
We indicated earlier that we could model various senses of pos
sibility and necessity by placing various conditions on the accessi
bility relation, or in our present parlance, on the sphere of acces
sibility. The sphere of accessibility for a given world could range
from the empty set to the entire set of possible worlds or anywhere
in between. Suppose that for a certain world u we have a choice of two
1 2
different spheres: and each determining a somewhat different


36
makes use of both possible worlds and the insights of the metalinguistic
account.
However, this recent history owes much to earlier developments in
modal logic, particularly the algebraic semantics of Lemmon [45, 46] and
other workers in this area. Modal logics were studied systematically by
C. I. Lewis [48], actually significantly earlier than the publication
date of the cited work. A survey of these systems bringing together many
strands of the treatment of modal logics is to be found in Zeman [104].
It is doubtful that the counterfactual conditional would have yielded at
all to logical analysis were it not for the groundwork laid in the study
of the strict conditional.
The concept of possible worlds, which makes its first appearance
in a formal semantics for modal logic with Kripke, is usually credited
to Leibnitz. Its use in semantics for modal logic was prefigured in
significant ways by other authors. A. N. Prior [81, 82] had made use of
possible worlds as moments of time in his study of tense logics. A full
bibliography of his works in this connection may be found in Zeman [104].
Earlier C. I. Lewis had identified possible worlds with the "comprehen
sion" of a proposition [47] in an attempt to explicate meaning. The
same tactic was followed by Carnap [10], who specifically identified pro
positions as sets of "Leibnitz possible worlds," or Wittgensteinian "pos
sible states of affairs."
One striking anticipation of possible world semantics for modal
logic occurs earlier than any of the works cited above. I refer to the
Existential Graphs of C. S. Peirce. The incomplete development by
Peirce of his system of Existential Graphs is traced in Roberts [84].
Of interest to us is the review of Roberts' book by Zeman [105] which


163
counterfactual conditional, and, indeed, that this failure was very
nearly a defining characteristic of counterfactuals. The notion of
the antecedent necessitating the consequent was not simply the notion
of a necessary material conditional.
Now one way of expressing the notion of the antecedent necessi
tating the consequent which we have not heretofore considered is to
assign a necessity operator to each sentence:
D4.41: Lab = ^ "b is a-necessary"
This notion of a "sententially indexed modality" is mentioned by Lewis
[51, p. 60], and is closely related to Chellas' [11] development of
conditional logic. The virtue of this as a starting point is that it
will enable us to make a rather easy transition from modal logics to
conditional logics.
Consider the characteristic rule of inference for classical modal
logics:
RE: Frctn Eab infer ELaLb.
Suppose the "L" is a sententially indexed modality. Then RE would ap
pear as:
RE': Fran Eab infer EL aL b.
c c
The force of RE in modal logics is that if two sentences express the
same proposition (and so are equivalent) then either they are both neces
sary or neither is. With respect to sententially indexed modalities the
analogous requirement is expressed by RE'.
However, with sententially indexed modalities a second considera
tion akin to that mentioned above arises. If a sentence c is a-necessary,
and a expresses the same proposition as b, is c also b-necessary? If so,


212
< U,f > is equivalent. To show (cb) holds we most show that either
R /X c R / (X U Y) or R /Y c R / (X U Y).
u u u u
Suppose by way of contradiction that v e Ru/X and v l R^/ (X U Y)
and w e R^/Y and w l RJ (X U Y). It can be tediously verified that one
of the diagrams of Figure 4.8.2 is a subset of X U Y.
v w v,w
I I
W V
I I
z,y zy z.y
Figure 4.8.2
But then in no case are v and w R^-minimal elements of X and Y, as was
assumed, whatever combination of z,y is in X or in Y. QED
T4.8.24: The logic determined by the class of weak-totally ordered com
parative order frames contains V.
Proof: Let F = < U,R > be a weak-totally ordered comparative order frame
and define the equivalent selection function frane as in T4.8.23. Our
proof of T4.8.20 can be modified to show that (id), (md), and (co) hold,
so we show that (cv) holds. We must show that R^/X c Y implies either
R /X c U Z or R /(X A Z) c Y.
u u
Suppose R^/X c Y and (R^/X) A Z / 0. By way of contradiction, sup
pose w e RU/(X A Z) and w l Y. Recall that by L4.8.22 RU/(X A Z) =
(v,w e X A Z : vR^w and wR^v}. Since Ru/X c. Y and w i Y, w i Ru/X, though
by (id) w e X A Z. Also suppose v e (Ru/X) A Z. By (id), v e X A Z.
So we have v,w e X, and as v e R /X and w l R /X, we have Ail w and w{l v,
u u u u
a contradiction. QED
In view of T4.8.21 and T4.8.24, we may conclude:
T4.8.25: CP, CA, SS are properly contained in V, VW, VC.
An alternative proof of part of T4.8.25 may be found in Nute [74, pp. 95-96].


228
cases that allow it to oppose CC. Having pared it down this far, we can
greatly simplify our logical task by discarding it entirely as an element
of a theory of truth. I am willing to retain it as part of a theory of
assertability.
In the expectation that these reasons are not absolutely persuasive,
we can continue to regard CC as somewhat problematic. If we do accept CB,
however, our logic will then be CA + CB, and thus correspond to a semi-
connected partial order of possible worlds. The presence of CC would
render u R^-least, rather than merely R^-minimal, and give us SS + CB.
Having accepted a partial order in any case, thesis CV is then
critical in distinguishing a partial order from a (weak) total order. If
we conpare CP and CV, we see that CP permits the strengthening of an ante
cedent with any counterfactual consequent of that antecedent. On the
other hand, CV permits strengthening the antecedent with any proposition
cotenable with that antecedent. The use of "cotenable" is appropriate in
this connection since Goodman [27, p. 15] defines "p is cotenable with q"
as NWpNq, which is just Vpq in our symbology. In fact, Loewer uses
Cot(p,q) precisely where we use Vpq [57, p. 102].
An alleged counterexample to CV has been presented by Pollock, and
we reproduce it below in our symbology [80, pp. 43-44]. Suppose p, q, and
r are three unrelated false statements: "My car is painted black," "My
garbage can blew over," and "My maple tree died." A substitution instance
of CV is:
E5.1.4: CKWApqNrVApqArqWKApqArqNr
From E5.1.4 we may derive:
E5.1.5: CKNWApqpNWAKprqqVApqr


231
once more by the dilemma Loewer [57] poses for those analyses of the
counterfactual conditional that utilize a relation of comparative simi
larity:
E5.2.1:
(a) If similarity ordering of possible worlds is primitive
to a semantics for the conditional, then it cannot support
an analysis.
(b) If similarity ordering is not antecedently well-enough
understood, then it cannot support an analysis.
(c) Similarity ordering is either primitive or not well
understood.
(d) . Similarity ordering cannot support an analysis.
Ultimately, we shall indicate how one may go between the horns of this
dilemma, as some of our prior comments have suggested.
The dilemma should be taken seriously, however. The distinction be
tween a logic and an analysis can be summed up in two words: validity
and truth. A logic, even if presented in terms of a semantics providing
truth conditions, can only partition the set of sentences into the valid
and the nonvalid, and into the contingent and the noncontingent. Among
the contingent sentences, a purely logical "analysis" will not provide a
partition into the true and the false.
The right kind of truth conditions must not only provide necessary
and sufficient conditions for validity, but must also explain our identi
fying certain contingent sentences as true and others as false. We do ac
cept certain sentences and reject others, and to the extent that we are
systematic in doing so, an analysis must provide a theoretical basis.
We have argued previously that what is required is an explanatory analysis.


213
We prefer the above proof showing CB is not contained in SS to Nute's
proof showing CV is not contained in SS. First, our proof is simpler,
and second, it reveals another family of logics which are partially, but
not totally, ordering. This follows from the fact that we can add CB to
CP without producing the weak total order that characterizes V.
First, we observe that the conditions on a selection function frame
determining CP suffice to define an equivalent comparative order frame
with vRw iff f(u,{v,w}) = {v}, and that the conditions on a selection
function frame determining V suffice to define an equivalent comparative
order frame with vR w iff either f(u,{v,w}) = {v} or f(u,{v,w}) = {v,w}.
Basically the proofs of T4.8.20 and T4.8.25 are reversed to show this.
T4.8.26: A selection function frame satisfying (id), (md), (co), and (ca)
can be partially ordered. That is, there is an equivalent partially or
dering comparative order frame.
Proof: Let F = < U,f > be a selection function frame satisfying (id),
(md), (co), and (ca). Define R c U x U x U by:
vR^w iff f (u, {v,w}) = {v}.
That the resulting frame < U,R > is equivalent given what we show below
is not difficult to show. We note that is antisynmetric by definition.
By (id), either f(u,{v}) = 0 or f(u,{v}) = (v). If the former,
then by (md), f(u,Y) A {v} = 0 for all Y c U. Hence v l Dorn R If the
u
latter, vR^v, hence R is reflexive.
For transitivity, assume vR^w and wR^z and show vR z. So we have
f(u,{v,w}) = {v} and f(u,{w,z}) = {w}. Note that as v e Dorn R ,
u
f(u,{v}) = {v}. By two applications of (ca), f(u,{v,w,z}) c f(u,{v,w}) U
f(u,{z}) and f(u,{v,w,z}) c f(u,{v}) U f(u,{w,z}). Hence, f(u,{v,w,z}) c
{v} U f (u, {z}) and f(u,{v,w,z}) c {v} U (w). Therefore, f(u,{v,w,z}) c


30
It is commonplace that laws (or more properly, lawlike state
ments, of which the true ones are laws) are generalizations. It is
apparently equally commonplace that they are not material generaliza
tions, many of the latter being clearly accidental rather than law-
like in nature, e.g.,
El.2.3: All the coins now in my pocket are silver
as opposed to
El.2.4: All pulsars are neutron stars.
Thus one of the problems of laws is to distinguish in a noncircular way
between accidental and lawlike generalizations. Now it is clear that
laws support counterfactuals, but this cannot be the distinguishing
characteristic of laws, or if it is, then we have placed laws squarely
in the analytic circle with counterfactuals (if we continue to analyze
counterfactuals in terms of laws).
A material generalization is conclusively confirmed in virtue of
the vacuity of its antecedent or by exhaustive enumeration. Such is
not the case with laws. A law, for example Newton's second law of mo
tion, may indeed have a vacuous antecedent, but it is not true in
virtue of that. Other examples could be cited, but this may be beating
a dead horse. It is generally admitted that laws are not material gen
eralizations .
However, it is equally obvious that laws are generally conditional
statements of some sort as well as generalizations of some sort, partic
ularly the causal laws usually taken to be intimately related to coun
terfactuals. In fact, considering the match example again, the "fa
vored" law II is a generalized conditional prediction, while its aber
rant relatives are conditional postdictions. Indeed Stalnaker suggests


126
The key element of Stalnaker's account is what he calls a selec
tion function, but what we shall call a world-selection function (see
[96, p. 171]):
D3.2.6: Let U be a set of possible worlds and K the absurd world at
which every proposition is true. A world-selection function is any
function f: U x P(U) -* (U U K) such that for any propositions p,q e P(U)
(we identify a proposition with a set of possible worlds) and any world
w e U:
(a) p @ f(w,p).
(b) If there is no u e U at which p is true, then f(w,p) = K.
(c) If p @ w, then f(w,p) = w.
(d) If p @ f(w,q) and q @ f(w,p), then f(w,p) = f(w,q).
The truth condition for conditional Wpq can then be stated in terms of
f as follows:
D3.2.7: Wpq @ w iff q @ f (w,p).
Now, as at any world w, a given proposition q is either true or false
exclusively, Stalnaker's semantics validates conditional excluded middle:
CM: AWpqWpNq
as we have noted previously. Thus the "might"-conditional cannot be de
fined as in Lewis' account.
We may further conpare Stalnaker's account to Lewis' by noting what
variety of order Stalnaker's system requires.
D3.2.8: Let U,K,f be as in D3.2.6. We define S c U x U x (UUK) as:
vSuw iff either w = K or for some p such that p @ v
and p @ w, f (u,p) = v.
If we fix our attention on one world u e U, then S establishes a well-
u
order of U U K with u the S^-least and K the S^-greatest elements.


117
been mnrp frequently followed by C-events than D-events in the most
similar worlds, including time t, so (b) will be judged true, erron
eously.
Jackson is certainly right in supposing the closest worlds will
have the sane laws and that A-events will have been more frequently
followed by C-events than by D-events in the most similar worlds, but
he is certainly wrong in supposing that in the most similar worlds C-
events will have occurred at time t. Let us suppose that the most
similar worlds have exactly one more A-event than the actual world,
and it occurs at time t. Otherwise, the most similar worlds have pre
cisely the same A-C and A-D combinations at every time an A-event oc
curred. Jackson would have it that because A-events are actually more
frequently followed by C-events, that this extra A-event will be followed
by a C-event in the closest worlds to maximize similarity. But if the
closest worlds do have the same laws, and no hidden ones that we do not
know about, then this assumption is unwarranted.
To see that this is the case, consider a more prosaic, but suf
ficiently analogous example. Suppose that in fairly flipping a fair
coin I produce a string of ten heads and then quit. If I had flipped
the coin again, then certainly it would have been either heads or tails,
but would it have been heads? Unless I irrationally believe in runs of
luck, I would not bet on it. Let us apply similarity considerations to
this example. Presumably the closest worlds all have the same laws, in
cluding the laws governing fair coin flipping. Among the closest worlds
will be the worlds where ten beads occurred in the first ten flips. Now
consider two of those worlds: in one the eleventh flip (not made in the
actual world) is heads and in the other tails. Otherwise they are as


17
Pollock [80] observes that if all that is required for inclusion
in F, as Goodman appears to believe, is truth and cotenability, then
this implies an even stronger requirement on the truths in F. To show
this we require acceptance of two obvious principles regarding counter-
factual s :
(A) If Wpq is true and LCqr is true, then Wpr is true.
(B) If WpCpq is true, then Wpq is true.
We postulate the following in accordance with Goodnan's proposals:
(a) Wpq is a counterfactual to be evaluated and p is false.
(b) Cotenability and truth are sufficient for inclusion of
r in set F; i.e., r is true and NWpNr is true.
Noting that LCKpNrNr is true and using (A) contrapositively, it follows
that NWpKpNr is true. Noting that LEKpNrNCpr is true and using (A)
contrapositively again, it follows that NWpNCpr is true. Since p is
false, Cpr is true, hence by (b) Cpr is included in F as it is true
and cotenable with p. Since F together with p logically implies any
thing included in F, Goodnan's proposal validates WpCpr. Hence by (B),
Wpr is true. So "r is cotenable with p" amounts to Wpr is true on
Goodman's assumptions [80, p. 11].
If F is to include everything cotenable with p, then F includes
the consequents of all true statements of the form Wpr. That is, F in
cludes everything that would be the case if p were true. Counterf ac
tuals are analyzed in terms of cotenability (and other elements), but
then cotenability is analyzed in terms of counter factual s, and so our
analysis is circular. Because he did not see a way out of this vicious
circle in analyzing counterfactuals, Goodman shifted his concern to a
weaker notion, that of dispositions.


181
LA.5.17: CM = CE + OI is complete wrt Cm-
Proof: Though no proper canonical model, satisfies (cm), every sup
plemented model does. Hence the supplementation of any proper canonical
model, say the largest, satisfies (cm), so by E4.5.4, the lemma follows. QED
L4.5.18: CR = CE + OI + CR is complete wrt C .
mr
Proof: We need only show some proper canonical model for CR satisfies
(cr). Let be the largest proper canonical model for CR. Then MR
satisfies (cr) whenever X ^ |a| for every formula a, since in those cases
we have set N(u,X) = P(U). Suppose X = |a| for some formula a. Let
|b|, 1 c| £ N(u, |a|). Then Wab,Wac e u and so KWabWac e u, by the properties
of maximally L-consistent sets. By closure under MP, and using the instance
of axiom CR in u of the form CKWabWacWaKbc, WaKbc e u. Consequently,
|Kbc| e N(u,|a|). So |b| A |c| eN(u,|a|). QED
L4.5.19: CK=CE + CM + CR + CNis complete wrt C
Proof: In view of the proof of L4.5.18, we need only show the largest
proper canonical model for CK satisfies (cn). It obviously does if
X / |aj for every formula a, so assume X = |a| for some formula a. Since
CN holds, for every u, Wal e u. Hence |1| e N(u, |a|), but |1| = U, so
U e N(u,|a|). QED
The completeness of CE + CR and CE + CN are obvious in view of the
proofs of the above. The completeness of CE + OQ offers an interesting
twist, as no proper canonical model satisfies (cq).
L4.5.20: CE + OQ is complete wrt C .
Proof: Let be the supplementation of the largest proper canonical
model for QE + OQ. Then for X i= | aJ for every formula a, N(u,X) = P(U),
so (cq) holds. Assume X = |a| for some formula a. Since OQ holds, it


73
Figure 2.5.2
E2.5.3 and E2.5.4(c) present an interesting contrast. Each can
be considered half of a principle of semisubstitutivity of the counter-
factual conditional with respect to the strict conditional. The in
validity of the first is essential if we are to avoid the fallacy of
strengthening the antecedent, since from LCKprp and Wpq, WKprq follows


123
C3.2.1: For all x,y,z e U,
(a) xR^x (reflexive).
(b) if xR^y and yR^z, then xR^z (transitive).
(c) either xR^y or yR^x (connected).
(d) not xR u for all x f u.
u
Conditions (a) and (b) follow iirmediately from D3.2.1, while the towering
of the spheres leads to (c) and the centering condition to (d). That R
is a weak order follows from the fact that we may have both xR^y and
yR x without x = y.
However, if we define
D3.2.2: [x] = {y : both xR^y and yRux}
then [x] is an equivalence class of worlds, all equally similar to u, and
the set of equivalence classes is totally ordered by the induced relation
R defined as R on any representatives of two equivalence classes. Then
[u] consists of a single element, u, and is R least. Since Lewis ex
pressly wishes to permit ties in comparative similarity, in general, for
world w, [w] will not be a singleton set.
Using the comparative similarity relation we may then state the
truth condition for a counterfactual conditional as:
D3.2.3: Wpq @ u e U iff either
(a) there is no p-world in U, or
(b) for some w e U, w is a p-world and for all v such that
vR^w, v is a Cpq-world.
This is Lewis' definition [51, p. 49], though incorporating the restric
tion that U$u = U. Lewis shows that the resulting semantics is equiva
lent to the sphere function semantics.


195
nonconditional formulas, and whatever else is required are as before.
(The truth condition for Vab is derived from that of Wab and D4.7.1.)
D4.8.1: Selection function semantics: Let U be a set of possible
worlds:
(a) A selection function is any function f:U x p(U) -> P(U).
(b) A selection function frame F = < U,f > is an ordered pair
where U is a set of possible worlds and f a selection function.
(c) A selection function model M = < U,f ,V > is an ordered
triple where < U,f > is a selection function frame and V:P -* P(U)
(P is the set of atomic sentences of CW) is a valuation.
(d) |= Wab iff f(u,||aj|M) c ||b||M.
(e) |g Vab iff f(u,||a||M) A ||b||W + 0.
Classes of frames will be specified by stating the conditions, if
any, which apply to the selection function. The following list comprises
the conditions we shall be using:
C4.8.1: For all u e U and all X,Y,Z e P(U):
(id)
f(u,X) c
X.
(mp)
If u e X,
then u e f (u,X).
(cc)
If u e X,
then f(u,X) = {u}.
(md)
If f (u,X)
= 0, then f(u,Y) A
X = 0.
(co)
If f(u,X)
c Y and f(u,Y) c X,
then f(u,X)
= f (u,Y).
(ca)
f (u,X U Y) c f (u,X) U f (u,Y).
(cb)
f(u,X) c :
f(u,X U Y) or f (u,Y)
c f (u,X U Y)

(cv)
If f(u,X)
c Y, then either f(u,X) c U Z
or f(u,X A Z) c Y.
(cem)
f(u,X) is
a singleton or 0.
(11)
If X c Y <
and f(u,X) f 0, then
f(u,Y) ^ 0.
(12)
If X c Y and X A f(u,Y) ^ 0,
then f(u,x) =
f (u,Y) A X.


116
We would be justified in settling for a more limited analysis of
counterfactuals, limited to the nomological ones, say, if Lewis' ap
proach were to systematically yield incorrect truth values for some
class of counterfactuals. Substantially this is the point argued by
Barker and Jackson. I shall illustrate with one of Jackson's counter
examples to Lewis' analysis ([36, pp. 4-5]).
We are to assume that an event of type A catases the occurrence of
either an event of type C or type D, but randomly and with equal prob
ability, as is often the case in quantum mechanics. Now suppose that
in fact at time t no A-event occurred. Consider the following counter
factuals:
E3.1.1:
(a) If an A had occurred at t, then a C or a D would have occurred.
(b) If an A had occurred at t, then a C would have occurred.
(c) If an A had occurred at t, then a D would have occurred.
Jackson recognizes that this produces a situation similar to that of the
Bizet and Verdi examples discussed previously. That is, (a) is true, but
(b) and (c) are false. It should be further observed that if "might" is
substituted for "would" in (b) and (c), then the resulting counterfactuals
are true.
Jackson then argues that under certain circumstances similarity
considerations would lead to an erroneous evaluation of (b) as true.
Suppose that in fact A-events have frequently been followed by C-events
in the actual world, which is improbable, but possible. The worlds most
similar to the actual world will have the same laws as the actual world,
and maximizing similarity in terms of particular fact, A-events will have


48
sense of what is possible relative to u. Corresponding to these we
12 12
have two necessity operators, L and L Now if and are disjoint
1 2
or properly intersect, then L and L are not in any obvious way cornpa-
2 1 1 2
rabie; if, however, is a subset of then L p will imply L p for
any proposition p. If the containment is proper, it will not generally
be the case that L p implies L^p. Hence our two necessity operators
will be ordered. In this context Lewis [51, p. 12] describes one
operator (L^) as stricter than the other, and hence a conditional de
fined in terms of one as a stricter conditional than the other.
The difficulty of taking the counterfactual to be a strict con
ditional lies in the variations on strictness of the conditional. For
for any fixed degree of strictness of the conditional, it is always pos
sible to strengthen the antecedent:
E2.2.1:
LCpq
. LCKprq
is valid for any operator of fixed strictness, L, as the following argu
ment shows.
LCpq is true at u iff at every w e Cpq is true. But if Cpq is
true at any world w, then CKprq is true, since strengthening the antece
dent is valid for the material conditional. Hence the truth of LCpq
leads inexorably to the truth of LCKprq, with no particular conditions
of the function S, since with CKpqr true at every world in S LCKprq
will be true at u.
However, for any given counterfactual (or at least those with
contingent antecedents) it is possible to ''undermine'' the antecedent


125
of possible worlds alternative to Lewis' account may be seen as varia
tions on these order requirements. We shall consider those of
Stalnaker [96, 97], Pollock [80], and Nute [67, 68, 74] below.
All of these accounts, and most of the metalinguistic and belief-
based accounts as well, start from approximately the same rough model
of what is involved in counterfactual deliberation: to evaluate Wpq
we consider a hypothetical situation constructed more or less on the
basis of what is true by adding antecedent p and altering just what we
must of what is actually the case, and then we see if q is true in that
situation. We do not arbitrarily change things irrelevant to the truth
(assumed) of p. The classical problem had been to determine just what
changes were relevant, the problem of cotenability, as Goodman [26, 27]
would have it. All the above mentioned authors (excepting Goodman)
agree that possible worlds can help us produce a more precise account of
what is involved in this rough model. In doing so, they all inpose on
the set of possible worlds some variety of order.
Stalnaker's account is the simplest and imposes the strongest
order requirements. The refinement of the rough model on which he bases
his formal account is that we select a possible world w "which differs
minimally from the actual world" (my emphasis) at which antecedent p is
true. If consequent q is also true at w, then Wpq is true at the actual
world. In presenting his account Stalnaker makes use of the notion of
an "absurd world" and leaves open the question as to whether every world
is accessible to every other. The former is to handle cases where the
antecedent is impossible, and the latter is to allow for varieties of
modal logics depending upon the accessibility relation adopted. We shall
simplify by continuing to assume that accessibility is universal.


174
(cq) N(u,X)=P(U).
(cs) N(u,X) = 0.
Because N is a function whose domain is U x p(U), we cannot directly
classify worlds as, say, singular, as in Section 4.2, but will first have
to classify world-proposition pairs, then worlds, then frames.
D4.5.6:
(a) A proposition X is singular at u iff (cs) is satisfied.
(b) A proposition X is monotonic at u iff (cm) is satisfied.
(c) A proposition X is regular at u iff (cm) and (cr) are
satisfied.
(d) A proposition X is normal at u iff (cm), (cr), (cn), are
jointly satisfied.
D4.5.7: A world is singular (monotonic, etc.) iff every proposition is
singular (monotonic, etc.) at that world.
D4.5.8: A frame is singular (monotonic, etc.) iff every world in that
frame is singular (monotonic, etc.).
Note that the joint satisfaction of (cm), (cr), and (cn) amounts to re
quiring that N(u,X) be a filter.
For any model, clauses (a) through (g) of E4.2.2 hold, so that we
may work with the same extensions of our truth definitions to defined con
nectives. In addition, we may define the following connective, intended
to represent the "might"-conditional, and a corresponding truth condition:
D4.5.9:
(a) Vab = ^ NWaNb
(b) |= Vab iff | =/= WaNb
We also call attention to the following conditional axioms:


184
D4.6.4: Let M < U,R,Q > be a prepositional relational model. Then
we define truth in M at world u for conditional formulas by:
|= Wab iff u i Q||a|| and (v : uR||a||V} c ||b||.
Chellas [11, pp. 134-135, 138] discusses propositional accessibility func
tion frames for the cases where Q = 0, that is, no singularities are pre
sent, as "standard" frames, and mentions the corresponding relational
frames. Nute [74, chapter 2] conpares various semantics for conditional
logics and discusses both the sentential and propositional versions of
normal accessibility function and relational frames (where normal means
Q = 0). All of Nute's discussions are in terms of models and classes of
models, which is necessary for the sentential version. Nute terms normal
accessibility functions "class selection functions" and distinguishes the
propositional from the sentential version [74, p. 63].
Given a propositional accessibility function frame, we can define
a corresponding propositional neighborhood frame:
D4.6.5: Let F = < U,S,Q > be a propositional accessibility function frame.
Then the corresponding propositional neighborhood frame is the ordered
pair F = < U,N > where N: U x p(U) -* P(P(U)) defined by
[ 0, if u e 0 e Q
N(u,X) = \
l {Y : Y c U and {v : v e U and v e S(u,X)} c Y}, otherwise
It is clear that we can do the same for propositional relational frames
as well, and in one-to-one correspondence with the above: replace
v e S(u,X) by uR^v in the definition of N(u,X).
In view of our earlier completeness results and the above defini
tions, we may state the following theorems:


93
a closer world where it is between 1" and 1+x" long, for each positive
value of x. Granted this assumption, Pollock then claims that the fol
lowing sentence is true on Lewis' analysis for each positive value of
x [80, p. 19]:
E2.6.1: If the line were more than an inch long, it would not be
1+x" long.
For this to be true it must be the case that in sane antecedent-
permitting sphere every antecedent-world is a consequent-world. And
this for each x. That is, for each x, there is a sphere where'the line is
more than an inch long at some world, and at every world where it is
more than an inch long in that sphere, it is not 1+x" long. Let us as
sure this condition is met, though it is not clear that Lewis meant it,
and certainly does not need it to make his point.
With x going to zero, it follows that the line would not be 1+x"
long for all positive values of x, hence the line would not be more than
one inch long; for if it is, it is by some positive amount. So, Pollock
concludes, if the line were more than one inch long, it would not be
more than one inch long, a flat contradiction [80, p. 19].
All that saves Lewis' semantics from evident inconsistency is
that the key principle used above is not valid on that semantics [80,
p. 20]:
E2.6.2: The Generalized Consequence Principle (GCP):
If G is a set of sentences and for each q e G Wpq is true, and G |=r,
then Wpr is true.
GCP is the version of CP generalized to all sets of sentences, including,
as in Pollock's example, infinite sets. While CP is, as we have noted,
valid, and its finite generalization is valid since we can then take G


19
notions are all unproblematic, as are the rationality requirements not
presented here. What is of concern is the definition of the modified
belief system if constructed from B which serves to characterize the
conditional Wpq.
According to Ellis, belief system B^ "can be thought of as the
assumed basis of reasoning from the supposition that p" [19, p. 109].
For the counterfactual conditional, Ellis' definition of B^ may be
paraphrased as follows [19, p. 112]:
Dl.2,3:
(1) r is held true in B^ if either Lr or Wpr is held true in B.
(2) p is held true in B\
(3) Otherwise B^ is agnostic.
In Ellis' words B^ "includes not only what we should take to be neces
sarily true, but also what we think would still be or have been the case
if 'p' were . ." [19, p. 113].
In view of condition D1.2.3(l), as an analysis of the counterfac-
tual condition this account is circular. D1.2.3(l) replaces Goodman's
notion of cotenability, and Dl.2.2 replaces the requirement that the
conjunction of F, L, and p logically implies q. What then does Ellis'
account accomplish? And why, considering the family resemblance of
D1.2.3(l) to cotenability should this account be thought to advance the
theory of conditionals? To answer these questions requires a digression
on the subject of what constitutes an analysis.
"Analysis" can mean one of two things, not necessarily exclusive.
Both are routes for the clarification of a concept. One is to explicate
or articulate the concept in terms of other, presumably better under
stood, concepts. In this context an analysis is much like a definition;