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## Material Information- Title:
- Counterfactuals
- Creator:
- Mayer, John Clyde, 1945-
- Publication Date:
- 1980
- 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.
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- University of Florida
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- 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.
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- 023407697 ( ALEPH )
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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. 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 |

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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 Â£ 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 Â£ 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) Â£ ||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} Â£ 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 Â£ $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 Â£_ 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}Â£$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 Â£ ||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 AiÂ£ 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 Â£ 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 Â£ 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) = |pÂ¡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|) Â£ 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} Â£ v} c (v : (c : Wac e u} Â£ v} U {v : {c : Wbc e u} c v}. Therefore, f(u, |Aab|) Â£ 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|) Â£ |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|Â¡. 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 Â£ "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, ||pÂ¡|) 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 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 Â£ 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 Â£ 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||. 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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 Â£ 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 Â¡|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 ||bÂ¡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) Â£ (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 ||plÂ¡. 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 Â£ ||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| Â£ 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; |