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The Inexpressibility of Truth

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Title: The Inexpressibility of Truth
Physical Description: 1 online resource (150 p.)
Language: english
Creator: Badici, Emil E
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: chihara, expressibility, herzberger, inclosure, inconsistency, inexpressibility, liar, paradox, priest, semantics, tarski, truth
Philosophy -- Dissertations, Academic -- UF
Genre: Philosophy thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The main purpose of my study is to explore and defend what I call an inexpressibility account of the semantic paradoxes. I argue that contrary to the widely held belief that English (as well as every other natural language) is universal, or at least semantically universal (in the sense that all its semantic concepts are expressible in it), the lesson one should draw from the Liar paradoxes is that in fact English fails to express the concept of truth. Taking this view provides the foundation of a satisfying resolution of the Liar paradoxes, but the promise of the view has been underappreciated for reasons which, I argue, turn out to be not well-founded. For example, the view might appear to be self-defeating because, it might be objected, in defending the inexpressibility view one actually expresses the concept that is held to be inexpressible. I argue that this and related challenges are founded on a failure to make certain fundamental distinctions such as the distinction I introduce between intended meaning and linguistic meaning. The distinction enables one to explain why communication is unproblematic although the concept of truth is inexpressible. The main virtue of the inexpressibility view is that it offers a solution to the Liar paradoxes without postulating that truth is an inconsistent concept. Moreover, the account can be easily extended to apply to all semantic paradoxes.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Emil E Badici.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Ray, Greg B.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021167:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021167/00001

Material Information

Title: The Inexpressibility of Truth
Physical Description: 1 online resource (150 p.)
Language: english
Creator: Badici, Emil E
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: chihara, expressibility, herzberger, inclosure, inconsistency, inexpressibility, liar, paradox, priest, semantics, tarski, truth
Philosophy -- Dissertations, Academic -- UF
Genre: Philosophy thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The main purpose of my study is to explore and defend what I call an inexpressibility account of the semantic paradoxes. I argue that contrary to the widely held belief that English (as well as every other natural language) is universal, or at least semantically universal (in the sense that all its semantic concepts are expressible in it), the lesson one should draw from the Liar paradoxes is that in fact English fails to express the concept of truth. Taking this view provides the foundation of a satisfying resolution of the Liar paradoxes, but the promise of the view has been underappreciated for reasons which, I argue, turn out to be not well-founded. For example, the view might appear to be self-defeating because, it might be objected, in defending the inexpressibility view one actually expresses the concept that is held to be inexpressible. I argue that this and related challenges are founded on a failure to make certain fundamental distinctions such as the distinction I introduce between intended meaning and linguistic meaning. The distinction enables one to explain why communication is unproblematic although the concept of truth is inexpressible. The main virtue of the inexpressibility view is that it offers a solution to the Liar paradoxes without postulating that truth is an inconsistent concept. Moreover, the account can be easily extended to apply to all semantic paradoxes.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Emil E Badici.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Ray, Greg B.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021167:00001


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THE INEXPRESSIBILITY OF TRUTH


By

EMIL BADICI

















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007


































2007 Emil Bddici

































To all of those who struggled to tell the truth









ACKNOWLEDGMENTS

Ancestors of some of the chapters of this dissertation have been presented at the Logica

Conference, June 2005 (Hejnice, the Czech Republic), the Florida Philosophical Association

Conference, November 2005 (Cocoa Beach) and November 2006 (Tampa) and the annual

meeting of the Society for Exact Philosophy, May 2007 (Vancouver, Canada). I am indebted to

my audiences for their helpful comments. I also wish to thank John Biro, William Butchard,

Douglas Cenzer, Michael Jubien, Kirk Ludwig and Elka Shortsleeve for helpful comments and

support over the past five years. I cannot express my gratitude to Greg Ray for patience,

guidance, fruitful discussion and uncountably many comments on previous versions of this

dissertation. Finally, I want to acknowledge a special debt to my wife, Ana-Maria Andrei, for

insightful thoughts and encouragement.











TABLE OF CONTENTS


page

A CK N O W LED G M EN T S ................................................................. ........... ............. .....

L IST O F TA B L E S .......... .... .............. ................................................................... 8

ABSTRAC T ..........................................................................................

CHAPTER

1 IN TR O D U C TO R Y R EM A R K S ................................................................. .....................10

2 LIAR PARADOXES, INCONSISTENCY AND UNIVERSALITY .............. .................17

L iar P ara d o x e s ............................................................................ 17
Paradoxical Sentences .............................................. ............... 17
P athological Sentences ................. .. .... .. .... .. .... ........ ............ ........... 19
Other Kinds of Liars: Utterances, Mental Representations, Propositions.....................20
The Concept of Truth .................................. .. .. ........ .. ............21
S elf-R eferen ce ................................................................22
V icious C ircularity ........................ .. ......................... .... .................22
The "Tarskian" Hierarchy of Languages............... ........................................ 24
Paradoxes w without Self-Reference ............................................................................ 24
Truth-V alue G aps ....................................... .............. ..............................25
Strengthened Liars and the Principle of Bivalence .......................................................27
Martin's Simple Liar Argument ................. ................. ... ............... 30
Choice Negation versus Exclusion Negation ......... .............. ......... ................31
The Principle of Bivalence and Classical Logic........................................... ......36
Reductio ad Absurdum versus Reductio ad Gapsurdum..................... ................38
T h e T -S c h e m a ........................................................................................................... 4 3
U universality and Inconsistency ............................ ................. ........................... .................47

3 THE INEXPRESSIBILITY OF TRUTH ........................................ .......................... 49

U universality and E xpressibility ...................................................................... ...................50
T he Inexpressibility A rgum ent ............................ ............................................................ 57
H erzberger on U universality ............................ ................................................ ................... 59
No Language contains all its Semantic Concepts............................................... 59
Class Expressibility versus Concept Expressibility ................................ ............... 61
Two Objections against the Inexpressibility View ............................................................. 63
Intentions are Sufficient for Expressibility .......................................... ...............63
The Inexpressibility Account is Self-Defeating ................................... .................64
Intended M meaning versus Linguistic M meaning ............................................ ............... 65
T he Status of the T -B iconditionals.............................................................. .....................72









Is the Concept of a True Sentence of English Expressible in other Languages? ...................73

4 ON THE COHERENCE OF THE INCONSISTENCY VIEW OF TRUTH .........................76

T he Inconsistency V iew ............................................................................. ....................77
The Inconsistency of Natural Languages ............................................. ............... 79
M meaning Postulates......................... .... .. .. ...................... .................. 81
Inconsistent Languages and the Inconsistency View of Truth......................................84
The Inconsistency of the Concept of Truth ........... ............................... ...............85
A Priori or Em pirical Inconsistency? ............................... .. ................................ 86
The Inconsistency View and Classical Logic....................................... ............... 88
Skepticism w ith R respect to Inconsistency..................................... ........................ ........... 89
Intentional Inconsistency versus Linguistic Inconsistency................. ............................93
Tw o K inds of Inconsistency .................................... ....................... ............... .93
Intentional Inconsistency .................................................................... ........ ...............96
Inconsistency Entails Inexpressibility ............................................................................. 98

5 N O N -L IN G U ISTIC L IA R S ........................................................................ ...................100

M mental R presentations .............................................................. .. .............. ... .. 10 1
Thoughts and Beliefs .................. .................................... ................ 102
L ia r T h o u g h ts .......................................................................................................... 1 0 3
Gappy Thoughts ............... .... .......... ........ .... ....................104
Intentional states and their propositional content................................................ 106
L iar B beliefs .........................................................................................................111
Liar Propositions ................................. .......................... ..... ..... ......... 113
M entalese Liars ................................. .......................... ...... ..... ........ 115

6 AN EXTENSION OF THE ACCOUNT: SEMANTIC VERSUS LOGICAL
P A R A D O X E S ................................ .................................................................1 18

Semantic Paradoxes ........................................ .............. .... .... .............. 19
Grelling's Paradox............... ......... .......... .. ............... .... 19
P aradoxes of D efi ability ........... ......... ...... ......... ........................... ...................... 119
Richard' s paradox ...... ... ................... ...................... .. ........ .. 119
Berry's paradox ........... ... ......... ........ .......... 120
Konig's paradox ......................................... ................... .... ........ 121
A Denotation Paradox .................................................. ......... 121
The Inexpressibility of the Semantic Concepts ........................................ 122
Similarities between the Semantic and the Logical Paradoxes ......................................124
A L little B it of H history ............................................................... .. ...... ......... ....125
Priest's U uniform ity A account ............. ......................... ................. ............... ..127
The Semantic Version of Russell's Paradox ....................................................130
Refuting the U uniform ity A ccount...................................... ....... ................. ............... 132
A n Objection from Circularity ..................................................................... 132
The Liar and the Inclosure Schema ..................... ......... ..................................... 136
The Existence clause ............................................ ... .... .... .. ........ .... 137



6









T he diagonaliser and the L iar..................................................................... ...... 139
Sem antic L iars and L ogical L iars...................................................................... .. .... 144

L IST O F R E F E R E N C E S ..................................................................................... ..................146

B IO G R A PH IC A L SK E T C H ......................................................................... ... ..................... 150

















































7









LIST OF TABLES

Table page

1-1 ...................................................................................................... ..................... ....... 48

1-2 ........... .................................................... .................................................. ............... 48
















































8









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

THE INEXPRESSIBILITY OF TRUTH

By

EMIL BADICI

August 2007

Chair: Greg Ray
Major: Philosophy

The main purpose of my study is to explore and defend what I call an inexpressibility

account of the semantic paradoxes. I argue that contrary to the widely held belief that English (as

well as every other natural language) is universal, or at least semantically universal (in the sense

that all its semantic concepts are expressible in it), the lesson one should draw from the Liar

paradoxes is that in fact English fails to express the concept of truth. Taking this view provides

the foundation of a satisfying resolution of the Liar paradoxes, but the promise of the view has

been underappreciated for reasons which, I argue, turn out to be not well-founded. For example,

the view might appear to be self-defeating because, it might be objected, in defending the

inexpressibility view one actually expresses the concept that is held to be inexpressible. I argue

that this and related challenges are founded on a failure to make certain fundamental distinctions

such as the distinction I introduce between intended meaning and linguistic meaning. The

distinction enables one to explain why communication is unproblematic although the concept of

truth is inexpressible. The main virtue of the inexpressibility view is that it offers a solution to

the Liar paradoxes without postulating that truth is an inconsistent concept. Moreover, the

account can be easily extended to apply to all semantic paradoxes.









CHAPTER 1
INTRODUCTORY REMARKS

Consider the following imaginary dialogue between an innovative graduate student and his

more skeptical adviser:

A. I've heard this story many times. Tell me your "groundbreaking" solution to the old

paradox of the Liar.

B. No sophisticated reasoning. If the Liar sentence is true, then it is not true. If it is false,

then it has to be true. The only option that remains is that the Liar is neither true nor false.

A. Wouldn't it follow that the Liar is not true?

B. Indeed, we should think that the Liar is not true.

A. But this is what the Liar says. It would follow that the Liar is true after all. You are

contradicting yourself.

B. Things are puzzling indeed. However, I think that you are going too far when you say

that that's what the Liar says. I indeed think that the Liar is not true, but this is not what the Liar

says.

A. This strikes me as utter nonsense. You are saying that the Liar does not say what you

are just saying by using it. If it does not say that the Liar is not true, what does the Liar say?

B. It turns out that there is nothing that the Liar says, and I'm not strictly speaking saying

anything by using it.

A. In that case your sentences are meaningless. How do you expect one to make sense of

your sentences?

B. It would be unfair to say that my sentences are meaningless. You understand every

single word that I have been using. I grant, however, that many of the sentences I used are not

true.









A. You are very subtle indeed. It is enough for me that you acknowledge that they are not

true. End of the story.

B. This shouldn't be the end of the story. Although some of my sentences fail to say

anything, you know what I intended to say.

A. How can I know what you intended to say if your sentences fail to say it?

B. You know what my sentences are intended to say, even if they fail to say it.

You cannot deny that as a result of this conversation you know what I think about the Liar.

A. I wonder whether our exchange of sentences can indeed be called a conversation....

The purpose of this study is to explore and defend the position articulated by B, which I call the

inexpressibility account of the Liar paradox. Contrary to the widely held belief that English (as

well as every other natural language) is universal, or at least semantically universal (in the sense

that all its semantic concepts are expressible in it), the lesson one should draw from the Liar

paradoxes is that in fact English fails to express the concept of truth. Taking this view provides

the foundation of a satisfying resolution of the Liar Paradoxes, but the promise of the view has

been underappreciated for reasons which, I argue, turn out to be not well-founded. For example,

the view might appear to be self-defeating because, it might be objected, in defending the

inexpressibility view one actually expresses the concept that is held to be inexpressible. I argue

that this and related challenges are founded on a failure to make certain fundamental distinctions,

such as the distinction I introduce between intended meaning and linguistic meaning. The

distinction enables one to explain why communication is unproblematic although 'true' fails to

express the concept of truth.

Chapter one consists in an examination of some of the attempts to solve the Liar paradox,

whose outcome is that the main obstacle to reaching a solution is the assumption of the









universality of English. The attempts to avoid the Liar paradox by banning self-reference (such

as the appeal to the theory of types or to the distinction between object-language and

metalanguage) have been very fruitful when applied to formal languages, but they do not seem to

be appropriate for natural languages because the expressive power of natural languages does not

seem to be limited in this way. English allows one to combine predicates and referring

expressions in a sentence without any type restrictions, and it contains many self-referential

expressions which are perfectly meaningful. Moreover, the distinction between an object-

language and a metalanguage cannot be applied, because English is its own metalanguage (it

seems that whatever can be said about English can be said in English). Likewise, the attempts to

solve the Liar paradox by postulating truth-value gaps or any other distinction between three

categories of sentences stablyy true/pathological/stably false) fail because contradictions would

still follow as long as the semantic notions that have been used as part of the solution are

expressible in English. As long as English is held to be able to express its own semantic

concepts, there seems to be no way out of the inconsistency. Thus, there appear to be two main

alternatives: either one accepts that English is universal, in which case one is forced to endorse

some version or other of the inconsistency view, or one can deny the universality of English and

thus avoid the inconsistency view.

The thesis that English is universal (more specifically, the thesis that English is able to

express its own semantic concepts) is usually considered too obvious to be argued for. In chapter

three, which is the central chapter of this study, I argue that the thesis is actually false. In

particular, the concept of truth turns out to be inexpressible in English. The inexpressibility

argument is a reduction argument formulated in ordinary English: I show that the supposition that

there is a predicate of English that expresses the concept of truth leads to a contradiction.









Roughly, the contradiction is obtained by noticing that the Liar argument makes an implicit

appeal to the assumption that 'true' expresses the concept of truth and then exploiting this by

turning the argument into a reduction argument. This version of the inexpressibility argument is to

be preferred to another version that is due to Hans Herzberger [1970], because Herzberger

formulates his argument as an argument for the inexpressibility of a class rather than that of a

concept.

The remaining part of chapter three is devoted to answering two objections that might

readily come to mind. It might be thought that expressibility is only a matter of associating the

right intentions with a certain expression. In this case, the expressibility of truth would be trivial,

because speakers of English do intend to use 'true' to express the concept of true. The other

objection is that the inexpressibility view is self-defeating. In defending the inexpressibility view

of truth I employed the word 'true', which one might think means that I actually expressed

the concept that I held to be inexpressible. Both objections can be satisfactorily answered by

drawing a distinction between the intended meaning and the linguistic meaning of an expression.

Not all the intentions we have with respect to the use of a word are fulfilled. This means that the

intended meaning may fail to become linguistic meaning, which means that expressibility is not

trivially guaranteed. The mere fact that we use the word 'true' in English is not enough to

guarantee that it expresses the concept of truth (i.e., that it has a linguistic meaning), which

means that the inexpressibility view is not self-defeating. On the other hand, communication

remains unproblematic in spite of the fact that 'true' lacks linguistic meaning, because the type

of meaning that is central for communication is the intended meaning, not the linguistic meaning.

Speakers of English know what the intended meaning of 'true' is, and they do not need to know

its linguistic meaning in order to count as competent speakers of the language.









One might grant that the inexpressibility view of truth offers a coherent way to avoid an

inconsistency, but still argue that it would be preferable to endorse an inconsistency view rather

than say that truth is inexpressible in English. Therefore, chapter four is focused on the

inconsistency view. I argue that, properly understood, the inconsistency view of truth is coherent

but the arguments based on Liar sentences fail to establish that it holds. Moreover, an

inconsistency view of truth entails that truth is inexpressible (which also means that English is

not universal). For this reason, an inconsistency view of truth cannot offer the best explanation of

the Liar paradoxes. The inconsistency view of truth has been attacked by many (e.g., Herzberger

[1970], Soames [1998]) as an incoherent view. Nevertheless, most of these attacks are a

consequence of the failure to distinguish between two ways in which a languages can be said to

be inconsistent. I argue that one should draw a distinction between intentional inconsistency and

linguistic inconsistency. Tarski [1933] and Chihara [1979] should be interpreted as advocating

versions of an intentional inconsistency view. It is the meaning principles that are intended to be

true that are inconsistent (either inconsistent as a set, or inconsistent with some empirical facts).

The objectors have usually misinterpreted the view as a linguistic inconsistency view, which

would indeed be incoherent, because it requires that an inconsistent set of principles be all true.

On its turn, intentional inconsistency could be interpreted in two ways which should be carefully

distinguished from one another. The thesis that the intended meaning rules are inconsistent

should be distinguished from the thesis that the intended meaning rules are inconsistent with the

hypothesis that 'true' succeeds in expressing the concept of truth. The Liar arguments offer

enough support for the latter thesis, but not for the former.

Even if one can find other reasons (than Liar sentences) for thinking that the concept of

truth is inconsistent, this would not defeat the inexpressibility view but rather provide support for









it (because the inconsistency view of truth entails the inexpressibility view). Nevertheless, one

reason why the inexpressibility view is attractive is that it can offer a solution to the Liar

paradoxes while preserving the consistency of the concept of truth. The significance of the

inexpressibility account would be dramatically diminished if Liar paradoxes could still occur,

this time not at the level of sentences, but at the level of propositions or mental representations.

This would mean that the inexpressibility view cannot actually provide a way to save the

consistency of truth. In chapter five I argue that there are no Liar arguments at the level of

propositions or mental representations that could force us to adopt an inconsistency view of

truth. The Liar thought (understood as mental representations) can coherently be said to lack a

truth-value. Although it has a true propositional content, the Liar thought itself cannot be said to

be true, because a closer examination of its structure reveals that it fails to be an intentional state

with the mind-to-world direction of fit, which would be required for an intentional state to be

truth-evaluable. Moreover, there are good reasons to think that there could be no Liar

propositions.

The Liar paradoxes are members of the larger family of semantic paradoxes, therefore a

successful account of the Liar paradoxes is expected to show how to account for the other

semantic paradoxes. In chapter six I show how the inexpressibility argument can be extended to

prove that heterology, satisfaction, denotation and other semantic concepts are inexpressible in

English. The same account cannot be applied to the so-called logical paradoxes, which are more

properly handled by mathematical methods (for instance, by restricting the universe of sets by

some axiomatic set-theory, such as Zermelo-Fraenkel's). It has been argued (by Russell and,

more recently, by Graham Priest) that logical and semantic paradoxes have the same structure,

and that similar paradoxes should receive similar solutions. Graham Priest [2002] argues that









both logical and semantic paradoxes have the same underlying structure (which he calls "the

Inclosure Schema") which, in conjunction with the Principle of Uniform Solution (same kind of

paradox, same kind of solution), suffices to 'sink virtually all orthodox solutions to the

paradoxes'. This would also suffice to sink the inexpressibility view, because it also fails to

provide a uniform account. I argue that Priest fails to provide a non-question-begging method to

impugn virtually all orthodox solutions, and that the Inclosure Schema cannot be the structure

that underlies the Liar paradox. Ramsey was right in thinking that logical and semantic

paradoxes are paradoxes of different kinds and that one should not expect them to receive the

same kind of solution.









CHAPTER 2
LIAR PARADOXES, INCONSISTENCY AND UNIVERSALITY

A Liar paradox is due to the fact that a number of intuitively true principles can be used to

derive an inconsistency by using only intuitively valid rules of inference. I will begin with a

survey of the most significant types of Liar paradoxes. Thereafter, I will examine some of the

main attempts that have been made to block the derivation of an inconsistency consisting either

in banning self-reference or in postulating truth-value gaps. The discussion is far from being

exhaustive, and is mainly aimed at identifying the principles that are commonly used to run a

Liar argument and explaining why it so hard to find a way to block it. It turns out that what

makes it difficult to avoid inconsistencies is some form or another of the principle of universality

that is commonly attributed to natural languages. Therefore, one is faced with a choice between

accepting the inconsistency of some intuitively true principles and denying the universality of

English and other natural languages.

Liar Paradoxes

Paradoxical Sentences

Natural languages are known to be capable of self-reference. Thus, one can talk in English

about the sentences of English. Moreover, given an arbitrary sentence of English, one can

introduce a name for it in English. As long as it belongs to the category of names, any expression

that has not already been assigned a referent could play this role. Thus, one can use 'Lo' to refer

to the following sentence-type (call it "the Simple Liar"):

(Lo) (Lo) is false.

(Lo) is a well-formed sentence of English: 'false' is a predicate of English that we understand,

'(Lo)' is a proper name that refers to the Simple Liar, and the sentence obeys the syntactic rules

of English. Nevertheless, it is easy to see that (Lo) is paradoxical. If (Lo) is true, then it is false,









because this is what it says. On the other hand, if (Lo) is false, since this is what (Lo) says, it

follows that (Lo) is true. In either case the conclusion is unacceptable, because no sentence can

be both true and false1.

Another Liar sentence that is very familiar in the literature on paradoxes is the

Strengthened Liar sentence:

(L) (L) is not true.

An informal version of the Liar argument for this sentence goes as follows:

If we respond to the strengthened liar sentence just the way we did to the simple liar, by
saying that the sentence is neither true nor false, then we will have to say, a fortiori, that
the strengthened liar sentence is not true. But that the strengthened liar sentence is not true
is precisely what the strengthened liar sentence says, and we are back in the briar patch.
[McGee, 1990, pp. 4-5]

The Simple Liar and the Strengthened Liar are not the only kinds of sentences that raise

problems for the concept of truth. There are many other sentences that are problematic in one

way or another. Consider first the Pair Liars:

S1 S2 is true.
S2 S1 is not true.

It is obvious that the two sentences are paradoxical. If S, is true, then S2 must be true, so S, is not

true. On the other hand, if S, is not true, then S2 would have to be true, which means that S1 is

true. The difficulty can be generalized to obtain Chain Liars of arbitrary length:

S1 S2 is true.
S2 S3 is true.




Sn S1 is not true.


1 Even the dialetheists, who deny that no sentence can be both true and false, acknowledge that this principle is at
least prima facie true.









Curry's paradox is a slightly different kind of paradox of truth. One version of the paradox

goes as follows. Consider the following sentence:

(1) If (1) is true then God exists.

Since (1) says that if(l) is true then God exists, one can infer (2):

(2) Sentence (1) is true iff if (1) is true then God exists.

Suppose now that (3) holds.

(3) Sentence (1) is true.

From (2) and (3) one can derive (4):

(4) If (1) is true then God exists.

From (1) and (4) one can get (5):

(5) God exists.

Since from the assumption of (3) one can derive (5), it follows that (6) must hold true.

(6) If (1) is true then God exists.

From (2) one can infer (7),

(7) Sentence (1) is true.

which together with (6) leads to the conclusion that (8) is true.

(8) God exists.

Obviously, the same argument pattern can be used to prove that God does not exist or any other

thesis.

Pathological Sentences

Besides the paradoxical sentences, there are also merely pathological sentences, such as the

Truth-Teller, the Truth-Teller Loops, or the infinite Truth-Teller. The Truth-Teller is the

following sentence:

(TT) (TT) is true.









The problem with these sentences is not that they lead to a contradiction, but that their meaning

together with the facts in the world fail to determine a truth-value. All one can say is that (TT) is

true if and only if (TT) is true. However, this is a mere tautology and offers no help whatsoever

in determining whether (TT) is true. Moreover, there does not seem to be any extra piece of

information that could determine its truth-value. Sentences of this sort raise a serious problem for

the concept of truth, because one normally thinks that the meaning of a sentence (plus the way

the world is) is enough to fix its truth-value. What makes things worse is the fact that the

assumption that (TT) fails to have a truth-value leads to a contradiction.

Other Kinds of Liars: Utterances, Mental Representations, Propositions

The Liar paradoxes that I enumerated above are paradoxes in which 'true' is applied to

sentences. However, there are many other kinds of entities that can play the role of truth-bearers.

Thus, one can talk about true utterances, statements, beliefs, propositions, etc. The debates

regarding which of these entities should be taken as the primary bearers of truth can be set aside.

What matters is that all of them can be legitimately said to be true or false. It is easy to see that

one can think of Liar paradoxes corresponding to each of these different types of truth-bearers.

Thus, there are paradoxical utterances, such as some of the utterances of (U):

(U) This utterance is not true.

If (U) is uttered in a context in which the demonstrative refers to the utterance itself, then if the

utterance is true, then it would have to be not true, and if it is not true, then it would have to be

true. Thus, both alternatives lead to a contradiction. Unlike the previous cases, self-reference in

(U) is achieved by using a demonstrative. Similarly, one can talk about Liar beliefs and Liar

propositions as well as Pair Liar beliefs, Pair Liar propositions, Chain Liar beliefs, and so on.

It will turn out to be convenient to distinguish between linguistic Liar paradoxes (such as

Liar sentences, Liar utterances and Liar statements) and non-linguistic Liar paradoxes (such as









Liar beliefs and Liar propositions). I will focus on linguistic Liars in chapters three and four, but

I will also consider the possibility of non-linguistic Liars in chapter five.

The Concept of Truth

A Liar sentence typically contains a predicate, 'true' or 'false', a referring expression and

possibly a form of negation. If falsity can be characterized in terms of truth and negation (a

sentence is false if and only if its negation is true), then the crucial notions involved in these

paradoxes are truth and negation. This does not give enough reasons to think that there is

something problematic with the notion of truth, as it could well be the case that it is the notion of

negation that is responsible for the paradoxes. Nevertheless, there are some reasons to doubt that

negation can be made responsible for the paradoxes. First of all, negation alone is not enough to

generate a paradox. One gets a paradox only when negation is associated with a semantic notion

(truth or another notion). Second, although negation is used in most of the Liar sentences, the

pathological sentences, such as the Truth-teller, do not contain any expression for negation. Thus

it would be fair to say that the Liar paradoxes reveal a difficulty with our notion of truth, a

difficulty which becomes even more troublesome when truth is thought of in conjunction with

the notion of negation.

The difficulties raised by the Liar paradoxes have implications for various dimensions of

the notion of truth. They raise problems for the property of truth, the concept of truth and for the

meaning of 'true'. There are various positions with respect to the difference between properties,

concepts and meanings. It would be useful not to start from the assumption that they are

identical. One thesis that follows from the view that I defend in this dissertation is that the

concept of truth cannot be identical with the meaning of any predicate of English.

However, I will mainly be concerned with the concept of truth, because this is the notion

that plays the crucial role in our scientific, philosophical and everyday thinking. The concept of









truth can be divided into some subconcepts in accordance with the types of truth-bearers. One

can talk about the concept of a true sentence of L, the concept of a true proposition or the

concept of a true belief. Notice that while the concepts of a true proposition or of a true belief are

non-relational, sentences can only be said to be true in a relative way. It appears that although

sentences are often said to be true (or not true), they can only be true relative to a certain

language: a sentence is true in L if and only if it expresses in L a true proposition. Likewise,

utterances can be true only relative to a certain language. However, one can always turn a

relational concept into a non-relational one by specifying the language. Thus, the concepts of a

true sentence of English, or of a true utterance of German, are non-relational. I will occasionally

talk about the concept of truth instead of the concept of a true sentence of English when the

relevant features of the restricted concept extend smoothly to the concept of truth per se.

Self-Reference

Vicious Circularity

It has been held that Liar paradoxes are the result of using expressions that are self-

referential or involve some circularity that is vicious. I will argue that self-reference cannot be

made responsible for the Liar paradoxes. Poincare thought that the paradoxes discovered in set-

theory have to do with definitions that are viciously circular2. Russell shared Poincare's belief

and argued that all paradoxes in semantics and in mathematics involve some sort of vicious

circularity. There actually are more phases of Russell's thought about paradoxes3. He came out

with the idea of developing a theory of types as early as in 1902-1903. Two years later, in 1905,

he proposes instead the no-class theory. In 1908 he returns to his previous project of developing

a theory of types that would avoid vicious circularity. Part of this project is trying to reject self-

2 See Chihara [1973: 3].

3 See Quine's introduction to Russell [1908a] invan Heijenoort [1967: 150-52].









referential expressions as meaningless. The theory of types distinguishes between different types

of expressions and introduces a number of restrictions that would rule out many expressions as

not well-formed. Self-referential expressions fail to meet the requirement of being well-formed,

and would have to be rejected as meaningless. Whatever the merits of this method for

constructing a formal language that is free of contradictions and adequate for the purposes of

science, it cannot be used for natural languages. First of all, not just any kind of self-

referentiality is vicious. Banning all self-referential expressions would amount to a significant

restriction of the expressive power of natural languages. It certainly would be illegitimate to

introduce '(L)' as the name of an expression containing it, if '(L)' already had a referent assigned

to it. The introduction of the new name would be incorrect, for the same reasons we take a

circular definition to be incorrect. However, as Kripke notes [1975: 693], there is no reason why

the introduction of 'Jack' as a name for the sentence 'Jack is short' is illegitimate, as long as

'Jack' has not already been assigned a role in the language. Moreover even if proper names were

not available, English allows one to achieve self-reference by using demonstratives (as in the

case of Liar utterances) or definite descriptions. There is nothing wrong with a self-referential

sentence of the form:

(9) The first sentence in this chapter which begins with 'The first sentence in this

chapter' belongs to English.

Even sentences such as (10)

(10) The first sentence uttered by Russell in 1905 is not true.

should count as legitimate, regardless of whether the first sentence Russell uttered in 1905 is (10)

itself or a different sentence. If (10) is paradoxical, this is a contingent matter, not an intrinsic









feature of the sentence4. The conclusion is that self-reference is not sufficient for paradoxicality,

and that we cannot ban paradoxical sentences for merely containing self-referential expressions.

The "Tarskian" Hierarchy of Languages

Very frequently the "Tarskian" hierarchy of languages is mentioned as one type of solution

to the Liar paradoxes. I use the quotation marks because although he talks about a hierarchy of

languages Tarski never proposes it as a solution to the Liar paradox, and he only thinks of it as

part of the project of devising semantic notions that are appropriate for the needs of science. The

idea behind the hierarchy of languages is to distinguish between an object-language and

metalanguage such that the semantic notions of the object language can only be expressed in the

metalanguage and not in the object-language itself. This way one can avoid self-referential

sentences of the paradoxical sort. Of course, to avoid other paradoxical sentences the semantic

notions of the metalanguage could only be part of a meta-metalanguage and so on. Although this

strategy can provide a useful alternative notion of truth, it does not shed much light on the

ordinary notion of truth. The hierarchy approach cannot be applied to natural languages because

in English, for instance, it seems to be possible to talk about the semantic concepts of English.

English appears to be its own metalanguage. The concept of a true sentence of English seems to

be expressible in English itself, not only in a metalanguage.

Paradoxes without Self-Reference

There are good reasons to think that pathology and paradoxicality are not due to self

reference. The following sentences are pathological, although they involve no self-reference5:

Sn Sn+1 is true.



4 See Kripke [1975] for other examples of contingent Liars.
5 Herzberger [1970: 150].









Each sentence in this infinite list says about the next sentence in the list that it is true. Although

these sentences are not paradoxical, they are pathological in the same way the Truth-Teller is.

The meaning of the sentences in this list, together with the way the world is, does not suffice to

determine their truth-values. As Herzberger puts it, these sentences involve a "vicious semantic

regress but no vicious circle".6

Moreover, Yablo provides an example of sentences that are paradoxical, although they do

not involve self-reference7. This means that self-reference is neither necessary nor sufficient for

paradoxicality. Consider an infinite list of sentences of the form8:

Si = 'For all k > i, Sk is untrue'.

One can run a Liar argument for each sentence in the list. If there is a sentence, Sn, in this list that

is true, it would follow that Sk is true, for all k > n. From this one can derive a contradiction,

because on the one hand, Sn+1 would have to be untrue, and, on the other hand, it would have to

be true, because all Sk for k > n+1 are untrue, and this is what Sn+1 says. If all sentences in the list

are untrue, then they would also have to be true, because for any sentence, all subsequent

sentences are untrue.

Truth-Value Gaps

Many attempts to solve the Liar paradox involve saying that the Liar sentence lacks a

truth-value: it is neither true nor false (it is customary to call such sentences gappy, because they

fall in the gap between truth and falsity). Gappy sentences are possible if the following principle,

the principle of bivalence, is false:


6 Herzberger [1970: 150].

7 It has been held that Yablo's paradox implicitly involves self-reference. Nevertheless, this can only be plausible if
self-reference is understood in a very loose way. No self-reference of the sort we are familiar with from the classical
Liar paradoxes is present in Yablo's paradox.

SYablo [1993: 251-52].









(Biv) Every sentence is either true or false9.

The rejection of Biv indeed succeeds in blocking the most common version of a Simple Liar

argument (although, as it emerges from the next sections, there are other arguments that show

that the Simple Liar remains paradoxical even in the absence of Biv). This argument involves,

besides the logical principles and the principle of bivalence, the following three principles:

(I) (Lo)= '(Lo) is false.'

(SNC) No sentence is both true and false.

(T) 'S' is true in English iff S.

(I) is an identity sentence that holds by stipulation, (SNC), the principle of semantic non-

contradiction, is an intuitively true principle that captures a relation between truth and falsity,

while (T) is a schema that is supposed to hold for any replacement of'S' with sentences of

English from an appropriately restricted class. 10 It is widely agreed that it is part of what we

mean by 'true' that the T-biconditionals should be true.11 The Simple Liar argument goes like

this:

1. '(Lo) is false' is true iff (Lo) is false.

2. (Lo) is true iff (Lo) is false.

3. [(Lo) is true and (Lo) is false] or [(Lo) is not true and (Lo) is not false]



4. -[(Lo) is not true and (Lo) is not false.]

5. (Lo) is true and (Lo) is false.


9 It is assumed that the principle is restricted to meaningful declarative sentences. Otherwise, it would be trivially
false.
10 It is not enough to restrict this class to meaningful declarative sentences. One should also exclude defective
sentences as well as context sensitive sentences.

1 Some hold that this is all that is meant by 'true'.









6. [(Lo) is true and (Lo) is false.] and -[(Lo) is true and (Lo) is false.]

The principles of logic that are employed in the argument are Material Equivalence (MatEquiv),

De Morgan's, Disjunctive Syllogism (DS) these are all familiar principles that can be found in

any introductory textbook in classical logic and (Subst), the principle of the intersubstitutivity

of identicals, which enables one to replace identicals for identicals and preserve the truth value:

(Subst) If A = B, then (p(A) (p(A/B),

where '(p(A/B)' is the result of replacing one or more occurrences of 'A' in '(p(A)' with 'B', for

any formula (p.

Since this Simple Liar argument appeals to the principle of bivalence, a defender of a truth-

value gap approach is able to block it by denying this principle. Unfortunately, the mere rejection

of the principle of bivalence is not enough to solve the Liar paradoxes. One reason is that there

are other Liar sentences, such as the Strengthened Liar, that allow one to run a Liar argument

that is similar to the one above, but does not involve the principle of bivalence. The other reason

is that even for the Simple Liar, one can run a Liar argument that is slightly more complex, but

does not appeal to the principle of bivalence. I will discuss each of the two kinds of argument in

the next two sections.

Strengthened Liars and the Principle of Bivalence

So far as I know, the expression 'The Strengthened Liar' has been introduced in the

literature on paradoxes by Baas van Fraassen, who uses it to apply to those sentences that have

been "designed especially for those enlightened philosophers who are not taken in by

bivalence".12 The idea is that if there are truth-value gaps, one can construct a sentence that




12van Fraassen [1968: 147].









closes off that gap. The Strengthened Liar, (L), is just one of the sentences that play this role.

Another sentence that will do is:

(L') (L') is either false or gappy.

A sentence that closes the gap explicitly is sometimes called a Revenge Liar, but the terminology

is far from being uniform. There is agreement in counting (Lo) as a Simple (or ordinary) Liar.

However, there is disagreement with respect to the appropriate label for (L) and (L'). Some take

(L) to be a Simple Liar13 which should be contrasted with (L'), which they call a Strengthened

Liar; others take (L) to be a Strengthened Liar, while (L') is either another Strengthened Liar or a

Revenge Liar. The important difference is between sentences that are paradoxical only in the

presence of the principle of bivalence, and sentences that remain paradoxical even if this

principle is dropped. This criterion presumably makes (Lo) a Simple Liar and both (L) and (L')

Strengthened Liars, because both (L) and (L') allow a straightforward way to run a Liar

argument in the absence of the principle of bivalence. I will keep naming (Lo) the Simple Liar

and (L) the SiN. igilitheieLiar (when there is no risk of ambiguity, I will call the latter simply the

Liar); as I argue in the next section, the criterion that has been proposed (no paradoxicality in the

absence of bivalence) fails to discriminate between the two types of sentences.

One version of a Liar argument for the Strengthened Liar goes as follows:

1. '(L) is not true' is true iff (L) is not true [from (T)]
2. (L) is true iff (L) is not true [from (1) and (Subst)]
3. (L) is true assumption
4. If (L) is true, then (L) is not true Mat. Equiv, 2
5. (L) is not true MP 3, 4
6. (L) is true and (L) is not true Conj. 3, 5
7. (L) is not true RA 3, 6

13 Gupta and Belnap [1993] take L* to be a Simple Liar.









8. If (L) is not true, then (L) is true Mat equiv. 2
9. (L) is true MP 7, 8
10. (L) is true and (L) is not true Conj. 7, 9

What this argument appeals to is inference by reduction ad absurdum14; this is a weaker principle

than the principle of bivalence. Since no appeal has been made to the principle of Bivalence, the

mere rejection of bivalence fails to block the argument. One could object that the first premise is

not available if (L) is neither true nor false on the grounds that the T-schema is not supposed to

hold for gappy sentences. Nevertheless, saying that (L) is gappy would entail that (L) is not true.

But this is what (L) says, so (L) should be true; once again, one can derive a contradiction. The

very fact that (L) expresses in English the thought that (L) is not true prevents one from solving

the paradox.

The idea of closing off the gap works as a general recipe for producing paradoxical

sentences that resist all the alleged solutions to the Liar paradoxes that divide sentences in three

categories (instead of two): true/neither true nor false/false (or true/gap/false);

true/ungrounded/false [Herzberger 1970]; stably true/pathological/stably false [Gupta 1993];

definitely true/unsettled/definitely not true [McGee, 1990]. In each of these cases, one can

construct in English a Liar sentence that says about itself that it belongs in either the second or

the third category. In all these cases, it seems that one cannot block the argument, because there

is a sentence of the language that must be true if the Liar sentence itself is not. This sentence

either says about itself that it is not true, or it says that it is either false or


14One can also run a Liar argument for the Strengthened Liar that used the principle of the Excluded Middle instead
of reduction ad absurdum.
1. '(L) is not true' is true iff (L) is not true [from (T)]
2. (L) is true iff (L) is not true [from (1) and (Subst)]
3. Either [(L) is true and (L) is not true] or [-((L) is true) and -((L) is not true)] [Mat.Equiv. 2]
4. Either [(L) is true and (L) is not true] or ~[(L) is true or (L) is not true] [DeMorgan's 3
5. -[(L) is true or (L) is not true] [DN, EM]
6. (L) is true and (L) is not true [DS 4,5]









gappy/ungrounded/pathological. Again, it is the expressive power of English that causes the

failure of these attempts to solve the paradoxes.

The fact that the Liar remains paradoxical in the absence of bivalence is enough to show

that the rejection of this principle cannot offer a general solution to the Liar paradoxes. An

inquiry regarding whether the Simple Liar paradox survives in the absence of bivalence would be

superfluous. Nevertheless, a discussion of a controversy surrounding the status of the Simple

Liar paradox will prove to be useful, because it brings to light some principles (the excluded

middle, RA) that are used also in the Strengthened Liar paradox and might be considered

unavailable in the absence of the principle of bivalence.

Martin's Simple Liar Argument

It is commonly thought that the mere rejection of the principle of bivalence is enough to

offer a solution to the Simple Liar paradox, but it fails to solve the Strengthened Liar paradox.

Robert Martin15 argues that, contrary to what is commonly thought, the rejection of that principle

does not offer a straightforward solution to the Simple Liar paradox. This means that the Simple

Liar "is actually just as independent of the principle of bivalence as its big brother".16

To defend this thesis, Martin shows how an argument that leads to inconsistency could be

built even in the absence of the principle of bivalence:

Let so be the ordinary Liar. First, we show that so is not false, as follows: suppose so is
false; then, since that is what it says, it is true, and hence not false. (Principle: no sentence
is both true and false.) Therefore, so is not false. But now we can see that so is false, since
so says something the negation of which (so is not false) is true. (Principle: a sentence is
false if its negation is true.) Thus a contradiction. [Martin 1984: 2]

Martin is explicit about some of the semantic principles involved in his argument. Thus, he is

explicitly committed to all the instances of the following schemas

15 A similar argument has been put forward by Burge [1984: 88, fn.8].

16 Martin [1984: 2].









(SNC) -(T(FA) & F(A))

(F) T(FA1) -> F(FA)

and, in a less explicit way, to the instances of the T-schema:

(T) T(FA1) <- A

In addition, there are some principles of logic that are not made explicit. Given these (logical and

semantic) premises, Martin's argument shows that the Simple Liar sentence is paradoxical.17

In order for a truth-value gap account to succeed in solving the Simple Liar, one would

have to argue that some of the principles involved in Marin's argument are not true. One could

reject some of the principles of logic that have been used, or one could reject principles such as

(F) and (T). I will discuss some of these alternatives below.

Choice Negation versus Exclusion Negation

Beall and Bueno [2002] argue that Martin's argument fails to establish the conclusion,

because (F) together with (T) and classical logic entail the principle of bivalence.18 I will discuss

this objection against Martin in the next section. In this section I will focus on the implicit

suggestion that one could avoid the Simple Liar paradox by rejecting (F). Martin's version of the

Simple Liar argument takes (F) to be a platitude derived from our notions of truth, falsity and

negation.

Do we have good reasons to think that (F) is true? If there are only two alternatives for a

sentence (true or false; i.e., Bivalence holds), then the principle of falsity holds trivially. If there

are three alternatives, it might not be immediately clear whether it holds or not. There is a

correlation between the falsity principle and the way negation is interpreted. Let us assume that


1 In fact, one can derive the contradiction from weaker principles. Thus, the right-to-left direction of the T-schema
would be enough.
18 My criticism against the objection raised by Beall and Bueno is an adaptation of [Badici 2005: 25-39].









the meaning of negation is determined by a truth-table. In cases in which there is a third

alternative for a sentence (besides truth and falsity), the truth table for negation has normally

been taken to be given by Kleene's three-valued schema (Table 1). This notion of negation is

usually called choice negation. According to Kleene's schema, if a sentence lacks a truth value

(is indeterminate), then the negation of that sentence also lacks a truth value. If negation is

interpreted as choice negation, then the principle of falsity clearly holds, because if A is true

(the last row), then A is false. It can also be noticed that classical logic does not hold. For

instance, the principle of the excluded middle is violated. If a sentence is neither true nor false,

then neither it, nor its negation holds. 19 On the other hand, all assumptions involved in Martin's

argument are consistent with this interpretation of negation. Thus, it is natural to think that

Martin interpreted negation as choice negation.

Nevertheless, it has been claimed that choice negation is not the unique way to interpret

negation and, moreover, that it does not reflect the way negation functions in English. The

alternative interpretation is called exclusion negation and is characterized by a truth-table (Table

2) which assigns truth to the negation of an indeterminate sentence. If negation is interpreted as

exclusion negation, then the principle of falsity does not hold (to be more specific, it does not

hold if there are more than two possibilities for a sentence). One cannot infer that a sentence is

false from the fact that its negation is true. This is shown by the second row of the corresponding

truth table: the negation of the sentence is true, but the sentence itself is indeterminate. What this

means is that if negation is interpreted as exclusion negation, one can no longer run a Martin-

style Liar argument for the Simple Liar. Either Bivalence holds, in which case one certainly can

run a Liar argument (however, this was not a matter of controversy), or it does not hold, in which

19 The principle of the excluded middle might hold under some non-standard semantics for the logical connectives.
For instance, it might be true under a supervaluationist semantics.









case (F) is no longer available. To put things differently, if negation is interpreted as exclusion

negation, (F) entails Bivalence. Thus, if negation is exclusion negation, Martin's argument fails

to support the claim that the Simple Liar remains problematic in the absence of Bivalence. He is

not allowed to use (F) as a premise, since, together with exclusion negation (one does not even

need to assume classical logic), it entails Bivalence and, thus, makes a "neither true nor false"

account impossible.

The question is which of the two alternative interpretations of negation corresponds to how

negation works in English? I think that an analysis of the main kinds of English sentences that

are candidates for being neither true nor false shows that negation functions in accordance with

Kleene's three-valued schema. Among the sentences that are candidates for being neither true

nor false one normally counts sentences containing vacuous names, sentences involving category

mistakes, sentences containing vague predicates, moral sentences (according to some views in

metaethics), etc. If one takes sentences of this sort to be gappy, it is hard to see what reasons one

could have to take their negations to be true, rather than gappy. Normally, a sentence that lacks a

truth value is said to be meaningless, or to be meaningful but to fail to express a proposition. If

negation were interpreted as exclusion negation, then the negation of a meaningless sentence

would have to become meaningful (and true); moreover, the negation of a sentence that fails to

express a proposition would have to express a true proposition20. Thus, the view that English

negation is exclusion negation is contrary to our intuitions. Keith Simmons21 tries to defend the

claim that English negation is exclusion negation by saying that if one thinks that (S) is

meaningless, from '(S) is meaningless' one wants to infer '(S) is not true'. This is possible,


20 The difficulty is more vivid in the case of moral sentences. If a moral sentence is taken to express an emotion, its
negation would have to express a true proposition.
21 Simmons [1993: 54].









according to him, only if 'not' is understood as exclusion negation, because 'not true', in the last

mentioned sentence, is not equivalent to 'false'. It is hard to see how this non-equivalence can

justify interpreting 'not' as exclusion negation. Contrary to what Simmons suggests, choice

negation does not take 'false' to be equivalent to 'not true'; it takes the claim that a certain

sentence is false to be equivalent to the claim that its negation is true.

The current understanding of the distinction between choice negation and exclusion

negation originates with van Fraassen:

Following Mannoury though narrowing his meaning somewhat we can draw the
following distinction:
(a) choice negation: (not-A) is true (respectively, false) if and only if A is false
(respectively, true);
(b) exclusion negation: (not-A) is true if and only if A is not true, and false otherwise.
...Of course if the principle of bivalence holds (that A is always either true or false), then
the distinction collapses. [van Fraassen 1969:69]

Throughout his paper van Fraassen uses only choice negation, as the most natural interpretation

of negation, and mentions exclusive negation merely as an alternative that is used in the

literature. It is interesting to notice that the meaning of exclusion negation has not only been

somewhat narrowed, but significantly changed and that there are few things in common between

Mannoury's distinction between exclusion negation and choice negation and the distinction that

is introduced by van Fraassen and has become familiar in the more recent literature. Mannoury22

is concerned with offering an account of those elements of mathematical thinking that are not

given in sensory intuition and cannot be derived from it. In particular, notions such as infinity or

parallelism cannot be given in sensory intuition. Mannoury's explanation of how we can acquire

this kind of notions involves exclusion negation which is distinguished from choice negation.

Beth explains the distinction between the two kinds of negation as follows:


22 The discussion on Mannoury's views of negation is based on [Beth 1965].









The choice negation presupposes a disjunction of possibilities; the negation of one of these
possibilities then implies the assertion that one of the remaining possibilities is realized.
The exclusion negation presupposes no such disjunction of possibilities; therefore it is,
unlike the choice negation, incapable of a positive interpretation. It merely excludes one
possibility without making any assertion regarding possible alternatives. According to
Mannoury, the exclusion negation, more than choice negation, has an emotional character.
[Beth, 1965: 20]

It is essential for exclusion negation that there is an emotional component that is part of its

meaning. This enables Mannoury to explain how the notion of infinity can be grasped even

though it cannot be given in empirical intuition. Infinity is the negation of the finite, but the

negation must be understood as exclusive negation. Choice negation would require infinity to be

the alternative possibility which is given independently of the act of negation. This would be

impossible. We acquire the notion of infinity by an (emotional) rejection of the finite. One can

see that the difference between choice and exclusion negation made by Mannoury has nothing to

do with how negation is interpreted when there are truth-value gaps. The distinction would apply

equally well to bivalent and to non-bivalent systems.

Although the two types of negation are often presented as the two main competitors in the

literature, it is only choice negation that has been employed as the natural interpretation. Kleene,

for instance, interprets negation as choice negation in his systems of 3-valued logics.23 Parsons

even went as far as to deny that there could be languages whose negation is exclusion negation.24

One can infer that an attempt to block Martin's argument by denying (F) fails because it would

depend on an interpretation of negation that does not have much in common with English

negation.



23 Exclusion negation would count for Kleene [1952] as an irregular connective (regular connectives are such that "a
given column (row) contains t in the u row (column), only if the column (row) consists entirely of t's; and likewise
for f' [1952: 334]. He was not interested in irregular connectives, because they are not partial recursive operations.
24 Parsons [1984].










The Principle of Bivalence and Classical Logic

The objection Beall and Bueno raise against the thesis that the Simple Liar remains

paradoxical even in the absence of the principle of bivalence is not that Martin's argument is

invalid, but that his assumptions are too strong. In order for Martin's thesis to be justified, it is

not enough to show that a contradiction can be derived without appealing to Bivalence. One

needs to show that the contradiction can be derived from premises that do not entail Bivalence.

The problem is that, as Beall and Bueno showed by a nine step proof, "(T) and (F), given

classical propositional logic (CPL), entail Bivalence".25

I will argue that neither Martin's Simple Liar argument nor the Strengthened Liar

argument presuppose classical logic. The contradiction can be obtained from principles that are

weaker than classical logic. In particular, the argumentation strategy employed by Beall and

Bueno is misguided because they assume that classical logic is a presupposition of Martin's

argument. It is true that if one wants to give an account of the Liar paradox by saying that the

Simple Liar is neither true nor false, one should not be committed to principles that entail

Bivalence. Nevertheless, there is nothing in what Martin says that suggests a commitment to

classical logic. It is true that the principles of classical logic have at least prima facie plausibility

and that they cannot be rejected unless there is some pressure in this direction. However, a

"neither true nor false" account of the Liar paradoxes usually comes together with the denial of


25Here is the proof, quoted from Beall and Bueno [2002: 24] (I formulated the proof in a footnote only, because it is
not its validity that I am concerned with).
0. T A [T]
1. -T
~ -A [0, MTT]
2. T<~A> ~A [T]
3. ~T
T<~A> [1,2, Transitivity]
4. -T
[Premiss, for CP]
5. T<~A> [3,4, MPP]
6. T<~A> F
[F]
7. F
[5,6, MPP]
8. ~T
F [4,7, CP]
9. T
V F [8, CPL equivalence and DNE]









classical logic.26 One reason to deny classical logic when the principle of bivalence is rejected is

offered by Alfred Tarski who argues that (T) and classical logic alone imply a version of the

principle of bivalence.

... we can deduce from our definition [Tarski's definition of truth] various laws of a general
nature. In particular, we can prove with its help the laws of contradiction and of excluded
middle ... These semantic laws should not be identified with the related logical laws of
contradiction and excluded middle... ; the latter belong to the sentential calculus ... and do
not involve the term "true" at all. [Tarski 1944: 354]

Tarski's point is that his definition (in particular, the T-biconditionals that are entailed by his

definition), together with classical logic, are enough to prove the semantic laws of non-

contradiction and excluded middle. More precisely, given the (logical) principle of the excluded

middle

(EM) AV -A

and the following two biconditionals

(T) T(FA1) <+ A

(T*) T(FAl) <+ -A

one can infer

(SEM) T(FA) V T(FA1)

In fact, only the right-to-left directions of (T) and (T*) are needed to infer (SEM). (SEM)

expresses the semantic law of the excluded middle. This is taken by Tarski to be one of the laws

"which are so characteristic of the Aristotelian conception of truth" and is considered by many to

be one way to express the principle of bivalence.27 If (SEM) is accepted as an alternative



26 One should not be misled by the fact that Kripke thought acceptance of truth value gaps to be compatible with
classical logic. Classical logic was understood by him to be a set of principles that apply to propositions, while here
it is supposed to apply to sentences. See Kripke [1984: 64-5, fn. 19].
27 See McGee [1990: 104 & 179] and Gupta and Belnap [1993: 224].









formulation of Bivalence, then this already shows that those "who are not taken in by Bivalence"

(to use van Fraassen's formula quoted by Beall and Bueno28) must reject classical logic.

Certainly, Tarski's remarks fall short of showing that classical logic and (T) entail Bivalence,

because (SEM) is different from Bivalence. In order to derive Bivalence, one needs, in addition

to classical logic and (T), a principle connecting the predicates of truth and falsity. One principle

that would do the trick is (F) itself, a principle which is well supported by the way we think of

truth, falsity and negation, and which Martin rightly appeals to in his argument. It can be noticed

that Bivalence follows immediately from (SEM) and (F).29 It is unlikely that Martin was not

aware that Bivalence follows from these principles. Given his explicit commitment to (F), as

well as his commitment to the Truth Principle, it follows that he could not have been committed

to (EM); this means that he could not have been committed to classical logic either.30

Reductio ad Absurdum versus Reductio ad Gapsurdum

It remains possible to save the truth-value gap account by saying that although Martin's

argument and the strengthened Liar argument do not assume classical logic, they are nonetheless

using laws of logic that are strong enough to entail the principle of Bivalence. Notice that the

mere denial of classical logic is not enough to save Martin's argument from troubles. One could


28 Beall and Bueno [2002: 23].

29 Thus, one can formulate a shorter proof meant to prove the same thing as the nine step proof offered by Beall and
Bueno:

1.AV -A (EM)
2. A T(A) (T)
3. -A T-A) (T*)
4. (A T(Ai)) & (~A T(-A)) (Addition)
5. T( A) V T( Al) (Constructive dilemma)
6. T(A1) V F(A1) by (F*)

30 It is possible to reject bivalence and keep classical logic, as van Fraassen's work on presuppositions illustrates.
However, van Fraassen was able to do this because he denied the T-schema. See the end of this chapter for more
details about this approach.









argue that, even though he is not explicitly committed to classical logic, the logical principles he

appeals to in his argument entail, together with (T) and (F), the principle of bivalence. Martin did

not make all principles of logic used in his argument explicit, so there are more ways to

reconstruct the argument. One quite plausible way to reconstruct it would involve an application

of reduction ad absurdum:

1. (Lo) is false. Assumption
2. '(Lo) is false' is true. from (T)
3. (Lo) is true from 2 and (Intersubstitutivity)
4. (Lo) is not false. from 3 and (SNC)
5. (Lo) is false and (Lo) is not false. from 1 and 4
6. (Lo) is not false RA
7. '(Lo) is not false' is true from 6 and (T)
8. '(Lo) is false' is false from 7 and (F)
9. (Lo) is false from 8 and (Intersubstitutivity)
10. (Lo) is false and (Lo) is not false from 6 and 9

Line (6) in this argument is inferred by an application of reduction ad absurdum (RA). It is true

that a commitment to RA does not presuppose a commitment to classical logic, or, at least, to the

principle of the excluded middle. In intuitionist logic, for instance, the principle of the excluded

middle fails, even though RA is valid.31 Nevertheless, the appeal to RA remains problematic in a

context in which one drops Bivalence and allows sentences that are neither true nor false. If a

certain assumption leads to a contradiction, it seems illegitimate to infer the negation of that

assumption, because it might happen that both that assumption and its negation lack a truth

value. It is natural for the defender of a truth-value gap view to also reject RA. It is important to

notice that this is a point that applies equally well to the Simple and the Strengthened Liar


31 I am indebted to Graham Priest for pointing this out to me.









arguments because both involve applications of RA. Thus, a Liar sentence is problematic (in a

context in which truth-value gaps are allowed) only if an inconsistency can be derived from even

weaker principles of logic. In particular, it should be derivable from principles that do not

include EM or RA.

I think that what makes the inconsistency hard to avoid is our commitment to some

semantic principles rooted in the intuitions we have about the use of 'true' and 'false' which help

one run a Liar argument from principles of logic that do not include EM and RA. One rule that

can replace RA in the Liar argument is a rule that allows one to infer from the fact that a certain

assumption leads to a contradiction the claim that the assumption is not true; I will call this rule

reduction ad gapsurdum (RG).32 If reduction ad absurdum sanctions as valid inferences of the form

A Assumption


B & -B
-A RA

reduction ad gapsurdum sanctions as valid inferences of the following form:

A Assumption


B & -B
-T(A1) RG

The new rule accommodates the possibility that the assumption that leads to a contradiction is a

gappy sentence (neither true nor false). Whether a sentence is false or gappy, we still want to say

that it is not true. Given that the truth predicate plays an essential role in RG, this is not a logical

rule of inference, but rather a semantic principle. In fact, RG can be seen as a consequence of

one of two semantic rules that are rooted in the way we think of the predicates of truth and


32 The expression occurred during a conversation with Greg Ray.









falsity. The two rules can be called the T-closing-off33 rule (TC) and the F-closing-off rule (FC)

and they apply as follows:

TC-rule: T(FAl) Assumption


FC-rule:


B & -B
-T(A1)

F(A1)


Assumption


B & -B
-F(FA1) FC

TC and FC can be thought of as restrictions of RA to attributions of truth and falsity. However,

they are semantic rules of inference, because the truth-preservation is not guaranteed by the

meaning of logical terms alone; one also needs to take the meanings of the truth and falsity

predicates into account.

The new version of the Strengthened Liar argument would be34:

1. '(L) is not true' is true iff (L) is not true [from (T)]
2. (L) is true iff (L) is not true [from (1) and (Subst)]
3. (L) is true assumption
4. If (L) is true, then (L) is not true Mat. Equiv, 2
5. (L) is not true MP,3,4
6. (L) is true and (L) is not true Conj, 3,5

33 The name is inspired by Kripke's 'closing off locution, used to refer to an interpretation of the truth predicate
according to which if a sentence is gappy, then the sentence that attributes truth to it is false, and its negation is true.
See Kripke [1984: 80-81].

34Revenge Liar argument for the Strengthened Liar (appeals to TEM instead of EM):
1. '(L) is not true' is true iff (L) is not true [from (T)]
2. (L) is true iff (L) is not true [from (1) and (Subst)]
3. Either [(L) is true and (L) is not true] or [-((L) is true) and -((L) is not true)] Mat.Equiv.,2
4. Either [(L) is true and (L) is not true] or ~[(L) is true or (L) is not true] DeMorgan's, 3
5. -[(L) is true or (L) is not true] DN, TEM
6. (L) is true and (L) is not true DS, 4,5









7. '(L) is true'is not true
8. '(L) is true' is true iff (L) is true
9. '(L) is true' is not true iff(L) is not true
10. (L) is not true
11. If (L) is not true, then (L) is true
12. (L) is true
13. (L) is true and (L) is not true


RG 3,6 (Reductio ad gapsurdum)
[from (T)]
Contraposition, 8
[from 7 and 9]
[from 2]
MP,10,11
Conj., 10,12


One can also reformulate Martin's argument such that it no longer appeals to RA, but rather to

FC:


1. (L) is false.
2. '(L) is false' is true.
3. (L) is true
4. (L) is not false.
5. (L) is false and (L) is not false
6. (L) is not false
7. '(L) is not false' is true
8. '(L) is false' is false
9. (L) is false
10. (L) is false and (L) is not false


Assumption
from (T)
from (2) and (Intersubstitutivity)
from (3) and (SNC)
from (1) and (4)
FC
from (6) and (T)
from (7) and (F)
from (8) and (Intersubstitutivity)
from (6) and (9)


In fact, the only difference between the former and the latter version of the argument is that line

6 is this time inferred by an application of FC, instead of RA. Do we have adequate reason to

think that TC and FC hold? I think that not only closing-off rules of inference, but also closing-

off principles like


T(A1) V -T(FA1)
F(A1) V -F(FA1)


[every sentence is either true or not true]
[every sentence is either false or not false]


TC*
FC*









are grounded in the way we think of the concepts of truth and falsity. Unlike 'bald' or 'smidget',

the predicates of truth and falsity are neither vague35 nor only partially defined.36 Of course,

when TC* and FC* are held to be true, it is assumed that English is able to express the closing-

off principles. Their denial would conflict with the thesis that English is universal.

Thus, as long as one is committed to the universality of English, both the Simple and the

Strengthened Liar allow one to construct a paradoxical argument from very weak premises that

do not entail the principle of bivalence.

The T-Schema

Another way one could try to block the Liar argument is to reject the T-schema. In fact, it

is well known that the schema cannot work in its full generality. There are context sensitive

sentences that require some adjustment to the general schema. There are also good reasons to

think that the schema, in particular the right-to-left direction does not hold for gappy sentences.37

The antecedent of the right-to-left conditional is gappy, while the consequent is false. Even

though there might be weak systems of logic that would still make these instances of the

conditional true, it seems that it would be more plausible to say that they are not true. There are

two reasons why denying the T-schema is not very appealing. First of all, even though some

instances of (T) might not be true, this does not mean that the principle does not have initial

plausibility, at least when it is applied to sentences that are not semantically defective. There is

no good reason to think that Liar sentences are semantically defective, other than the fact that

35 There have been attempts to treat 'true' as a vague predicate. See, for instance, McGee [1990]. I think however
that it would be a mistake to think that 'truth' inherits the vagueness of the vague predicates of English. A sentence
like "Harry is bald' is true', where Harry is a borderline case, is more appropriately treated as false rather than as
neither true nor false.
36 Soames [1990: 164] introduced 'smidget' as an example of a partially defined predicate. Its meaning is defined,
roughly, by the following stipulative clauses: i) 'smidget' applies to all adults less than four feet tall; ii) 'smidget'
fails to apply to any adults more than five feet tall; iii) 'smidget' fails to apply to nonadults.

7 I am grateful to Kirk Ludwig for bringing this possibility to my attention.









they are paradoxical. Second, some weaker principles suffice to enable one to run a Liar

argument. I made the assumption (granted by Beall and Bueno) that Martin is committed to the

T-schema, even though this commitment is not made explicit in his argument. However, a couple

of paragraphs before, Martin takes the Truth Principle, repeated here,

(Truth Principle) A sentence is true if, and only if, what it says is the case.

to be part of the assumptions that make the Liar arguments possible. I think Martin took it to

entail that the instances of (T) are true.38 Anyway, when he runs the Liar arguments, Martin in

fact employs a rule of inference that is weaker than (T):

(Ri) 'S' says that A
(R2) A
(R3) Therefore, 'S' is true.

This rule of inference holds even if 'A' is replaced by a gappy sentence, and is enough to run the

Simple Liar argument.

One account that denies the T-schema (although it remains committed to some slightly

weaker rules of inferences) is the presuppositional account proposed by Bas van Fraassen. van

Fraassen denies the principle of bivalence, but he wants to stay committed to the principles of

classical logic, including the principle of excluded middle; he is also committed to the principle

of falsity, although he only thinks of it as a more convenient way to think of negation, truth and

falsity. Inspired by Strawson's work on presuppositions, van Fraassen develops a semantic

account according to which some sentences have presuppositions that need to be met in order for

them to be true or false. He is able to reject bivalence and preserve classical logic because he




38 One can add some restrictions to exclude context sensitive sentences. The Simple Liar will not be affected by a
restriction of this sort.









denies the T-schema. Although the T-schema does not hold, there are some other principles that

capture the intuitions speakers have with respect to the meaning of 'true':

The argument from A to It is true that A is of course valid, as is the converse argument.

But we shall in general reject the material equivalence of the two sentences, in order to block the

inference from (A or not-A) to (True
or True), because the former is valid even

when neither A nor not-A is true.39

Thus, the T-schema fails, because the right-to-left direction fails. Nevertheless, the

following two inferences are valid:

A

Therefore, T
.

and

T


Therefore, A

Paradoxical arguments would have the form:

T
; hence A; hence -T

-T
; hence A; hence T

The Simple Liar paradox is solved by the rejection of bivalence. Since both it and its negation

entail a contradiction, it follows that the Simple Liar presupposes a contradiction. Given that its

presupposition fails, the Simple Liar would have to be neither true nor false. There is a lesson

that needs to be drawn from the Simple Liar:

This I shall call the basic lesson of the Liar paradox: even assertions of truth or falsity do
not in general satisfy the law of bivalence. [van Fraassen 1968: 148]




39van Fraassen [1978: 15].









The Strengthened Liar teaches an additional lesson. Not only would the Strengthened Liar have

to be gappy, but also T<(L)> because T<(L)> presupposes (L). One would have to deny not only

bivalence, but also a principle such as:

(*) T is true, or T<-Y> is true, or -T & ~T<~Y> is true.

Thus, one thing the Strenghtened Liar paradox shows is that although (L) is neither true nor

false, the truth or non-truth of (L) is not expressed by T<(L)>, but by some other true sentence of

the language, ~T>. Although T<(L)> is gappy, the sentence T> can have a

truth-value (in particular, it is false, and ~T> is true). Thus, van Fraassen links the

Strengthened Liar with the issue of expressibility: some truths about (L) fail to be expressed by

the sentence that appears to have been designed for this job, but is expressed by some other

sentence. Moreover, the problem can be generalized. There are Strengthened Strengthened Liars

(such as -T>, Strengthened Strengthened Strengthened Liars, etc. van Fraassen manages

to argue that there are also infinitely paradoxical sentences, that is, sentences whose truth or non-

truth cannot be expressed by any sentence in the language. Thus, the solution to the Liar

paradoxes is possible by acknowledging some limits to the expressive power of the language. I

think that this is a fruitful idea, although van Fraassen did not show how his remarks can be

applied to natural languages. The proofs he constructs involve formal languages, but he wants

the results to hold for natural languages too, because he is interested in those formal languages

that mirror the relevant features on natural languages:

A solution to the semantic paradoxes should presumably have two distinguishable parts: an
analysis of the logically relevant features of the paradoxes as stated in natural language,
and a formal construction in which corresponding sentences play roles roughly similar to
those which our analysis ascribes to the paradoxical statements. [van Fraassen 1978: 13]

If one tries to extend van Fraassen solution to natural languages, one would notice that it

conflicts with the intuitions according to which natural languages are universal, van Fraassen









suggests that even though -T<(L)> fails to express the non-truth of the Liar, -T> might

very well express it. Nevertheless, the latter could only express the non-truth of the Liar in a

metaphorical sense, because it actually expresses the different thought: that the sentence that (L)

is true is non-true. Thus, if the view is to be applied to English, one consequence of it would be

that the non-truth of the Strengthened Liar is inexpressible in English. This would be in conflict

with the universality of English.

Universality and Inconsistency

The universality of English should be acknowledged as the main obstacle against offering

a solution to the paradoxes. It is the universality that is attributed to natural languages that

precludes a solution in terms of some type restrictions, or by distinguishing between object-

language and metalanguage; it is again universality that makes it difficult to solve the paradoxes

by denying the principle of bivalence, the excluded middle or reduction ad absurdum; the

universality makes the account proposed by van Fraassen difficult to apply to natural languages.

It seems that as long as natural languages are considered universal, there is no way to avoid

the inconsistencies generated by the Liar paradoxes. Keith Simmons [1993] believes that

paradoxes should be understood as a conflict between the possibility of running diagonal

arguments that lead to a contradiction and the universality of the language. Thus, there appear to

be two main alternatives that one has to consider: either English is universal, in which case there

appears to be no other option than to endorse one version or another of an inconsistency view of

truth, or one tries to preserve consistency by denying the universality of English. I chapter three I

explore and endorse the latter alternative. The universality of English has normally been taken as

too obvious to deserve any special attention. Nevertheless, I argue that English is not universal,

and that the Liar paradoxes are best solved by dropping the assumption that truth is expressible

in English. Chapter four is concerned with the former alternative, and I argue that universality









cannot be saved by endorsing an inconsistency view of truth because an inconsistency view of

truth which remains committed to the law of non-contradiction entails that truth is inexpressible.


Table 1-1.

A -A
T F
I I
F T





Table 1-2.
A -A
T F
I T
F T









CHAPTER 3
THE INEXPRESSIBILITY OF TRUTH

In the previous chapter I argued that the main strategies that have been used to avoid the

inconsistency generated by Liar paradoxes conflict with the thesis that natural languages are

universal. Therefore, one is faced with a choice: either to adopt an inconsistency view of truth or

to deny the universality of English. Most philosophers have taken the universality of English as

granted and tried to find ways to live with inconsistencies. The lesson that they have drawn from

the Liar paradoxes is either that English is inconsistent [Tarski 1933], or that the postulates that

give the meaning of 'true' are inconsistent [Chihara 1979], or, maybe, that there are true

contradictions [Priest 1998].

I will examine the inconsistency view of truth in chapter four. The purpose of this chapter

is to explore and defend the thesis that what the Liar paradox shows is that natural languages are,

appearances to the contrary notwithstanding, not semantically universal. In particular, I argue

that truth is inexpressible in English. This is established by an argument that can be adequately

characterized in English and which consists in showing that the supposition that truth is

expressible in English leads to a contradiction.

The main challenge for the inexpressibility view is to answer some objections against it

that may readily come to mind. It might be objected that the expressibility of a concept is quite

easy to secure, because it is only a matter of what intentions speakers in a linguistic community

have with respect to the use of a predicate. It has also been objected that the inexpressibility view

is self-defeating, because in defending the view one actually expresses the concept that is held to

be inexpressible. Moreover, one may worry that if 'true' fails to express in English the concept

of truth, then it would be hard to explain the fact that non-problematic sentences in which 'true'









is used are successfully deployed in communication. I will argue that all these challenges can be

met if one takes into account the distinction between intended meaning and linguistic meaning.

Universality and Expressibility

To see how universality and expressibility are involved in the semantic paradoxes, one

needs a more precise characterization of these notions. Although it is a widely held view that

English is universal, the characterization of universality has remained vague. The strongest

version of universality one can think of is the following:

(U-universality) A language is U-universal if and only if every concept/thought is

expressible in it1.

One standard objection against the thesis that English is U-universal is that English has only

countably many expressions, while there are uncountably many concepts. There are some more

or less satisfactory answers to this objection2, but I will set these debates aside, because U-

universality is needlessly strong for my purposes in this chapter.

Tarski argues for a slightly different universality thesis:

A characteristic feature of colloquial language (in contrast to various scientific languages)
is its universality. It would not be in harmony with the spirit of this language if in some
other language a word occurred which could not be translated into it; it could be claimed
that if we can speak meaningfully about anything at all, we can also speak about it in
colloquial language. [Tarski 1956: 164]

He never offered a precise definition of universality. Nevertheless, it is plausible that the

following notion of universality comes pretty close to what he had in mind:




1 I take thoughts to be the kind of entities that are the contents of declarative sentences, regardless of whether they
are conceived as propositions, Fregean thoughts, or in some other way. Concepts should be understood as
constituents of thoughts.
2 Although only countably many concepts can be expressed in English, there are uncountably many that can be
expressed in some extension or other of English (but no single extension can express more than countably many
concepts).









(T-universality) An interpreted language3, L, is T-universal iff for every interpreted

language, L', and for every meaningful expression, E', in L', there is an

expression, E, in L, such that E in L has the same meaning as E' in L'.

It is universality in this sense that is presumably taken by Tarski to guarantee that colloquial

languages must contain their own semantic predicates (in particular, English must be able to

express the concept of a true sentence of English).

T-universality is in one respect unnecessarily strong and in another respect too weak4. It is

too strong because English does not have to be able to express all that can be expressed in other

languages in order for there to be an obstacle to reaching a solution to the semantic paradoxes.

All that is needed is that English express its own semantic concepts. T-universality is also too

weak, because the thesis that English is universal is relevant to the problem of semantic

paradoxes only if it entails that the concept of a true sentence of English is expressible in

English.

The thesis that English is T-universal does not by itself entail that English is able to

express this concept. However, it entails it in conjunction with the additional thesis that there is a

language that is able to express the concept of a true sentence of English. This additional thesis is

true if the following principle of expressibility advocated by Searle [1969: 20] holds:

(PE) For any meaning X and any speaker S, whenever S means (intends to convey,

wishes to communicate in an utterance, etc.) X, then it is possible that there is

some expression E such that E is an exact expression of or formulation of X.




3 A language counts as interpreted only if besides the syntax rules, there are also semantic rules that determine the
meaning of its expressions. Natural languages are interpreted languages.
4 Similar things can be said about U-universality.










Note that X here cannot be understood as the meaning of a certain expression (this would make

the principle trivial5), but rather as a content (such as a concept or proposition) that can be

characterized independently of any linguistic entity. If we can grasp the semantic concepts of

English, (PE) guarantees that there is some language in which they are expressible, which

together with the T-universality of English would guarantee that they are also expressible in

English.

For the purposes of this study it would be enough to focus on the following universality

notion:

(S-universality) A language is semantically universal (S-universal) if and only if all its

semantic concepts are expressible in it.

On the one hand, S-universality is more restricted than T-universality and, on the other, one does

not need a commitment to (PE) in order for the S-universality of a language to entail that it is

able to express its own semantic notions. If English is S-universal, it would have to be able to

express all its semantic concepts.6 Notice that the notion of semantic universality (as well as the

other notions of universality) is defined in terms of the notion of expressibility, which is often

assumed to be clear enough and not to require explicit characterization. However, it is important




5 I am indebted to Michael Jubien for this remark.
6 Martin [1976] distinguishes between two notions of universality: universality in the sense of Tarski (characterized
in a way similar to T-universality above) and universality in the sense of Fitch, which is characterized as "a
language's capability to say much, if not everything, about every language, including itself' [Martin 1976: 274].
Alternatively, languages are characterized as universal in the sense of Fitch if "they can be used to talk about all
languages, including themselves, and in particular to express much, if not all, of their own semantic theories" [1976:
271]. I prefer to use S-universality for two reasons. One is that Martin's notion of universality in the sense of Fitch is
not precise enough, the other is that it is not at all clear that Fitch intended to offer a notion of universality different
from universality in the sense of Tarski. Fitch [1964] argues that there are universal formal languages. In order to
show this, he argues that such a language would be able to say everything there is to say about its own semantics.
This does not mean that he works with a notion of universality different from that used by Tarski; it means only that
Fitch believes that the difficulties raised by the semantic concepts can be overcome and, consequently, that there are
universal languages in the sense of Tarski.









to clarify what 'expressibility' means, because it is often understood in different ways. The first

step in characterizing expressibility is this:

(Ec) A concept is expressible in L if and only if there is a predicate of L that (given an

appropriate context) expresses it.

(Et) A thought is expressible in L if and only if there is a sentence of L that (given an

appropriate context) expresses it.7

The next step would be to offer a (non-extensionalist) characterization of expressions of the form

expresses in L'. This can be done as follows:

(EC) P expresses C in L (in context c) if and only if C is the concept determined by the

meaning of P in L (and context c).

(ET) S expresses T in L (in context c) if and only if T is the thought determined by the

meaning of S in L (and context c).8

The relation between the meaning of an expression and the concept or thought expressed by it

can be conceived of in various ways depending on one's favorite view of semantics. Some views

might fail to draw a distinction between the meaning of an expression and the concept or thought

expressed by that expression. In this case, the concept or the thought determined by (associated

with) a certain meaning is the meaning itself. Other semantic views would distinguish between

the meaning of a predicate and the concept expressed by that predicate. Similarly, there would be

a distinction between the meaning of a sentence and the thought expressed by that sentence.9



I characterized expressibility as a relation between thoughts/concepts and languages. It is not difficult to extend
this to a relation between beliefs (or thoughts understood as mental representations) and languages. A sentence
would be said to express a belief if it expresses the content of that belief.
8 (ET) is supposed to make more explicit the intuition that S expresses T in L if and only if T is what S says.
9 Note, however, that I take concepts here to be constituents of thoughts that correspond to predicates. Fregean
semantics distinguishes between the sense of a predicate and the concept denoted by it. The sense of a predicate is
the mode of presentation of a concept (the referent of the predicate). Fregean concepts are semantic values of









Regardless of how exactly the relation between meanings and concepts is understood, (EC) puts

one in a position to formulate an adequacy condition that needs to be met in order for a predicate

to express a certain concept. 10 If one thinks of a concept as being associated with a set of

application rules, then the Expressive Adequacy condition would require that the concept

determined by the meaning of the predicate is the concept governed by the corresponding

application rules. More specifically, if P is a predicate of L and C a concept, then P expresses C

in L only if the following condition is met:

(Expressive Adequacy) For any a, P applies in L to a iff the application rules for C entail

that a falls under C.

In particular, in order for a predicate of English to express the concept of truth, the Expressive

Adequacy condition must be met. This is normally understood as an extensional adequacy

condition, but I will argue that it is more general because it must also be satisfied when there is

no extension associated with the concept.

There are two difficulties that might suggest that there is something wrong with this way of

thinking of expressibility because it would fail to provide a satisfactory criterion to measure the

expressive power of a language: the first has to do with context-sensitive expressions, the second

with the fact that natural languages do not have a well-defined syntax and semantics. I will argue

that none of them poses a serious threat to my project.





predicates. If the concept is understood as the semantic value of a predicate, then it cannot be part of the thought
expressed by a sentence containing that predicate, because the thought is a composite of senses, not of referents.
Since I take concepts to be parts of thoughts, they should be identified with Fregean senses of predicates rather than
with Fregean concepts. What an expression expresses is a sense or a mode of presentation, and it does express it
when its meaning determines the mode of presentation.

10 For reasons of simplicity, I omit context sensitivity. I will assume that the semantic predicates are not context
sensitive expressions.









Consider first the case of context-sensitive expressions. It may seem that the presence of

demonstratives or indexicals in a language makes that language universal in a very strong sense.

Hofweber [2006] distinguishes between two kinds of expressibility: language expressibility and

loose speaker expressibility.11 Language expressibility has to do with what can be expressed by

using only the context-insensitive expressions of a language. English is not universal in this

sense, because there are things that are not language expressible in English. Nevertheless,

Hofweber argues that every natural language is universal in the sense that everything is loosely

speaker expressible in it. Something is loosely speaker expressible in a language, if it can be

expressed by using either a context-sensitive or a context-insensitive expression of that language.

Hofweber uses as an example the property of tasting better than Diet Pepsi, which although not

language expressible in Ancient Greek, is, according to him, loosely speaker expressible in it. It

can be expressed by the Ancient Greek equivalent of

(1) tasting better than this.

in a context in which there is Diet Pepsi in front of the speaker.

However, Hofweber's example does not succeed in showing that everything is loosely

speaker expressible in Ancient Greek (or in any other natural language). 12 The addition of

demonstrative expressions does indeed increase the expressive power of a language.

Nevertheless, what referring devices contribute to the thought expressed by a sentence is



11 Hofweber is concerned with expressibility of properties, but the discussion can be rephrased in terms of
expressibility of concepts. Moreover, his distinction can be extended to account for the expressibility of propositions
or thoughts.
12 There are other problems for Hofweber's argument, in addition to the objection formulated above. It is not at all
obvious that what 'tasting better than Diet Pepsi' expresses is the same as what is said by using the Ancient Greek
equivalent of (1) uttered in the appropriate context. One problem is that 'Diet Pepsi' works as a mass term rather
than as a singular referring term. Even if it is assumed that it plays the same role as a demonstrative, the thoughts
expressed are not necessarily the same. They are different on a Fregean view of semantics. The thoughts are the
same if all that 'Diet Pepsi' and 'this' contribute to the thought expressed is the referent.









different from the contribution made by predicative devices. In particular, a distinction needs to

be drawn between a concept being expressible in a language and a concept being denotable in a

language. Given any concept, one can certainly say something about it by uttering (2)

(2) That is a concept.

in a context in which the demonstrative refers to that concept. However, the fact that a certain

referring expression of English denotes a certain concept does not guarantee that that concept is

expressible in English. 13 Thus, the fact that a certain language contains context-sensitive

expressions by no means shows that everything is loosely speaker expressible in it; in particular,

it fails to show that the language is semantically universal.

The other difficulty consists in the fact that natural languages do not have a well-defined

syntax and semantics. Expressions that did not belong to English a century ago are frequently

used today, and expressions that do not belong to English today will be added to it in the future.

Thus, the question whether a certain concept is expressible in English may not have a definite

answer. The issue of what is and what is not expressible in a certain natural language is indeed

very imprecise, but I think that the imprecision is due to the fact that natural languages are not

precisely defined, rather than to some defect in the characterization of expressibility. If there is a

need to make things more precise, one can replace the question whether a certain concept is





13 Denotability is much easier to secure than expressibility. Although English does not have enough names for all
concepts, one could argue that any concept whatsoever has a name in some extension of English. Moreover, any
concept can be referred to by using a demonstrative in some appropriate context. Similarly, there may be thoughts
which cannot be expressed in a certain language, even if they can be denoted by some expression of that language.
Nevertheless, it is one thing for an expression to denote a certain concept (or thought) in L, and quite another for it
to express that concept (or thought) in L. To show that everything is loosely speaker expressible in a certain
language one should also prove that it contains context sensitive predicative expressions that are able to express
(rather than denote) any concept, given the appropriate context. Although there is room for arguing that there are
context-sensitive predicative expressions, such as 'tall' (even 'true' is a candidate, according to Burge [1984] and
Simmons [1993]) it is hard to think of one that could express any concept whatsoever given the appropriate context.









expressible in English with the question whether it is expressible in English at t.14 Moreover, it

turns out that the lack of a well-defined vocabulary has no bearing on whether semantic notions

such as the concept of a true sentence of English are expressible in English.

The Inexpressibility Argument

Contrary to the widely held view that English is semantically universal, I will argue that

the concept of a true sentence of English is not expressible in it. This is, I think, the lesson that

should be drawn from the Liar paradoxes. Consider the following frequently used version of the

Liar argument:

1. The following sentence (The Liar) is a sentence of English:

(L) (L) is not true.

2. '(L) is not true' is true if and only if (L) is not true.15 [instance of the T-schema]

3. (L) is true if and only if (L) is not true. [intersubstitutivity of identicals, 1, 2]

4. Either (L) is true or (L) is not true. 16 [every sentence is either true or not true]

5. (L) is true and (L) is not true. [truth-functional consequence of 3 and 4]

Notice that this argument (as well as many other versions of the Liar argument) makes no

explicit appeal to the notions of universality or expressibility. Nevertheless, it involves an

implicit appeal to the assumption that 'true' expresses the concept of truth. The instances of the



14 Notice that even if a language, L, survives the addition of any new predicate, this does not mean that it is S-
universal. If one thinks of the syntax and semantics of L at different times as stages of L, then L could indeed be said
to be universal in the sense that every semantic concept of L is expressible in some stage of L. Nevertheless, S-
universality is not guaranteed if it is understood as the thesis that every semantic concept of L at t is expressible in L
at t. In addition to this, there is the possibility of concepts that cannot be expressed by any predicate of any language.
15 It may be objected that the T-schema does not hold for sentences that are neither true nor false, and that (L) may
be one of these gappy sentences. This move cannot prevent the paradox, because if (L) is neither true nor false, it
would follow that it is not true. Therefore, (L) would have to be true after all.
16 Line 4 of the argument is not needed if the laws of classical logic hold. The fact that there is a semantic rule
(rather than a rule of logic) that justifies 4 is important because it shows that the Liar argument only needs very
weak principles of logic.










T-schema are held to be true because 'true' is assumed to express the concept of truth. The

assumption either has escaped notice, or has been considered to be too obvious to need

mentioning.17 Nevertheless, it needs to be acknowledged as a premise in the Liar argument.

Moreover, by making the premise explicit one becomes able to turn the Liar argument into an

inexpressibility argument. More specifically, the Liar argument can be turned into a reduction

argument whose conclusion is that 'true' does not express the concept of truth. By generalizing

the Liar argument one obtains the following argument that shows that the concept of truth is

inexpressible in English.

The inexpressibility argument

1. Suppose that the concept of a true sentence of English is expressible in English and that

T is a predicate of English that expresses this concept.

2. Let (L*) be [(L*) is not TI.

3. The instances of the T-schema for predicate T hold. 18 [from 1 and the Expressive

Adequacy condition]

4. [F(L*) is not TI is T if and only if(L*) is not TI is true. [from 2 and 3]19



17 The assumption is more explicit in the informal version of the Liar argument presented at the beginning of this
chapter. A reformulation of the argument is this:
1. The following sentence (The Liar) is a sentence of English:
(L) (L) is not true.
2. (L) expresses the thought that (L) is not true. [from 1 and the Expressive Adequacy condition]
3. (L) is true. [assumption]
4. (L) is not true. [from 2, 3 and the Descent-rule]
5. (L) is not true. [RA from 3 and 4]
6. (L) is true. [from 2, 5, and the Ascent-rule]
7. (L) is true and (L) is not true. [from 5 and 6]
The assumption that 'true' expresses the concept of truth is needed not only for (2), but also to justify the application
of the two rules of inference (the Ascent- and the Descent-rule). This version of the argument has the additional
virtue that the possibility that (L) is gappy requires no special discussion. I have chosen the other version of the
argument, because it is important to show that the expressibility assumption is made even in arguments that do not
mention it explicitly.

18 One needs to add restrictions to the T-schema to account for semantically defective sentences.









5. [(L*) is T if and only if (L*) is not TI is true. [from 2 and 4]

6. Either (L*) is true or (L*) is not true.20 [every sentence is either true or not true.]

7. Either [(L*) is TI is true or [(L*) is not TI is true. [from 6 and the T-schema]

8. r(L*) is T and (L*) is not TI is true. [truth-functional consequence of 5 and 7]

9. Therefore, there is no predicate of English that expresses the concept of a true sentence

of English. reductiono ad absurdum]

Generalizing a little more, one can prove that no language that meets some minimal expressivity

requirements is semantically universal.

The inexpressibility argument is quite straightforward and involves no sophisticated proof

techniques. The main challenge for the inexpressibility view is to answer the objections that

might be (and have been) raised against it. Before answering the objections against the

inexpressibility view of truth I just sketched, I want to explain why my version of the

inexpressibility argument is to be preferred to another version that is due to Hans Herzberger.

Herzberger on Universality

No Language contains all its Semantic Concepts

Hans Herzberger [1970] challenges what he takes to be the thesis that languages can be

universal in the sense of Tarski. In fact, he argues only for the claim that no language contains all

its semantic concepts; he does not show that those concepts are expressible in other languages.

Thus, although Herzberger takes himself to be arguing against Tarski's universality thesis,

strictly speaking, he argues only that no language is S-universal. Nevertheless, if it is true that

one can grasp semantic concepts such as the concept of a true or of a grounded sentence of such

19 (1) may also be needed to infer (4), because it guarantees that (L*) is not meaningless. The hypothesis that (L*) is
gappy does not solve the problem, because it would follow that (L*) is not true.
20 Line 6 is not needed if the laws of classical logic hold (see also fn. 16). Moreover, if sentences such as that
mentioned in 5 cannot be true, this would be enough for a reduction argument.









a language, and if Searle's principle of expressibility holds, it would also follow that there are no

T-universal languages. In particular, there would be semantic concepts that cannot be expressed

in English.

According to Herzberger, the semantic paradoxes have to do with the limitations inherent

in our languages, and they should be understood as analogous to the paradoxes of set theory.

Both types of paradoxes are consequences of some very general theorems that can be established

in the theory of relations. Herzberger proves that formal languages of a certain sort cannot

express all of their semantic concepts. The proof appeals to the technique of diagonalization,

which has been made popular by Cantor and Godel, and is more or less explicitly employed in

most of the semantic paradoxes, as well as in the paradoxes of set-theory. Herzberger takes a

concept to be inexpressible "if the semantic rules assign its extension to no terms of that system"

[Herzberger 1970: 159]. This does not mean that the concept is identified with its extension, but

only that extensional adequacy is a necessary condition for a predicate to express a certain

concept.

The argument that L cannot express all of its semantic concepts is formulated in a

metalanguage and begins by specifying a class which is the extension of some semantic concept

(the class of all grounded sentences of L, or the class of all true sentences of L). One then proves

that that class cannot be represented in L, in the sense that there is no predicate of L that has that

class as its extension. The proof is conceived of as a reduction argument. Assuming that there is a

predicate that has that class as its extension, diagonalization enables one to find an object for

which it can be proved that it is a member of that class if and only if it is not a member of that

class. The contradiction forces one to deny the assumption that the class can be represented in L.

Herzberger is mainly concerned to show that no language is able to express all its grounding









concepts. However, he argues that the same strategy can be extended to other concepts, such as

the concepts of truth and satisfaction.21

Herzberger argues that the inexpressibility result that he was able to obtain for relatively

simple formal languages carries over to substantively more complex languages, such as natural

languages, because the increase in complexity has no essential bearing on what is needed to run

an inexpressibility argument.

The idea of using semantic paradoxes to prove that there are some limits to the expressive

power of any language is, as I have argued in the previous section, correct. Nevertheless,

Herzberger's defense of the inexpressibility view remains unsatisfactory. The problem with his

strategy is that questions about the expressibility of a concept cannot be settled by investigating

the expressibility of a class.

Class Expressibility versus Concept Expressibility

First of all, Herzberger assumes that a predicate expresses a certain concept only if the

extension of the predicate coincides with the extension of that concept. However, this

extensional adequacy requirement that, according to Herzberger, must be met by a predicate in

order to express a concept is too strong: it assumes that there is a class of objects that fall under

the concept (its extension). In particular, for each language, L, he considers, Herzberger assumes

that its semantic concepts do have an extension (the class of true sentences of L, the class of

grounded sentences of L, etc.). Nevertheless, there may be concepts that do not have an

extension.22 Good candidates are the concepts expressed by vague predicates; it is not at all clear


21Herzberger takes his proofs of inexpressibility to be similar to a certain diagonal proof offered by Tarski [1953:
46]. However, the two proofs are quite different, because Tarski was not concerned with the inexpressibility of a
concept, but with the indefinability of a certain class (the class of codes of true sentences of the language).
Moreover, what Tarski's theorem establishes is not that the class is indefinable, but that it is indefinable in a
language ifthe language is consistent.
22 Herzberger tackles this issue only in a footnote [Herzberger 1979: 161, fn. 17]. An example he offers of a concept
that lacks an extension is the concept of a Russellian class, namely the concept of a class that is not member of itself.









that there is a set or a class of bald persons. This could also be the case of semantic concepts.

One might argue that there is no extension determined by the concepts of groundedness or truth,

although here the problem no longer has to do with the lack of a sharp borderline but with the

existence of some cases that are paradoxical. Ray [2002], for instance, argues that the concept of

truth (together with the facts in the world) fails to determine a class of objects that fall under it,

because it is incoherent. If the concept fails to determine an extension, one certainly cannot

require the predicate to have the right extension. One can resist Herzberger's inexpressibility

arguments by denying that the concepts he wants to prove inexpressible have an extension.

The other shortcoming of Herzberger's strategy is that there is no obvious way to extend it

to natural languages. He argues that the strategy works for all languages in which the principle of

the bivalence holds. The problem is that the strategy cannot be applied for languages with gappy

sentences such as the language constructed by Kripke [1975]. Languages of this sort contain

their own truth-predicate ( in the sense that there is a predicate whose extension coincides with

the extension of the concept of a true sentence of L). No contradiction would follow from the

assumption that there is a predicate of L whose extension is precisely the class of true sentences

of L. Therefore, Herzberger's strategy cannot establish that no languages can contain their own

truth-predicates. Moreover, there are good reasons to think that if there is a class of true

sentences of English, then there is a predicate of English (one could define it) which has it as its





The predicate 'Russellian class' has a sense but no extension. Here, an extension is taken to be a set, as opposed to a
proper class (or ultimate class, to use Quine's terminology). The issue needs to be handled, Herzberger argues, by
associating terms not with classes, but with virtual classes, as they are conceived in Quine [1969]. He suggests that
an adequate revision of the principles he used would enable one to extend the inexpressibility result to apply also to
the case of concepts without an extension. The language of virtual classes is indeed useful in showing that diagonal
arguments can be used even in the absence of a platonistic commitment to the existence of sets or classes.
Nevertheless, it is harder to use the same apparatus for predicates that do not even determine a proper class (be it
understood as real or merely virtual).









extension. This would not be enough for the predicate to express the concept of truth. Therefore,

one needs to distinguish the notions of class expressibility and concept expressibility.

The version of the inexpressibility argument that I have offered does not assume that there

is a class of true sentences of English. This way the inexpressibility argument remains successful

regardless of whether the concept of truth determines a class. Moreover, the argument does not

depend on whether the sentences of the language obey the principles of classical logic.

Two Objections against the Inexpressibility View

Intentions are Sufficient for Expressibility

The first objection against the inexpressibility view that I will consider is that the

expressibility of a concept is guaranteed by merely having the right intentions with respect to the

use of the predicate. Expressibility, it may be argued, is a matter of what intentions speakers in a

linguistic community associate with a certain expression. So, the third line in my inexpressibility

argument would not be justified. What should be required for T to express the concept of truth is

that the T-schema govern the meaning of T (i.e., Material Adequacy), not that the T-schema for

T be true (which is the requirement of Expressive Adequacy). There is good evidence that

Tarski's material adequacy requirement is a requirement that the predicate be governed by the

corresponding T-schema; this requirement is trivially met if the speakers have the right

intentions with the use of the predicate. If this requirement is all that is needed, the objection

goes, then one cannot infer a contradiction from the assumption that truth is expressible in

English, because one cannot infer the third premise of the inexpressibility argument. One can

only infer (as Tarski did) that the language is inconsistent. This can deliver the conclusion that

the concept of truth is inexpressible in English only if one can prove that the inconsistency of

English leads to a contradiction.









The Inexpressibility Account is Self-Defeating

The second objection is that the view that truth, groundedness or other semantic concepts

are not expressible in English is self-defeating. The objection has been raised against

Herzberger's view, but it is a challenge to any inexpressibility view. Simmons argues that

Herzberger's thesis that groundedness is inexpressible in English is undermined by the very fact

that Herzberger's paper, in which he uses the word 'grounded' and explains what it means, is

written in English23. A similar objection has been put forward by Martin:

The claim was that, no matter what language I chose, a concept could be defined
(expressed) which the language I chose could not express. So I said: okay, I choose the
language in which you are arguing. If you succeed you fail. So you fail. [Martin 1976: 282]

He takes the argument to involve a definition of the concept that is then proved to be

inexpressible. This definition amounts to specifying "the set about which it is to be shown that it

cannot be the extension of any term"24 of the language. Martin agrees that if the definition were

given in a metalanguage, Herzberger's strategy might work. It does not work if it is formulated

in the same language whose semantic concepts are supposed to be inexpressible, because this

amounts to proving that the concept that has just been expressed is inexpressible.25

Martin proposes a better way to formulate the inexpressibility argument. One could prove

the inexpressibility thesis by an inductive argument that consists in the following two steps:

(i) If all other languages are expressively incomplete, then so is this language (the

language of the argument).





23 Simmons [1993: 58-61].
24 Martin [1976: 282].
25 Notice that Martin's objection does not depend on the fact that Herzberger begins his argument by specifying a
class. Thus, the objection is also a challenge to my version of the argument.









(ii) For every language L other than this one, there exists a concept which is

inexpressible in L.26

This move enables one to avoid the charge of being self-defeating, although the argument

remains inconclusive. As Martin points out, even though this version of the argument is harder to

reject, there are some "unsettling consequences"27 that raise doubts about the plausibility of the

view. Although we are presented with an apparently good argument for the conclusion that the

concept of a grounded term of English is inexpressible in English, we feel that we can express it

in English: we can express it by the predicate 'grounded term of English'. If it is denied that this

predicate expresses the concept of a grounded term of English, how do we know what concept is

inexpressible ("how do we know what set of terms Herzberger is talking about"28)?

Intended Meaning versus Linguistic Meaning

I will argue that both objections fail. It is true that a necessary condition for a predicate to

express a concept is that the members of the linguistic community have the right intentions with

respect to the use of the predicate. Yet this condition cannot be sufficient. Some of these

intentions may remain unfulfilled. This could happen, for instance, if their fulfillment is blocked

by empirical or semantic facts, or if the intended rules are simply inconsistent. One needs to

distinguish the linguistic meaning of an expression from its intended meaning. The members of a

linguistic community intend to use an expression to express a certain content. The linguistic

meaning of an expression is supposed to be a measure of the extent to which these intentions are



26 Hofweber [2006] argues that the inductive argument for inexpressibility fails, because, given the appropriate
context, everything can be expressed in a language that contains context-sensitive expressions. This would mean that
the induction basis is no longer available. I have argued already that the presence of demonstrative or indexical
expressions in a language does not guarantee that everything can be expressed in that language.
27 Martin [1976: 284-85].

28 Martin [1976: 284].









fulfilled by using sentences in which the expression occurs. The linguistic meaning of an

expression can also be thought of as that notion of meaning that (together with how the world is)

determines the semantic value of the expression. If the intentions associated with an expression

cannot be fulfilled, then the intended meaning (together with how the world is) fails to determine

a semantic value.29 The match between intended meaning and concept is indeed trivially

guaranteed (this would also be enough to guarantee material adequacy in Tarski's sense).

However, the match between linguistic meaning and the concept is not trivially guaranteed,

because such a match requires in addition that speaker's intentions can be fulfilled. If an

expression expresses a certain content, then it is in virtue of that content that the expression

acquires its semantic value. In particular, a sentence is true or false only insofar as the thought

expressed by it is true or false. Therefore, the notion of meaning that should be part of the correct

characterization of expressibility is linguistic meaning rather than intended meaning. The relation

between a predicate and the concept expressed by it can now be more precisely characterized in

the following way:

(EC*) P expresses C in L (in context c) if and only if C is the concept determined by

the linguistic meaning of P in L (and context c).

For the same reason, the Expressive Adequacy condition that must be met by a predicate in order

to express a certain concept requires that there should be a match between the concept and the

linguistic meaning of the predicate.

The import of the Expressive Adequacy condition is better understood by comparison with

Tarski's Material Adequacy requirement. Tarski [1944] argues that a truth predicate for a

29 The distinction between intended meaning and linguistic meaning does not match the distinction between speaker
meaning and sentence meaning. The latter is a distinction between a notion based on the intentions of a particular
speaker with respect to a particular utterance of a sentence and one based on the intentions of the entire community
with respect to that sentence. The former is a distinction between two notions that are both concerned with the
intentions of the entire linguistic community.









language L is indefinable in languages that meet a certain set of requirements. In such languages

there can be no definition of truth for L that is materially adequate and formally correct. Tarski

did not talk about concepts at all. However, if one wants to reformulate his material adequacy

condition as a relation between a predicate and a concept, one would formulate it along the

following lines:

(Material Adequacy) P is intended to be such that for any a, P applies in L to a iff the

application rules for C entail that a falls under C.

The difference between Material Adequacy and Expressive Adequacy is that while the former

only requires that members of a linguistic community have the right intentions with respect to

the use of the predicate, Expressive Adequacy requires in addition to this that the intentions be

fulfilled. More specifically, Material Adequacy requires that the meaning of 'true' be governed

by the T-schema, while Expressive Adequacy requires that the instances of the T-schema be true.

The comparison between the two adequacy conditions also sheds light on the relation

between the inexpressibility argument and Tarski's argument for the inconsistency of languages

of a certain sort. Part of Tarski's indefinability argument is an argument that if a definition of the

notion of a true sentence of L in L' is materially adequate, then it cannot be formally correct. A

consequence of this fact is that languages that contain their own truth predicate would have to be

inconsistent. Since English is assumed to be T-universal, it contains its own truth predicate.

Therefore, Tarski [1933] takes English to be one of these inconsistent languages.30




30 Tarski [1944] is more cautious and refrains from calling English inconsistent on the grounds that it does not have
an exactly specified semantics. The issue of inconsistency occurs only in connection with languages with an exactly
specified semantics. However, languages that are close enough to English but have an exactly specified semantics
are, even according to Tarski [1944], inconsistent. If Tarski was justified in restricting the claim to languages with
an exactly specified semantics, then I should also restrict the inexpressibility of truth in the same way. Nevertheless,
I think that the lack of an exactly specified semantics should not prevent Tarski from arguing that English is
inconsistent, and should not prevent one from proving that truth is inexpressible in English.









It may appear that the inexpressibility argument is related to the inconsistency argument in

the same way as modus tollens is related to modus ponens. After a significant amount of

oversimplification, the argument for the inconsistency of English can be said to have the

following form:

I1. If truth is expressible in English, then English is inconsistent.

12. Truth is expressible in English.

13. Therefore, English is inconsistent.

It may be thought that the inexpressibility argument merely turns the above argument into a

modus tollens. This is incorrect, since the inexpressibility thesis is not derived from Ii together

with the thesis that English is consistent, but, rather, from the fact that the supposition that truth

is expressible leads to a contradiction. Thus, the inexpressibility argument requires one to

establish Ei, which is a much stronger thesis than Ii:

Ei. If truth is expressible in English, then a contradiction is true.

Material Adequacy enables one to establish Ii, but it is not enough to establish El. In order to

establish Ii one needs to appeal to Expressive Adequacy, which is a stronger condition. Material

Adequacy helps to establish that if truth is expressible in English, then a certain set of sentences

is inconsistent. It cannot be used for a reduction argument, unless one proves that the

inconsistency of that set of sentences (i.e., the inconsistency of English) is absurd. On the other

hand, Expressive Adequacy enables one to establish that if truth is expressible in English, then

the sentences that belong to a certain inconsistent set of sentences are true, which can be used in

a reduction argument, because inconsistent sentences cannot all be true.

The distinction between intended meaning and linguistic meaning is also crucial for

answering the second objection. Although one could try to support the inexpressibility thesis by









an inductive argument of the sort suggested by Martin, it would be better to find another way to

answer the objection that the inexpressibility view is self-defeating. One reason is that an

inductive argument avoids the problem only if the concepts that are proved to be inexpressible in

a language are expressible in a metalanguage. If there is no language in which they can be

expressed, the inductive argument remains as problematic as the original version, because no

metalanguage will have the resources that are required to run the argument. Fortunately, the

inexpressibility view does not depend on semantic concepts of English being expressible in some

language other than English. There is a way to answer the objection that the inexpressibility view

is self-defeating that does not make use of an inductive argument. At the same time, it is possible

to explain away the universality intuitions.

Speakers of English certainly intend to use 'true sentence of English' to express the

concept of a true sentence of English. However, an expression may fail to express its intended

content and, consequently, may lack linguistic meaning. Indeed, it turns out that not only

paradoxical sentences but any sentence in which 'true' is used fails to express the intended

content. Nevertheless, expressions can be used to communicate successfully even if they fail to

express the intended content. In general, it is the intended meaning of an expression rather than

its linguistic meaning that plays the crucial role in communication. What is required for one to be

a competent speaker of a language is to know the intended meaning of the expressions in the

basic vocabulary of that language. One could be a competent user of a predicate without

knowing what its linguistic meaning is. Therefore, the default interpretation of a certain sentence

takes the speaker to convey the content determined by the intended meaning of the sentence. In

particular, all that is required for one to be a competent user of 'true' is that one know its









intended meaning31. Therefore, in the vast majority of cases, the use of 'true' is unproblematic:

the hearers know what the speakers intend to express, because both are competent speakers of

English.

There are indeed cases in which what the speaker wants to communicate is not the content

determined by the intended meaning. This is not surprising, because it is widely accepted that

communication is frequently achieved by pragmatic means, rather than by relying on semantic

rules. This is the central idea of Grice's theory about the semantics and pragmatics of natural

languages. Grice argues that there are many cases in which the speaker's meaning is different

from the literal meaning of an expression. The distinction I draw between intended meaning and

linguistic meaning should be understood as a distinction between two aspects of what Grice calls

the literal meaning of an expression. One could easily imagine cases of sentences in which 'true'

is used, such that the speaker's meaning is different from the intended meaning. These cases are

no different from the cases described by Grice, and communication can be explained by

appealing to some conversational implicatures that enable the hearer to figure out what the

speaker wanted to convey.

Difficulties might occur only in a conversation between astute puzzle lovers who notice

that there is a conflict between the intended meaning of a Liar sentence and the commitments

that would result from asserting it. Typically, when one makes an assertion, one is committed to

the truth of the asserted sentence. The worry is that, for Liar sentences, even if the speaker's

meaning is the intended meaning of the Liar, the conflict between the intended meaning and the

commitment to the truth of the Liar sentence might make the hearer think that this is another case

in which the speaker's meaning and the intended meaning are, in fact, different. Although there

31 One can be a competent user of 'true' even if one does not realize that it lacks linguistic meaning. Nevertheless,
one who does not know that true is intended to capture the concept of truth is not a competent user of 'true'.









could be contexts in which someone who utters a Liar sentence wants to convey something

different from the content determined by its intended meaning or does not want to convey

anything (this could only happen in the context of a seminar on the Liar paradox in which the

speaker wants to draw the attention to the notion of linguistic meaning which, for Liar sentences,

is lacking), in most contexts the hearer is able to tell what the speaker intended to convey. When

I am uttering the Liar to say what I think about the status of the Liar, I expect the hearers to be

able to figure out that what I want to say is that the Liar is not true, even though the sentence I

am using is devoid of linguistic meaning.32 Thus, the successful use of 'true' does not entail that

truth is expressible in English. Therefore, the inexpressibility view is not self-defeating.

The universality intuitions can now be explained by distinguishing between two kinds of

semantic universality:

(IU) A language is intentionally semantically universal iff for every semantic concept,

S, there is a predicate of that language that is intended to express S.

(LU) A language is linguistically semantically universal iff for every semantic concept,

S, there is a predicate of that language that (linguistically) expresses S.

Only the linguistic semantic universality of natural languages would entail that truth is

expressible in natural languages. Natural languages, however, are not universal in this sense. The

universality intuitions with respect to natural languages are derived from their intentional

semantic universality.






32 Although sentences in which 'true' is used lack linguistic meaning, as long as there is an intended meaning that
the members of a linguistic community associate with them, it would be improper to say that they are meaningless.









The Status of the T-Biconditionals

If the inexpressibility account is correct, then the T-schema does not hold. Not only are

instances of

(T-schema) 'S' is true iff S.

obtained by replacing 'S' by paradoxical sentences not true; none of the instances of this schema

are true, because they fail to have linguistic meaning. This may seem to be problematic, because

the T-biconditionals have been taken to capture an essential aspect of the concept of truth. It is

true that there are notorious problems with some instances of the T-schema, such as those

obtained from sentences containing context-sensitive expressions or empty names. Nevertheless,

there still seems to be some pressure to require that the T-schema hold true for a restricted class

of well-behaved sentences (maybe declarative sentences that contain no context-sensitive

expressions and are not defective for reasons that do not have to do with truth).

A defender of an inexpressibility view has an easy answer. There certainly is something to

this intuition that needs to be preserved, but this is something different from requiring that the T-

schema be true. For instance, it must be the case that the thought that snow is white is true if and

only if snow is white. For sentential truth, we want to say that a sentence such as 'snow is white'

is true in L if and only if it expresses in L the thought that snow is white, and snow is indeed

white. Nevertheless, it would be wrong to think that the concept of truth requires that the T-

biconditionals be true. What the concept of truth can be held to require is that ifthere is a

predicate that expresses the concept of a true sentence of English, then the corresponding T-

biconditionals for well-behaved sentences should be true. It turns out that the T-biconditionals

are not true, but this is what one should expect, because 'true' fails to express the concept of

truth.









Is the Concept of a True Sentence of English Expressible in other Languages?

Tarski [1933] argues that for any open language that has an exactly specified semantics

one can define its concept of truth in a metalanguage that is essentially richer than it.33 If an

inexpressibility view of truth is correct, and the concept of a true sentence of English is

inexpressible in English, can one express this concept in a metalanguage?34 Herzberger suggests

that the grounding concepts of English, although not expressible in English, are expressible in

German or French. I think there are good reasons to resist this thesis. I will argue that the concept

of a true sentence of English cannot be expressed in German. It cannot be expressed by 'wahrer

englischer Satz', because this would presuppose that 'wahr' expresses in German the concept of

truth. Suppose 'twahr' is a predicate of German that expresses the concept of a true sentence of

English and applies to all and only the true sentences of English. Thus

(A) 'Snow is white' ist twahr gdw schnee ist weiss.

would be true in German. There is no reason to think that there is something special about the

expressive power of German. We should also assume that there is a predicate of English that

expresses the concept of a true sentence of German. Suppose 'wtrue' is a predicate of English

that applies to all and only the true sentences of German. Thus

(B) 'Schnee ist weiss' is wtrue iff snow is white.

would be true in English. Consider now the following pair of sentences

(G) (E) ist twahr.

(E) (G) is not wtrue.




33 Languages of infinite order may be an exception.
34 English does not have an exactly specified semantics, and for this reason it may not be clear what it would mean
for a language to be richer than it.









(G) would be a perfectly well-formed sentence of German, and (E) a perfectly well-formed

sentence of English. However, the pair of sentences leads to a paradox. If (G) is a true sentence

of German, then one can infer that (E) is a true sentence of English. If (E) is a true sentence of

English, then (G) is not a true sentence of German. On the other hand, if (G) is not a true

sentence of German, one can infer that '(G) is not wtrue' is true in English, therefore, (E) is true

in English. If would follow that '(E) is twahr' is true in German, therefore, (G) is a true sentence

of German. Thus, both alternatives lead to a contradiction. This means that either 'twahr' fails to

express the concept of a true sentence of English, or 'wtrue' fails to express the concept of a true

sentence of German, or both fail.35 Since there are no reasons to think that the two languages are

different in this respect, we should conclude that both fail, and thus we need to abandon the

assumption that the concept of a true sentence of English can be expressed in German.

Notice that if the concept of a true sentence of English is inexpressible in other similar

natural languages, then the appeal to an inductive argument would not help an advocate of the

inexpressibility view. There is no way to avoid defending the inexpressibility view in English

itself by using a vocabulary that does not actually succeed in expressing the content that is meant

to be conveyed.

There are additional reasons that speak in favor of the inexpressibility view, but which go

beyond the scope of this chapter. Thus, in chapter four I argue that the expressibility of the

concept of truth cannot be saved by endorsing an inconsistency view of truth because the latter

account actually entails that truth is inexpressible. This also shows that the inexpressibility view

offers a better explanation of the Liar paradoxes than the inconsistency view of truth. Moreover,

in chapter six I argue that the inexpressibility argument can be smoothly extended to provide a

35 The same holds for any other pair of predicates of English and German that are assumed to express the concept of
a true sentence of German and the concept of a true sentence of English, respectively.









uniform solution to all semantic paradoxes. Thus the inexpressibility view should not be

dismissed as self-defeating and counterintuitive, as it has been in the past, but rather

acknowledged as an account that fares better than all other alternatives.









CHAPTER 4
ON THE COHERENCE OF THE INCONSISTENCY VIEW OF TRUTH

It might be argued that although the inexpressibility view is a coherent position, the price

paid for blocking the Liar argument is too high. One might hope that it would still be preferable

to adopt an inconsistency view which is also a coherent view, but does not have the

counterintuitive consequence that truth is inexpressible in English. One huge advantage of the

inconsistency account is that it appears to accommodate all of the ordinary intuitions that lie

behind the Liar argument, so it becomes pointless to search for a flaw in the Liar argument in

order to avoid the inconsistency.

Therefore, one should investigate whether the inconsistency view of truth fares any better.

There are different formulations of the inconsistency views of truth. Charles Chihara [1979]

characterizes an inconsistency view of truth as a view according to which a complete statement

of what 'true' means is inconsistent with all known facts (in particular, it is inconsistent with a

certain undeniably true sentence of reference). Tarski [1933] endorses the view that natural

languages are inconsistent, which is in fact a corollary of an inconsistency view of truth. Finally,

the inconsistency view can be understood as the view that the concept of truth is incoherent.

[Ray 2002] These views should be distinguished from dialetheism, which in addition to the

thesis that a statement of what 'true' means is inconsistent with the facts holds that there could

be true contradictions. I will mainly be concerned with inconsistency views which are committed

to the principle of non-contradiction.

I argue that the inexpressibility view and the inconsistency view are not inconsistent with

one another. Moreover, unless one adopts an extreme position, such as dialetheism, the

inconsistency view entails inexpressibility. The inconsistency view of truth has been frequently

misunderstood. Thus, it has been argued that the idea that languages can be inconsistent is of









dubious coherence on the grounds that the semantic principles seem to be guaranteed to be true

by fiat while the defenders of an inconsistency view of truth, with the exception of dialetheists,

think that some might fail to be true. Hans Herzberger [1967] and Scott Soames [1998], for

instance, argue that the only plausible ways to articulate the idea of an inconsistent language

suggest that inconsistent languages are impossible.

I argue that the misunderstandings are due to the failure to distinguish between intentional

inconsistency and linguistic inconsistency. Most of the attacks against the inconsistency view of

truth miss their target, because what they reject is the notion of linguistic inconsistency (which is

indeed incoherent), while the defenders of an inconsistency view thought of it as a thesis of

intentional inconsistency.

Although the inconsistency view of truth is coherent, the arguments based on linguistic

Liar paradoxes offered by Tarski and Chihara fail to establish that it is true. There are two ways

to think of what it means for the meaning postulates to be inconsistent, which are quite different,

but are normally confused with one another. What the Liar arguments can be used to show is that

the rules which would be underwritten by the intended meaning of 'true' if 'true' succeeded in

expressing the concept of truth are inconsistent. Thus, they do not establish that the intended

meaning of true is inconsistent, but that it is inconsistent with the thesis that 'true' succeeds in

expressing its intended meaning. Even though the inconsistency view cannot be established by

appealing to Liar sentences, it remains possible to prove it on independent grounds.

Nevertheless, even if the inconsistency view holds, truth would be inexpressible. Therefore the

inexpressibility remains the simplest account of the linguistic Liar paradoxes.

The Inconsistency View

Many philosophers have argued that the best way to account for the Liar paradox is to

endorse an inconsistency view of truth. Recall that a Liar paradox consists in an argument that,









starting from some principles that are prima facie true, leads to a contradiction. The premises of

the argument are semantic principles governing the meaning of truth (instances of the T-schema

or some slightly different principles that are supposed to capture the same semantic intuitions)

and certain (semantic or empirical) reference facts about one or more Liar sentences. If such an

argument is valid, then the set of premises is inconsistent. The difficulty stems from the fact that

all semantic premises that are involved in the argument are prima facie true (they appear to be

derived from the meaning we associate with a semantic term, in particular, with 'true'), while the

rules of inference are prima facie valid (they are commonly accepted as valid in every other area

of human thought). Moreover, the reference fact is obviously true. Nevertheless, if the Liar

argument is valid, unless one wants to reject the principle of non-contradiction, the premises

cannot all be true.

Chihara [1979] draws a distinction between two kinds of approach to the Liar paradoxes:

the consistency and the inconsistency views of truth. A consistency view of truth holds that

an accurate statement of what 'true' means will be logically consistent with all known
facts, and in particular with all known facts of reference. [Chihara 1979: 607]

Most of the accounts of the Liar paradoxes that have been offered are, according to Chihara,

different versions of the consistency view. Given that what a Liar argument seems to show is that

some statements grounded in the meaning of the word 'true' are inconsistent with some facts of

reference, either there is an illegitimate step in the argument or some of the semantic principles

fail to correctly capture the meaning of 'true'. An account of the semantic paradoxes is supposed

to point out either the illegitimate rule of inference1 or the illegitimate semantic principle. On the

other hand, an inconsistency view acknowledges the validity of the Liar argument and the


1 A good example of such a strategy is Skyrms' attempt to show that the use of the intersubstitutivity principle is
illegitimate in that context. See Skyrms' paper in Martin [1984].









legitimacy of all its semantic premises. The fact that a valid argument can be used to derive a

contradiction is taken to show that a set of sentences is derivablyy) inconsistent. More

specifically, according to the inconsistency view, an accurate statement of what 'true' means is

logically inconsistent with the facts, in particular with some facts of reference. The semantic

principle Chihara appeals to is an informal version of the T-schema:

(Tr) A sentence is true if, and only if, what is said to be the case by the sentence is in fact

the case. [Chihara 1979: 605]

The Liar argument shows that (1) holds.

(1) (Tr) together with a certain reference fact yields a contradiction.

Sentence (1) together with (2)

(2) (Tr) is part of an accurate statement of what 'true' means.

yields the inconsistency thesis:

(3) An accurate statement of what 'true' means is inconsistent with all known facts.

One can see that the attitude of an advocate of the inconsistency view with respect to the

semantic paradoxes is quite different from that of a defender of a consistency view. A Liar

argument is valid and establishes that an accurate statement of what 'true' means together with

some facts of reference leads to a contradiction. No attempt is made to block the derivation of a

contradiction. One rather tries to offer a diagnosis of this contradiction, which correctly captures

our semantic intuitions.

The Inconsistency of Natural Languages

One version of the inconsistency view of truth has been articulated by Tarski in some (but

not all) of his writings. Even though he was mainly concerned with the notion of truth in a

language and the languages he dealt with are mainly formalized languages, some of his remarks

about natural languages can be read as an endorsement of the inconsistency view of truth. Tarski









was engaged in the project of formulating a definition of the notion of a true sentence of a

language. He introduced two requirements that need to be met by any candidate for a definition:

it must be materially adequate and formally correct. The requirement of material adequacy is,

roughly, the requirement that the definition capture the actual meaning of the ordinary notion of

truth. In particular, the definition should entail the T-biconditionals. Tarski does not say

explicitly what the requirement of formal correctness amounts to, but it is reasonable to assume

that it requires the truth definition to meet some general constraints on definitions (such as

consistency, non-circularity and conservativeness). Thus, an inconsistent definition would not be

formally correct. Although there are languages for which the notion of a true sentence of that

language can be defined, a definition of this sort is not available for all languages. If the

language has a certain degree of expressive power, then no definition of the required sort can be

formulated in the language itself. More specifically, if a language L has enough resources to

allow one to construct a Liar argument for language L', then, in L, there can be no definition of a

true sentence of L' that is both materially adequate and formally correct. This is, roughly, the

content of Tarski's indefinability theorem.

Tarski distinguishes between semantically closed and semantically open languages. A

language is semantically closed if, in addition to its expressions, it contains the names of these

expressions, and semantic terms such as 'true' that refer to sentences of the language and meet

the requirement of material adequacy. If a language is closed and has enough expressive power

to enable one to construct a Liar argument, then there is no way to formulate a definition of a

true sentence of that language, regardless of whether one tries to do it in the language itself or in

a metalanguage. On the other hand, for open languages a definition of truth can be done in a









metalanguage. The metalanguage itself cannot contain its own truth-predicate. One could define

the notion of a true sentence of the metalanguage in a meta-metalanguage, and so on.

Part of the indefinability argument is an argument that the principles that are required for

material adequacy (the T-biconditionals), together with some facts of reference, lead to a

contradiction. Therefore, Tarski takes it to be a corollary of the indefinability theorem that

languages of a certain sort are inconsistent. In fact, his views about the scope of his results

changed. Thus, Tarski [1935] takes the inconsistency result to apply to natural languages.

English, for instance, cannot contain a materially adequate and formal correct definition of the

notion of a true sentence of English. Moreover, since English contains a predicate, 'true', that

meets the requirements for material adequacy, it follows that it is an inconsistent language.

Tarski [1944] is more cautious and claims that the inconsistency theorem can only be established

for languages with an exactly specified semantics. The issue of inconsistency occurs only in

connection with languages with an exactly specified semantics. Since English does not have an

exactly specified semantics, Tarski refrains from saying that it is an inconsistent language.

However, languages that are close enough to English but have an exactly specified semantics are,

even according to Tarski [1944], inconsistent. Moreover, one can argue that the lack of an

exactly specified semantics should not prevent English from being inconsistent. In fact, English

still allows one to run the Liar argument, because all the principles involved in it hold.

Herzberger [1967] interprets Tarski's view as entailing that English is inconsistent.

Meaning Postulates

Given that the Liar argument is accepted as valid and leads to a contradiction, a defender

of the inconsistency view would have to either drop the principle of non-contradiction or accept

that some semantic principles are not true. Both Tarski and Chihara are committed to the laws of

classical logic and try to explain why it is plausible to think that some semantic principles are not









true. This position is in conflict with the widely held view that definitions or meaning postulates

are true by fiat. Therefore, a challenge for the defenders of an inconsistency view is to explain

how it is possible for meaning postulates to fail to be true and still provide a word with meaning.

Moreover, one needs to explain how it is possible for expressions whose meanings are governed

by inconsistent principles to be used in successful communication.

Tarski thinks of the expressions of English as having a meaning captured by a number of

meaning postulates. These meaning postulates might happen to be inconsistent with one another

or inconsistent with other true sentences. In particular, the meaning postulates that determine the

meaning of 'true' turn out to be inconsistent. Since he is committed to the principle of non-

contradiction, Tarski draws the conclusion that not all meaning postulates are true and some of

them must be rejected. What is needed in order for a set of postulates to capture the meaning of a

word is not the truth of those postulates but rather the fact that those postulates are intended to

govern the use of the word. Recall that unlike the Expressive Adequacy requirement formulated

in chapter three, Material Adequacy does not require that the T-biconditionals be true.

Chihara also thinks of meanings in terms of meaning postulates, and tries to explain how it

is possible for some meaning postulates not to be true. He argues that in the case of truth there

are some "generally accepted conventions which give the meaning of 'true"'.2 A lesson that he

wants to draw from the paradoxes is that these conventions or meaning postulates are

inconsistent and some of them are not true.3 To show that the inconsistency of the meaning

postulates does not preclude them from providing a word with a meaning, Chihara constructs the



2 See [Chihara 1979: 611].

3 Matti Eklund [2002] also argues that the notion of an inconsistent language can be made intelligible by observing
the distinction between semantic competence and semantic value.









case of the Secretary Liberation club. The founding members of the Secretary Liberation club

(Sec Lib) have decided that the eligibility to join Sec Lib is determined by the following rule:

(R) A person is eligible to join this club if, and only if, he (she) is secretary of a club

which he (she) is not eligible to join. [1979: 594]

The rule is good enough to ensure the well-functioning of the club. Problems occur only when

Sec Lib hires Ms. Fineline, who happens to be secretary of no other club, as a secretary.

According to (R), Ms. Fineline is eligible to join Sec Lib if and only of she is not eligible. To be

more specific, (R) is inconsistent with the following two claims:

(Fi) Ms. Fineline is secretary of Sec Lib.

(F2) Ms. Fineline is not secretary of any other club.

Given that the truth of (Fi) and (F2) can be established empirically, the only option is to deny the

truth of (R). This shows that not all the meaning postulates can be made true by fiat. (R) cannot

be made true by fiat. This does not mean that they cannot provide the word with a meaning if

these inconsistencies occur only in some isolated cases. Since inconsistencies only occur in some

isolated cases (Ms.Fineline), (R) can still be used as an effective criterion of eligibility.

Similarly, not all the T-biconditionals can be made true by fiat. They cannot all be made true

because they are either inconsistent or inconsistent with some empirical facts.

The Sec Lib case offers a nice model of how one should think of inconsistent meaning

postulates, but there are additional worries that need to be addressed. On the face of it, it seems

that a consequence of the inconsistency view of truth is that no sentence in which 'true' functions

as a predicate can make a statement. Chihara is aware of this difficulty which he formulates as

follows:

An explicating formula, ..., determines whether a predicate applies to an object x only if
there is no valid argument from true statements and the explicating formula itself to the









conclusion both that the predicate does apply to x and that the predicate does not apply to
x. [Chihara 1983: 225]

Given that the explicating formula (the T-schema) is inconsistent with some known facts, it

follows that everything can be inferred from them. This would mean that no sentences containing

'true' as a predicate can be used to make a statement. Chihara thinks that the difficulty can be

solved by saying that even though there is a way to derive a contradiction, it would not be

reasonable to do so: "in "real life" situations, one doesn't simply accept blindly the logical

consequences of whatever one may initially have reason to believe" [1983: 226]. It would also be

wrong to think that an inconsistency view forces one to endorse a contradiction. Although the

meaning postulates enable one to infer that a Liar sentence is both true and not true, it is not

reasonable to make this inference. Once one realizes that the meaning postulates are

inconsistent, one realizes that it does not make sense to ask whether the Liar is true or not4

Inconsistent Languages and the Inconsistency View of Truth

There is a sense in which the view that English is inconsistent and the inconsistency view

of truth are different. The thesis that the language is inconsistent is, in fact, weaker than the

thesis that truth is inconsistent with the facts. The inconsistency of the language might not have

anything in particular to do with the concept of truth. It might be related with notions that occur

in paradoxes other than the Liar paradox. Thus, the inconsistency might be caused by principles

having to do with the notion of satisfaction, with the notion of reference, or it might be the case

that no particular notion is responsible for the inconsistency, but the language as a whole. In fact,



4 Gupta and Belnap raise an objection against this answer provided by Chihara:
By putting the entire burden of the theory on the notion of "reasonable inference" a notion of which no
theoretical or even intuitive account is given it obscures completely the contribution of the T-biconditionals
to the meaning of 'true'. [Gupta & Belnap 1993: 15]
The problem, according to Gupta and Belnap, is that one needs to offer an account of what it means for an inference
to count as reasonable, but it is hard to see how such an account can be developed without losing the simplicity that
was emphasized as one of the main reasons to prefer the inconsistency account over its competitors.









Tarski's argument for the inconsistency of English relies on a version of Grelling's paradox

rather than the Liar paradox. Nevertheless, he thought of the Grelling paradox as a paradox that

is, just like the Liar paradox, based on the semantic principles that govern the meaning of 'true'.

Thus, a defender of the inconsistency of English is normally also committed to an inconsistency

view of truth.

The Inconsistency of the Concept of Truth

Tarski and Chihara talk about the meaning of 'true' and potential definitions of it, but they

do not talk explicitly about the concept of truth. Nevertheless, the inconsistency story can be and

has been extended to the level of concepts. Ray [forthcoming: 8] argues that in fact "Tarski is

best understood as having held an incoherence view of the concept of truth". The T-

biconditionals (or, if one prefers, Chihara's (Tr) principle) are principles that capture part of the

content (or maybe the entire content) of the concept of sentential truth. It makes sense to talk

about inconsistent concepts if concepts are understood as coming supplied with explicit

application rules associated with a predicate. Certainly, this is not the only way one can think of

concepts, so one can reject the whole strategy if one can argue that this is not the correct way to

think of concepts5. Anyway, if concepts can be conceived of in this way, one can offer a pretty

clear picture of what it means for a concept to be inconsistent. The concept is inconsistent if the

application rules that are associated with it turn out to be inconsistent. These rules can offer full

conceptual warrant to some sentences in the language, but they cannot guarantee that those

sentences are true. The idea of thinking of concepts as coming supplied with application rules is

also useful in characterizing the notion of an inconsistent language. A language M is inconsistent

5 One can worry that concepts should not be tied up with a predicate, but they should rather be thought of in terms of
conditions for objects to fall under it. This could still enable one to talk about inconsistent concepts. They would be
concepts such that the conditions for objects to fall under them are inconsistent. This way one could talk about a
concept as being inconsistent even if no predicate has ever been intended to express it.









if "a deductively inconsistent sentence is derivable by the rules of inference of M from the

assertible sentences of M". [Ray 2002: 170] 6

A Priori or Empirical Inconsistency?

The inconsistency view of truth does not depend on any empirical facts. The argument for

inconsistency depends on a fact of reference which could be either empirical or non-empirical.

An empirical Liar would appeal to a contingent identity sentences such as (4).

(4) The first sentence on p. 6 in Quine's first published book is identical to 'The first

sentence on p. 6 in Quine's first published book is not true'.

The truth of the reference premise can be verified, for instance, by a mere inspection of a page in

a book. Tarski's argument for the inconsistency of the language consisted initially in a Liar

argument based on an empirical fact of reference. Nevertheless, he argued that the thesis that the

language is inconsistent can also be derived without appealing to any empirical premise. One

way to do this is to consider Grelling's paradox instead of the Liar paradox. Tarski shows how

one can run a Grelling argument that involves only semantic premises and no empirical premise.

Ray (forthcoming) argues that the version of the Grelling argument sketched by Tarski falls short

of offering the desired result. This is because the argument assumes that the definition of self-

applicability is formally correct. This means that the thesis that is delivered by the Grelling

argument is not that truth is indefinable, but that either truth or self-applicability is an indefinable

notion. Nonetheless, Ray shows how one can modify the Grelling argument so that it yields an

indefinability theorem that depends neither on empirical facts of reference nor on the formal

correctness of some other notion, such as self-applicability. Moreover, Tarski could have made


6 The assertible sentences of the language are those that either have full conceptual warrant or have been confirmed
empirically.
7 See [Ray forthcoming: 10].









the same point by using a non-empirical Liar paradox instead of the Grelling paradox. Instead of

using an empirical sentence such as (4), one could use instead (5)

(5) (L) = '(L) is not true'.

which is not a merely contingent truth, because the expressions on both sides of the identity sign

are rigid designators. In the case of non-empirical Liars all the reference facts involved in the

argument are semantic facts. The corresponding identity premise is true by fiat (and, in addition,

necessarily true), because English is a flexible language that allows one to choose any name

whatsoever for a certain sentence.

The fact that natural languages have all the features needed for running a Liar argument for

a non-empirical Liar shows that both theses can be formulated in the strong version as claims of

inconsistency per se, not only inconsistency with empirical facts. In what follows I will use the

inconsistency thesis to refer to the strong version, unless I explicitly indicate that it is the weaker

version that I have in mind.

The fact that there is also a non-empirical way to support the inconsistency view can be

used to show that the incoherence of the concept of truth is a purely conceptual matter: it can be

established a priori. In fact, to show that the incoherence is a purely conceptual matter, one

would not need to use a non-empirical Liar: one could use a modified version of the empirical

argument. Instead of arguing that there is a Liar sentence, one could argue that there could be a

Liar sentence. Thus,

(S) The first sentence on p. 6 in Quine's first published book is not true.

could be used in a Liar argument even if the first sentence on p. 6 in Quine's book is not (S)

itself. It is enough that it could have been (S). Given that the T-biconditionals must hold

necessarily, one can infer from them









0(S & ~S),

from which one can derive

O(S & -S) & O~(S & ~S),

which is an explicit contradiction. This means that even for a language that does not have the

resources to refer to the Liar sentence via a rigid designator, one could still prove a priori that the

concept of truth is incoherent.

The Inconsistency View and Classical Logic

Both Tarski and Chihara assume that the laws of classical logic hold for sentences of

natural language. It is important to notice that the inconsistency view of truth does not actually

depend on the assumption of classical logic. In chapter two I argued that one who rejects the

principle of bivalence is normally also tempted to reject the principle of the excluded middle or

reduction as absurdum. This way one might hope that although (Tr) together with the reference

premise for (L) still allow one to deductively infer (C),

(C) (L) is true iff (L) is not true.

one no longer has resources to derive a contradiction of the form A & ~A. According to some

very weak systems of logic, (C) would count as true if both sides of the biconditional lack a

truth-value. If this is the case, then (Tr) is not inconsistent with all known facts. Other systems of

logic, such as the system based on Kleene's three-valued schema, take (C) to lack a truth value if

both sides of the biconditional lack a truth-value. In this case (Tr) is still inconsistent with all

known facts as long as inconsistency is understood as a semantic notion. Nevertheless, they are

not inconsistent in the sense that one can derive a contradiction. Chihara took the inconsistency

to hold in both senses. If one takes (Tr) to be a complete statement of what 'true' means8, then


8 Chihara seems to suggest that they indeed exhaust the meaning of 'true'.









by rejecting classical logic one can indeed weaken the strength of the inconsistency thesis; at

least it would not have the worrisome consequence that the mere acceptance of the principles

governing the meaning of 'true' commits one to accepting the truth of a contradiction.

Nevertheless, there are good reasons to think that (Tr) cannot exhaust the meaning that we

associate with the predicate. In chapter two, I argued that there are other principles that capture

other aspects of the meaning of 'true' and these principles together with (Tr) can be used to

derive a contradiction on the basis of some very weak principles of logic. For instance, TC* and

FC*, repeated here,

TC*: T( Al) V ~T( Al) [every sentence is either true or not true]

FC*: F( Al) V ~F( Al) [every sentence is either false or not false]

are principles that capture part of the meaning that we associate with the predicates of truth and

falsity. If TC* is added to the list of meaning postulates, then they can be proved to be

inconsistent (in the sense that one can derive from them a contradiction) from logical laws that

are much weaker than classical logic. Thus, the inconsistency view of truth does not depend on

the assumption of classical logic.9

Skepticism with Respect to Inconsistency

The inconsistency view has been treated by many with skepticism. There are some prima

facie reasons that have been offered to show that a view of this sort is incoherent. The

consistency of natural languages, as well as the consistency of the meaning of 'true', seems to be

a matter that is postulated rather than an issue that needs some investigation in order to be

confirmed or informed.



9 Of course, the inconsistency view does not depend on the assumption of the principle of non-contradiction.
Dialetheists have explored the possibility of solving the paradoxes by denying the principle of non-contradiction, but
I will not discuss this view in this study.









It has been argued that there is no interesting way to articulate the notion of an inconsistent

language. First of all, for uninterpreted languages the problem of consistency cannot make sense;

there can be no sentences that can be evaluated as true or false and there are no axioms that

would allow one to infer a contradiction. 10 There is no set of sentences that could be tested for

consistency. So one must assume that when the issue of consistency occurs, languages are

understood as interpreted languages; when Tarski talks about inconsistent languages, he certainly

has interpreted languages in mind. Scott Soames [1998] lists a number of possible ways to

understand the notion of an inconsistent language, none of which are acceptable:11

A. A language is inconsistent iff there are inconsistent or contradictory sentences in

it.

B. A language is inconsistent iff some sentence of the language and its negation are

both true (iff at least one contradiction is true)

C. A language is inconsistent iff there is some theory, T, formulated in that language,

such that T is inconsistent.

The first proposal does not work, because "any language with negation contains

inconsistent sentences" and "any language with both negation and conjunction contains

contradictory sentences" [Soames 1998: 53]. The second characterization, attributed by Soames

to Salmon, fares no better: as long as the principle of non-contradiction is true, no language can

be inconsistent in this sense. Finally, C cannot be the characterization one is looking for, because

the inconsistency of a language cannot be characterized as the inconsistency of any old theory




10 There might be inconsistent rules of syntax associated with a certain vocabulary, but it is hard to think of a
language in the absence of a consistent system of rules of syntax.

1 Soames [1998: 53]









formulated in that language. The inconsistency of a particular theory formulated in a certain

language does not necessarily make the language itself inconsistent.

Although C fails to provide the desired characterization, it remains helpful in that it

indicates that a correct characterization should have to do not with some arbitrary theory that can

be formulated in the language, but with a theory that has to do with the semantics of that

language (formulated either in the language itself or in a metalanguage). One option that one

might want to consider is the theory that consists in all analytic sentences of that language,

because the analytic sentences are those that are true in virtue of the meanings of the words. An

inconsistent language could then be characterized by the following clause:

(*) A language is inconsistent iff its analytic sentences are inconsistent.

Unfortunately, there cannot be inconsistent languages in this sense. 12 This is the conclusion of

the following argument offered by Herzberger (from "the Logical Consistency of Language"):

P1. The language L is not logically consistent in its analytic sentences.

P2: L contains a nonempty set A of sentences such that: [from P1]

P2a: Every sentence in A is analytic; and

P2b: The set A is a logically inconsistent set of sentences.

P3. Every sentence in A is true. [from P2a]

P4. At least one sentence in A is not true. [from P2b]

This means that the assumption that there is a language that is inconsistent in this sense

leads to a contradiction. Thus, if one still wants to talk about inconsistent languages, one should

characterize this notion in a different way. Soames believes that there is no acceptable

characterization of the notion and rejects it as incoherent.


12 Priest [in Semantic Closure] argues that there are, but this is only because he denies LNC.









The claim that the meaning of a term is inconsistent also faces prima facie difficulties.

Traditionally, a definition has been taken as successfully conferring meaning to a term only if it

was consistent. Thus, a set of postulates used as a definition is either consistent, in which case all

postulates in the set are true by fiat, or it is inconsistent, in which case it fails to provide a

definition. It is true that not all expressions in English have been introduced by a definition.

However, there are semantic rules that govern the use of these expressions. As long as these

expressions are held to be meaningful, those rules should be true by fiat.

The notion of an inconsistent concept lends itself to similar criticism. In general, it is not

clear whether and in what sense one can talk about inconsistent concepts. There is room for

denying that there are inconsistent concepts. The notion of an inconsistent concept should be

distinguished from the notion of a concept with an empty extension or with a necessarily empty

extension. Normally, an expression like 'round square' is taken to have an empty extension

necessarily. One can find more or less reasonable grounds to deny that it actually expresses a

concept. However, having a necessarily empty extension does not offer enough grounds for

denying that there is a concept expressed by the predicate. It is hard to deny that 'prime number

bigger than 23 but smaller than 29' expresses a concept, even though its extension (assuming that

mathematical truths are necessary) is also necessarily empty. In any case, inconsistent concepts

are not merely empty concepts, concepts whose extension is (contingently or necessarily) empty.

In their case there is no class or set that can be associated with the concept. If the extension of a

concept is understood as a class or set of objects to which the concept applies, then there are no

inconsistent concepts, because they would have no extension. There could still be objects to

which the concept unproblematically applies or fails to apply; thus, unlike the concept of a round

square which applies to no object, the concept of truth applies without difficulties to many









sentences, such as 'Snow is white', and it fails to apply to some other sentences. Nevertheless,

this is not enough to determine an extension. Thus, one could argue against the idea that there are

inconsistent concepts by saying that having an extension is a necessary condition for something

to count as a concept.

Intentional Inconsistency versus Linguistic Inconsistency

The distinction between intended meaning and linguistic meaning enables a better

understanding of the inconsistency view and shows, at the same time, in what sense the view has

been misunderstood by some of the opponents of the view. The inconsistency thesis, (3), can be

read either as a thesis of intentional inconsistency or as a thesis of linguistic inconsistency. I will

argue that the thesis is properly understood as the thesis that the intended meaning of 'true' is

inconsistent. Tarski's thesis should be understood as the thesis that languages of a certain sort are

intentionally inconsistent. Most of the objections that have been raised against inconsistency

views are misguided because they stem from a misinterpretation of the inconsistency view as a

thesis of linguistic inconsistency, which would indeed be an incoherent view.

Nevertheless, I will argue that the arguments offered by Tarski, Chihara and others fail to

establish that truth is intentionally inconsistent, because there are two ways in which one could

think of the meaning postulates which have not been properly distinguished from one another.

Two Kinds of Inconsistency

One can notice that Chihara's thesis of the inconsistency of 'true' can be read in two ways:

as the thesis that a statement of what is the linguistic meaning of 'true' is inconsistent, or as the

thesis that the principles that characterize the intended meaning of 'true' are inconsistent. Under

the first reading, the inconsistency thesis requires that the rules that characterize the linguistic

meaning of 'true' (the T-biconditionals) are inconsistent. Under the second reading, the

inconsistency view is more modest. It only requires that the principles that are intended to govern









the meaning of 'true' are inconsistent; they do not have to be true. It is obvious that what Chihara

is committed to is the view that the intended meaning of 'true' is inconsistent. He allows the

possibility that some of the meaning postulates (some instances of the T-schema) are not true.

Similarly, there are two ways in which a language can be said to be inconsistent. Tarski's

indefinability theorem establishes that closed languages are inconsistent. Nevertheless, he does

not use this result to run a reduction argument to show that no language is in fact closed. He takes

the inconsistency of a language to be a perfectly acceptable result. There is no contradiction

involved in claiming that a language is inconsistent. It only means that the set of rules that are

intended to govern the use of the expressions of the language is inconsistent.

On the other hand, Herzberger and others like him (Kripke, for instance) think that the

indefinability theorem can be used to run a reduction argument to prove that there are no

semantically closed languages. According to Herzberger, as long as one sticks with the T-

biconditionals, "Tarski's definition [the definition of a semantically closed language] leaves too

little room for any such languages semanticallyy closed languages]" [Herzberger 1982: 481]. To

support this idea he quotes the following remark by Kripke:

the result should rather be formulated as such: no interpreted language in the ordinary first-
order predicate calculus containing number theory and so on, can be semantically closed;
not that there are semantically closed ones but they are inconsistent". [Herzberger 1982:
481]

It is clear that Tarski, on the one hand, and Herzberger and Kripke, on the other, do not talk

about the same notion of language inconsistency. Thus, one can distinguish two senses in which

a language can be said to be inconsistent13:



13 There is a parallel distinction between two senses in which a language can be closed:
- a language is ii,,. ..illi closed if it has a predicate that expresses the concept of truth as part of its linguistic
meaning.
-a language is intentionally closed if there is a predicate in the language that is intended to express the concept of
truth.









(Intentional Inconsistency) The principles that are intended to govern the use of the

expressions of the language are inconsistent.

(Linguistic Inconsistency) The principles underwritten by the linguistic meanings of the

expressions of the language are inconsistent.

Herzberger and Kripke (and Soames as well) talk about linguistic inconsistency and correctly

argue that there cannot be languages of this sort. The thesis that a language is linguistically

inconsistent comes down to the thesis that its analytic sentences are inconsistent, which is false

because analytic sentences are true by definition. Nevertheless, what Tarski argues for is that

natural languages are intentionally inconsistent. Herzberger and Soames miss the target, because

they misinterpret Tarski as endorsing a thesis of linguistic inconsistency.

The view that the linguistic meaning of 'true' is inconsistent is indeed false. In order for an

expression to have linguistic meaning a certain adequacy condition, Expressive Adequacy, must

be met. This condition requires that the principles that are intended to be true are true and

determine the linguistic meaning of the expression. Therefore, the rules that determine the

linguistic meaning of an expression cannot be inconsistent. In particular, the linguistic meaning

of 'true' cannot be inconsistent. This means that the inconsistency view of truth is consistent

only if it is understood as the thesis that the intended meaning of 'true' is inconsistent. The

objections raised by Herzberger and Soames against the inconsistency view miss the target,

because they are objections against a view that nobody endorsed (the view that linguistic

meaning is inconsistent).





It is undeniable that there is a predicate in English that is intended to express the concept of truth. Tarski's thesis that
English is inconsistent is supposed to be a consequence of the fact that English is intentionally closed. If one wants
to reject the thesis that English is intentionally inconsistent, what one needs to show is that intentional closure does
not in fact lead to a contradiction.









I will argue that although the thesis that meanings can be intentionally inconsistent is

coherent, the paradox based arguments such as those offered by Tarski and Chihara fail to

establish that the intended meaning of truth is inconsistent or that English is inconsistent.

Intentional Inconsistency

I think that the thesis that the intended meaning of a predicate is inconsistent is ambiguous.

Suppose that one needs an expression to capture a certain content, which can be characterized by

a certain set of application rules. When a predicate is assigned the role of capturing that content,

then it is supposed to be used in accordance with a corresponding set of principles. This

corresponding set of principles could be identified with the application rules of the concept (the

intended meaning rules) or with the rules that would characterize the predicate in the hypothesis

that it successfully expresses the concept (the hypothetical meaning rulesl4). In most cases, the

hypothetical meaning rules are consistent. Nevertheless, they could also be inconsistent. This

does not necessarily mean that the intended meaning is inconsistent. What it means is that if the

predicate succeeds in playing the role that has been assigned to it, then the intended meaning

rules are inconsistent. There are two ways in which this situation could occur: either the intended

meaning is inconsistent, or it is inconsistent with the hypothesis that the predicate successfully

plays the role that has been assigned to it. Take, for instance, the predicate 'true'. This predicates

is intended to capture the concept of truth, which means that it is supposed to apply to a sentence

just in case that sentence is true (the intended meaning rules). If the predicate succeeds in playing

this role, then its meaning is determined by the T-biconditionals (the hypothetical meaning

rules), which can be proved to be inconsistent by using a Liar argument. This could be taken to

mean that the intended meaning of 'true' is inconsistent or only that the intended meaning of

14 When one wants to express a certain concept in L, on is looking for a predicate of L which, if it is successful,
would behave in the way prescribed by the concept.









'true' is inconsistent with the hypothesis that 'true' successfully plays the role that has been

assigned to it. An inconsistency theorist chooses the former option, while a defender of the

inexpressibility view chooses the latter. The difference between the two options can be better

emphasized by comparing the meanings of 'true' and 'true*'. Chihara [1979] introduces a new

predicate, 'true*', which is implicitly defined by the following T-biconditionals:

(T*) 'S' is true* if and only if S.

The meaning of 'true*' is by definition exhausted by the above T*-biconditionals. Therefore, in

this case the T*-biconditionals constitute the intended meaning rules, not only the hypothetical

meaning rules. The intended meaning of 'true*' is characterized by the T*-biconditionals, which

are inconsistent. On the other hand, 'true' is not defined by the T-biconditionals. It is a predicate

intended to capture the concept of truth, which means that if it succeeds, then its meaning would

be characterized by the T-biconditionals. The T-biconditionals constitute the hypothetical

meaning rules, not also the intended meaning rules. Both 'true' and 'true*' fail to express the

concepts that they are intended to express. The difference is that the concept of a true* sentence

of English is incoherent (the intended meaning of 'true*' is inconsistent), while the concept of a

true sentence of English could very well be consistent (the intended meaning of 'true' could very

well be consistent) because Liar sentences do not offer any good reason to think that it is not

consistent. Chihara's concept of an eligible member of Sec Lib would indeed be an incoherent

concept (i.e., the intended meaning of 'eligible member of Sec Lib' is inconsistent), but the case

is more similar to the 'true*' case rather than 'true', because the application rules of the concept

are inconsistent regardless of whether the predicate succeeds in playing the role that has been

assigned to it.









What the above remarks show is not that the inconsistency view is false. The thesis that

some concepts are incoherent is indeed coherent. Nevertheless, they show that the arguments

offered by Tarski and Chihara fail to establish the inconsistency view, because they leave open

the possibility that the intended meaning is consistent, but the predicate fails to express it.

Moreover, it can be argued that the whether the concept of truth is coherent or not, it remains

inexpressible. Therefore, the thesis that truth is inexpressible offers the best explanation of the

Liar paradoxes and the thesis that the concept is incoherent is unjustified.

Inconsistency Entails Inexpressibility

The inexpressibility argument that I formulated in chapter three does not appeal to the

assumption that truth is a consistent concept. Therefore, the same argument can be used to argue

that the concept is inexpressible even if the inconsistency view holds. However, one can

establish a more general result: inconsistent concepts are inexpressible. In order for a predicate to

express a concept it must have a linguistic meaning that would have to match its intended

meaning. If one thinks of concepts, the way Ray suggested, in terms of application rules

associated with a predicate, then the notion of a concept can be assimilated with the notion of

intended meaning of a predicate: both are thought of in terms of some intended rules of

application of a predicate. I already argued that if the hypothetical meaning rules are

inconsistent, then they fail to provide the predicate with linguistic meaning, because in order for

there to be a match between the rules that characterize the linguistic meaning and the

hypothetical meaning rules, the hypothetical meaning rules must be consistent. Nevertheless, if

the intended meaning rules are inconsistent, then the hypothetical meaning rules would be

inconsistent as well. Thus, if the intended meaning rules are inconsistent, the predicate fails to

have a linguistic meaning.









Chihara recommends the following criterion that can be used to choose between

competitor theories:

One consideration that moves reasonable people to prefer a theory to a competitor is
whether the acceptance of the one would require less revision of our presently accepted
scientific theories than would acceptance of the other. [Chihara 1979: 604]

According to the inconsistency view, the Liar paradoxes are explained by assuming that the

principles that characterize the intended meaning of 'true' are inconsistent, a view that entails

that the language cannot express the concept of truth. However, if this is how paradoxes are

explained, one could also explain them by just denying that the language can express the concept

of truth without also having to assume that the concept is inconsistent. Thus, the inexpressibility

account offers the simplest account of the semantic paradoxes.









CHAPTER 5
NON-LINGUISTIC LIARS

In chapter three I argued that the linguistic Liar paradoxes (the paradoxes generated by

Liar sentences, Liar statements or Liar utterances) can be solved by saying that natural

languages, appearances to the contrary notwithstanding, are unfit to express the concept of truth.

This account enables one to argue that none of these Liar paradoxes give us good reason to think

that the concept of truth is inconsistent. However, there are non-linguistic Liar paradoxes that are

still unaccounted for which could provide such a reason. The inexpressibility view is compatible

with the inconsistency view of truth. In fact, one of the theses that I argued for in chapter four is

that all inconsistent concepts are inexpressible. Nevertheless, the inexpressibility view would

lose much of its appeal if inconsistencies can be proved to persist at the non-linguistic level. In

chapter four, I argued that the inexpressibility view can be used to show that linguistic Liars do

not suffice to show that the concept of truth is inconsistent. If it can be argued that an

inconsistency survives at the level of non-linguistic Liars even if one drops the assumption that

there is a predicate of English that expresses the concept of truth, then the inexpressibility view

would fail to provide a way to save the consistency of truth.

In this chapter I argue that all Liar arguments generated by non-linguistic Liars can be

blocked. There are two kinds of non-linguistic Liar paradoxes that need to be accounted for, each

of them correlated with a particular kind of truth-bearer: propositions and mental representations.

First, let us notice that non-linguistic Liars that are still specified by means of a sentence do not

raise special difficulties for the inexpressibility view. Consider, for instance, the following non-

linguistic Liar:

(LP) The proposition expressed by (LP) is not true.









It is clear that a defender of the inexpressibility view would have no difficulty blocking the Liar

argument by denying that (LP) expresses a proposition (because it fails to have a linguistic

meaning). The difficult cases which I will focus on in this chapter are the potential Liars which

are not specified via a sentence. I will first argue that no inconsistency can be derived at the level

of mental representations, and then show that there are no Liar propositions.

Mental Representations

Consider first the possibility of Liar thoughts, understood as mental representations. Could

one think that one's thought is not true? On the face of it, it appears that there should be Liar

thoughts. Just like sentences, mental states can have the property of being about themselves. It is

hard to deny that one could think at t that the thought one has at t is not true. This thought is

paradoxical, because both the assumption that it is true and the assumption that it is not true seem

to lead to a contradiction. It seems also possible for there to be Pair Liar thoughts. If A thinks at t

that B's thought at t' is not true, and B's thought at t' is that A's thought at t is true, then we

seem to land again in paradox. If A's thought is true, then B's thought is not true. Since what B

thinks at t' is that A's thought is true, it would follow that A's thought is not true. On the other

hand, if A's thought is not true, it follows that B's thought at t' is not true. Since this is the

content of A's thought, it would follow that A's thought at t is true after all. Again, both

alternatives appear to lead to a contradiction.

I used 'thought' in its ordinary sense. Thoughts are very often understood in a very broad

sense, to cover any mental state or psychological experience. Descartes, for instance, takes all

intentional state (belief, desire, hope, etc.) to be a form of thinking. Nevertheless, one needs to

distinguish the various senses in which one could talk about Liar thoughts, and see exactly which

of them indeed generates a contradiction. Some of them do not. For instance, the Liar thought

would be really unproblematic if it were understood as the mere act of entertaining a proposition.









Frege distinguishes between merely entertaining a proposition and the act of judging, which is an

act of assenting to that proposition or of holding-it-true.1 If my thought at t is the act of merely

entertaining the proposition that my thought at t is not true, it would be a completely harmless

Liar thought. The act of merely entertaining a content is not the kind of entity that has a truth-

value. If at t' I merely entertain the proposition that my thought at t is not true, then my having

that thought at t' would not be enough to infer that my thought at t is not true. Likewise, no

contradiction can be derived from my later judgment that the Liar thought is not true. If at t' I

judge that my thought at t is not true, then my judgment at t' is true. However, this does not

entail that the thought I have at t is true, because my belief at t' and my thought at t differ in

precisely that respect that entitles the former but not the latter to have a truth-value: the

psychological mode.

Thoughts and Beliefs

The Liar thought paradoxes become less trivial when the structure of the mental state

includes an assenting psychological mode in addition to the grasp of the propositional content.

When one believes something, one does not merely entertain a certain content but also assents to

that propositional content. It is customary to distinguish between dispositional and occurrent

beliefs. Normally, dispositional beliefs are considered to be those that I might not be actually

aware of (I might have never had them present in my mind), but I have the disposition to assent

to if asked. Occurrent beliefs are those that I am currently aware of. According to this picture, I

have an indefinite number of dispositional beliefs, only a few of them (if any) being occurrent at

a certain instant of time. Whether this is the right way to draw the distinction is controversial. It


1 What exactly Frege meant by a judgment is controversial. Ricketts [1996], for instance, argues that the act of
judging is to be understood as the act of acknowledging the truth of the content, and that Frege would want to say
that one can judge only true propositions. I assumed here the more standard understanding, according to which the
content that is judged does not have to be true. See also Kremer [2000] and Ricketts [1986].









has been argued plausibly that the way we use 'belief in English suggests that it only applies to

dispositional states2. If so, the proper distinction should be between tacit and explicit beliefs. I

will not engage in this controversy here, but from now on I will use 'belief to refer to

dispositional states and 'thought' to refer to what sometimes are called occurrent beliefs3

A complete account of the mental representation Liars would have to deal with both Liar

beliefs and Liar thoughts. I think that the notion of thought is more basic, because dispositional

beliefs are characterized in terms of thoughts. Therefore, I will first give an account of Liar

thoughts and then I will apply the account to Liar beliefs.

Liar Thoughts

One way in which one could block the Liar thought argument is to deny that there are Liar

thoughts. It is a fact that I have direct psychological evidence that I have a mental state at t,

which I have good reasons to characterize as a thought. However, one could argue that although

it might seem to me that I have a thought at t, what the Liar argument shows is that I actually do

not have one. The difficulty with denying that there are Liar thoughts can be brought to light

when considering empirical Liar thoughts. Adapting a line of reasoning from Kripke, we can

make the case as follows. It is implausible to argue that the mental states that are commonly

described as empirical Liar thoughts are, in fact, not thoughts because they are paradoxical. In

their case paradoxicality is an empirical matter, while being a thought is an intrinsic feature of

the mental state. Had the empirical circumstances been different, then there would have been no

problem calling that mental state a thought. This means that the mental state should also count as

a thought under the current circumstances. Therefore, it would be a mistake to deny that there are


2 See Bach [1981] and Lycan [1986].

3 1 will distinguish later between thoughts and judgments and between acts of thinking and acts of judging. Not all
thoughts constitute judgments.









empirical Liar thoughts or Pair Liar thoughts. Notice that this view does not assume an internalist

view about thought content. It might seem that the externalist thesis that the content of a mental

state might depend on external factors undermines the argument I used to support the thesis that

the mental state I have at t is a thought. Nevertheless, even an externalist would have to admit

that the mental state's being a thought cannot depend on external factors. Therefore, it is safe to

conclude that there could be Liar thoughts.

Gappy Thoughts

I will assume now that the mental state I have at t is a Liar thought, T. If the Liar thought is

true, it is hard to find a way to avoid a contradiction. The question is whether one can coherently

maintain that the Liar thought is not true. The Liar argument shows that there are prima facie

reasons to think that one cannot. Suppose that at t' I think that the Liar thought is not true, and

suppose that the thought I have at t', T', is true. The problem is that the thought I have at t and

the thought I have at t' appear to have the same content. This would mean that in order to avoid

the contradiction, one would have to deny the attractive principle that the truth-value of a thought

is a function of its content and how the world is.

I will argue that there are good reasons to think that the two thoughts, T and T', do not

have the same content, but this will not be quite enough to solve the problem. The challenge will

be to argue that the difference in content is sufficient to determine a difference in truth-value.

The idea that the two thoughts have the same content derives from the observation that in

both cases, the same thing is thought about the same entity. Various traditional accounts of the

relation between thought and thought content indeed support the content identity thesis. One

could take the two thoughts to be singular thoughts whose contents have the object of thought as

a subpart. Since both thoughts are about the same object (the thought at t) and the other belief

ingredients appear to be the same, then the beliefs would have to have the same content.










Alternatively, one could also think that it is not the object of thought that is part of the thought

content, but a mode of presentation that is insensitive to contextual features such as the thinker

and the time at which the thought occurs. For instance, the mode of presentation could be the

content of a purely qualitative description. This view would also support the idea that T and T'

have the same content.

Both Fregean and Russellian semantics have enough resources to support the idea that, in

fact, T and T' have different contents. The difference in content can be explained if the

contribution made by contextual factors such as the thinker and the time of the thinking event

- to the content of the thought is taken seriously4. Unfortunately, this is not yet enough to block

the Liar Thought argument. What needs to be argued is that there is a difference in content that is

sufficient to determine a difference in truth-value. It appears to be difficult to explain why two

thoughts that differ only in that they occur at different instances of time differ in truth-value. For

example, one could argue that my thought that Brutus killed Caesar and another person's thought

that Brutus killed Caesar do not have exactly the same content, due to some contextual elements


4 For instance Traditional interpretations of Fregean semantics took modes of presentations to be contents of purely
qualitative definite descriptions. However, it has been argued (Evans [1997] and Perry [1997]) that a proper
understanding of the Fregean semantics is one that makes the mode of presentation sensitive to the thinker, the time
of the thinking event and the object of thought. According to this picture, T and T' would have to have different
contents although they are thoughts of the same person and about the same object, because the mode of presentation
of T at t is different from the mode of presentation of T at t'.
A similar story can be told within a non-Fregean semantic framework. Even if the possession of a thought such as T
or T' could be reported by using a proper name in the that-clause, as in (1) and (2),
(1) At t I was thinking that T was not true.
(2) At t' I was thinking that T was not true.
the contents of these thoughts are not singular propositions that contain T as a proper part. Leaving aside the fact
that 'true' fails to have a linguistic meaning, there are good reasons to think that the content of T or T' is not
identical with the content of the sentence following the that-clause in the thought report, but rather with the content
of a sentence that contains a description of the thought. Ludwig [1996] argues that if the description of the object is
purely qualitative, then it might fail to pick out an object. If the universe we live in turns out to be a symmetrical or
an infinitely repetitive universe, then the description would apply to more than one entity, so it would fail to pick out
a particular object. Therefore, in order for a description to successfully pick out an object, it must be anchored in the
self and the present time. This suggests that the content of a thought such as T or T' should be identified with the
content of a sentence containing a description that is anchored in the self and the present time. If this is the case,
then it is clear that T and T' will have different contents, because they contain different descriptions, anchored in
different times.









(such as the thinker and the time when the thinking event occurs) that are part of the thought

content. Nevertheless, that difference in content does not make for a difference in truth-value.

The principle that requires that these thoughts have the same truth-value is not the thesis that

sense (plus the world) determines reference, but a principle corresponding to the principle of

intersubstitutivity at the level of thoughts. Although T and T' have different contents, they

involve different modes of presentations of the same entities, so the principle of

intersubstitutivity would require that the modes of presentations can be interchanged salva

veritate. In the next section I argue that there is more to the difference between the two thoughts

than the differences in the description or the mode of presentation of the object. One can explain

why the two thoughts differ in truth-value without violating any intersubstitutivity principle. The

difference in truth-value is not the result of a difference in propositional content.

Intentional states and their propositional content

I will argue that the difference in content between the two thoughts is sufficient to

determine a difference in truth-value. The structure of a propositional attitude consists in a

propositional content and a certain psychological mode that determines the type of the

intentional state. One can distinguish between thoughts, beliefs, desires, commands or promises

which are different attitudes one could have towards the same propositional content. Thus, it is

clear that the propositional content is not enough to determine the semantic value of a thought.

Moreover, I will argue that the conjunction of a propositional content and a psychological mode

might also fail to determine a semantic value for the thought. It could happen for a certain

thought to fail to be an act of judging although it is an act of thinking. In particular, although it is

a thought, the Liar thought fails to have what I call an assentive content, so it fails to have a

truth-value. When one has a thought, one performs an action: the action of thinking a certain

content. Therefore, one needs to draw a distinction between what one thinks and what one does









in thinking it, which is parallel to the distinction drawn in speech act theory between what one

says and what one does in saying it. I will argue that parallel to the way one can talk about

infelicitous speech acts, one can also talk about infelicitous thought acts. In particular, the Liar

thought turns out to be an infelicitous thought act, and for this reason lacks a truth-value.

This type of solution to the Liar thought paradox can be better explained by comparing it

with the corresponding attempt to solve the Liar statement paradox. One strategy that has been

proposed for solving the linguistic Liar paradox is to say that truth and falsity apply to statements

rather than sentences and deny that there is a Liar statement. Thus, Martinich [1983] proposes

that Liar paradoxes should be treated not as semantic paradoxes but rather as speech act

paradoxes. According to him, the paradox generated by the Liar statement belongs to the same

family of paradoxes as the paradoxes of commands, the paradoxes of promises and the paradoxes

of bets. Utterances (3) (5)5

(3) I order you not to obey any of my orders.

(4) I bet that I do not win this bet.

(5) I promise you that I will not keep any promises.

are paradoxical, according to him, in the same way as an utterance of (6)

(6) This statement is not true.

All these paradoxes can be solved, according to Martinich, by saying that the speech act is

infelicitous. In particular, when one utters (6), one fails to make a statement (i.e., one fails to

make an assertion), because there is an essential condition for a speech act to count as an

assertion which is not met. This essential condition, pointed out by Searle, requires that "the


5 See Martinich [1983]. There is a similar paradox of interrogative speech acts, proposed by Lappin [1982]:
(Q) Is there a negative answer to this question?
where a negative answer is any answer "which answers a question, and which contains either negation of its main
VP or negation of the main sentence" [Lappin 1982: 574]. For desires, a candidate would be:
(D) I desire that my current desire remain unsatisfied.









speaker intends that the audience will take his utterance as representing how things are"6

[Martinich 1983: 64]. However, Martinich claims, "a speaker cannot have this intention if he

utters (6) and knows what it means" [1983: 64]. In the same way, utterances of (3), (4) and (5)

are infelicitous speech acts, because their corresponding essential condition is not met.7

In fact, I think that this strategy fails to offer a satisfactory account of the linguistic Liar

paradoxes. For one thing, the strategy cannot be extended to other linguistic Liar paradoxes.

Martinich was content with the idea that his solution does not apply to contingent Liars on the

grounds that there is no good reason to expect that contingent and non-contingent Liar paradoxes

receive the same kind of solution.8 Nevertheless, there are also non-contingent Liar paradoxes

which cannot be solved in this way. In particular, the strategy does not apply to Pair Liar

statements, because the essential condition could be met in their case, so one cannot deny that the

two speech acts constitute statements. Moreover, the strategy fails to account for other

paradoxical statements such as:

(7) (7) is either not true or fails to be a statement.

If one says that (7) fails to be a statement, a simple rule of logic would force one to saying that

(7) is either not true or fails to be a statement, which would mean that (7) is true after all.

The type of strategy proposed by Martinich is more appropriate to account for the Liar

paradoxes that occur at the level of thought acts than for those at the level of speech acts. One

can explain the Liar thought paradoxes as thought act paradoxes, and one can solve them by

noticing that there is a certain infelicity that can be attributed to certain thoughts.

6 This might be spell out as the requirement that the mental state is identified as one with a mind-to-world direction
of fit and with conditions of satisfaction.
7 Martinich [1983] characterizes his account as a pragmatic solution, but I think that it would be more appropriately
called semantic.

SMartinich [1983: 64-65].









In particular, it can be argued that the Liar thought is in a certain sense infelicitous.9 It

cannot be denied that when I think the Liar thought I am in a particular psychological mode.

Nevertheless, the propositional content and the psychological mode fail to determine a truth-

value for the thought. Normally, the notion of content is thought of as determining the semantic-

value. Therefore, I will introduce a new notion, the semantic content of a thought act, which is

supposed to determine the truth-value of the thought and consists in its propositional content and

what I call its assentive content. The assentive content is normally identified with the

psychological mode of the thought, but sometimes as it happens for the Liar thought the

two come apart.

For Searle, in order for a mental state to count as a belief it must have a mind-to-world

direction of fit and a satisfaction condition. It is normally thought that these requirements are met

if one is in a state with a certain type of psychological mode. Nevertheless, in some rare cases,

such as the paradoxical thoughts, there is a conflict between the psychological mode and the

propositional content. This conflict precludes the thought from having a direction of fit and

satisfaction conditions.

It would be helpful to draw a distinction between an act of thinking and an act of judging10.

The former only requires the right psychological state. The latter requires that the act has an

assentive content. The Liar thought fails to be a judgment. The Liar thought has in common with

the mental state in which one merely entertains a proposition the fact that neither is an act of

judging or of holding something true. They differ in that the former has a psychological


9 One could plausibly argue that there could be no Liar thought if at the time when the thought occurs I know what
my Liar thought is about and I know what its propositional content is. It was for this kind of reason that Martinich
thought that there cannot be a Liar statement. Nevertheless, it is hard to deny that one could have a Liar thought if
one does not know at the time when the thought occurs what the content of the thought he is thinking about is.
10 'Act of judging' is a technical term.









component that the latter (the act of merely entertaining a proposition) does not have. The Liar

argument can be blocked by noticing that the thought I have at t does not determine a judgment.

One might wonder why one should care about the semantic content and the assentive

content of a thought. Why is it not enough to look at its propositional content and psychological

mode? The answer is that the notion of content is normally understood as determining

satisfaction condition. This is how Searle thinks of the content of an intentional state.11 The Liar

Thought argument is run on the premise that psychological mode and the propositional content

(together with the facts in the world) are enough to determine the semantic value of the thought.

This view can accommodate the internalist intuition that the content of an intentional state

is an intrinsic feature of it. This intuition would require that if two possible worlds differ only in

what B's thought at t' is, then A's thought that B's thought at t' is not true would have to have

the same content in the two worlds. The view I articulated is compatible with saying that A's

thought would have the same propositional content. The propositional content of the intentional

state could be the same regardless of how the world is. However, the assentive content of the

intentional state is not an intrinsic property of the state, and it could be different in different

possible circumstances.

There is a reason why the strategy described above is more successful for thought act Liar

paradoxes. For both speech acts and thought acts one can distinguish between the propositional

content and the type of the act (its force, to use Frege's vocabulary). However, there is an

important difference. Typically, the aspect of the speech act that determines its type can be

separated from the aspect that contributes the propositional content (the meaning of the sentence

in the given context). Most frequently there is a syntactical force indicator (the syntactical


1 See [Searle 2004: 169].









structure of the sentence, the tense, the presence of some expressions such as 'promise' or

'please', etc.), but these are not reliable indicators. 12 A more reliable force indicator is how the

speaker intends his act to be recognized by the hearer. Moreover, the act "succeeds, and the

intention with which it is performed is fulfilled, if the audience recognizes that intention" [Bach

1998]. One can easily imagine Liar statements that are not only intended by the speaker to be

recognized as assertions, but are also recognized as such by the audience. 13 One cannot deny that

the speech act is infelicitous because there is a conflict between its force and its propositional

content, because the felicity conditions are met. Therefore, in their case the strategy proposed by

Martinich fails.

On the other hand, a thought act is felicitous if it succeeds in being an act of holding

something true. For propositional attitudes, the psychological state that determines the type of

the state is inseparable from the attitude itself. In particular, the assentive part (the holding-

something-true) cannot be separated from the thought act. Therefore the assentive content of the

thought cannot be determined independently of its propositional content. This makes it possible

to explain the infelicity of a thought act as a result of a conflict between the two aspects.

Liar Beliefs

I will now turn to the case of Liar beliefs, where beliefs are understood as dispositional

states. It is hard to imagine a non-contingent Liar belief that is not specified via a sentence or a

thought. Nevertheless, there might well be contingent Liar beliefs or Pair Liar beliefs. Suppose

that A believes that B's favorite belief is not true and that B believes that A's favorite belief is



12 To use an example offered by Bach [1998], section 1, by uttering 'I will call a lawyer' one could perform speech-
acts of various types: predictions, promises or warnings.
13 This happens for Liar thoughts whose paradoxicality depends on some empirical facts that neither the speaker nor
the hearer is aware of.









true. Assume that it happens that the two are A's and B's favorite beliefs. The two beliefs are

paradoxical.

The solution to this paradox will rely on the solution I proposed for the Liar thought

paradoxes. Beliefs can be spelled out in terms of thoughts (or, if one prefers, occurrent beliefs),

so a Liar belief paradox comes down to some variation of a Liar thought paradox. Having a

dispositional nature, beliefs cannot be said to be infelicitous. Only acts can be said to be

infelicitous. Nonetheless, beliefs could display a dispositional infelicity if they could have

thought occurrences which are infelicitous. Take, for instance, A's belief that B's favorite belief

is not true. Saying that A has this belief amounts to saying that if asked, A would think that B's

favorite belief is not true. This thought of A would be infelicitous in the same sense as the Pair

Liar thoughts. 14 This can be seen from the following inferences:

(8) If asked, A's thought is true.

(9) B's favorite belief is not true.

(10) B's belief that A's favorite belief is true is not true.

(11) The thought B would have if asked (which is the thought that A's favorite belief is

true) is not true.

(12) A's favorite belief is not true.

(13) If asked, A's thought is not true.

This shows that the Liar belief paradoxes are actually derived from a paradox at the level

of thoughts. This paradox can be solved by saying that A's thought would lack a truth-value

because it is infelicitous. Therefore, the two Liar beliefs also lack a truth-value, because they

display a dispositional infelicity.


14 I take the notion of a true thought as basic, and treat the notion of a true belief as derivative.









Liar Propositions

Consider now the case of propositions, understood in a broad enough sense to include both

Russellian propositions and Fregean thoughts. If there is yet a Liar argument at the level of

propositions, then we still have a reason to think that there is something wrong with the concept

of truth or with the property of truth. We might be forced to saying that there is no property of

truth, and so nothing to be expressed. I will argue that there are after all no Liar propositions that

would force us to such a position.

If there is a Liar proposition, it is a proposition that represents itself as not being true.

Could there be a Liar proposition? Of course, the issue arises only if one grants that there are

propositions, so I will assume that there are abstract entities of this sort. I will argue that none of

our grounds for admitting propositions forces us to admit Liar propositions. Propositions are

generally admitted in our ontology for two reasons: we need some entities that, first, play the role

of truth bearers better than sentences and, second, serve the role of objects of our propositional

attitudes. It is easy to see that the existence of a Liar proposition could not be motivated by the

first of the above mentioned reasons, because no sentence of English could express a Liar

proposition, for truth is inexpressible in English. It could be the case that no sentence in any

language can express a Liar proposition.

Could a Liar proposition be the propositional content of a thought? Suppose that I think at t

that the propositional content of my thought at t is not true. 15 The propositional content of the

thought I have at t appears to be paradoxical, because it can be used to run a Liar argument. If the

propositional content is true, then it would also have to be not true. If it is not true, then it would

follow that it is in fact true. However, there are ways to block the derivation of this contradiction.

15 1 could also consider the case in which I merely entertain at t the proposition that the proposition I am merely
entertaining at t is not true.









The Liar argument assumes that my thought at t has a propositional content, which means that it

can be turned into a reduction argument whose conclusion is that my thought at t in fact has no

propositional content. This conclusion might strike one as implausible on the grounds that if my

thought at t is an empirical thought, then we would have had no reason to deny that it had a

propositional content had the circumstances been different. Nevertheless, there is no good reason

to believe that if a thought act has a propositional content it must have a propositional content in

all counterfactual circumstances in which it is performed. A further complaint is that although

one might deny that the thought has a propositional content, it nonetheless has a content which

falls short of being a proposition. If this is granted, the challenge would be to account for the

content of a slightly modified thought that I might have at t: I might think that the content16 of

my thought at t is not true. If this content is not a proposition, then it has to be neither true nor

false. Therefore, at t' I would be justified in thinking that the content of my thought at t is not

true. I want to say that my thought at t' has a propositional content, which is true. The difficulty

is to explain why the thought at t' has a propositional content while the thought at t does not have

one, although in both cases I seem to be thinking the same thing about the same entity. This

might seem surprising, but it has to be accepted as a fact. Something can be called a proposition

only in so far as it has a representational power. It turns out that the content of the thought at t' is

representational, while the content of the thought at t is not. The difference between the two

contents might be explained by the observation that in one case the content is part of what it is

supposed to represent.







16 As opposed to the propositional content.









What remains to be investigated is whether there could be a Liar proposition such that

there could be no thought or sentence has it as its propositional content.17 Although none of the

traditional grounds for admitting propositions would motivate the existence of a Liar thought of

this sort, one who takes abstract entities such as propositions and their ingredients seriously

might think that there are propositions that are both inexpressible and unthinkable. Although

there could be propositions that are both inexpressible and unthinkable, I think that there are

good reasons to think that they cannot include Liar propositions. The ingredients of propositions

(concepts, senses of descriptions, etc.) could be recombined to obtained new propositions.

Nevertheless, one gets a proposition only if the new abstract entity is a representation. No

arrangement of the ingredients would result in a Liar proposition. What is thought to be a Liar

proposition cannot be a representation, therefore it is not a proposition. One might find it

puzzling why some arrangements of the concept of truth and other ingredients fail to constitute a

proposition, but it is hard to see why the concept of truth could be made responsible for this.

Therefore, there are no good Liar arguments at the level of propositions that motivate the view

that the concept of truth is inconsistent.

Mentalese Liars

I will end this chapter with a brief discussion of Mentalese Liar paradoxes. According to

the representational view of thoughts and beliefs, having a thought or a belief is having a mental

or physical representation with a certain propositional content. What the Language of Thought

Hypothesis adds to this picture is the idea that these representations constitute a mental language

(Mentalese) that has a syntax and semantics just like natural languages. If this hypothesis is true,



1 One way to imagine a Liar proposition would be to think of some propositional ingredients that could be arranged
in the appropriate order to generate the appropriate proposition. One ingredient is, certainly, the concept of truth.
The nature of the remaining ingredients would depend on one's favorite view of semantics.









then Mentalese appears to have all the ingredients that are necessary to construct a Liar belief.

The Liar belief would be a representation that represents itself as not true. The question is

whether the Mentalese Liar paradoxes should be accounted for the way linguistic Liar paradoxes

have been accounted for, or like the non-linguistic Liar paradoxes. The fact that we are dealing

with a language, albeit a language of thought, might suggest that Mentalese Liar paradoxes

should be solved by applying the inexpressibility account. I will argue that this strategy would be

unsuccessful, and that the paradoxes should rather be treated like the other non-linguistic

paradoxes.

To apply the inexpressibility account to the Mentalese Liars would involve saying that the

concept of truth cannot be the content of any predicative part of a representation, and that the

proposition that the Mentalese Liar is not true cannot be the content of any representation. The

difficulty is that once one admits that one can think that the Mentalese Liar is not true,

representationalism about thoughts forces one to accept that there is a representation of that

thought in Mentalese. Moreover, to say that there is nothing that corresponds in Mentalese to the

concept of truth (i.e., the concept of truth is unthinkable) and that the Mentalese Liar thought

fails to have a content is highly implausible, because it is hard to deny that one can grasp the

concept of truth. Moreover, one cannot draw a distinction between an intended content and a

linguistic content of a thought. Intentionality is an intrinsic feature of a thought, although it is not

an intrinsic feature of sentences of English. Therefore, the intended content of a thought cannot

fail to be its content. Notice that this view does not involve an internalist assumption. Even for

an externalist view about mental content, according to which the content of a belief is not an

intrinsic feature of the belief, intentionality must be an intrinsic feature.









Therefore, I think that the Mentalese Liar paradoxes should be handled alongside other

non-linguistic Liar paradoxes. What one should say is that the Mentalese Liar is defective in

some sense, because it fails to constitute a judgment. This would amount to saying that the

thought fails to represent something as being the case, because it has a self-undermining nature.

The psychological evidence cannot guarantee that the Mentalese Liar is a successful

representation. The content of the thought cannot be reduced to the propositional content and the

psychological mode.









CHAPTER 6
AN EXTENSION OF THE ACCOUNT: SEMANTIC VERSUS LOGICAL PARADOXES

I have already argued that the Liar paradoxes show that truth is inexpressible in English.

The purpose of this chapter is to investigate to what extent the inexpressibility account can be

extended to other paradoxes. There is a plethora of paradoxes that do not involve the concept of

truth, including semantic paradoxes, logical paradoxes, paradoxes of vagueness, speech-act

paradoxes, etc. I will only be concerned with those paradoxes that are similar in some significant

respect with the Liar paradoxes. If the following principle proposed by Graham Priest [1994: 32]

is true

(1) Same kind of paradox, same kind of solution. (The Principle of Uniform Solution)

then a correct account for a particular paradox should be applicable to the paradoxes of the same

kind. Priest uses this principle as a test of any proposed solution to a paradox: if a solution of the

same type cannot be offered to the paradoxes of the same kind, then the proposed solution is

unsuccessful.

By all accounts, paradoxes involving semantic notions are similar enough to the Liar

paradoxes to demand a similar kind of solution. In the first part of this chapter I argue that the

inexpressibility account can be extended smoothly to apply to all semantic paradoxes.

Nevertheless, the account cannot be extended to the so-called logical paradoxes. It has been

argued that semantic paradoxes and logical paradoxes are paradoxes of the same kind, so the

Principle of Uniform Solution would require that they have the same kind of solution. If this is

the case and the inexpressibility account indeed does not apply to logical paradoxes, then one has

a reduction argument against the inexpressibility account of the semantic paradoxes. In the second

part of the chapter I argue that semantic paradoxes and logical paradoxes are not similar in a

sense that would require that they have the same kind of solution. Therefore, the fact that the









inexpressibility account cannot be extended to account for the logical paradoxes is not

problematic.

Semantic Paradoxes

I will start with a brief survey of the main paradoxes that occur in connection with

semantic notions other than truth, such as heterology, satisfaction, definability, etc. Thereafter, I

argue that all these paradoxes can be solved by an inexpressibility account.

Grelling's Paradox

One can distinguish between two categories of adjectives: those that are true of themselves

and those that are not. By definition, those in the first category are called autological, while those

in the latter category heterological. Thus, adjectives such as 'short' or 'polysyllabic' are

autological, while 'long', 'German', or 'monosyllabic' are heterological. Grelling's paradox

arises with the question: is 'heterological' heterological? If it is heterological, then it is not true

of itself, so it is not heterological. On the other hand, if it is not heterological, then it is true of

itself, so it is heterological.

The concept of satisfaction leads to a paradox in a way that resembles Grelling's paradox.

One can ask the question: does the expression 'x does not satisfy itself satisfy itself? If it

satisfies itself, then it does not satisfy itself. If it does not satisfy itself, then it would follow that

it satisfies itself.

Paradoxes of Definability

The following three paradoxes (Richard's paradox, Berry's paradox and Konig's paradox)

are usually grouped together as paradoxes of definability.

Richard's paradox

List in alphabetical order all permutations of the twenty-six letters of the English alphabet

taken two at a time, followed by all permutations of the same letters taken three at a time, again









in alphabetical order (repetitions of the same letter are allowed), etc. Everything that can be

named by finitely many words is named by some permutation in the list. Eliminate from the list

all those permutations that do not denote numbers. The set of numbers that are definable by

finitely many words is the set containing the first number defined by a permutation, the second

number defined by a permutation, the third, and so on. Thus, the set of numbers that are

definable by finitely many words, call it E, is a denumerably infinite set. The contradiction is

obtained, using Richard's own words, as follows:

We can form a number not belonging to this set. "Let p be the digit in the nth decimal
place of the nth number of the set E; let us form a number having 0 for its integral part and,
in its nth decimal place, p+l if p is not 8 or 9, and 1 otherwise." This number N does not
belong to the set E. If it were the nth number of the set E, the digit in its nth decimal place
would be the same as the one in the nth decimal place of that number, which is not the
case. I denote by G the collection of letters between quotation marks. The number N is
defined by the words of the collection G, that is, by finitely many words; hence it should
belong to the set E. But we have seen that it does not. Such is the contradiction. [Richard
1905: 143]

Berry's paradox

Berry's paradox was first published by Russell, who gives the credit for it to "Mr. G. G.

Berry of the Bodleian Library"1. The number of syllables in English is finite. Therefore the

number of descriptions or names under 19 syllables is finite. Since there are infinitely many

natural numbers, it follows that there are numbers that cannot be named or described in less than

19 syllables. The number two can be named, for instance, by using the one syllable expression

'two'. Consider then the least integer not nameable in fewer than nineteen syllables. It turns out

that this number has just been named in 18 syllables.







1 Russell [1908: 153 fn. 3].









Kinig's paradox

The reasoning behind Konig's paradox has initially been offered as part of a proof that the

continuum is not well-ordered. Konig notices that only a denumerable set of real numbers is

finitely definable. If the set of real numbers were well-ordered, there would be a real number

which is the least not finitely definable real number. The problem is that that number has just

been defined by the finite italicized expression. Therefore, there would be a number that is both

finitely definable and not finitely definable. Konig's conclusion was that the set of real numbers

cannot be well-ordered2. Shortly after Konig made his proof public, due primarily to Zermelo's

proof that every set can be well-ordered, but also to Richard's paradox that does not appeal to the

notions of ordinal or well-ordening, Konig's conclusion had to be rejected. His argument

towards a contradiction turned into a paradox which is sometimes known as the Zermelo-Konig

paradox.

A Denotation Paradox

There are other paradoxes that belong to the paradoxes of definability. It is worth mentioning

Simmons' denotation paradox that resembles the paradoxes above, but is meant to make the self-

referential structure more explicit.3 Suppose the following expressions are written on the board

in room 1014

The ratio of the circumference of a circle to its diameter.

The positive square root of 36.

The sum of the numbers denoted by expressions on the board in room 101.


2 KOnig [1905: 147-48].

3 In addition to this, Simmons offers an example of a denotation paradox that does not have a self-referential
structure. This paradox is supposed to stand to the self-referential denotation paradox as Yablo's paradox stands to
the Liar. See Simmons [2002]
4 Simmons [2002].









The question is what is the denotation of the third definite description?

The Inexpressibility of the Semantic Concepts

The above semantic paradoxes can be solved in the same way as the Liar paradox.

Consider, for instance, heterology. Unlike the concept of truth, the concept of a heterological

term is not one that is used in everyday thoughts. It is a concept that has been introduced by

mathematicians and logicians for certain purposes. However, heterology is normally thought of

as a concept that is easily grasped by ordinary speakers.

Just as in the case of truth, the argument that leads to an inconsistency involves the

assumption that there is a predicate of English, 'H', that expresses the concept of heterology. If

'H' expresses the concept of heterology, then at least part of the concept of heterology should be

captured by the following H-schema,

(H-schema) 'P' is H iff 'P' is not P.

where 'P' is replaced by predicates of English. Given that 'H' is supposed to be a meaningful

predicate of English, one can infer the following biconditional

(2) 'H' is H iff 'H' is not H.

which is an instance of the H-schema. Since 'H' is neither vague, nor partially defined, (2) could

also be turned into an explicit contradiction. Therefore, the assumption that 'H' expresses the

concept of truth leads to a contradiction and must be rejected. Moreover, no predicate of English

can express the concept of heterology. Consequently, the predicate 'heterological' fails to have a

linguistic meaning. Since it lacks linguistic meaning, strictly speaking it does not apply to

anything, not even to itself, therefore it is heterological. Nevertheless, the sentence

"'Heterological' is heterological" is not true but rather lacks linguistic meaning. Of course, one

needs again an answer to a potential self-defeat objection, but the explanation would be the same

as the one I offered in chapter three for truth.









There is no need for thinking that the concept of heterology is inconsistent, as there was no

reason to say that the concept of truth is inconsistent. By analogy with truth and truth*, one can

distinguish between heterology and heterology*. If one allows concepts to be inconsistent, then

heterology* would be an inconsistent concept and, at the same time, inexpressible (as it also is

the case with truth*). The meaning of 'heterology*' is implicitly defined by the following

schema

(H*-schema) 'P' is heterological* iff 'P' is not P

where 'P' is replaced by predicates of English.

The same explanation can be extended smoothly to the groundedness, satisfaction and

definability paradoxes as well as to any other semantic paradoxes, so there is no need to go over

each particular inexpressibility argument. Nonetheless, some comments are worth being made in

connection with the definability paradoxes. In their case it is not obvious what semantic concepts

are involved. This is due, at least in part, to the lack of a uniform terminology. There are various

semantic expressions that are used in the formulation of the definability paradoxes, most of them

in a misleading way: 'definable', finitelyy definable', nameablee', 'denote' and 'can be written

with finitely many words'. Which of the corresponding concepts is the one that should be

claimed to be inexpressible? In order to be able to answer this question, one first needs to

investigate which semantic concept (or concepts, if there are more) is in fact intended to be

captured by those rather misleading semantic expressions. Notice that the entities that are said to

be definable are numbers. Moreover, a number is said to be definable in English if there is an

expression of English that denotes or refers to that number. Definability is more commonly

thought of as a property of expressions (or concepts): an expression of L is definable in L' if and

only if there is an expression of L' that is synonymous with it. Thus, the paradoxes of









definability are improperly called so. It would also be wrong to call them paradoxes of

nameability, because they involve the notion of reference in a broader sense, to include definite

descriptions. On the other hand, unrestricted reference or denotability would not do, because

demonstratives, or contingent definite descriptions such as 'Harry's favorite number', would

trivialize the issue. The property that seems to be relevant to the definability paradoxes is being

denotable by some non-contingent singular referring term. Let us call the property and the

corresponding concept specifiability. The definability paradoxes are solved by saying that the

concept of specifiability cannot be expressed in English. Expressions such as 'the smallest

number not specifiable in less than 19 syllables' lack linguistic meaning and fail in fact to

specify a number. Peano's idea that paradoxes such as Richard's paradox belong to linguistics

rather than mathematics is correct.5

Similarities between the Semantic and the Logical Paradoxes

The inexpressibility account does not seem to be applicable to Russell's paradox. The

account would only enable one to deny that the concept of a class is expressible in English.

However, even if the concept were not expressible in English, there would probably still be an

extension of the concept. There would also have to be an extension of the concept of not being a

member of itself. This would be enough to generate a paradox. Therefore, Russell's paradox

does not depend on any expressibility assumptions.

This might appear to be in conflict with the fact that there are some similarities between

the semantic paradoxes and the logical paradoxes. Nevertheless, I will argue that these

similarities are not as strong as to require that the two groups of paradoxes should receive the

same kind of solution. I will start with a brief review of Russell and Ramsey's accounts of the


5 See also Ramsey [1990: 184].









relation between the semantic and logical paradoxes. Thereafter, I will present Priest's argument

in favor of a uniform treatment of the paradoxes, followed by my rejection of the uniformity

account.

A Little Bit of History

Russell believes that all paradoxes of self-reference share a common structure:

In each contradiction something is said about all cases of some kind, and from what is said
a new case seems to be generated, which both is and is not of the same kind as the cases of
which all were concerned in what was said. [Russell 1908a: 154]

Consequently, he proposes that they should receive the same kind of solution, one that relies on

his well-known vicious circle principle. Russell introduces the vicious circle principle in order to

prevent the construction of expressions that involve vicious circularity, which leads him to the

development of the ramified type theory.

Ramsey argues that Russell is wrong in thinking that the two groups of paradoxes have the

same structure, and claims that they belong, in fact, to two 'fundamentally distinct groups' of

paradoxes (the label that was current at that time is 'contradictions') [Ramsey 1990: 183]:

A. 1. The class of all classes which are not members of themselves.

2. The relation between two relations when one does not have itself to the other.

3. Burali Forti's contradiction of the greatest ordinal.

B. 4. 'I am lying'

5. The least integer not nameable in fewer than nineteen syllables.

6. The least indefinable ordinal.

7. Richard's Contradiction.

8. Weyl's contradiction about 'heterologisch'.6



6 It seems that Ramsey has misattributed Grelling's paradox to Weyl.









The first paradox in group A is Russell's paradox, the well known paradox he communicated to

Frege in 1903. The class of all classes which are not members of themselves either is or is not a

member of itself. However, if it is a member of itself, then it is not. If it is not a member of itself,

then it is. Therefore, both options lead to a contradiction. The second paradox in group A is an

extension of Russell's paradox from sets to relations. As for the Burali-Forti paradox, it has been

stated by Russell as follows:

It can be shown that every well-ordered series has an ordinal number, that the series of
ordinals up to and including any given ordinal exceeds the given ordinal by one, and (on
certain very natural assumptions) that the series of all ordinals (in order of magnitude) is
well-ordered. It follows that the series of all ordinals has an ordinal number, Q say. But in
that case the series of all ordinals including Q has the ordinal number Q + 1, which must
be greater that Q. Hence Q is not the ordinal number of all ordinals. [Russell, 1908, p. 154]

Paradoxes in group A are usually called logical paradoxes (in fact, they would more properly be

called mathematical or set-theoretical paradoxes), while those in group B semantic paradoxes,

although this is not Ramsey's terminology. The difference between them consists in the fact that

[The contradictions of group A] involve only logical or mathematical terms such as class
and number, and show that there must be something wrong with our logic or mathematics.
But the contradictions of group B are not purely logical, and cannot be stated in logical
terms alone; for they all contain some reference to thought, language, or symbolism, which
are not formal but empirical terms. So they may be due not to faulty logic or mathematics,
but to faulty ideas concerning thought and language. [Ramsey 1990: 183-4]

Ramsey believes that the motivation Russell has for developing a ramified type theory is that he

wants his theory to also provide a solution to the semantic paradoxes. He argues that there is no

point in trying to offer a unified account, because the two groups of paradoxes have a

fundamentally different structure. Thus, the logical paradoxes should rather be solved by a

simple type theory, while the semantic paradoxes would have to be accounted for in a totally

different way. In any case, according to Ramsey, solving the semantic paradoxes should not be

the business of mathematics, and should not require a solution to the logical paradoxes.









Priest's Uniformity Account

Leaving aside the difficulties related to the notions of kind, both Russell and Ramsey

appear to be committed to the Principle of Uniform Solution, but they disagree about whether

paradoxes in the two classes share the same underlying structure. Priest argues that 'Russell was

right and Ramsey was wrong' [Priest 1994: 25]. Contrary to Ramsey's suggestion that logical

and semantic paradoxes are two different kinds of paradoxes, Graham Priest [1994; 2002] argues

that they have in fact the same underlying structure. More specifically, he argues that

(3) The Inclosure Schema is the structure that underlies both logical and semantic

paradoxes.

Priest uses (3) together with the Principle of Uniform Solution and (4),

(4) Same underlying structure, same kind of paradox.

which is supposed to be intuitively true, to show that the solutions that have been proposed in the

past for both the semantic and the logical paradoxes are wrong. He contends that since the

orthodox solutions to the paradoxes of one kind are not applicable to the paradoxes of the other

kind, the principles above are 'sufficient to sink virtually all orthodox solutions to the paradoxes'

[Priest 1994: 33]. Since the inexpressibility account is not applicable to the logical paradoxes, it

would follow that it is not even appropriate as a solution to the semantic paradoxes. Only Priest's

own dialetheist account is supposed to pass the test because it is the only one that applies equally

well to both kinds of paradoxes.

(3) is supposed to be established by an examination of all major semantic and logical

paradoxes and by noticing that they all share an underlying structure. Although paradoxes in

group A involve the notion of a set, while those in group B involve a reference to thought and

language, this is, according to Priest, only a superficial distinction. Upon investigation, one can

identify a common underlying structure, which is a slight modification of a schema discovered









by Russell. Priest takes Russell to have already identified a structure that is common to all

paradoxes in group A. They all display what Priest [2002: 129] calls Russell's Schema:

For given properties (p, and function 6:
1. Q = {y : qp(y)} exists
2. ifx is a subset of Q:
a) 6(x) x
and b) 6(x) e
Therefore, 6(Q) V Q and 6(Q) e Q.
The first clause is the Existence clause, and guarantees that there is a totality of objects of a

certain sort, Q, while the second requires that Transcendence (2a) and Closure (2b) hold for

every subset of the totality guaranteed by the Existence clause. The function 6(x) is called a

diagonaliser, because it is constructed in a way that resembles the technique of diagonalization

made popular by Cantor's theorem. Transcendence guarantees that the value the diagonaliser

assigns to a subset, x, of Q does not belong to x, while Closure ensures that this value remains a

member of the totality guaranteed by the Existence clause. When the subset is Q itself, a

contradiction can easily be derived. For Russell's paradox, qp(x) is 'x V x', Q is the Russell set,

{y; y V y}, and 6 is the identity function. Priest [2002: 134] argues that although paradoxes in

group B fail to satisfy Russell's Schema, both paradoxes of type A and B satisfy what he calls

the Inclosure Schema (in [Priest 1994] it is called the Qualified Russell's Schema):

For given properties (p and x, and (possibly partial) function 6:
1. Q = {x : qp(x)} exists and y(92)
2. ifx is a subset of Q such that y(x):
a) 6(x) x
and b) 6(x) e
Therefore, 6(Q) V Q and 6(Q) e Q.









The function \ has been added to the schema in order to ensure that Closure holds for paradoxes

in group B. For semantic paradoxes, the mere fact that x is a subset of Q does not guarantee that

6(x) belongs to Q. In particular, for Konig's paradox qp(x) is 'x is a definable ordinal', while 6(x)

is 'the smallest ordinal that is not in x'. Ifx is not definable, then there is no guarantee that 6(x)

is a definable ordinal (i.e., there is no guarantee that Closure holds for x). Thus, the diagonaliser

must be restricted in some way (to definable subsets of Q, to sets that are definable in 19 words,

etc.), which is why the additional function \ is needed. For the Liar paradox, qp(x) is 'x is true',

y(x) is 'x is definable', and Q is the set of true sentences. 6 is a function a constructed by

diagonalization: o(a) = a, were a =
(the sentence 'c(a)' is supposed to say about itself

that it is not in a). No similar restriction is needed for the logical paradoxes, but one can easily

adjust them to fit the Inclosure Schema. For Russell's paradox x is the universal property 'x =

x'. Priest also shows how various choices of p, x, and 6 capture precisely the structures

displayed by other paradoxes in groups A and B.

It should be noticed that the need to add a restriction to definable sets for semantic

paradoxes raises a worry with respect to Priest's uniform account: the universal property appears

to be a mere ad-hoc addition that does not play any role in Russell's paradox, while the

definability requirement plays a genuine role in the semantic paradoxes. Priest takes this to be

only an insignificant difference that has to do with the particular way in which the schema of the

paradox is obtained, rather than with the schema itself. There is only a difference in the class of

objects the function 6 ranges over. The fact that for semantic paradoxes one has to restrict the

function to a subset of the power set of Q is, in his view, only an accidental feature.

In fact, according to Priest [2002], the Inclosure Schema does not only underlie the

semantic and logical paradoxes. It is supposed to capture the structure of a much wider class of









contradictions at the limits of thought that can be found in the history of Western philosophy

(from Cratylus to Kant, Hegel, Russell, Heidegger and Derrida), and in the Eastern tradition as

well (Nagarjuna).

The Semantic Version of Russell's Paradox

Priest is not the only one who has misgivings with respect to Ramsey's position. Simmons

[2002] also makes an attempt to blur the distinction between semantic and logical paradoxes.

However, while Priest tries to reconstruct type B paradoxes in such a way that they reproduce the

structure of the logical paradoxes, Simmons' proposal is quite opposite. He tries to reconstruct

Russell's paradox in a manner that is slightly different from its original form, and make it similar

to the semantic paradoxes. Simmons argues that Russell's paradox is more appropriately treated

as a semantic paradox. It is true that the paradox involves in an essential way the notion of

extension, but extension should not be conceived of, according to Simmons, as reducible to the

notions of a set or class. Extension should rather be related to the notion of predication. This is

the reason why the paradox should count as a semantic paradox. For Simmons, Liar-like

paradoxes, definability paradoxes and Russell's paradoxes belong to the category of semantic

paradoxes. The three categories are subcategories of the semantic paradoxes, and can be

distinguished by the type of semantic value that is essentially involved in the paradox: an

extension, an object, or a truth value. Although there are important differences between them,

they are subcategories of the same category of paradoxes, which suggests that they should be

solved in the same way. Simmons believes that they all can be solved by a contextualist

approach, in particular, his singularity theory of the paradoxes.

I think that Simmons' view does not pose a serious threat to the inexpressibility account.

The handling of the extensional version of Russell's paradox would depend on how exactly the

notion of extension is understood. If extensions are understood as totalities, albeit different from









the traditional totalities of set-theory, then the inexpressibility view would indeed not apply to

Russell's paradox, but there is no good reason to expect it to apply, because Russell's paradox

would deal with totalities (and could be solved by denying that there are totalities of a certain

sort), while the semantic paradoxes do not. On the other hand, if the notion of extension can be

thought of in terms of predication only, independently of the notion of a totality, then there might

be good reason to require a uniform solution, but the inexpressibility account would be

applicable to Russell's paradox. I think that Simmons is rather committed to the former view. He

seems to think that extension should still be thought of as a totality (the collection of objects that

satisfy a certain predicate), although this totality is no longer part of to the

iterative/combinatorial universe of sets. Simmons' own characterization of the alternative notion

of extension is this:

to any predicate that denotes a well-determined condition or concept (in Frege's sense),
such as the predicates 'abstract' or 'set' or 'class', there corresponds the collection of those
things to which the predicate applies the collection of abstract things, the collection of
sets, the collection of classes. Call this the predicative conception and call the collection
of things to which a given predicate applies the extension of the predicate. [Simmons,
2000, p. 116]

Anyway, it is still assumed that for any meaningful predicate there is an entity (a collection of

objects) that is the extension of that predicate. This is why Russell's paradox survives even after

the assumptions that there is a class of all classes, a set of all sets or of all self-membered sets is

dropped. Russell's paradox reoccurs as a paradox of extensions. Notice that the reason why the

paradox survives is that one still assumes that there is an entity that is the extension of the

predicate. The contradiction no longer occurs if the assumption that 'extension' or 'self-

membered extension' have an extension is dropped. Of course, the fact that some predicates fail

to have an extension might still strike us as surprising. However, it is hard to think that it is part

of the concept of a predicate that all predicates have an extension. If one thinks of predication in









purely semantic terms, then one can avoid the contradiction. Thus, Simmons' idea that Russell's

paradox of extensions should count as a semantic paradox remains implausible. Even if one

accepted Russell's paradox of extension as a semantic paradox, one should still count Russell's

paradox of sets or classes as a logical one. This suggests that some paradoxes might be amenable

to two different formulations, such that there is both a semantic and a logical version of the

paradox.

Refuting the Uniformity Account

The goal of this section is threefold. I first argue that Priest fails to show that (3) provides a

non-question-begging method to sink virtually all orthodox solutions to the paradoxes. This is

because he fails to prove that one could uncover the structure underlying a certain paradox in the

absence of a solution to it. Second, I argue that it is not the case that the Inclosure Schema is the

schema that underlies the semantic paradoxes (in particular, it is not the schema that underlies

the Liar paradox). Third, I argue that the temptation to treat semantic and logical paradoxes as

sharing the same underlying structure is, at least in part, due to the illegitimate identification of

semantic paradoxes with their logical counterparts. I will first present Priest's arguments for the

uniformity of semantic and logical paradoxes.

An Objection from Circularity

In this section, I argue that Priest fails to show that (3) can be used to sink virtually all

orthodox solutions to the paradoxes. Unless one can provide a criterion for establishing (3)

independently of having found a solution to both paradoxes, the use of (3) to refute a certain type

of solution to a paradox is question begging. Priest does not provide a criterion of this sort.

Smith [2000] argues that Priest's argument that logical and semantic paradoxes should

receive the same kind of solution fails because of an equivocation on the notion of kind. The

difficulty stems from the fact that 'two objects can be of the same kind at some level of









abstraction and of different kinds at another level of abstraction' [Smith 2000: 118]. At some

level of abstraction the logical paradoxes and the semantic paradoxes are of different kinds: the

former mention sets, the latter do not. At a different level of abstraction they are of the same

kind, because both display the same Inclosure Schema. However, Priest 'describes the orthodox

solutions [...] at a far more concrete level than the level at which [...] he describes paradoxes'

[Smith 2000: 118]. At the level of abstraction that corresponds to the Inclosure Schema the

orthodox solutions can be said to be of the same kind: both are Inclosure Schema circumventers.

Priest agrees that when one talks about paradoxes being of the same kind, 'the notion of a

kind is relative to a degree of abstraction' [Priest 2000: 123]. However, he points out that when

he talks about paradoxes being of the same kind, he has a certain level of abstraction in mind:

'the level of the underlying structure that generates and causes the contradictions' [Priest 2000:

125]. Thus, the structure he is interested in is not an arbitrary structure, but the underlying

structure of a paradox, i.e., the structure that generates or causes the paradox. As he further

qualifies, causation should not be understood as physical causation, but rather as explanation.

Thus, paradoxes are said to be of the same kind in the sense that they have the same explanation.

Priest acknowledges that it is not an easy task to characterize the notion of explanation, but

claims that an intuitive grasp of this notion is enough for the points he wants to make. By

choosing the structure to be not any old structure but an underlying structure that generates and

explains the paradox, Priest wants to make sure that paradoxes of the same kind indeed have the

same kind of solution.

The problem for Priest's account of an underlying structure is that there is a sense of

'explanation' that makes it the case that in order for one to know the explanation of a certain









paradox, one would need to know its solution.7 Chihara, for instance, takes the problem of

explaining a paradox to be nothing else than the problem of finding a diagnostic (solution):

The problem of pinpointing that which is deceiving us and, if possible, explaining how and
why the deception was produced is what I wish to call 'the diagnostic problem of the
paradox'. [Chihara 1979: 590]

Priest himself seems to think that having an explanation of a paradox presupposes having a

solution to that paradox. Evidence for this is the fact that for him (1) is 'little more than a truism'

[Priest 2002: 287]. It is hard to see why the principle would be trivial if having the same

explanation does not entail having the same solution. Another piece of evidence is Priest's

discussion of the following three paradoxes of infinity [Priest 2002: 288]:

(i) If the world is infinite in time past then the number of days before today is equal
to the number of days before yesterday. But there are obviously fewer of these.

(ii) If the world is infinite in time past then the number of months before now must be
twelve times the number of years before now, but this is already infinite, and so is as large
as it can be.

(iii) There are more natural numbers than even natural numbers; yet there must be the
same number of each, since they can be put into one-to-one correspondence.

Here is Priest's comment on these three paradoxes:

With the wisdom of hindsight, however, we can now see that they are the same kind of
paradox. This is because they are all examples of a single phenomenon, namely, that an
infinite set can have a proper subset that is the same cardinal size as itself. This fact also
provides the solution to all the paradoxes. [Priest 2002: 289]

Thus, for him, the explanation of the paradoxes offers at the same time a solution to them.

It is easy to see why this notion of an explanation is problematic for Priest's claim that

virtually all orthodox solutions to the paradoxes can be proved wrong by establishing (3). One

could not establish that two paradoxes are of the same kind and require a uniform solution,

7 'Solution' here must be understood in a wide enough sense: it should cover not only accounts that indicate a way to
block the argument toward contradiction, but also those accounts that take the derivation of a contradiction to be
inescapable. I will keep using (following Priest) the word 'solution', although Chihara's notion of diagnosis would
be more apt in this context.









unless one already established that they have the same kind of solution. Using (3) to reject a non-

uniform solution to a certain paradox (i.e., a solution that cannot be extended to the paradoxes in

the other category) would amount to begging the question against the proponent of that solution,

because one who entertains a non-uniform solution to a certain kind of paradoxes would not

acknowledge that logical and semantic paradoxes are both generated by the same underlying

structure. This is actually a general objection that applies to any schema that one might consider

to be the structure of a paradox.

In particular, in order for (3) to enable one to sink virtually all orthodox solutions it must

be possible to establish that the Inclosure Schema underlies a certain paradox without having a

solution to it. Priest's explanation of why the Inclosure Schema is the schema that explains and

generates the logical and semantic paradoxes fails to meet this condition. The misleading

suggestion that sharing the Inclosure Schema is enough to guarantee that the solutions are of the

same kind comes from reading too much into the comparison between the diagonaliser and a

mechanism that generates an effect. Presumably, in order to prevent the effect from occurring,

one would have to remove the mechanism that generates it. Priest draws heavily on this

comparison to motivate (3). According to him, 'all inclosure contradictions are generated by the

same underlying mechanism'; also, the diagonaliser 'manages to "lever itself out" of a totality'

[Priest 2002: 289] or 'to operate on a totality of objects of a certain kind to produce a novel

object of the same kind' [Priest 2002: 136]. Nevertheless, the comparison becomes illegitimate if

one puts too much weight into the words 'generates' or 'produces'. The diagonaliser would be

conferred the magic power of making inconsistent principles that are otherwise innocuous and to

bring novel objects into existence. Of course, none of these powers can be attributed to the

diagonaliser. The fact that there is a technique that enables one to derive a contradiction from a









number of intuitively true principles does not show that that technique is responsible for (or the

cause of) the contradiction, but only shows that it can be used to bring into light the

inconsistency of those principles. One needs to provide additional reasons for thinking that the

Inclosure Schema is the structure that generates the paradoxes. In the absence of these additional

reasons, it might be the case that the technique is no more responsible for the contradiction than a

thermometer is responsible for a patient's high fever. This provides us no reason to think that

two paradoxes that share the Inclosure Schema could not be solved in different ways: one by

rejecting Existence and the other by rejecting Closure for Q, or, if one chooses to go that far, by

embracing dialetheism.

It is not my intention to deny that there are criteria for establishing that two paradoxes

should have the same kind of solution in the absence of a solution to each. The point is that Priest

fails to provide a non-question-begging criterion. Although it is not the purpose of this paper to

identify non-trivial criteria for paradoxes having solutions of the same kind, it seems more

promising to relate them with the principles that appear to be inconsistent rather than with the

mechanism that is used to bring to light the inconsistency.

The Liar and the Inclosure Schema

In this section I argue that the Inclosure Schema is not the structure that underlies the Liar

paradox and does not capture in any significant sense the essence of the phenomenon. Regardless

of what the precise characterization of an explanation is, it should capture the 'essence of the

phenomenon' [Priest 2000: 24], not simple accidents. It turns out that the Inclosure Schema

contains elements that are not essential to the Liar paradox and that there are essential elements

of the Liar paradox that are not represented in the Inclosure Schema. This is enough to show that

the Inclosure Schema fails to explain or to capture in any significant sense the essence of the

paradox. Moreover, it follows that the distinction between the two groups of paradoxes is not









merely superficial, and that Ramsey was actually right in drawing a sharp distinction between

them. There is no good reason to expect them to receive the same kind of solution.

In order to get clear on the structure of the Liar paradox, it is useful to consider an informal

version of it. Let (L) be the following sentence:

(L) (L) is not true.

The informal argument goes as follows. If (L) is true, then it follows that (L) is not true, because

(L) says that (L) is not true. Therefore, one has to infer that (L) is not true. But, since this is what

(L) says, (L) would have to be true after all. It can be noticed that this formulation makes no

explicit appeal to there being a set of true sentences. Nor does it appeal to the thesis that certain

subsets of the set of true sentences of English, including the set of true sentences of English

itself, are definable in English (in the sense that there is a referring term or a predicate of English

which refers to those subsets). This suggests that none of these theses should be made part of the

explanation of the Liar paradox, because the Liar paradox survives even if one or both theses are

rejected. On the other hand, other elements that are essential in the generation of any version of a

Liar paradox, such as the T-schema and the assumption that the Liar sentence says that the Liar

is not true, do not show up in the Inclosure Schema.

The Existence clause

Part of the structure that generates the Liar paradox is, according to Priest, the Existence

clause, namely, the assumption that there is a set of true sentences of English. It is undeniable

that the existence of a set of true sentences of English has no less prima facie intuitive support

than the thesis that there is a set of all sets that are not members of themselves. Nevertheless,

there is an important difference: the Liar paradox survives after the Existence thesis is dropped,

while the logical paradoxes do not. The orthodox solution to Russell's paradox is to deny that

there is a set of all non-self-membered sets. The contradiction vanishes once the assumption is









dropped. This does not mean that the orthodox solution is unproblematic, but at least in principle

the argument is blocked if the set existence is denied. Unlike Russell's paradox, the Liar paradox

survives after the Existence thesis is dropped. The informal version of the Liar paradox, as well

as the most familiar formal versions of it, do not actually make use of this assumption. As

Tennant [1998: 28-29] notices, the Liar paradox appeals to a predicate, 'true', not to a set of true

sentences. The fact that the contradiction is derived from weaker principles, and that the

assumption of a set is not needed, is what makes semantic paradoxes particularly difficult to

solve.

Priest attempts to answer this objection by saying that once one is willing to talk about true

sentences of English, one is thereby committed to there being a set of true sentences of English.

For him, 'there is a conceptual connection between satisfying a condition being true and

being a member of a certain totality being one of the totality of true things' [Priest 2002: 279].

Moreover, the reason that is commonly used to explain why there is no set of all non-self-

membered sets (the fact that the collection is too big) cannot be applied to deny the existence of a

set of true sentences of English, because there can only be countably many true sentences of

English.

Priest's defense remains unsatisfactory for two reasons. First, the idea of a conceptual

connection between satisfying a condition and being a member of a certain totality should not be

taken as granted. Even if there could not be more than countably many objects that satisfy a

certain condition (a meaningful predicate), there are ways in which the condition could fail to

determine a totality. This might happen, for instance, if the predicate turns out to be vague or if

the meaning postulates associated with it are inconsistent with the facts. Second, even if one

grants that there is a conceptual connection between a condition and there being a totality, it is









not clear why the thesis that there is a totality of true sentences of English should be made part of

the schema. In general, it does not seem to be the case that if A is part of the explanation of a

certain paradox, and A entails B, then B is also part of the explanation of that same paradox.

The diagonaliser and the Liar

In this section I will assume, for the sake of the argument, that there is a set of all true

sentences of English and argue that the diagonaliser fails to play the role Priest ascribes to it in

the Liar paradox. Priest presents the Liar paradox as if there were a general mechanism (the

diagonaliser) that for any definable subset of Q (the set of all true sentences of English) makes

both Transcendence and Closure true. When this general mechanism is applied to the set Q itself,

a contradiction is obtained. Tennant [1998: 29] points out that the general mechanism is relevant

only at the limit, i.e. at the level of Q. Whether Transcendence and Closure obtain for other

subsets of Q is completely irrelevant to the paradox. It is undeniable that the limit case is enough

to obtain a contradiction. Nevertheless, Priest counters Tennant's objection by saying that by

focusing only on the limit case one would miss completely 'an essential part of the story' [Priest

2002: 279]. The reason is that the paradoxes are produced by a certain mechanism, and only

'when one understands this, one understands why contradictions of this kind arise' [Priest 2002:

279]. In a previous section, I argued that it would be illegitimate to support the idea that the

Inclosure Schema generates the paradoxes by assimilating the role played by the diagonaliser

with that of a causal mechanism. Priest tries to defend the idea that it is the Inclosure Schema

that generates (and thus explains) the paradoxes by arguing that the diagonalisers are such that









'there is a genuine functional dependence of the value of the function on its argument: the

argument is actually used in computing the value of the function' [Priest 2002: 136, fn. 18].8

I will first argue that in the case of the Liar paradox the diagonaliser fails to provide the

genuine functional dependency that would justify the thesis that the Inclosure Schema is the

underlying structure that generates the Liar paradox. There is no good reason to make the

requirement that Transcendence and Closure hold for subsets of the set of true sentences of

English part of the underlying structure of the Liar paradox. Second, I will argue that the

Inclosure Schema does not even hold true for the Liar paradox.

A closer look at the Liar paradox reveals that it is hard to identify a genuine functional

dependency that could be made responsible for the contradiction. Understood in a certain way,

functional dependency is quite easy to get. Anderson & al. [1992] take one-to-one functions to

offer the paradigmatic case of genuine mathematical functional dependency. One-to-one

functions are easy to construct, but it is obvious that more is needed in order to make the

function in any sense responsible for generating the contradiction. In order to give credibility to

the idea that the function is indeed responsible for generating the contradiction, one should take

seriously Priest's idea that the argument should actually be used in computing the value of the

function.9 However, appearances to the contrary notwithstanding, the diagonaliser does not

provide a genuine functional dependency, because the argument is not used in computing its

value. Recall that in the case of the Liar paradox 6 is supposed to be the function o(a) = a, were a



8 Priest [2002: 136, fn. 18] refers to [Anderson et al. 1992] for more details about functional dependency. To his
credit, he acknowledges that the problem of being able to tell when a functional dependency is genuine is a 'tricky
and unresolved' problem.

9 What Anderson et al. [1992] call syntactic dependency might come closer to what Priest has in mind. A function
(formula) M depends syntactically on a variable x 'just in case M can be seen by syntactic inspection to be
semantically strict in the variable x' [Anderson et al. 1992: 397]. A function (formula) M is semantically strict in the
variable x 'just in case, if the value of x is undefined, so is the value of M' ibidd. 397].









=
. It should be noticed that Priest's characterization of the function is ill-formed,

because it involves a use-mention confusion: the first occurrence of 'a' is in the material mode,

while the second is in the formal mode. Therefore, it is not clear which function Priest has in

mind. One would have a genuine functional dependency if the function were understood as

ranging over expressions that refer to sets of true sentences, rather than over sets of true

sentences. However, in that case the structure of the Liar paradox would clearly be different from

the Inclosure Schema, because the Inclosure Schema requires that 6 range over a subset of P(Q).

The alternative is to interpret 6 as a function that takes a set of true sentences, s, into a sentence

that contains an expression that refers to s. That there is such a sentence for each set is

guaranteed by the fact that the sets are restricted to definable sets of true sentences. The axiom of

choice guarantees that there is a function that takes every definable set of true sentences, s, into a

sentence o(s) = a, where a =
and 'a' is an expression that refers to the set s. 6 would be

in this case a function that ranges over a subset of P(Q), the way Priest wants, but it can be

noticed that the argument is not actually used in computing the value of the function. This shows

that the diagonaliser cannot be made responsible for the contradiction; no good reason has been

provided for thinking that the Inclosure Schema underlies in any significant sense the Liar

paradox or that it captures an essential part of the story.

The second point is that, far from being the underlying schema that generates the Liar

paradox, the Inclosure Schema does not even hold true for the Liar paradox. Although the mere

definability of a set is enough to guarantee Transcendence, it is not enough to guarantee Closure.

It is hard to deny that Transcendence and Closure hold for the set of true sentences itself, Q, but

this does not follow only from the fact that Q is definable. According to Priest's own definition,

'something is definable iff there is a (non-indexical) noun-phrase that refers to it' [1994: 28]. If









'refer' is used in a wide enough sense, to also cover the relation between a predicate and its

extension, then the definition is compatible with a set being definable if there is a predicate that

has that set as its extension. Nevertheless, Priest seems to think of a set to be definable only if

there is a name or a definite description in English that has that set as its referent. Let s be a set

of true sentences and o(s) = a, where a =
and 'a' is an expression that refers to s.

Transcendence would hold, because one can prove that o(s) cannot belong to s. If it did, then it

would be true, so it would follow that it does not belong to the set 'a' refers to, which is s. I will

argue that Closure, namely the thesis that o(s) is true, cannot be derived unless one makes some

additional assumptions, such as the assumption of bivalence, or the assumption that o(s) really

says that o(s) does not belong to s.

One lesson that can be drawn from Kripke [1975] is that there can be languages that

contain their own truth predicate. The price that must be paid for there being such a predicate is

that some things cannot be said in that language, because some sentences would lack a truth

value. There is no good reason to think that English could not contain a similar predicate, 'true*',

that represents the set of true sentences of English. If there is no such predicate in English, one

could certainly add it to the language. In this case, 'the set of true* sentences' would ensure the

definability of the set of true sentences of English. Consider now the following sentence:

(L*) (L*) does not belong to the set of true* sentences.

(L*) would not belong to the set of true sentences of English (Transcendence holds), but it would

be inappropriate to infer that (L*) is true (Closure is not guaranteed). Thus, the definability of a

subset of true sentences, a, is not enough to guarantee that Closure holds (i.e., that o(a) is true).

What this shows is that the diagonaliser does not actually guarantee that both

Transcendence and Closure hold for all definable subsets of Q, which means that the Inclosure









Schema is not an accurate representation of the Liar paradox. In order to save the Inclosure

Schema, one would have to further restrict x to an even smaller subset of P(Q). It might be held

that if x is restricted to sets that are definable by proper names, Closure could be saved on the

grounds that expressions of the form 'a is not a member of b', where 'a' and 'b' are proper

names, could not lack a truth-value. The problem with this strategy is that Q, the set of true

sentences of English, as well as subsets of Q, would have to be defined by proper names of

English, but there is no intuitive support for this thesis. One could certainly add new names for

these sets by stipulation, but it is clear that the Liar paradox should not be dependent on the

existence of the stipulative names. Alternatively, one could try to restrict x to those sets that

guarantee that Transcendence and Closure hold, but this would be an explicit acknowledgement

of the fact that it is not the general mechanism that should be made part of the structure of the

Liar paradox, but only what makes Transcendence and Closure at the limit case possible.

It can now be inferred that all that really matters for there to be a Liar paradox is that

Transcendence and Closure hold for the set of true sentences of English. Moreover, definability

alone is not enough to establish that the two conditions hold at the limit case. Two additional

elements emerge as essential aspects of the Liar paradox: the T-schema, which is needed to

prove Transcendence, and the fact that the Liar (the sentence that says about itself that it is not a

member of the set of true sentences of English) expresses in English the thought that the Liar is

not true, which is needed (together with the T-schema) to establish Closure. These elements do

not show up in the Inclosure Schema, but they are essential parts of the Liar. Moreover, there are

no similar elements in the structure of logical paradoxes.









Semantic Liars and Logical Liars

One source of the idea that semantic and logical paradoxes have the same underlying

structure is the fact that each semantic paradox can be associated and sometimes confused with a

logical paradox. The Liar Paradox is a semantic paradox because it deals with truth, which is a

semantic notion. However, it is common practice to use the set-theoretical apparatus to analyze

semantic notions. As Priest points out, this tendency has been encouraged by the successes of

Tarski's and Godel's work in semantics. The set-theoretical apparatus enables one to reframe the

Liar paradox as a paradox in set-theory. Instead of talking about the meaning of 'true' or about

the concept of truth, it is often more convenient to talk about the set of true sentences. On the one

hand, it is intuitively true that there is a set of true sentences of English, which is represented in

English by the predicate 'true sentence of English'. On the other hand, principles that are

intuitively true enable one to prove that the set of true sentences of English cannot be represented

in English. This paradox, call it the Logical Liar, has indeed a structure that is more similar to the

structure of logical paradoxes. As such, it has all the appearances of a logical paradox, but it

would be different from the original semantic Liar paradox.

The temptation to identify the Liar paradox with the Logical Liar is encouraged by the

tendency to take the advances in semantics to show that there is no clear borderline between set-

theory and semantics. Priest argues that although at the time when Ramsey wrote his paper it was

commonly held that the vocabularies of mathematics and of semantics are indeed different, in the

meantime things have changed:

In particular, both syntactic and semantic linguistic notions became quite integral parts of

mathematics. Indeed, in a sense, the work of Godel and Tarski showed how these notions could

be reduced to other parts of mathematics (number theory and set theory, respectively) [Priest

1994: 26].









Nevertheless, although some aspects of the semantic notions can be properly captured in

set-theoretic terms, I think that the analogies should not be pushed too far. Tarski showed indeed

how the concept of truth for formalized languages can be extensionally characterized in

mathematical terms, but this does not mean that the concept can be reduced to mathematical

notions. Moreover, it is the concept of a true sentence of English that gives rise to a paradox, not

that of a true sentence of a certain formalized language, and Tarski did not actually claim that the

definition of truth can be extended to natural languages.

One way to see that the two Liars should be sharply distinguished from one another is to

notice that the intuitions that lay behind the Logical Liar are parasitic upon those that lay behind

the Liar paradox. Absent the intuitions that lead to the Liar paradox, the Logical Liar could be

solved by denying that there is a set of true sentences of English (the way one can block other

logical paradoxes), or by denying that the set is representable in English.

Thus, the Liar paradox should not be identified with the logical Liar. For the same reasons,

the structure of the Liar paradox, as well as the structure of other semantic paradoxes, is different

from the structure of logical paradoxes. Consequently, it is perfectly reasonable to expect that the

two kinds of paradoxes have different kinds of solutions. The structural differences between the

two kinds of paradoxes indicate that the lessons that should be drawn from them might be quite

different. At the end of the day, logical paradoxes suggest that there are fewer sets in the

universe than one normally is tempted to allow (which is the idea behind the orthodox solutions

to the logical paradoxes based on the axiomatization of set-theory), while the semantic paradoxes

suggest that fewer concepts are expressible in English than one normally believes.









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Burge, Tyler 1984. Semantical Paradox, in Recent Essays on Truth and the Liar Paradox, ed. R.
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Chihara, Charles 1973. Ontology and the Vicious-Circle Principle, Cornell University Press.

Chihara, Charles 1979. The Semantic Paradoxes: A Diagnostic Investigation, The Philosophical
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Chihara, Charles 1984. The Semantic Paradoxes: Some Second Thoughts, Philosophical Studies
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Evans, Gareth 1997. Understanding Demonstratives, in Readings in the Philosophy ofLanguage,
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Fitch, Friedrich 1964. Universal Metalanguages for Philosophy, Review of Metaphysics 17: 396-
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Frege, Gottlob 1997. Begriffsschrift, in The Frege Reader, ed. M. Beaney, Blackwell Publishers:
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Frege, Gottlob 1997. Thought, in The Frege Reader, ed. M. Beaney, Blackwell Publishers: 325-
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Gupta, Anil and Nuel Belnap 1993. The Revision Theory of Truth, Cambridge: MIT Press.

Herzberger, Hans 1967. The Truth-Conditional Consistency of Natural Languages, Journal of
Philosophy 64.2: 29-35.









Herzberger, Hans 1970. Paradoxes of Grounding in Semantics, Journal of Philosophy 67: 145-
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Hofweber, T. 2006. Inexpressible Properties and Propositions, in Oxford Studies in Metaphysics,
vol. 2, ed. D. Zimmerman, Oxford University Press.

Kleene, S.C. 1952. Introduction toi lti[,,uitheiul, utic' New York: Van Nostrand.

Konig, J. 1967 (1905). On the Foundations of Set Theory and the Continuum Problem, in From
Frege to Godel, ed. van Heijenoort, Cambridge MA: Harvard University Press.

Kremer, M. 2000. Judgment and Truth in Frege, Journal of the History of Philosophy 38.4: 549-
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Kripke, Saul 1975. Outline of a Theory of Truth, Journal of Philosophy 72: 690-716.

Lappin, S. 1982. On the Pragmatics of Mood, Linguistics and Philosophy 4: 559-78.

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BIOGRAPHICAL SKETCH

I completed my undergraduate studies in Romania, where I received a BS in Mathematics

and a BA in Philosophy from Bucharest University. I moved to Florida for the graduate school in

August 1999, and I received an MA in philosophy form the University of Florida in December

2002. After completing and defending the dissertation, I will receive my PhD in philosophy from

the same university, in August 2007.

My research has mainly been focused on some issues in philosophical logic. In particular, I

spent a fair amount of energy trying to find a solution to the semantic paradoxes and to better

understand the nature of truth and the role it plays in human inquiry. The study of paradoxes is

even more frustrating than the study of the origins of the universe because it is very easy to get

seduced by a path which turns out to be a dead end. I found consolation in the thought that many

beautiful minds of the past have been tricked by the paradoxes in a similar way. Truth is an

inexhaustible philosophical topic, so I plan on continuing my research on the Liar paradox and

on some implications of the inexpressibility view, and I also intend to spend more time reflecting

on the debate between deflationists and the defenders of a more robust notion of truth.

In addition to philosophical logic, I have strong interests in other areas of analytic

philosophy, such as logic, metaphysics, philosophy of language and the philosophy of

mathematics. Although I am trying to keep up with the current philosophical literature, I have a

vicious inclination toward outdated views.





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1 THE INEXPRESSIBILITY OF TRUTH By EMIL B DICI A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

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2 2007 Emil B dici

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3 To all of those who struggled to tell the truth

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4 ACKNOWLEDGMENTS Ancestors of some of the chapters of this di ssertation have been presented at the Logica Conference, June 2005 (Hejnice, the Czech Repub lic), the Florida Philosophical Association Conference, November 2005 (Cocoa Beach) and November 2006 (Tampa) and the annual meeting of the Society for Exact Philosophy, May 2007 (Vancouver, Canada). I am indebted to my audiences for their helpful comments. I also wish to thank John Biro, William Butchard, Douglas Cenzer, Michael Jubien, Kirk Ludwig a nd Elka Shortsleeve for helpful comments and support over the past five years. I cannot expr ess my gratitude to Greg Ray for patience, guidance, fruitful discussion and uncountably ma ny comments on previous versions of this dissertation. Finally, I want to acknowledge a special debt to my wife, Ana-Maria Andrei, for insightful thoughts and encouragement.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 ABSTRACT.....................................................................................................................................9 CHAP TER 1 INTRODUCTORY REMARKS............................................................................................ 10 2 LIAR PARADOXES, INCONSIS TENCY AND UNIVERSALITY .................................... 17 Liar Paradoxes........................................................................................................................17 Paradoxical Sentences..................................................................................................... 17 Pathological Sentences.................................................................................................... 19 Other Kinds of Liars: Utterances, Mental Representations, Pro positions....................... 20 The Concept of Truth...................................................................................................... 21 Self-Reference........................................................................................................................22 Vicious Circularity..........................................................................................................22 The Tarskian Hierarchy of Languages......................................................................... 24 Paradoxes without Self-Reference.................................................................................. 24 Truth-Value Gaps............................................................................................................... ....25 Strengthened Liars and the Principle of Bivalence ......................................................... 27 Martins Simple Liar Argument...................................................................................... 30 Choice Negation versus Exclusion Negation.................................................................. 31 The Principle of Bivalence and Classical Logic.............................................................. 36 Reductio ad Absurdum versus Reductio ad Gapsurdum................................................. 38 The T-Schema.................................................................................................................43 Universality and Inconsistency............................................................................................... 47 3 THE INEXPRESSIBILITY OF TRUTH............................................................................... 49 Universality and Expressibility............................................................................................... 50 The Inexpressibility Argument...............................................................................................57 Herzberger on Universality..................................................................................................... 59 No Language contains all its Semantic Concepts............................................................ 59 Class Expressibility versus Concept Expressibility ........................................................ 61 Two Objections against the Inexpressibility View................................................................. 63 Intentions are Sufficient for Expressibility...................................................................... 63 The Inexpressibility Acc ount is Self-Defeating .............................................................. 64 Intended Meaning versus Linguistic Meaning....................................................................... 65 The Status of the T-Biconditionals......................................................................................... 72

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6 Is the Concept of a True Sentence of English Expressible in other Languages? ................... 73 4 ON THE COHERENCE OF THE INCO NSISTENCY VIEW OF TRUTH ......................... 76 The Inconsistency View......................................................................................................... 77 The Inconsistency of Natural Languages........................................................................ 79 Meaning Postulates..........................................................................................................81 Inconsistent Languages and the Inconsistency View of Truth ........................................ 84 The Inconsistency of the Concept of Truth.....................................................................85 A Priori or Empiri cal Inconsistency? .............................................................................. 86 The Inconsistency View and Classical Logic.................................................................. 88 Skepticism with Respec t to Inconsistency .............................................................................. 89 Intentional Inconsistency versus Linguistic Inconsistency..................................................... 93 Two Kinds of Inconsistency............................................................................................93 Intentional Inconsistency.................................................................................................96 Inconsistency Entails Inexpressibility.................................................................................... 98 5 NON-LINGUISTIC LIARS.................................................................................................100 Mental Representations........................................................................................................101 Thoughts and Beliefs.....................................................................................................102 Liar Thoughts................................................................................................................103 Gappy Thoughts............................................................................................................104 Intentional states and th eir propositional content ...................................................106 Liar Beliefs....................................................................................................................111 Liar Propositions...................................................................................................................113 Mentalese Liars................................................................................................................ .....115 6 AN EXTENSION OF THE ACCOUNT: SEMANTIC VE RSUS LOGICAL PARADOXES......................................................................................................................118 Semantic Paradoxes..............................................................................................................119 Grellings Paradox......................................................................................................... 119 Paradoxes of Definability.............................................................................................. 119 Richards paradox..................................................................................................119 Berrys paradox......................................................................................................120 Knigs paradox.....................................................................................................121 A Denotation Paradox...................................................................................................121 The Inexpressibility of the Semantic Concepts.................................................................... 122 Similarities between the Seman tic and the Logical Paradoxes ............................................ 124 A Little Bit of History................................................................................................... 125 Priests Uniformity Account.........................................................................................127 The Semantic Version of Russells Paradox................................................................. 130 Refuting the Uniformity Account.........................................................................................132 An Objection from Circularity...................................................................................... 132 The Liar and the Inclosure Schema............................................................................... 136 The Existence clause.............................................................................................. 137

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7 The diagonaliser and the Liar.................................................................................139 Semantic Liars and Logical Liars.................................................................................. 144 LIST OF REFERENCES.............................................................................................................146 BIOGRAPHICAL SKETCH.......................................................................................................150

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8 LIST OF TABLES Table page 1-1 .............................................................................................................................................48 1-2 .........................................................................................................................................48

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9 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE INEXPRESSIBILITY OF TRUTH By EMIL B DICI August 2007 Chair: Greg Ray Major: Philosophy The main purpose of my study is to explore and defend what I call an inexpressibility account of the semantic paradoxes. I argue that cont rary to the widely held belief that English (as well as every other natural language) is universal, or at least sema ntically universal (in the sense that all its semantic concepts are expressible in it) the lesson one should draw from the Liar paradoxes is that in fact Englis h fails to express the concept of truth. Taking this view provides the foundation of a satisfying reso lution of the Liar paradoxes, but the promise of the view has been underappreciated for reasons which, I argue, turn out to be not well-founded. For example, the view might appear to be self-defeating because, it might be objected, in defending the inexpressibility view one actually expresses the concept that is held to be inexpressible. I argue that this and related challenges are founded on a fa ilure to make certain fu ndamental distinctions such as the distinction I in troduce between intended meaning and linguistic meaning. The distinction enables one to explain why communi cation is unproblematic although the concept of truth is inexpressible. The main vi rtue of the inexpressibility view is that it offers a solution to the Liar paradoxes without postulating that trut h is an inconsistent concept. Moreover, the account can be easily extended to apply to all semantic paradoxes.

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10 CHAPTER 1 INTRODUCTORY REMARKS Consider the following im aginary dialogue betw een an innovative gra duate student and his more skeptical adviser: A. Ive heard this story many times. Tell me your groundbreaking solution to the old paradox of the Liar. B. No sophisticated reasoning. If the Liar sentence is true, then it is not true. If it is false, then it has to be true. The only option that remains is that the Liar is neither true nor false. A. Wouldnt it follow that the Liar is not true? B. Indeed, we should think th at the Liar is not true. A. But this is what the Liar says. It would fo llow that the Liar is true after all. You are contradicting yourself. B. Things are puzzling indeed. However, I think that you are going too far when you say that thats what the Liar says. I indeed think that the Liar is not true, but this is not what the Liar says. A. This strikes me as utter nonsense. You ar e saying that the Liar does not say what you are just saying by using it. If it does not say that the Liar is no t true, what does the Liar say? B. It turns out that there is nothing that th e Liar says, and Im not strictly speaking saying anything by using it. A. In that case your sentences are meaningle ss. How do you expect one to make sense of your sentences? B. It would be unfair to say that my sentences are meaningless. You understand every single word that I have been us ing. I grant, however, that many of the sentences I used are not true.

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11 A. You are very subtle indeed. It is enough for me that you acknowledge that they are not true. End of the story. B. This shouldnt be the end of the story. Although some of my sentences fail to say anything, you know what I intended to say. A. How can I know what you intended to sa y if your sentences fail to say it? B. You know what my sentences are intended to say, even if they fail to say it. You cannot deny that as a result of this conve rsation you know what I think about the Liar. A. I wonder whether our exchange of senten ces can indeed be called a conversation. The purpose of this study is to explore and defend the position articulated by B, which I call the inexpressibility account of the Liar paradox. Contrary to the widely held belief that English (as well as every other natural language) is universal, or at least sema ntically universal (in the sense that all its semantic concepts are expressible in it) the lesson one should draw from the Liar paradoxes is that in fact Englis h fails to express the concept of truth. Taking this view provides the foundation of a satisfying reso lution of the Liar Paradoxes, but the promise of the view has been underappreciated for reasons which, I argue, turn out to be not well-founded. For example, the view might appear to be self-defeating because, it might be objected, in defending the inexpressibility view one actually expresses the concept that is held to be inexpressible. I argue that this and related challenges are founded on a fa ilure to make certain fu ndamental distinctions, such as the distinction I in troduce between intended meaning and linguistic meaning. The distinction enables one to explain why communica tion is unproblematic a lthough true fails to express the concept of truth. Chapter one consists in an examination of so me of the attempts to solve the Liar paradox, whose outcome is that the main obstacle to reaching a solution is the assumption of the

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12 universality of English. The attempts to avoid the Liar paradox by banning self-reference (such as the appeal to the theory of types or to the distinction betw een object-language and metalanguage) have been very fruitful when applie d to formal languages, but they do not seem to be appropriate for natural langua ges because the expressive powe r of natural languages does not seem to be limited in this way. English allo ws one to combine predicates and referring expressions in a sentence without any type restrictions, and it contains many self-referential expressions which are perfectly meaningful. Moreover, the distinc tion between an objectlanguage and a metalanguage cannot be applied, because English is its own metalanguage (it seems that whatever can be said about English can be said in English). Likewise, the attempts to solve the Liar paradox by postula ting truth-value gaps or any other distinction between three categories of sentences (stably true/pathological/s tably false) fail because contradictions would still follow as long as the semantic notions that have been used as part of the solution are expressible in English. As long as English is held to be able to express its own semantic concepts, there seems to be no way out of the in consistency. Thus, there appear to be two main alternatives: either one accepts th at English is universal, in which case one is forced to endorse some version or other of the inconsistency view, or one can deny the universality of English and thus avoid the inconsistency view. The thesis that English is universal (more speci fically, the thesis that English is able to express its own semantic concepts ) is usually considered too obvious to be argued for. In chapter three, which is the central chapter of this study, I argue that the thesis is actually false. In particular, the concept of truth turns out to be inexpressible in English. The inexpressibility argument is a reductio argument formulated in ordi nary English: I show th at the supposition that there is a predicate of English that expresses the concept of truth leads to a contradiction.

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13 Roughly, the contradiction is obtained by noticin g that the Liar argument makes an implicit appeal to the assumption that true expresses the concept of tr uth and then exploiting this by turning the argument into a reductio argument. This version of the inexpressibility argument is to be preferred to another versi on that is due to Hans Herzberger [1970], because Herzberger formulates his argument as an argument for the ine xpressibility of a class rather than that of a concept. The remaining part of chapter three is devot ed to answering two objections that might readily come to mind. It might be thought that e xpressibility is only a ma tter of associating the right intentions with a certain expression. In this case, th e expressibility of tr uth would be trivial, because speakers of English do intend to use t rue to express the concept of true. The other objection is that the inexpressibil ity view is self-defea ting. In defending the inexpressibility view of truth I employed the word true, which one might think means that I actually expressed the concept that I held to be inexpressible. Bo th objections can be satisfactorily answered by drawing a distinction between th e intended meaning and the linguistic meaning of an expression. Not all the intentions we have with respect to th e use of a word are fulfilled. This means that the intended meaning may fail to become linguistic m eaning, which means that expressibility is not trivially guaranteed. The mere fact that we use the word true in English is not enough to guarantee that it expresses the concept of trut h (i.e., that it has a li nguistic meaning), which means that the inexpressibility view is not self-defeating. On the other hand, communication remains unproblematic in spite of the fact that true lacks linguistic meaning, because the type of meaning that is central for communication is the intended meaning, no t the linguistic meaning. Speakers of English know what the intended meaning of true is, and they do not need to know its linguistic meaning in order to count as competent speakers of the language.

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14 One might grant that the inexpres sibility view of truth offers a coherent way to avoid an inconsistency, but still argue that it would be preferable to endorse an inconsistency view rather than say that truth is inexpressible in English. Therefor e, chapter four is focused on the inconsistency view. I argue that, properly underst ood, the inconsistency view of truth is coherent but the arguments based on Liar sentences fail to establish that it holds. Moreover, an inconsistency view of truth entails that truth is inexpr essible (which also means that English is not universal). For this reason, an inconsistency view of truth cannot offer the best explanation of the Liar paradoxes. The inconsistency view of truth has been attacked by many (e.g., Herzberger [1970], Soames [1998]) as an in coherent view. Nevertheless, most of these attacks are a consequence of the failure to distinguish between two ways in which a languages can be said to be inconsistent. I argue that one should draw a distinction between intentional inconsistency and linguistic inconsistency. Tarski [1933] and Chih ara [1979] should be inte rpreted as advocating versions of an intentional inconsistency vi ew. It is the meaning principles that are intended to be true that are inconsistent (either inconsistent as a set, or inconsistent with some empirical facts). The objectors have usually misint erpreted the view as a linguis tic inconsistency view, which would indeed be incoherent, because it requires that an inconsistent set of principles be all true. On its turn, intentional inconsistency could be in terpreted in two ways wh ich should be carefully distinguished from one another. The thesis that the intended meaning rules are inconsistent should be distinguished from the thesis that the intended meaning rules are inconsistent with the hypothesis that true succeeds in expressing the concept of truth. The Liar arguments offer enough support for the latter thesis but not for the former. Even if one can find other reasons (than Liar sentences) for thinking that the concept of truth is inconsistent, this would not defeat the inexpressibility view but rather provide support for

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15 it (because the inconsistency view of truth entail s the inexpressibility view). Nevertheless, one reason why the inexpressibility view is attractiv e is that it can offer a solution to the Liar paradoxes while preserving the consistency of th e concept of truth. The significance of the inexpressibility account would be dramatically diminished if Liar paradoxes could still occur, this time not at the level of sentences, but at th e level of propositions or mental representations. This would mean that the inexpressibility vi ew cannot actually provide a way to save the consistency of truth. In chapter five I argue that there are no Li ar arguments at the level of propositions or mental representa tions that could force us to adopt an inconsistency view of truth. The Liar thought (understood as mental repres entations) can coherently be said to lack a truth-value. Although it has a true propositional cont ent, the Liar thought itself cannot be said to be true, because a closer examinati on of its structure reveal s that it fails to be an intentional state with the mind-to-world direction of fit, which w ould be required for an intentional state to be truth-evaluable. Moreover, th ere are good reasons to think th at there could be no Liar propositions. The Liar paradoxes are members of the larger family of sema ntic paradoxes, therefore a successful account of the Liar paradoxes is e xpected to show how to account for the other semantic paradoxes. In chapter six I show how th e inexpressibility argument can be extended to prove that heterology, satisfacti on, denotation and other semantic concepts are inexpressible in English. The same account cannot be applied to th e so-called logical para doxes, which are more properly handled by mathematical methods (for in stance, by restricting th e universe of sets by some axiomatic set-theory, such as Zermelo-Fr aenkels). It has been argued (by Russell and, more recently, by Graham Priest) that logical an d semantic paradoxes have the same structure, and that similar paradoxes should receive similar solutions. Grah am Priest [2002] argues that

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16 both logical and semantic paradoxes have the sa me underlying structure (which he calls the Inclosure Schema) which, in conjunction with the Principle of Uniform Solution (same kind of paradox, same kind of solution), suffices to s ink virtually all orthodox solutions to the paradoxes. This would also suffice to sink the in expressibility view, beca use it also fails to provide a uniform account. I argue that Priest fails to provide a non-que stion-begging method to impugn virtually all orthodox solutions, and that the Inclosure Schema cannot be the structure that underlies the Liar paradox. Ramsey was ri ght in thinking that logical and semantic paradoxes are paradoxes of different kinds and that one should not expect them to receive the same kind of solution.

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17 CHAPTER 2 LIAR PARADOXES, INCONS IST ENCY AND UNIVERSALITY A Liar paradox is due to the fact that a number of intuitively tr ue principles can be used to derive an inconsistency by using only intuitively valid rules of inferen ce. I will begin with a survey of the most significant t ypes of Liar paradoxes. Thereafter, I will examine some of the main attempts that have been made to block th e derivation of an incons istency consisting either in banning self-reference or in postulating truth-value gaps. The discussion is far from being exhaustive, and is mainly aimed at identifying the principles that are commonly used to run a Liar argument and explaining why it so hard to fi nd a way to block it. It turns out that what makes it difficult to avoid inconsistencies is some form or another of the principle of universality that is commonly attributed to natural languages. Therefore, one is faced with a choice between accepting the inconsistency of some intuitively true principles and denying the universality of English and other natural languages. Liar Paradoxes Paradoxical Sentences Natural languages are kn own to be capable of self -referen ce. Thus, one can talk in English about the sentences of English. Moreover, given an arbitrary sentence of English, one can introduce a name for it in English. As long as it belongs to the category of names, any expression that has not already been assi gned a referent could play th is role. Thus, one can use L0 to refer to the following sentence-type (call it the Simple Liar): (L0) (L0) is false. (L0) is a well-formed sentence of English: false is a predicate of English that we understand, (L0) is a proper name that refers to the Simple Liar, and the sentence obeys the syntactic rules of English. Nevertheless, it is easy to see that (L0) is paradoxical. If (L0) is true, then it is false,

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18 because this is what it sa ys. On the other hand, if (L0) is false, since this is what (L0) says, it follows that (L0) is true. In either case the conclusi on is unacceptable, because no sentence can be both true and false1. Another Liar sentence that is very familiar in the literature on paradoxes is the Strengthened Liar sentence: (L) (L) is not true. An informal version of the Liar argument for this sentence goes as follows: If we respond to the strengthened liar sentence just the way we did to the simple liar, by saying that the sentence is neither true nor fals e, then we will have to say, a fortiori, that the strengthened liar sentence is not true. But that the strengthe ned liar sentence is not true is precisely what the strengthe ned liar sentence says, and we are back in the briar patch. [McGee, 1990, pp. 4-5] The Simple Liar and the Strengt hened Liar are not the only kinds of sentences that raise problems for the concept of truth. There are many other sentences that are problematic in one way or another. Consider first the Pair Liars: S1 S2 is true. S2 S1 is not true. It is obvious that the two se ntences are paradoxical. If S1 is true, then S2 must be true, so S1 is not true. On the other hand, if S1 is not true, then S2 would have to be true, which means that S1 is true. The difficulty can be generalized to obtain Chain Liars of arbitrary length: S1 S2 is true. S2 S3 is true. Sn S1 is not true. 1 Even the dialetheists, who deny that no sentence can be bot h true and false, acknowledge that this principle is at least prima facie true.

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19 Currys paradox is a slightly different kind of paradox of truth. One version of the paradox goes as follows. Consider the following sentence: (1) If (1) is true then God exists. Since (1) says that if (1) is true then God exists, one can infer (2): (2) Sentence (1) is true iff if (1) is true then God exists. Suppose now that (3) holds. (3) Sentence (1) is true. From (2) and (3) one can derive (4): (4) If (1) is true then God exists. From (1) and (4) one can get (5): (5) God exists. Since from the assumption of (3) one can derive (5), it follows that (6) must hold true. (6) If (1) is true then God exists. From (2) one can infer (7), (7) Sentence (1) is true. which together with (6) leads to the conclusion that (8) is true. (8) God exists. Obviously, the same argument pattern can be used to prove that God does not exist or any other thesis. Pathological Sentences Besides the paradoxical sentences, there are also m erely pathol ogical sentences, such as the Truth-Teller, the Truth-Teller L oops, or the infinite Truth-Telle r. The Truth-Teller is the following sentence: (TT) (TT) is true.

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20 The problem with these sentences is not that they lead to a contradiction, but that their meaning together with the facts in the world fail to determin e a truth-value. All one can say is that (TT) is true if and only if (TT) is true. However, this is a mere tautology and o ffers no help whatsoever in determining whether (TT) is true. Moreover, there does not seem to be any extra piece of information that could determine its truth-value. Se ntences of this sort raise a serious problem for the concept of truth, because one normally thinks that the meaning of a sentence (plus the way the world is) is enough to fix its truth-value. Wh at makes things worse is the fact that the assumption that (TT) fails to have a truth-value leads to a contradiction. Other Kinds of Liars: Utterances, Men tal Rep resentations, Propositions The Liar paradoxes that I enumerated above ar e paradoxes in which true is applied to sentences. However, there are many other kinds of entities that can play the role of truth-bearers. Thus, one can talk about true utterances, statements, beliefs, propositions, etc. The debates regarding which of these entities should be taken as the primary bearers of truth can be set aside. What matters is that all of them can be legitimately said to be true or false. It is easy to see that one can think of Liar paradoxes corresponding to each of these different types of truth-bearers. Thus, there are paradoxical utterances, such as some of the utterances of (U): (U) This utterance is not true. If (U) is uttered in a c ontext in which the demonstrative refers to the utterance itself, then if the utterance is true, then it would have to be not true, and if it is not true, then it would have to be true. Thus, both alternatives lead to a contradiction. Unlike the pr evious cases, self-reference in (U) is achieved by using a demonstrative. Simila rly, one can talk about Liar beliefs and Liar propositions as well as Pair Liar beliefs, Pair Liar propositions, Chain Liar beliefs, and so on. It will turn out to be convenient to disti nguish between linguistic Liar paradoxes (such as Liar sentences, Liar utterances and Liar statements) and non-li nguistic Liar paradoxes (such as

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21 Liar beliefs and Liar propositions). I will focus on linguistic Liars in chap ters three and four, but I will also consider the possibility of non-linguistic Liars in chapter five. The Concept of Truth A Liar sentence typically contains a predicate, true or fal se, a referring expression and possibly a form of negation. If fa lsity can be characterized in terms of truth and negation (a sentence is false if and only if its negation is true), then the crucial notions involved in these paradoxes are truth and negation. This does not give enough reasons to think that there is something problematic with the noti on of truth, as it could well be th e case that it is the notion of negation that is responsible for the paradoxes. Ne vertheless, there are some reasons to doubt that negation can be made responsible for the paradoxes First of all, negati on alone is not enough to generate a paradox. One gets a paradox only when negation is associated with a semantic notion (truth or another notion). Second, although negation is used in most of the Liar sentences, the pathological sentences, such as the Truth-teller do not contain any expression for negation. Thus it would be fair to say that the Liar paradoxes reveal a difficulty with our notion of truth, a difficulty which becomes even more troublesome when truth is thought of in conjunction with the notion of negation. The difficulties raised by the Liar paradoxes ha ve implications for various dimensions of the notion of truth. They raise problems for the pr operty of truth, the concept of truth and for the meaning of true. There are various positions with respect to the difference between properties, concepts and meanings. It woul d be useful not to start from the assumption that they are identical. One thesis that follows from the view that I defend in this dissertation is that the concept of truth cannot be identical with the meaning of any predicate of English. However, I will mainly be concerned with the concept of truth, because this is the notion that plays the crucial role in our scientific, philos ophical and everyday thinking. The concept of

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22 truth can be divided into some subconcepts in accordance with the types of truth-bearers. One can talk about the concept of a true sentence of L, the concept of a true proposition or the concept of a true belief. Notice that while the concepts of a true pr oposition or of a true belief are non-relational, sentences can only be said to be true in a relative way. It appears that although sentences are often said to be true (or not true ), they can only be true relative to a certain language: a sentence is true in L if and only if it expresses in L a true proposition. Likewise, utterances can be true only relative to a cer tain language. However, one can always turn a relational concept in to a non-relational one by specifying th e language. Thus, the concepts of a true sentence of English, or of a true utterance of German, ar e non-relational. I will occasionally talk about the concept of truth instead of the concept of a true sentence of English when the relevant features of the restricted concept extend smoothly to the c oncept of truth per se. Self-Reference Vicious Circularity It has been held that Liar paradoxes are th e result of using expre ssions that are selfreferential o r involve some circularity that is vicious. I will argue that self-reference cannot be made responsible for the Liar paradoxes. Poincar e thought that the paradoxes discovered in settheory have to do with definitions that are viciously circular2. Russell shared Poincares belief and argued that all paradoxes in semantics and in mathematics involve some sort of vicious circularity. There actually are more pha ses of Russells thought about paradoxes3. He came out with the idea of developing a th eory of types as early as in 1902-1903. Two years later, in 1905, he proposes instead the no-class theory. In 1908 he returns to his previous project of developing a theory of types that would avoid vicious circularit y. Part of this project is trying to reject self2 See Chihara [1973: 3]. 3 See Quines introduction to Russell [1908a] in van Heijenoort [1967: 150-52].

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23 referential expressions as meani ngless. The theory of types dis tinguishes between different types of expressions and introduces a number of restrictions that would rule out many expressions as not well-formed. Self-refe rential expressions fail to meet th e requirement of being well-formed, and would have to be rejected as meaningle ss. Whatever the merits of this method for constructing a formal language that is free of contradictions and adequa te for the purposes of science, it cannot be used for natural languages. First of all, not ju st any kind of selfreferentiality is vicious. Banni ng all self-referential expression s would amount to a significant restriction of the expressive pow er of natural languages. It cer tainly would be illegitimate to introduce (L) as the name of an expression containing it, if (L) already had a referent assigned to it. The introduction of the new name would be incorrect, for the same reasons we take a circular definition to be incorre ct. However, as Kripke notes [1975: 693], there is no reason why the introduction of Jack as a name for the senten ce Jack is short is illegitimate, as long as Jack has not already been assign ed a role in the language. More over even if proper names were not available, English allows one to achieve se lf-reference by using demonstratives (as in the case of Liar utterances) or defi nite descriptions. There is noth ing wrong with a self-referential sentence of the form: (9) The first sentence in this chapter which begins with The first sentence in this chapter belongs to English. Even sentences such as (10) (10) The first sentence uttered by Russell in 1905 is not true. should count as legitimate, regardless of whether the first sentence Russell uttered in 1905 is (10) itself or a different sentence. If (10) is paradoxical, this is a co ntingent matter, not an intrinsic

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24 feature of the sentence4. The conclusion is that self-referenc e is not sufficient for paradoxicality, and that we cannot ban paradoxical sentences for me rely containing self-ref erential expressions. The Tarskian Hierarchy of Languages Very frequently the Tarskian hierarchy of la nguages is m entioned as one type of solution to the Liar paradoxes. I use the quotation marks because although he talks about a hierarchy of languages Tarski never proposes it as a solution to the Liar paradox, and he only thinks of it as part of the project of devising semantic notions th at are appropriate for the needs of science. The idea behind the hierarchy of languages is to distinguish between an object-language and metalanguage such that the semantic notions of the object language can only be expressed in the metalanguage and not in the object-language itse lf. This way one can avoid self-referential sentences of the paradoxical sort. Of course, to avoid other paradoxical sentences the semantic notions of the metalanguage could only be part of a meta-metalanguage and so on. Although this strategy can provide a useful a lternative notion of truth, it doe s not shed much light on the ordinary notion of truth. The hierarchy approach cannot be applied to natural languages because in English, for instance, it seems to be possible to talk about th e semantic concepts of English. English appears to be its own metalanguage. The c oncept of a true senten ce of English seems to be expressible in English itself, not only in a metalanguage. Paradoxes without Self-Reference There are good reasons to th ink that pathology and paradoxi cality are not due to self reference. T he following sentences are pathol ogical, although they i nvolve no self-reference5: Sn Sn+1 is true. 4 See Kripke [1975] for other examples of contingent Liars. 5 Herzberger [1970: 150].

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25 Each sentence in this infinite list says about the next sentence in the list that it is true. Although these sentences are not paradoxical they are pathological in the same way the Truth-Teller is. The meaning of the sentences in this list, togeth er with the way the world is, does not suffice to determine their truth-values. As Herzberger puts it, these sentences involve a vicious semantic regress but no vicious circle.6 Moreover, Yablo provides an example of sentences that are pa radoxical, although they do not involve self-reference7. This means that self-reference is neither necessary nor sufficient for paradoxicality. Consider an infinite list of sentences of the form8: Si = For all k > i, Sk is untrue. One can run a Liar argument for each sentence in the list. If there is a sentence, Sn, in this list that is true, it would follow that Sk is true, for all k > n. From this one can derive a contradiction, because on the one hand, Sn+1 would have to be untrue, and, on the other hand, it would have to be true, because all Sk for k > n+1 are untrue, and this is what Sn+1 says. If all sentences in the list are untrue, then they would also have to be true, because for any sentence, all subsequent sentences are untrue. Truth-Value Gaps Many attempts to solve the Liar paradox invol ve saying that the Liar sentence lacks a truth -value: it is neither true nor false (it is custom ary to call such sentences gappy, because they fall in the gap between truth and falsity). Gappy se ntences are possible if the following principle, the principle of bivalence, is false: 6 Herzberger [1970: 150]. 7 It has been held that Yablos parado x implicitly involves self-reference. Nevert heless, this can only be plausible if self-reference is understood in a very loose way. No self-re ference of the sort we are familiar with from the classical Liar paradoxes is present in Yablos paradox. 8 Yablo [1993: 251-52].

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26 (Biv) Every sentence is either true or false9. The rejection of Biv indeed su cceeds in blocking the most co mmon version of a Simple Liar argument (although, as it emerges from the next sections, there are other arguments that show that the Simple Liar remains pa radoxical even in the absence of Biv). This argument involves, besides the logical principles and the principle of bivalence, the following three principles: (I) (L0) = (L0) is false. (SNC) No sentence is both true and false. (T) S is true in English iff S. (I) is an identity sentence that holds by sti pulation, (SNC), the principle of semantic noncontradiction, is an intuitively true principle that captures a relation be tween truth and falsity, while (T) is a schema that is supposed to hold for any replacement of S with sentences of English from an appropriately restricted class.10 It is widely agreed that it is part of what we mean by true that the T-biconditionals should be true.11 The Simple Liar argument goes like this: 1. (L0) is false is true iff (L0) is false. 2. (L0) is true iff (L0) is false. 3. [(L0) is true and (L0) is false] or [(L0) is not true and (L0) is not false] 4. ~[(L0) is not true and (L0) is not false.] 5. (L0) is true and (L0) is false. 9 It is assumed that the principle is restricted to meanin gful declarative sentences. Othe rwise, it would be trivially false. 10 It is not enough to restrict this class to meaningful declarative sentences. One s hould also exclude defective sentences as well as context sensitive sentences. 11 Some hold that this is all that is meant by true.

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27 6. [(L0) is true and (L0) is false.] and ~[(L0) is true and (L0) is false.] The principles of logic that are employed in th e argument are Material Equivalence (MatEquiv), De Morgans, Disjunctive Syllogism (DS) these are all familiar principles that can be found in any introductory textbook in classical logic and (Subst), the principle of the intersubstitutivity of identicals, which enables one to replace identica ls for identicals and preserve the truth value: (Subst) If A = B, then (A) (A/B), where (A/B) is the result of replacing one or more occurrences of A in (A) with B, for any formula Since this Simple Liar argument appeals to th e principle of bivalence, a defender of a truthvalue gap approach is able to block it by denying this principle. Unfortunately, the mere rejection of the principle of bivalence is not enough to solve the Liar para doxes. One reason is that there are other Liar sentences, such as the Strengthene d Liar, that allow one to run a Liar argument that is similar to the one above, but does not i nvolve the principle of bivalence. The other reason is that even for the Simple Liar, one can run a Liar argument that is slightly more complex, but does not appeal to the principle of bivalence. I will discuss each of the two kinds of argument in the next two sections. Strengthened Liars and the Principle of Bivalence So far as I know, the expression The Stre ng thened Liar has been introduced in the literature on paradoxes by Baas van Fraassen, who us es it to apply to thos e sentences that have been designed especially for those enlightened philosophers who are not taken in by bivalence.12 The idea is that if there are truth-valu e gaps, one can construct a sentence that 12 van Fraassen [1968: 147].

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28 closes off that gap. The Strengthened Liar, (L), is just one of the sentences that play this role. Another sentence that will do is: (L) (L) is either false or gappy. A sentence that closes the gap explicitly is so metimes called a Revenge Liar, but the terminology is far from being uniform. There is agreement in counting (L0) as a Simple (or ordinary) Liar. However, there is disagreement with respect to th e appropriate label for (L) and (L). Some take (L) to be a Simple Liar13 which should be contrasted with (L ), which they call a Strengthened Liar; others take (L) to be a Strengthened Liar, while (L) is eith er another Strengt hened Liar or a Revenge Liar. The important difference is between sentences that are paradoxical only in the presence of the principle of bivalence, and se ntences that remain paradoxical even if this principle is dropped. This criterion presumably makes (L0) a Simple Liar and both (L) and (L) Strengthened Liars, because both (L) and (L) allow a straightforward way to run a Liar argument in the absence of the principl e of bivalence. I will keep naming (L0) the Simple Liar and (L) the Strengthened Liar (when there is no risk of ambiguity, I will call the latter simply the Liar ); as I argue in the next section, the criterion that has been proposed (n o paradoxicality in the absence of bivalence) fails to discriminate between the two types of sentences. One version of a Liar argument for th e Strengthened Liar goes as follows: 1. (L) is not true is true iff (L ) is not true [from (T)] 2. (L) is true iff (L) is not true [from (1) and (Subst)] 3. (L) is true assumption 4. If (L) is true, then (L) is not true Mat. Equiv, 2 5. (L) is not true MP 3, 4 6. (L) is true and (L) is not true Conj. 3, 5 7. (L) is not true RA 3, 6 13 Gupta and Belnap [1993] take L* to be a Simple Liar.

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29 8. If (L) is not true, then (L) is true Mat equiv. 2 9. (L) is true MP 7, 8 10. (L) is true and (L) is not true Conj. 7, 9 What this argument appeals to is inference by reductio ad absurdum14; this is a weaker principle than the principle of bivalence. Since no appeal has been made to the principle of Bivalence, the mere rejection of bivalence fails to block the argument. One could object that the first premise is not available if (L) is neither true nor false on the grounds that the T-schema is not supposed to hold for gappy sentences. Neverthele ss, saying that (L) is gappy would entail that (L) is not true. But this is what (L) says, so (L) should be tr ue; once again, one can deri ve a contradiction. The very fact that (L) expresses in English the thought that (L) is not true preven ts one from solving the paradox. The idea of closing off the gap works as a general recipe for producing paradoxical sentences that resist all the alle ged solutions to the Liar paradoxes that divide sentences in three categories (instead of two): true/neither true nor false/false (or true/gap/false); true/ungrounded/false [Herzber ger 1970]; stably true/pathol ogical/stably false [Gupta 1993]; definitely true/unsettled/defin itely not true [McGee, 1990]. In each of these cases, one can construct in English a Liar senten ce that says about itself that it belongs in either the second or the third category. In all these ca ses, it seems that one cannot bl ock the argument, because there is a sentence of the language that must be true if the Liar sentence itself is not. This sentence either says about itself that it is not true, or it says that it is either false or 14 One can also run a Liar argumen t for the Strengthened Liar that used the principle of the Excl uded Middle instead of reductio ad absurdum. 1. (L) is not true is true iff (L ) is not true [from (T)] 2. (L) is true iff (L) is not true [from (1) and (Subst)] 3. Either [(L) is true and (L) is not true ] or [~((L) is true) and ~((L) is not true)] [Mat.Equiv. 2] 4. Either [(L) is true and (L) is not true] or ~[(L) is true or (L) is not true] [DeMorgans 3 5. ~~[(L) is true or (L) is not true] [DN, EM] 6. (L) is true and (L) is not true [DS 4,5]

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30 gappy/ungrounded/pathological. Again, it is the e xpressive power of English that causes the failure of these attempts to solve the paradoxes. The fact that the Liar remains paradoxical in the absence of bivale nce is enough to show that the rejection of this prin ciple cannot offer a general solu tion to the Liar paradoxes. An inquiry regarding whether the Simple Liar paradox survives in the absence of bivalence would be superfluous. Nevertheless, a discussion of a cont roversy surrounding the status of the Simple Liar paradox will prove to be useful, because it brings to light some principles (the excluded middle, RA) that are used also in the Strengt hened Liar paradox and might be considered unavailable in the absence of the principle of bivalence. Martins Simple Liar Argument It is comm only thought that the mere rejecti on of the principle of bivalence is enough to offer a solution to the Simple Liar paradox, but it fails to solve the Strengthened Liar paradox. Robert Martin15 argues that, contrary to what is commonl y thought, the rejection of that principle does not offer a straightforward solution to the S imple Liar paradox. This means that the Simple Liar is actually just as i ndependent of the principle of bivalence as its big brother.16 To defend this thesis, Martin shows how an ar gument that leads to inconsistency could be built even in the absence of the principle of bivalence: Let s0 be the ordinary Liar. First, we show that s0 is not false, as follows: suppose s0 is false; then, since that is what it says, it is true, and hence not false. (Principle: no sentence is both true and false.) Therefore, s0 is not false. But now we can see that s0 is false, since s0 says something the negation of which (s0 is not false) is true. (Principle: a sentence is false if its negation is true.) Thus a contradiction. [Martin 1984: 2] Martin is explicit about some of the semantic pr inciples involved in his argument. Thus, he is explicitly committed to all the instances of the following schemas 15 A similar argument has been put forward by Burge [1984: 88, fn.8]. 16 Martin [1984: 2].

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31 (SNC) ~(T(A) & F(A)) (F) T(~A) F(A) and, in a less explicit way, to the instan ces of the T-schema: (T) T(A) A In addition, there are some principl es of logic that are not made explicit. Given these (logical and semantic) premises, Martins argument shows th at the Simple Liar sentence is paradoxical.17 In order for a truth-value gap account to succe ed in solving the Simple Liar, one would have to argue that some of the principles involved in Marins argument are not true. One could reject some of the principles of logic that have been used, or one could reject principles such as (F) and (T). I will discuss some of these alternatives below. Choice Negation versus Exclusion Negation Beall and B ueno [2002] argue th at Martins argument fails to establish the conclusion, because (F) together with (T) and classica l logic entail the principle of bivalence.18 I will discuss this objection against Martin in the next sec tion. In this section I will focus on the implicit suggestion that one could avoid th e Simple Liar paradox by rejecting (F). Martins version of the Simple Liar argument takes (F) to be a platitude derived from our notions of truth, falsity and negation. Do we have good reasons to thi nk that (F) is true? If there are only two alternatives for a sentence (true or false; i.e., Biva lence holds), then the principle of falsity holds trivially. If there are three alternatives, it might not be immediately clear whethe r it holds or not. There is a correlation between the falsity principle and the wa y negation is interpreted. Let us assume that 17 In fact, one can derive the contradiction from weaker pr inciples. Thus, the right-to-left direction of the T-schema would be enough. 18 My criticism against the objection raised by Beall and Bueno is an adaptation of [Badici 2005: 25-39].

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32 the meaning of negation is determined by a trut h-table. In cases in which there is a third alternative for a sentence (besides truth and falsity), the truth table for negation has normally been taken to be given by Kleen es three-valued schema (Table 1). This notion of negation is usually called choice negation According to Kleenes schema, if a sentence lacks a truth value (is indeterminate), then the negation of that se ntence also lacks a truth value. If negation is interpreted as choice negation, then the principle of falsity clearly holds, because if ~A is true (the last row), then A is fals e. It can also be noticed that classical logic does not hold. For instance, the principle of the excluded middle is viol ated. If a sentence is neither true nor false, then neither it, nor its negation holds.19 On the other hand, all assump tions involved in Martins argument are consistent with this interpretation of negation. Thus, it is natural to think that Martin interpreted negation as choice negation. Nevertheless, it has been claimed that choice negation is not the unique way to interpret negation and, moreover, that it does not reflec t the way negation func tions in English. The alternative interpretation is called exclusion negation and is characterized by a truth-table (Table 2) which assigns truth to the negation of an inde terminate sentence. If negation is interpreted as exclusion negation, then the princi ple of falsity does not hold (to be more specific, it does not hold if there are more than two possibilities for a sentence). One cannot in fer that a sentence is false from the fact that its negation is true. This is shown by the second row of the corresponding truth table: the negation of the se ntence is true, but the sentence itself is indeterminate. What this means is that if negation is interpreted as exclusion negation, one can no longer run a Martinstyle Liar argument for the Simple Liar. Either Bivalence holds, in which case one certainly can run a Liar argument (however, th is was not a matter of controvers y), or it does not hold, in which 19 The principle of the excluded middle might hold under some non-standard semantics for the logical connectives. For instance, it might be true under a supervaluationist semantics.

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33 case (F) is no longer available. To put things differently, if nega tion is interpreted as exclusion negation, (F) entails Bivalence. Thus, if negation is exclusion negation, Ma rtins argument fails to support the claim that the Simple Liar remains problematic in the absen ce of Bivalence. He is not allowed to use (F) as a premise, since, toge ther with exclusion negation (one does not even need to assume classical logic) it entails Bivalence and, thus, ma kes a neither true nor false account impossible. The question is which of the two alternative interpretations of negation corresponds to how negation works in English? I think that an analysis of the ma in kinds of English sentences that are candidates for being neither true nor false shows that negation functions in accordance with Kleenes three-valued schema. Am ong the sentences that are candidates for being neither true nor false one normally counts sentences containing vacuous names, sentences involving category mistakes, sentences containing vague predicates, moral sentences (according to some views in metaethics), etc. If one takes sent ences of this sort to be gappy, it is hard to see what reasons one could have to take their negations to be true, rather than gappy. Normally, a sentence that lacks a truth value is said to be meaningless, or to be meaningful but to fail to express a proposition. If negation were interpreted as ex clusion negation, then the nega tion of a meaningless sentence would have to become meaningful (and true); moreover, the negation of a sentence that fails to express a proposition would have to express a true proposition20. Thus, the view that English negation is exclusion negation is contra ry to our intuitions. Keith Simmons21 tries to defend the claim that English negation is exclusion negati on by saying that if one thinks that (S) is meaningless, from (S) is meaningless one wants to infer (S) is not true. This is possible, 20 The difficulty is more vivid in the case of moral sentence s. If a moral sentence is taken to express an emotion, its negation would have to express a true proposition. 21 Simmons [1993: 54].

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34 according to him, only if not is understood as exclusion negation, because not true, in the last mentioned sentence, is not equivalent to false. It is hard to see how this non-equivalence can justify interpreting not as ex clusion negation. Contrary to wh at Simmons suggests, choice negation does not take false to be equivalent to not true; it takes the claim that a certain sentence is false to be equivalent to the claim that its negation is true. The current understanding of the distinction between choi ce negation and exclusion negation originates with van Fraassen: Following Mannoury though narrowing his m eaning somewhat we can draw the following distinction: (a) choice negation: (not-A) is true (respectively, false) if and only if A is false (respectively, true); (b) exclusion negation: (not-A) is true if a nd only if A is not true, and false otherwise. Of course if the principle of bivalence holds (t hat A is always either true or false), then the distinction collapses. [van Fraassen 1969:69] Throughout his paper van Fraassen us es only choice negation, as th e most natural interpretation of negation, and mentions exclusive negation mere ly as an alternative that is used in the literature. It is interesting to notice that the meaning of exclusion negation has not only been somewhat narrowed, but significantly changed and that there are few things in common between Mannourys distinction between ex clusion negation and choice nega tion and the distinction that is introduced by van Fraassen and has become fa miliar in the more recent literature. Mannoury22 is concerned with offering an account of those el ements of mathematical thinking that are not given in sensory intuition and cannot be derived from it. In particular, notions such as infinity or parallelism cannot be given in sensory intuiti on. Mannourys explanation of how we can acquire this kind of notions involves exclusion negation which is di stinguished from choice negation. Beth explains the distinction between the two kinds of negation as follows: 22 The discussion on Mannourys views of negation is based on [Beth 1965].

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35 The choice negation presupposes a disjunction of possibilities; the negation of one of these possibilities then implies the assertion that one of the remaining possibilities is realized. The exclusion negation presupposes no such disj unction of possibilities; therefore it is, unlike the choice negation, incapab le of a positive interpretati on. It merely excludes one possibility without making any assertion rega rding possible alternat ives. According to Mannoury, the exclusion negation, more than choi ce negation, has an em otional character. [Beth, 1965: 20] It is essential for excl usion negation that there is an emoti onal component that is part of its meaning. This enables Mannoury to explain how th e notion of infinity can be grasped even though it cannot be given in empiri cal intuition. Infinity is the negation of the finite, but the negation must be understood as exclusive negation. Choice negation would require infinity to be the alternative possibility which is given indepe ndently of the act of negation. This would be impossible. We acquire the notion of infinity by an (emotional) rejection of the finite. One can see that the difference between choice and ex clusion negation made by Mannoury has nothing to do with how negation is in terpreted when there are truth-valu e gaps. The distinction would apply equally well to bivalent a nd to non-bivalent systems. Although the two types of negati on are often presented as the two main competitors in the literature, it is only choice nega tion that has been employed as th e natural interpretation. Kleene, for instance, interprets negation as choice ne gation in his systems of 3-valued logics.23 Parsons even went as far as to deny that there could be languages whose negation is exclusion negation.24 One can infer that an attempt to block Martin s argument by denying (F) fails because it would depend on an interpretation of negation that does not have much in common with English negation. 23 Exclusion negation would count for Kleene [1952] as an ir regular connective (regular connectives are such that a given column (row) contains t in the u row (column), only if the column (row) consists entirely of ts; and likewise for f [1952: 334]. He was not interest ed in irregular connectives, because they are not partial recu rsive operations. 24 Parsons [1984].

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36 The Principle of Bivalence and Classical Logic The objectio n Beall and Bueno raise against th e thesis that the Simple Liar remains paradoxical even in the absence of the principle of bivalence is not that Martins argument is invalid, but that his assumptions are too strong. In order for Martins thesis to be justified, it is not enough to show that a contradiction can be derived without appeali ng to Bivalence. One needs to show that the contradiction can be derived from premises that do not entail Bivalence. The problem is that, as Beall and Bueno showed by a nine step proof, (T) and (F), given classical propositional logic (CPL), entail Bivalence.25 I will argue that neither Martins Simple Liar argument nor the Strengthened Liar argument presuppose classical logic. The contradic tion can be obtained from principles that are weaker than classical logic. In particular, the argumentati on strategy employed by Beall and Bueno is misguided because they assume that classical logic is a presupposition of Martins argument. It is true that if one wants to give an account of the Liar paradox by saying that the Simple Liar is neither true nor false, one should not be commi tted to principl es that entail Bivalence. Nevertheless, there is nothing in what Martin says that suggests a commitment to classical logic. It is true that the principles of classical logic have at leas t prima facie plausibility and that they cannot be rejected unless there is some pressure in this direction. However, a neither true nor false account of the Liar paradoxes usually come s together with the denial of 25 Here is the proof, quoted from Beall and Bueno [2002: 24] (I formulated the pr oof in a footnote only, because it is not its validity that I am concerned with). 0. T
A [T] 1. ~T ~A [0, MTT] 2. T<~A> ~A [T] 3. ~T T<~A> [1,2, Transitivity] 4. ~T [Premiss, for CP] 5. T<~A> [3,4, MPP] 6. T<~A> F [F] 7. F [5,6, MPP] 8. ~T F [4,7, CP] 9. T V F [8, CPL equivalence and DNE]

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37 classical logic.26 One reason to deny classical logic when th e principle of bivalence is rejected is offered by Alfred Tarski who ar gues that (T) and classical logi c alone imply a version of the principle of bivalence. we can deduce from our definition [Tarskis de finition of truth] various laws of a general nature. In particular, we can prove with its he lp the laws of contradiction and of excluded middle These semantic laws should not be identified with the related logical laws of contradiction and excluded middle ; the latt er belong to the senten tial calculus and do not involve the term true at all. [Tarski 1944: 354] Tarskis point is that his definition (in partic ular, the T-biconditionals that are entailed by his definition), together with classical logic, are enough to prove the semantic laws of noncontradiction and excluded middle. More precisel y, given the (logical) pr inciple of the excluded middle (EM) A V ~A and the following two biconditionals (T) T(A) A (T*) T(~A) ~A one can infer (SEM) T(A) V T(~A) In fact, only the right-to-left di rections of (T) and (T*) are needed to infer (SEM). (SEM) expresses the semantic law of the excluded middle. This is taken by Tarski to be one of the laws which are so characteristic of the Aristotelian conception of truth and is considered by many to be one way to express the principle of bivalence.27 If (SEM) is accepted as an alternative 26 One should not be misled by the fact that Kripke thou ght acceptance of truth value gaps to be compatible with classical logic. Classical logic was understood by him to be a set of principles that apply to propositions, while here it is supposed to apply to sentences. See Kripke [1984: 64-5, fn. 19]. 27 See McGee [1990: 104 & 179] and Gupta and Belnap [1993: 224].

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38 formulation of Bivalence, then this already show s that those who are not taken in by Bivalence (to use van Fraassens formula quoted by Beall and Bueno28) must reject classical logic. Certainly, Tarskis remarks fall short of showing that classical logic and (T) entail Bivalence, because (SEM) is different from Bivalence. In or der to derive Bivalence, one needs, in addition to classical logic and (T), a principle connecting the predicates of truth and falsity. One principle that would do the trick is (F) itself, a principle wh ich is well supported by the way we think of truth, falsity and negation, and which Martin rightly appeals to in his argu ment. It can be noticed that Bivalence follows immediately from (SEM) and (F).29 It is unlikely that Martin was not aware that Bivalence follows from these princi ples. Given his explicit commitment to (F), as well as his commitment to the Truth Principle, it follows that he could not have been committed to (EM); this means that he could not have been committed to classical logic either.30 Reductio ad Absurdum versus Reductio ad Gapsurdum It rem ains possible to save the truth-valu e gap account by saying that although Martins argument and the strengthened Liar argument do not assume classical logic, they are nonetheless using laws of logic that are strong enough to entail the principle of Biva lence. Notice that the mere denial of classical logic is not enough to save Martins argument from troubles. One could 28 Beall and Bueno [2002: 23]. 29 Thus, one can formulate a shorter proof meant to prove the same thing as the nine step proof offered by Beall and Bueno: 1. A V ~A (EM) 2. A T(A) (T) 3. ~A T(~A) (T*) 4. (A T(A)) & (~A T(~A)) (Addition) 5. T(A) V T(~A) (Constructive dilemma) 6. T(A) V F(A) by (F*) 30 It is possible to reject bivalence and keep classical l ogic, as van Fraassens work on presuppositions illustrates. However, van Fraassen was able to do this because he deni ed the T-schema. See the end of this chapter for more details about this approach.

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39 argue that, even though he is not ex plicitly committed to classical l ogic, the logical principles he appeals to in his argument entail, together with (T ) and (F), the principle of bivalence. Martin did not make all principles of logic used in his argument explicit, so there are more ways to reconstruct the argument. One quite plausible way to reconstruct it would involve an application of reductio ad absurdum: 1. (L0) is false. Assumption 2. (L0) is false is true. from (T) 3. (L0) is true from 2 and (Intersubstitutivity) 4. (L0) is not false. from 3 and (SNC) 5. (L0) is false and (L0) is not false. from 1 and 4 6. (L0) is not false RA 7. (L0) is not false is true from 6 and (T) 8. (L0) is false is false from 7 and (F) 9. (L0) is false from 8 and (Intersubstitutivity) 10. (L0) is false and (L0) is not false from 6 and 9 Line (6) in this argument is inferred by an applica tion of reductio ad absurdum (RA). It is true that a commitment to RA does not presuppose a commitme nt to classical logic, or, at least, to the principle of the excluded middle. In intuitionist logic, for instan ce, the principle of the excluded middle fails, even though RA is valid.31 Nevertheless, the appeal to RA remains problematic in a context in which one drops Bivalence and allows se ntences that are neither true nor false. If a certain assumption leads to a contradiction, it se ems illegitimate to infer the negation of that assumption, because it might happen that both th at assumption and its negation lack a truth value. It is natural for the defender of a truth-valu e gap view to also reject RA. It is important to notice that this is a point that applies equally well to the Simple and the Strengthened Liar 31 I am indebted to Graham Priest for pointing this out to me.

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40 arguments because both involve applications of RA. Thus, a Liar sentence is problematic (in a context in which truth-value gaps are allowed) only if an inconsistency can be derived from even weaker principles of logic. In particular, it should be derivable from principles that do not include EM or RA. I think that what makes the inconsistency ha rd to avoid is our commitment to some semantic principles rooted in the intuitions we ha ve about the use of true and false which help one run a Liar argument from principles of logi c that do not include EM and RA. One rule that can replace RA in the Liar argument is a rule that allows one to infer from the fact that a certain assumption leads to a contradiction the claim that the assumption is not true; I will call this rule reductio ad gapsurdum (RG).32 If reductio ad absurdum sanctions as valid inferences of the form A Assumption B & ~B ~A RA reductio ad gapsurdum sanctions as valid inferences of the following form: A Assumption B & ~B ~T(A) RG The new rule accommodates the possi bility that the assumption that leads to a contradiction is a gappy sentence (neither true nor false). Whether a se ntence is false or gappy, we still want to say that it is not true. Given that the truth predicate plays an essential role in RG, this is not a logical rule of inference, but rather a semantic principl e. In fact, RG can be seen as a consequence of one of two semantic rules that are rooted in the way we think of the predicates of truth and 32 The expression occurred during a conversation with Greg Ray.

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41 falsity. The two rules can be called the T-closing-off33 rule (TC) and the F-closing-off rule (FC) and they apply as follows: TC-rule: T(A) Assumption B & ~B ~T(A) TC FC-rule: F(A) Assumption B & ~B ~F(A) FC TC and FC can be thought of as restrictions of RA to attributi ons of truth and falsity. However, they are semantic rules of in ference, because the truth-pres ervation is not guaranteed by the meaning of logical terms alone; one also needs to take the meanings of the truth and falsity predicates into account. The new version of the Streng thened Liar argument would be34: 1. (L) is not true is true iff (L) is not true [from (T)] 2. (L) is true iff (L) is not true [from (1) and (Subst)] 3. (L) is true assumption 4. If (L) is true, then (L) is not true Mat. Equiv, 2 5. (L) is not true MP,3,4 6. (L) is true and (L) is not true Conj, 3,5 33 The name is inspired by Kripkes closing off locution, used to refer to an interpretation of the truth predicate according to which if a sentence is gappy, then the sentence that attr ibutes truth to it is false, and its negation is true. See Kripke [1984: 80-81]. 34 Revenge Liar argument for the Strengthened Liar (appeals to TEM instead of EM): 1. (L) is not true is true iff (L) is not true [from (T)] 2. (L) is true iff (L) is not true [from (1) and (Subst)] 3. Either [(L) is true and (L) is not tr ue] or [~((L) is true) and ~((L) is not true)] Mat.Equiv.,2 4. Either [(L) is true and (L) is not true] or ~[(L) is true or (L) is not true] DeMorgans, 3 5. ~~[(L) is true or (L) is not true] DN, TEM 6. (L) is true and (L) is not true DS, 4,5

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42 7. (L) is trueis not true RG 3,6 (Reductio ad gapsurdum) 8. (L) is true is true iff (L) is true [from (T)] 9. (L) is true is not true iff (L) is not true Contraposition, 8 10. (L) is not true [from 7 and 9] 11. If (L) is not true, then (L) is true [from 2] 12. (L) is true MP,10,11 13. (L) is true and (L) is not true Conj., 10,12 One can also reformulate Martins argument such that it no longer appeals to RA, but rather to FC: 1. (L) is false. Assumption 2. (L) is false is true. from (T) 3. (L) is true from (2) and (Intersubstitutivity) 4. (L) is not false. from (3) and (SNC) 5. (L) is false and (L) is not false from (1) and (4) 6. (L) is not false FC 7. (L) is not false is true from (6) and (T) 8. (L) is false is false from (7) and (F) 9. (L) is false from (8) and (Intersubstitutivity) 10. (L) is false and (L) is not false from (6) and (9) In fact, the only difference between the former a nd the latter version of the argument is that line 6 is this time inferred by an application of FC instead of RA. Do we have adequate reason to think that TC and FC hold? I thi nk that not only closing-off rules of inference, but also closingoff principles like TC* T(A) V ~T(A) [every sentence is either true or not true] FC* F(A) V ~F(A) [every sentence is either false or not false]

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43 are grounded in the way we think of the concepts of truth and falsity. Unlik e bald or smidget, the predicates of truth and falsity are neither vague35 nor only partially defined.36 Of course, when TC* and FC* are held to be true, it is assu med that English is able to express the closingoff principles. Their denial would conflict w ith the thesis that English is universal. Thus, as long as one is committed to the universality of English, both the Simple and the Strengthened Liar allow one to construct a parad oxical argument from very weak premises that do not entail the prin ciple of bivalence. The T-Schema Another way one could try to bloc k the Lia r argument is to reje ct the T-schema. In fact, it is well known that the schema cannot work in its full generality. Ther e are context sensitive sentences that require some ad justment to the general schema. There are also good reasons to think that the schema, in partic ular the right-to-lef t direction does not hold for gappy sentences.37 The antecedent of the right-to-left conditional is gappy, while the consequent is false. Even though there might be weak systems of logic th at would still make these instances of the conditional true, it seems that it would be more plausible to say th at they are not true. There are two reasons why denying the T-schema is not very appealing. First of all, even though some instances of (T) might not be true, this does not mean that the principle does not have initial plausibility, at least when it is applied to sentences that are not semantically defective. There is no good reason to think that Liar sentences are sema ntically defective, other than the fact that 35 There have been attempts to treat true as a vague pr edicate. See, for instance, McGee [1990]. I think however that it would be a mistake to think that truth inherits th e vagueness of the vague predicates of English. A sentence like Harry is bald is true, where Harry is a borderline case, is more appropri ately treated as false rather than as neither true nor false. 36 Soames [1990: 164] introduced smidget as an example of a partially defined predicate. Its meaning is defined, roughly, by the following stipulative clauses: i) smidget applies to all adults less than four feet tall; ii) smidget fails to apply to any adults more than five feet tall; iii) smidget fails to apply to nonadults. 37 I am grateful to Kirk Ludwig for bringing this possibility to my attention.

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44 they are paradoxical. Second, some weaker principles suffice to enable one to run a Liar argument. I made the assumption (granted by Beall and Bueno) that Martin is committed to the T-schema, even though this commitment is not made explicit in his argument. However, a couple of paragraphs before, Martin takes the Truth Principle, repeated here, (Truth Principle) A sentence is true if and only if, what it says is the case. to be part of the assumptions that make the Li ar arguments possible. I think Martin took it to entail that the instances of (T) are true.38 Anyway, when he runs the Liar arguments, Martin in fact employs a rule of inferen ce that is weaker than (T): (R1) S says that A (R2) A (R3) Therefore, S is true. This rule of inference holds ev en if A is replaced by a gappy se ntence, and is enough to run the Simple Liar argument. One account that denies the T-schema (although it remains committed to some slightly weaker rules of inferences) is the presuppositional account proposed by Bas van Fraassen. van Fraassen denies the principle of bivalence, but he wants to stay committed to the principles of classical logic, including the principle of excluded middle; he is also committed to the principle of falsity, although he only thinks of it as a more convenient way to think of negation, truth and falsity. Inspired by Strawsons work on presuppo sitions, van Fraassen develops a semantic account according to which some sentences have pr esuppositions that need to be met in order for them to be true or false. He is able to reject bivalence and preserve classical logic because he 38 One can add some restrictions to exclude context sensitiv e sentences. The Simple Liar will not be affected by a restriction of this sort.

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45 denies the T-schema. Although the T-schema does not hold, there are some ot her principles that capture the intuitions speakers have w ith respect to the meaning of true: The argument from A to It is true that A is of course valid, as is the converse argument. But we shall in general reject the material equivalence of the two sentences, in order to block the inference from (A or not-A ) to (True
or True ), because the former is valid even when neither A nor not-A is true.39 Thus, the T-schema fails, because the rightto-left direction fails. Nevertheless, the following two inferences are valid: A Therefore, T. and T Therefore, A Paradoxical arguments would have the form: T; hence A; hence ~T ~T; hence A; hence T The Simple Liar paradox is solved by the reject ion of bivalence. Since both it and its negation entail a contradiction, it follows that the Simple Liar presupposes a contradiction. Given that its presupposition fails, the Simple Liar would have to be neither true nor false. There is a lesson that needs to be drawn from the Simple Liar: This I shall call the basic lesson of the Liar paradox: even asse rtions of truth or falsity do not in general satisfy the law of bivalence. [van Fraassen 1968: 148] 39 van Fraassen [1978: 15].

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46 The Strengthened Liar teaches an additional less on. Not only would the Strengthened Liar have to be gappy, but also T<(L)> because T<(L)> presupposes (L). One would have to deny not only bivalence, but also a principle such as: (*) T is true, or T<~Y> is true, or ~T & ~T<~Y> is true. Thus, one thing the Strenghtened Liar paradox sh ows is that although (L) is neither true nor false, the truth or non-truth of (L) is not expres sed by T<(L)>, but by some other true sentence of the language, ~T>. Although T<(L)> is gappy, the sentence T> can have a truth-value (in particular, it is false, and ~T > is true). Thus, van Fraassen links the Strengthened Liar with the issue of expressibility: some truths about (L) fail to be expressed by the sentence that appears to have been designe d for this job, but is expressed by some other sentence. Moreover, the problem can be generaliz ed. There are Strengthened Strengthened Liars (such as ~T>, Strengthene d Strengthened Strengt hened Liars, etc. van Fraassen manages to argue that there are also infi nitely paradoxical sentences, that is, sentences whose truth or nontruth cannot be expressed by a ny sentence in the language. Thus, the solution to the Liar paradoxes is possible by acknowledgi ng some limits to the expre ssive power of the language. I think that this is a fruitful idea, although van Fraassen did not show how his remarks can be applied to natural languages. The proofs he constructs involve fo rmal languages, but he wants the results to hold for natural languages too, because he is interested in those formal languages that mirror the relevant f eatures on natural languages: A solution to the semantic paradoxes should pres umably have two distin guishable parts: an analysis of the logically relevant features of the paradoxes as stated in natural language, and a formal construction in which correspond ing sentences play ro les roughly similar to those which our analysis ascribes to the pa radoxical statements. [van Fraassen 1978: 13] If one tries to extend van Fraassen solution to natural languages, one would notice that it conflicts with the intuitions according to whic h natural languages are universal. van Fraassen

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47 suggests that even though ~T<(L)> fails to expr ess the non-truth of the Liar, ~T> might very well express it. Nevertheless, the latter could only express the non-truth of the Liar in a metaphorical sense, because it actua lly expresses the different though t: that the sentence that (L) is true is non-true. Thus, if th e view is to be applied to Engl ish, one consequence of it would be that the non-truth of the Strengthe ned Liar is inexpressible in E nglish. This would be in conflict with the universality of English. Universality and Inconsistency The universality of English should be acknowle dged as the m ain obstacle against offering a solution to the paradoxes. It is the universality that is attr ibuted to natural languages that precludes a solution in terms of some type restrictions, or by distinguishing between objectlanguage and metalanguage; it is again universality that makes it difficult to solve the paradoxes by denying the principle of bivalence, the ex cluded middle or reductio ad absurdum; the universality makes the account prop osed by van Fraassen difficult to apply to natural languages. It seems that as long as natural languages are considered universal, there is no way to avoid the inconsistencies generated by the Liar para doxes. Keith Simmons [1993] believes that paradoxes should be understood as a conflict between the possibility of running diagonal arguments that lead to a contradi ction and the universality of the language. Thus, there appear to be two main alternatives that one has to consider: either English is universal, in which case there appears to be no other option than to endorse one version or another of an inconsistency view of truth, or one tries to preserve c onsistency by denying the universal ity of English. I chapter three I explore and endorse the latter alte rnative. The universal ity of English has normally been taken as too obvious to deserve any special attention. Nevert heless, I argue that En glish is not universal, and that the Liar paradoxes are best solved by dropping the assump tion that truth is expressible in English. Chapter four is concerned with the fo rmer alternative, and I argue that universality

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48 cannot be saved by endorsing an in consistency view of truth beca use an inconsistency view of truth which remains committed to the law of non-cont radiction entails that truth is inexpressible. Table 1-1. Table 1-2. A ~A T I F F I T A ~A T I F F T T

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49 CHAPTER 3 THE INEXPRESSIBILITY OF TRUTH In the previous chapter I argued that the main strategies that have been used to avoid the inconsistency generated by Liar paradoxes conflict with the thesis that natural languages are universal. Therefore, one is faced with a choice: e ither to adopt an inconsistency view of truth or to deny the universality of English. Most philosophe rs have taken the univer sality of English as granted and tried to find ways to live with inconsistencies. The le sson that they have drawn from the Liar paradoxes is either that English is inconsistent [Tarski 1933], or th at the postulates that give the meaning of true are inconsistent [Chihara 1979], or, maybe, that there are true contradictions [Priest 1998]. I will examine the inconsistency view of truth in chapter four. The purpose of this chapter is to explore and defend the thesis that what the Liar paradox shows is that natural languages are, appearances to the contrary notwithstanding, not semantically universal. In particular, I argue that truth is inexpressible in English. This is established by an argument that can be adequately characterized in English and wh ich consists in showing that the supposition that truth is expressible in English leads to a contradiction. The main challenge for the inexpressibility vi ew is to answer some objections against it that may readily come to mind. It might be objected that the expressibility of a concept is quite easy to secure, because it is only a matter of wh at intentions speakers in a linguistic community have with respect to the use of a predicate. It has also been object ed that the inexpressibility view is self-defeating, because in defe nding the view one actually expresse s the concept that is held to be inexpressible. Moreover, one may worry that if true fails to expre ss in English the concept of truth, then it would be hard to explain the fact that non-problem atic sentences in which true

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50 is used are successfully deployed in communication. I will argue that all these challenges can be met if one takes into account the distinction between intended meaning and linguistic meaning. Universality and Expressibility To see how universality and expressibility ar e involved in the sem antic paradoxes, one needs a more precise characteri zation of these notions Although it is a widely held view that English is universal, the characterization of universality has remained vague. The strongest version of universality one can think of is the following: (U-universality) A language is U-universal if and only if every concept/thought is expressible in it1. One standard objection against the thesis that En glish is U-universal is that English has only countably many expressions, while there are unc ountably many concepts. There are some more or less satisfactory answ ers to this objection2, but I will set these debates aside, because Uuniversality is needlessly strong fo r my purposes in this chapter. Tarski argues for a slightly different universality thesis: A characteristic feature of coll oquial language (in contrast to various scientific languages) is its universality. It would not be in harmony with the spirit of this language if in some other language a word occurred which could not be translated into it; it could be claimed that if we can speak meaningfully about anythi ng at all, we can also speak about it in colloquial language. [Tarski 1956: 164] He never offered a precise definition of universality. Neverthele ss, it is plausible that the following notion of universality comes pre tty close to what he had in mind: 1 I take thoughts to be the kind of entities that are the contents of declarative sentences, regardless of whether they are conceived as propositions, Fregean thoughts, or in some other way. Concepts should be understood as constituents of thoughts. 2 Although only countably many concepts can be expressed in English, there are uncountably many that can be expressed in some extension or other of English (but no single extension can express more than countably many concepts).

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51 (T-universality) An interpreted language3, L, is T-universal i ff for every interpreted language, L, and for every meaningful expression, E, in L, there is an expression, E, in L, such that E in L has the same meaning as E in L. It is universality in this sens e that is presumably taken by Tars ki to guarantee that colloquial languages must contain their own semantic predicat es (in particular, English must be able to express the concept of a true sentence of English). T-universality is in one respect unnecessar ily strong and in another respect too weak4. It is too strong because English does not have to be able to express all that can be expressed in other languages in order for there to be an obstacle to reaching a solution to the semantic paradoxes. All that is needed is that English express its ow n semantic concepts. T-un iversality is also too weak, because the thesis that E nglish is universal is relevant to the problem of semantic paradoxes only if it entails that the concept of a true sentence of English is expressible in English. The thesis that English is T-universal does not by itself entail that English is able to express this concept. However, it entails it in conjunction with the additional thesis that there is a language that is able to express th e concept of a true sentence of English. This additional thesis is true if the following principl e of expressibility advocated by Searle [1969: 20] holds: (PE) For any meaning X and any speaker S, whenever S means (intends to convey, wishes to communicate in an utterance, et c.) X, then it is possible that there is some expression E such that E is an ex act expression of or formulation of X. 3 A language counts as interpreted only if besides the syntax rules, there are also semantic rules that determine the meaning of its expressions. Natural languages are interpreted languages. 4 Similar things can be said about U-universality.

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52 Note that X here cannot be understood as the m eaning of a certain expression (this would make the principle trivial5), but rather as a content (such as a concept or proposit ion) that can be characterized independently of a ny linguistic entity. If we can gr asp the semantic concepts of English, (PE) guarantees that there is some la nguage in which they are expressible, which together with the T-universality of English woul d guarantee that they are also expressible in English. For the purposes of this study it would be enough to focus on the following universality notion: (S-universality) A language is semantically uni versal (S-universal) if and only if all its semantic concepts are expressible in it. On the one hand, S-universality is more restricted than T-universality and, on the other, one does not need a commitment to (PE) in order for the S-universality of a language to entail that it is able to express its own semantic notions. If English is S-universal, it would have to be able to express all its semantic concepts.6 Notice that the noti on of semantic univers ality (as well as the other notions of universality) is defined in terms of the notion of expressibility, which is often assumed to be clear enough and not to require explicit characterization. However, it is important 5 I am indebted to Michael Jubien for this remark. 6 Martin [1976] distinguishes between two notions of univers ality: universality in the sense of Tarski (characterized in a way similar to T-universality a bove) and universality in the sense of Fitch, which is characterized as a languages capability to say much, if not everything, ab out every language, including itself [Martin 1976: 274]. Alternatively, languages are characterized as universal in the sense of Fitch if they can be used to talk about all languages, including themselves, and in particular to expres s much, if not all, of their own semantic theories [1976: 271]. I prefer to use S-universality for two reasons. One is th at Martins notion of universality in the sense of Fitch is not precise enough, the other is that it is not at all clear th at Fitch intended to offer a notion of universality different from universality in the sense of Tarski. Fitch [1964] argues that there are universal formal languages. In order to show this, he argues that such a language would be able to say everything there is to say about its own semantics. This does not mean that he works with a notion of universa lity different from that used by Tarski; it means only that Fitch believes that the difficulties raised by the semantic co ncepts can be overcome and, consequently, that there are universal languages in the sense of Tarski.

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53 to clarify what expressibility means, because it is often understood in different ways. The first step in characterizing expressibility is this: (Ec) A concept is expressible in L if and only if there is a predicate of L that (given an appropriate context) expresses it. (Et) A thought is expressible in L if and only if there is a sentence of L that (given an appropriate context) expresses it.7 The next step would be to offer a (non-extensiona list) characterization of expressions of the form __ expresses __ in L. This can be done as follows: (EC) P expresses C in L (in context c) if an d only if C is the concept determined by the meaning of P in L (and context c). (ET) S expresses T in L (in context c) if and only if T is the thought determined by the meaning of S in L (and context c).8 The relation between the meaning of an expression and the concept or thought expressed by it can be conceived of in various ways depending on ones favorite vi ew of semantics. Some views might fail to draw a distinction between the mean ing of an expression an d the concept or thought expressed by that expression. In this case, the concept or the thought determined by (associated with) a certain meaning is the meaning itself. Other semantic views would distinguish between the meaning of a predicate and the concept expres sed by that predicate. Similarly, there would be a distinction between the meaning of a senten ce and the thought expressed by that sentence.9 7 I characterized expressibility as a relation between thought s/concepts and languages. It is not difficult to extend this to a relation between beliefs (or thoughts understood as mental representations) and languages. A sentence would be said to express a belief if it expresses the content of that belief. 8 (ET) is supposed to make more explicit the intuition that S expresses T in L if and only if T is what S says. 9 Note, however, that I take concepts here to be constitu ents of thoughts that correspond to predicates. Fregean semantics distinguishes between the sense of a predicate an d the concept denoted by it. The sense of a predicate is the mode of presentation of a concept (the referent of the predicate). Fregean concepts are semantic values of

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54 Regardless of how exactly the re lation between meanings and concepts is understood, (EC) puts one in a position to formulate an adequacy condition that needs to be met in order for a predicate to express a certain concept.10 If one thinks of a concept as being associated with a set of application rules, then the Expressive Ade quacy condition would requ ire that the concept determined by the meaning of the predicate is the concept govern ed by the corresponding application rules. More specificall y, if P is a predicate of L and C a concept, then P expresses C in L only if the following condition is met: (Expressive Adequacy) For any a, P applies in L to a iff the application rules for C entail that a falls under C. In particular, in order for a predicate of Englis h to express the concept of truth, the Expressive Adequacy condition must be met. This is no rmally understood as an extensional adequacy condition, but I will argue that it is more general because it must also be satisfied when there is no extension associated with the concept. There are two difficulties that might suggest that there is something wrong with this way of thinking of expressibility because it would fail to provide a satisf actory criterion to measure the expressive power of a language: th e first has to do with contextsensitive expressions, the second with the fact that natu ral languages do not have a well-define d syntax and semantics. I will argue that none of them poses a serious threat to my project. predicates. If the concept is understood as the semantic value of a predicate, then it cannot be part of the thought expressed by a sentence containing that predicate, because the thought is a co mposite of senses, not of referents. Since I take concepts to be parts of thoughts, they should be identified with Fregean senses of predicates rather than with Fregean concepts. What an expre ssion expresses is a sense or a mode of presentation, and it does express it when its meaning determines the mode of presentation. 10 For reasons of simplicity, I omit cont ext sensitivity. I will assume that the semantic predicates are not context sensitive expressions.

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55 Consider first the case of context-sensitive expressions. It may seem that the presence of demonstratives or indexicals in a language makes that language universal in a very strong sense. Hofweber [2006] distinguishes between two kinds of expressibility: langua ge expressibility and loose speaker expressibility.11 Language expressibility has to do with what can be expressed by using only the context-insensitiv e expressions of a language. Eng lish is not universal in this sense, because there are things that are not language expressible in English. Nevertheless, Hofweber argues that every natural language is un iversal in the sense that everything is loosely speaker expressible in it. Someth ing is loosely speaker expressibl e in a language, if it can be expressed by using either a context-sensitive or a context-insensitive expression of that language. Hofweber uses as an example the property of ta sting better than Diet Pepsi, which although not language expressible in Ancient Gr eek, is, according to him, loosel y speaker expressible in it. It can be expressed by the Ancient Greek equivalent of (1) tasting better than this. in a context in which there is Diet Pepsi in front of the speaker. However, Hofwebers example does not succeed in showing that ev erything is loosely speaker expressible in Ancient Gree k (or in any other natural language).12 The addition of demonstrative expressions does indeed incr ease the expressive power of a language. Nevertheless, what referring de vices contribute to the thought expressed by a sentence is 11 Hofweber is concerned with expre ssibility of properties, but the discussion can be rephrased in terms of expressibility of concepts. Moreover, his distinction can be extended to account for the expressibility of propositions or thoughts. 12 There are other problems for Hofwebers argument, in addition to the objection formulated above. It is not at all obvious that what tasting better than Diet Pepsi expresse s is the same as what is sa id by using the Ancient Greek equivalent of (1) uttered in the appropriate context. One problem is that Diet Pepsi works as a mass term rather than as a singular referring term. Even if it is assumed that it plays the same role as a demonstrative, the thoughts expressed are not necessarily the same. They are different on a Fregean view of sema ntics. The thoughts are the same if all that Diet Pepsi and this contri bute to the thought expressed is the referent.

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56 different from the contribution made by predicative devices. In particular, a distinction needs to be drawn between a concept being expressible in a language and a concept being denotable in a language. Given any concept, one can certainly say something about it by uttering (2) (2) That is a concept. in a context in which the demonstrative refers to that concept. However, the fact that a certain referring expression of English denotes a certain concept does not gua rantee that that concept is expressible in English.13 Thus, the fact that a certain la nguage contains context-sensitive expressions by no means shows that everything is loosely speaker expr essible in it; in particular, it fails to show that the langua ge is semantically universal. The other difficulty consists in the fact that natural languages do not have a well-defined syntax and semantics. Expressions that did not belong to English a century ago are frequently used today, and expressions that do not belong to English today will be added to it in the future. Thus, the question whether a certain concept is e xpressible in English may not have a definite answer. The issue of what is and what is not ex pressible in a certain na tural language is indeed very imprecise, but I think that the imprecision is due to the fact that natural languages are not precisely defined, rather than to some defect in th e characterization of expre ssibility. If there is a need to make things more precise, one can replace the question whether a certain concept is 13 Denotability is much easier to secure than expressibility. Although English does not have enough names for all concepts, one could argue that any concept whatsoever has a name in some extension of English. Moreover, any concept can be referred to by using a demonstrative in some appropriate context. Similarly, there may be thoughts which cannot be expressed in a certain language, even if they can be denoted by some expression of that language. Nevertheless, it is one thing for an expression to denote a certain concept (or thought) in L, and quite another for it to express that concept (or thought) in L. To show th at everything is loosely spea ker expressible in a certain language one should also prove that it contains context sensitive predicative expressions that are able to express (rather than denote) any concept, given the appropriate co ntext. Although there is room for arguing that there are context-sensitive predicative expressions, such as tall (e ven true is a candidate, according to Burge [1984] and Simmons [1993]) it is hard to think of one that could expr ess any concept whatsoever give n the appropriate context.

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57 expressible in English with the question wh ether it is expressible in English at t.14 Moreover, it turns out that the lack of a well-defined vocabulary has no bearing on whether semantic notions such as the concept of a true sentence of English are expressible in English. The Inexpressibility Argument Contrary to the widely he ld view that Englis h is semantically universal, I will argue that the concept of a true sentence of English is not ex pressible in it. This is I think, the lesson that should be drawn from the Liar paradoxes. Consider the following frequently used version of the Liar argument: 1. The following sentence (The Liar ) is a sentence of English: (L) (L) is not true. 2. (L) is not true is true if and only if (L) is not true.15 [instance of the T-schema] 3. (L) is true if and only if (L) is not true. [intersubstitutivity of identicals, 1, 2] 4. Either (L) is true or (L) is not true.16 [every sentence is either true or not true] 5. (L) is true and (L) is not true. [truth-functional consequence of 3 and 4] Notice that this argument (as well as many ot her versions of the Liar argument) makes no explicit appeal to the notions of universality or expressibility. Neverthele ss, it involves an implicit appeal to the assumption that true expresses the concept of truth. The instances of the 14 Notice that even if a language, L, survives the additio n of any new predicate, this does not mean that it is Suniversal. If one thinks of the syntax and semantics of L at different times as stages of L, then L could indeed be said to be universal in the sense that every semantic concept of L is expressible in some stage of L. Nevertheless, Suniversality is not guaranteed if it is und erstood as the thesis that every semantic concept of L at t is expressible in L at t. In addition to this, there is the possibility of concepts that cannot be expressed by any predicate of any language. 15 It may be objected that the T-schema does not hold for se ntences that are neither true nor false, and that (L) may be one of these gappy sentences. This move cannot prevent the paradox, because if (L) is neither true nor false, it would follow that it is not true. Therefore, (L) would have to be true after all. 16 Line 4 of the argument is not needed if the laws of classical logic hold. The fact that there is a semantic rule (rather than a rule of logic) that justifies 4 is importa nt because it shows that the Liar argument only needs very weak principles of logic.

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58 T-schema are held to be true because true is assumed to express the concept of truth. The assumption either has escaped notice, or has b een considered to be too obvious to need mentioning.17 Nevertheless, it needs to be acknowledge d as a premise in the Liar argument. Moreover, by making the premise explicit one becomes able to turn the Liar argument into an inexpressibility argument. More specifically, the Liar argument can be turned into a reductio argument whose conclusion is that true does not express the c oncept of truth. By generalizing the Liar argument one obtains th e following argument that shows that the concept of truth is inexpressible in English. The inexpressibility argument 1. Suppose that the concept of a true sentence of English is expressible in English and that T is a predicate of English that expresses this concept. 2. Let (L*) be (L*) is not T 3. The instances of the T-schema for predicate T hold.18 [from 1 and the Expressive Adequacy condition] 4. (L*) is not T is T if and only if (L*) is not T is true. [from 2 and 3]19 17 The assumption is more explicit in the informal version of the Liar argument presente d at the beginning of this chapter. A reformulation of the argument is this: 1. The following sentence (The Liar) is a sentence of English: (L) (L) is not true. 2. (L) expresses the thought that (L) is not true. [from 1 and the Expressive Adequacy condition] 3. (L) is true. [assumption] 4. (L) is not true. [from 2, 3 and the Descent-rule] 5. (L) is not true. [RA from 3 and 4] 6. (L) is true. [from 2, 5, and the Ascent-rule] 7. (L) is true and (L) is not true. [from 5 and 6] The assumption that true expresses the concept of truth is needed not only for (2), but also to justify the application of the two rules of inference (the Ascentand the Descent-rule). This version of the argument has the additional virtue that the possibility that (L) is gappy requires no special discussion. I have chosen the other version of the argument, because it is important to show that the expressibility assumption is made even in arguments that do not mention it explicitly. 18 One needs to add restrictions to the T-schema to account for semantically defective sentences.

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59 5. (L*) is T if and only if (L*) is not T is true. [from 2 and 4] 6. Either (L*) is true or (L*) is not true.20 [every sentence is either true or not true.] 7. Either (L*) is T is true or (L*) is not T is true. [from 6 and the T-schema] 8. (L*) is T and (L*) is not T is true. [truth-functiona l consequence of 5 and 7] 9. Therefore, there is no predicat e of English that expresses th e concept of a true sentence of English. [reductio ad absurdum] Generalizing a little more, one can prove that no language that meets some minimal expressivity requirements is semantically universal. The inexpressibility argument is quite stra ightforward and involves no sophisticated proof techniques. The main challenge for the inexpressi bility view is to answer the objections that might be (and have been) raised against it. Before answering the objections against the inexpressibility view of trut h I just sketched, I want to explain why my version of the inexpressibility argument is to be preferred to another version that is due to Hans Herzberger. Herzberger on Universality No Language contains all its Semantic Concepts Hans Herzberger [1970] challenges what he take s to be the thesis th at languages can be universal in the sense of Tarski. In fact, he argues only for the clai m that no language contains all its semantic concepts; he does not show that those concepts are expressible in other languages. Thus, although Herzberger takes himself to be arguing against Tarskis universality thesis, strictly speaking, he argues only th at no language is S-universal. Neve rtheless, if it is true that one can grasp semantic concepts such as the concep t of a true or of a grounded sentence of such 19 (1) may also be needed to infer (4), because it guarantees th at (L*) is not meaningless. The hypothesis that (L*) is gappy does not solve the problem, because it would follow that (L*) is not true. 20 Line 6 is not needed if the laws of classical logic hold (see also fn. 16). Moreover, if sentences such as that mentioned in 5 cannot be true, this would be enough for a reductio argument.

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60 a language, and if Searles principl e of expressibility holds, it would also follow that there are no T-universal languages. In particul ar, there would be semantic concepts that cannot be expressed in English. According to Herzberger, the semantic paradoxe s have to do with the limitations inherent in our languages, and they should be understood as analogous to the paradoxes of set theory. Both types of paradoxes are consequences of some very general theorems that can be established in the theory of relations. Herzberger proves th at formal languages of a certain sort cannot express all of their semantic concepts. The pr oof appeals to the tech nique of diagonalization, which has been made popular by Cantor and Gde l, and is more or less explicitly employed in most of the semantic paradoxes, as well as in the paradoxes of set-theory. Herzberger takes a concept to be inexpressible if the semantic rule s assign its extension to no terms of that system [Herzberger 1970: 159]. This does not mean that the concept is identified with its extension, but only that extensional adequacy is a necessary condition for a predicate to express a certain concept. The argument that L cannot express all of it s semantic concepts is formulated in a metalanguage and begins by specifying a class whic h is the extension of some semantic concept (the class of all grounded sentences of L, or the class of all true sentences of L). One then proves that that class cannot be represente d in L, in the sense that there is no predicate of L that has that class as its extension. The proof is conceived of as a reductio argument. Assuming that there is a predicate that has that class as its extension, diagonalization enables one to find an object for which it can be proved that it is a member of that class if and only if it is not a member of that class. The contradiction forces one to deny the assumption that the class can be represented in L. Herzberger is mainly concerned to show that no language is able to express all its grounding

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61 concepts. However, he argues that the same stra tegy can be extended to other concepts, such as the concepts of truth and satisfaction.21 Herzberger argues that the inexpr essibility result that he was able to obtain for relatively simple formal languages carries over to substant ively more complex languages, such as natural languages, because the increase in complexity has no essential bearing on what is needed to run an inexpressibility argument. The idea of using semantic paradoxes to prove th at there are some limits to the expressive power of any language is, as I have argued in the previous s ection, correct. Nevertheless, Herzbergers defense of the inexpressibility vi ew remains unsatisfactory. The problem with his strategy is that questions about the expressibility of a concept cannot be settled by investigating the expressibility of a class. Class Expressibility versus Concept Expressibility First of all, Herzberger assum es that a pred icate expresses a certain concept only if the extension of the predicate coincides with the extension of that con cept. However, this extensional adequacy requirement that, according to Herzberger, must be met by a predicate in order to express a concept is too strong: it assumes that there is a class of objects that fall under the concept (its extension). In pa rticular, for each language, L, he considers, Herzberger assumes that its semantic concepts do have an extension (the class of true sentences of L, the class of grounded sentences of L, etc.). Nevertheless, there may be conc epts that do not have an extension.22 Good candidates are the concep ts expressed by vague predicat es; it is not at all clear 21 Herzberger takes his proofs of inexpressibility to be sim ilar to a certain diagonal proof offered by Tarski [1953: 46]. However, the two proofs are quite different, because Tarski was not conc erned with the inexpressibility of a concept, but with the indefinability of a certain class (the class of codes of true sentences of the language). Moreover, what Tarskis theorem establishes is not that th e class is indefinable, but that it is indefinable in a language if the language is consistent. 22 Herzberger tackles this issue only in a footnote [Herzberger 1979: 161, fn. 17]. An example he offers of a concept that lacks an extension is the concept of a Russellian class, na mely the concept of a class that is not member of itself.

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62 that there is a set or a class of bald persons. This could also be the case of semantic concepts. One might argue that there is no extension dete rmined by the concepts of groundedness or truth, although here the problem no longer has to do with the lack of a sharp borderline but with the existence of some cases that are paradoxical. Ray [2002], for instance, argues that the concept of truth (together with the facts in the world) fails to determin e a class of objects that fall under it, because it is incoherent. If the concept fails to determine an extension, one certainly cannot require the predicate to have th e right extension. One can resist Herzbergers inexpressibility arguments by denying that the concepts he wants to prove inexpressible have an extension. The other shortcoming of Herzbergers strategy is that there is no obvious way to extend it to natural languages. He argues th at the strategy works for all langua ges in which the principle of the bivalence holds. The problem is that the st rategy cannot be applied for languages with gappy sentences such as the language constructed by Kripke [1975]. Languages of this sort contain their own truth-predicate ( in the sense that ther e is a predicate whose extension coincides with the extension of the concept of a true sentence of L). No contradiction would follow from the assumption that there is a predicate of L whose extension is precisely the class of true sentences of L. Therefore, Herzbergers strategy cannot establish that no languages can contain their own truth-predicates. Moreover, there are good reasons to think that if there is a class of true sentences of English, then there is a predicate of English (one coul d define it) which has it as its The predicate Russellian class has a sense but no extension. Here, an extension is taken to be a set, as opposed to a proper class (or ultimate class, to use Quines terminology). The issue needs to be handled, Herzberger argues, by associating terms not with classes, but with virtual classes, as they are conceived in Quine [1969]. He suggests that an adequate revision of the principles he used would enable one to extend the inexpressibility result to apply also to the case of concepts without an extensio n. The language of virtual classes is indeed useful in showing that diagonal arguments can be used even in the absence of a platon istic commitment to the existence of sets or classes. Nevertheless, it is harder to use the same apparatus for pr edicates that do not even de termine a proper class (be it understood as real or merely virtual).

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63 extension. This would not be enough for the predicat e to express the concept of truth. Therefore, one needs to distinguish the not ions of class expressibility and concept expressibility. The version of the inexpressibili ty argument that I have offe red does not assume that there is a class of true sentences of English. This wa y the inexpressibility ar gument remains successful regardless of whether the concept of truth determines a class. Moreover, the argument does not depend on whether the sentences of the language obey the principles of classical logic. Two Objections against the Inexpressibility View Intentions are Sufficient for Expressibility The f irst objection against the inexpressibility view that I will consider is that the expressibility of a concept is guaranteed by merely having the right intentions with respect to the use of the predicate. Expressibilit y, it may be argued, is a matter of what intentions speakers in a linguistic community associate with a certain expr ession. So, the third line in my inexpressibility argument would not be justified. Wh at should be required for T to e xpress the concept of truth is that the T-schema govern the meaning of T (i.e., Material Adequacy), not that the T-schema for T be true (which is the require ment of Expressive Adequacy). There is good evidence that Tarskis material adequacy requirement is a re quirement that the predicate be governed by the corresponding T-schema; this re quirement is trivially met if the speakers have the right intentions with the use of the predicate. If this requirement is all that is needed, the objection goes, then one cannot infer a contradiction from the assumption that truth is expressible in English, because one cannot infer the third premis e of the inexpressibility argument. One can only infer (as Tarski did) that the language is inconsistent. This can deliver the conclusion that the concept of truth is inexpressi ble in English only if one can prove that the inconsistency of English leads to a contradiction.

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64 The Inexpressibility Account is Self-Defeating The second objection is that th e view that truth, groundedness or other sem antic concepts are not expressible in English is self-defeating. The objection has been raised against Herzbergers view, but it is a challenge to any inexpressibility view. Simmons argues that Herzbergers thesis that groundedne ss is inexpressible in English is undermined by the very fact that Herzbergers paper, in which he uses the word grounded and explains what it means, is written in English23. A similar objection has been put forward by Martin: The claim was that, no matter what language I chose, a concept could be defined (expressed) which the language I chose could not express. So I said: okay, I choose the language in which you are arguing. If you succee d you fail. So you fail. [Martin 1976: 282] He takes the argument to involve a definition of the concept that is then proved to be inexpressible. This definition amount s to specifying the set about whic h it is to be shown that it cannot be the extension of any term24 of the language. Martin agrees that if the definition were given in a metalanguage, Herzbergers strategy might work. It does not work if it is formulated in the same language whose semantic concepts ar e supposed to be inexpressible, because this amounts to proving that the concept that ha s just been expresse d is inexpressible.25 Martin proposes a better way to formulate th e inexpressibility argument. One could prove the inexpressibility thesis by an inductive argum ent that consists in the following two steps: (i) If all other languages are expressively incomplete, then so is this language (the language of the argument). 23 Simmons [1993: 58-61]. 24 Martin [1976: 282]. 25 Notice that Martins objection does not depend on the fact that Herzberger begins his argument by specifying a class. Thus, the objection is also a challenge to my version of the argument.

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65 (ii) For every language L other than this one, there exists a concept which is inexpressible in L.26 This move enables one to avoid the charge of being self-defeating, although the argument remains inconclusive. As Martin points out, even though this version of the argument is harder to reject, there are some unsettling consequences27 that raise doubts about the plausibility of the view. Although we are presented with an apparently good argument for the conclusion that the concept of a grounded term of English is inexpressi ble in English, we feel that we can express it in English: we can express it by the predicate grounded term of English. If it is denied that this predicate expresses the concept of a grounded term of English, how do we know what concept is inexpressible (how do we know what set of terms Herzberger is talking about28)? Intended Meaning versus Linguistic Meaning I will argue that bo th objections fail. It is true that a necess ary condition for a predicate to express a concept is that the members of the lingu istic community have the right intentions with respect to the use of the predicate. Yet this condition ca nnot be sufficient. Some of these intentions may remain unfulfilled. This could happen, for instance, if their fulfillment is blocked by empirical or semantic facts, or if the intend ed rules are simply inconsistent. One needs to distinguish the linguistic meaning of an expression from its intended meaning. The members of a linguistic community intend to use an expression to express a certain content. The linguistic meaning of an expression is supposed to be a meas ure of the extent to which these intentions are 26 Hofweber [2006] argues that the inductive argument for inexpressibility fails, because, given the appropriate context, everything can be expressed in a language that contains context-sensitive expressions. This would mean that the induction basis is no longer available. I have argued already that the presence of demonstrative or indexical expressions in a language does not guarantee that everything can be expressed in that language. 27 Martin [1976: 284-85]. 28 Martin [1976: 284].

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66 fulfilled by using sentences in which the expression occurs. The linguistic meaning of an expression can also be thought of as that notion of meaning that (together with how the world is) determines the semantic value of the expression. If the intentions associat ed with an expression cannot be fulfilled, then the intended meaning (toget her with how the world is) fails to determine a semantic value.29 The match between intended meaning and concept is indeed trivially guaranteed (this would also be enough to guara ntee material adequacy in Tarskis sense). However, the match between linguistic meaning and the concept is not trivially guaranteed, because such a match requires in addition that speakers intentions can be fulfilled. If an expression expresses a certain cont ent, then it is in virtue of that content that the expression acquires its semantic value. In particular, a sent ence is true or false onl y insofar as the thought expressed by it is true or false. Therefore, the not ion of meaning that should be part of the correct characterization of expre ssibility is linguistic meaning rather than intended meaning. The relation between a predicate and the concept expressed by it can now be more precisely characterized in the following way: (EC*) P expresses C in L (in context c) if and only if C is the concept determined by the linguistic meaning of P in L (and context c). For the same reason, the Expressive Adequacy cond ition that must be met by a predicate in order to express a certain concept requires that ther e should be a match betw een the concept and the linguistic meaning of the predicate. The import of the Expressive Adequacy condition is better understood by comparison with Tarskis Material Adequacy requirement. Tars ki [1944] argues that a truth predicate for a 29 The distinction between intended meaning and linguistic meaning does not match the distinction between speaker meaning and sentence meaning. The latter is a distinction be tween a notion based on the intentions of a particular speaker with respect to a particular utterance of a senten ce and one based on the intentions of the entire community with respect to that sentence. The former is a distinction between two notions that are both concerned with the intentions of the entire linguistic community.

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67 language L is indefinable in languages that meet a certain set of requireme nts. In such languages there can be no definition of truth for L that is materially adequate and formally correct. Tarski did not talk about concepts at a ll. However, if one wants to re formulate his material adequacy condition as a relation between a predicate and a concept, one would formulate it along the following lines: (Material Adequacy) P is intended to be such that for any a, P applies in L to a iff the application rules for C entail that a falls under C. The difference between Material Adequacy and E xpressive Adequacy is that while the former only requires that members of a linguistic commun ity have the right intentions with respect to the use of the predicate, Expressi ve Adequacy requires in addition to this that the intentions be fulfilled. More specifically, Materi al Adequacy requires that the meaning of true be governed by the T-schema, while Expressive Adequacy requires that the instan ces of the T-schema be true. The comparison between the two adequacy c onditions also sheds light on the relation between the inexpressibility argument and Tarski s argument for the inconsistency of languages of a certain sort. Part of Tarskis indefinability argument is an argument that if a definition of the notion of a true sentence of L in L is materially adequate, then it cannot be formally correct. A consequence of this fact is that languages that c ontain their own truth predicate would have to be inconsistent. Since English is assumed to be T-un iversal, it contains its own truth predicate. Therefore, Tarski [1933] takes English to be one of these inconsistent languages.30 30 Tarski [1944] is more cautious and refrains from calling English inconsistent on the grounds that it does not have an exactly specified semantics. The issue of inconsistency occurs only in connection with languages with an exactly specified semantics. However, languages that are close enough to English but have an exactly specified semantics are, even according to Tarski [1944], inconsistent. If Tars ki was justified in restricting the claim to languages with an exactly specified semantics, then I should also restrict the inexpressibility of truth in the same way. Nevertheless, I think that the lack of an exactly specified semantics should not prevent Tarski from arguing that English is inconsistent, and should not prevent one from proving that truth is inexpressible in English.

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68 It may appear that the inexpressibility argumen t is related to the inconsistency argument in the same way as modus tollens is related to modus ponens. After a significant amount of oversimplification, the argument for the inconsis tency of English can be said to have the following form: I1. If truth is expressible in English, then English is inconsistent. I2. Truth is expressible in English. I3. Therefore, English is inconsistent. It may be thought that the inexpressibility ar gument merely turns the above argument into a modus tollens. This is incorrect since the inexpressibility th esis is not derived from I1 together with the thesis that English is consistent, but, rather, from the f act that the supposition that truth is expressible leads to a cont radiction. Thus, the inexpressibi lity argument requires one to establish E1, which is a much stronger thesis than I1: E1. If truth is expressible in Englis h, then a contradiction is true. Material Adequacy enables one to establish I1, but it is not en ough to establish E1. In order to establish I1 one needs to appeal to Expressive Adequ acy, which is a stronger condition. Material Adequacy helps to establish that if truth is expr essible in English, then a certain set of sentences is inconsistent. It cannot be used for a reductio argument, unless one proves that the inconsistency of that set of sent ences (i.e., the inconsistency of E nglish) is absurd. On the other hand, Expressive Adequacy enables one to establish that if truth is expressible in English, then the sentences that belong to a certain inconsistent set of sentences are true, which can be used in a reductio argument, because inconsis tent sentences cannot all be true. The distinction between intended meaning and linguistic meaning is also crucial for answering the second objection. Although one could try to support the ine xpressibility thesis by

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69 an inductive argument of the sort suggested by Martin, it would be better to find another way to answer the objection that the inexpressibility view is self-defeating. One reason is that an inductive argument avoids the problem only if the c oncepts that are proved to be inexpressible in a language are expressible in a metalanguage. If there is no language in which they can be expressed, the inductive argument remains as pr oblematic as the original version, because no metalanguage will have the resources that are re quired to run the argument. Fortunately, the inexpressibility view does not depend on semantic concepts of English being expressible in some language other than English. There is a way to answ er the objection that the inexpressibility view is self-defeating that does not ma ke use of an inductive argument. At the same time, it is possible to explain away the universality intuitions. Speakers of English certainly intend to use true sentence of English to express the concept of a true sentence of English. However, an expression may fail to express its intended content and, consequently, may lack linguistic meaning. Indee d, it turns out that not only paradoxical sentences but any sent ence in which true is used fails to express the intended content. Nevertheless, expression s can be used to communicate succe ssfully even if they fail to express the intended content. In general, it is th e intended meaning of an expression rather than its linguistic meaning that plays the crucial role in communicati on. What is required for one to be a competent speaker of a language is to know th e intended meaning of the expressions in the basic vocabulary of that language. One could be a competent user of a predicate without knowing what its linguistic meaning is. Therefore, th e default interpretation of a certain sentence takes the speaker to convey the content determin ed by the intended meaning of the sentence. In particular, all that is required for one to be a competent user of true is that one know its

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70 intended meaning31. Therefore, in the vast majority of cases, the use of true is unproblematic: the hearers know what the speakers intend to express, because both are competent speakers of English. There are indeed cases in which what the sp eaker wants to communicate is not the content determined by the intended meaning. This is no t surprising, because it is widely accepted that communication is frequently achieved by pragmatic means, rather than by relying on semantic rules. This is the central idea of Grices theory about the semantics and pragmatics of natural languages. Grice argues that ther e are many cases in which the sp eakers meaning is different from the literal meaning of an expression. The di stinction I draw between intended meaning and linguistic meaning should be understood as a distin ction between two aspect s of what Grice calls the literal meaning of an expres sion. One could easily imagine cases of sentences in which true is used, such that the speakers meaning is different from the intended meaning. These cases are no different from the cases described by Gr ice, and communication can be explained by appealing to some conversational implicatures that enable the hearer to figure out what the speaker wanted to convey. Difficulties might occur only in a conversati on between astute puzzle lovers who notice that there is a conflict between the intended m eaning of a Liar sentence and the commitments that would result from asserting it. Typically, wh en one makes an assertion, one is committed to the truth of the asserted sentence. The worry is that, for Liar sentences, even if the speakers meaning is the intended meaning of the Liar, the conflict between the intended meaning and the commitment to the truth of the Liar sentence might make the hearer think that this is another case in which the speakers meaning and the intended meaning are, in fact, different. Although there 31 One can be a competent user of true even if one does not realize that it lacks linguistic meaning. Nevertheless, one who does not know that true is intended to capture the concept of truth is not a competent user of true.

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71 could be contexts in which someone who utte rs a Liar sentence want s to convey something different from the content determined by its intended meaning or does not want to convey anything (this could only happen in the context of a seminar on the Liar paradox in which the speaker wants to draw the attention to the notion of linguistic meaning whic h, for Liar sentences, is lacking), in most contexts th e hearer is able to tell what th e speaker intended to convey. When I am uttering the Liar to say what I think about th e status of the Liar, I ex pect the hearers to be able to figure out that what I want to say is that the Liar is not true, even though the sentence I am using is devoid of linguistic meaning.32 Thus, the successful use of true does not entail that truth is expressible in English. Therefore, the inexpressibility view is not self-defeating. The universality intuitions can now be expl ained by distinguishing between two kinds of semantic universality: (IU) A language is intentionally semantically universal iff for every semantic concept, S, there is a predicate of that langua ge that is intended to express S. (LU) A language is linguistically semantically universal iff for every semantic concept, S, there is a predicate of that langua ge that (linguistically) expresses S. Only the linguistic semantic universality of natural languages would entail that truth is expressible in natural languages. Natural languages, however, are not universa l in this sense. The universality intuitions with respect to natura l languages are derived from their intentional semantic universality. 32 Although sentences in which true is used lack linguistic mean ing, as long as there is an intended meaning that the members of a linguistic community associate with them, it would be improper to say that they are meaningless.

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72 The Status of the T-Biconditionals If the inexpressibility account is correct, then the T-sche m a does not hold. Not only are instances of (T-schema) S is true iff S. obtained by replacing S by paradoxi cal sentences not true; none of the instances of this schema are true, because they fail to have linguistic meaning. This may seem to be problematic, because the T-biconditionals have been taken to capture an essential aspect of the concept of truth. It is true that there are notorious problems with so me instances of the T-schema, such as those obtained from sentences containing context-sensitive expressi ons or empty names. Nevertheless, there still seems to be some pressure to require that the T-schema hold true for a restricted class of well-behaved sentences (maybe declarative sentences that contai n no context-sensitive expressions and are not defective for reasons that do not have to do with truth). A defender of an inexpressibility view has an easy answer. There certainly is something to this intuition that needs to be preserved, but this is something different from requiring that the Tschema be true. For instance, it must be the case th at the thought that snow is white is true if and only if snow is white. For sentential truth, we want to say that a sentence such as snow is white is true in L if and only if it expresses in L the thought that snow is white, and snow is indeed white. Nevertheless, it would be wrong to think that the concept of tr uth requires that the Tbiconditionals be true. What the concept of truth can be held to require is that if there is a predicate that expresses the c oncept of a true sentence of E nglish, then the corresponding Tbiconditionals for well-behaved sentences should be true. It turns out that the T-biconditionals are not true, but this is what one should expect, because true fails to express the concept of truth.

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73 Is the Concept of a True Sentence of En glish Expressible in other Languages? Tarski [1933] argues that for any open language that has an exactly specif ied semantics one can define its concept of truth in a metalanguage that is essentially richer than it.33 If an inexpressibility view of truth is correct, and the concept of a true sentence of English is inexpressible in English, can one express this concept in a metalanguage?34 Herzberger suggests that the grounding concepts of English, although not expressible in English, are expressible in German or French. I think there are good reasons to resist this thes is. I will argue that the concept of a true sentence of English cannot be expresse d in German. It cannot be expressed by wahrer englischer Satz, because this would presuppose th at wahr expresses in German the concept of truth. Suppose twahr is a predi cate of German that expresses th e concept of a true sentence of English and applies to all and only th e true sentences of English. Thus (A) Snow is white ist twahr gdw schnee ist weiss. would be true in German. There is no reason to think that there is something special about the expressive power of German. We should also assume that there is a predicate of English that expresses the concept of a true sentence of Ge rman. Suppose wtrue is a predicate of English that applies to all and only the true sentences of German. Thus (B) Schnee ist weiss is wtrue iff snow is white. would be true in English. Consider now the following pair of sentences (G) (E) ist twahr. (E) (G) is not wtrue. 33 Languages of infinite order may be an exception. 34 English does not have an exactly specified semantics, an d for this reason it may not be clear what it would mean for a language to be richer than it.

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74 (G) would be a perfectly well-formed sentence of German, and (E) a perfectly well-formed sentence of English. However, the pair of senten ces leads to a paradox. If (G) is a true sentence of German, then one can infer that (E) is a true sentence of English. If (E ) is a true sentence of English, then (G) is not a true sentence of German. On the ot her hand, if (G) is not a true sentence of German, one can infer th at (G) is not wtrue is true in English, therefore, (E) is true in English. If would follow that (E) is twahr is tr ue in German, therefore, (G) is a true sentence of German. Thus, both alternatives lead to a contradiction. This mean s that either twahr fails to express the concept of a true sentence of English, or wtrue fails to express the concept of a true sentence of German, or both fail.35 Since there are no reasons to th ink that the two languages are different in this respect, we s hould conclude that both fail, and thus we need to abandon the assumption that the concept of a true sentence of English can be expressed in German. Notice that if the concept of a true sentence of English is inexpressible in other similar natural languages, then the appeal to an induc tive argument would not help an advocate of the inexpressibility view. There is no way to avoi d defending the inexpressibility view in English itself by using a vocabulary that does not actually succeed in expressing the content that is meant to be conveyed. There are additional reasons that speak in favor of the inexpressibility view, but which go beyond the scope of this chapter. Thus, in chapter four I argue that the expressibility of the concept of truth cannot be saved by endorsing an inconsistency view of truth because the latter account actually entails that truth is inexpressible. This also show s that the inexpressibility view offers a better explanation of the Liar paradoxes than the inconsistency vi ew of truth. Moreover, in chapter six I argue that the inexpressibility argument can be smoothly extended to provide a 35 The same holds for any other pair of predicates of English and German that are assumed to express the concept of a true sentence of German and the concept of a true sentence of English, respectively.

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75 uniform solution to all semantic paradoxes. Thus the inexpressibility view should not be dismissed as self-defeating and counterintuitiv e, as it has been in the past, but rather acknowledged as an account that fares better than all other alternatives.

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76 CHAPTER 4 ON THE COHERENCE OF THE INCO NSIST ENCY VIEW OF TRUTH It might be argued that although the inexpressi bility view is a cohe rent position, the price paid for blocking the Liar argument is too high. On e might hope that it woul d still be preferable to adopt an inconsistency view which is al so a coherent view, but does not have the counterintuitive consequence that truth is inexpr essible in English. One huge advantage of the inconsistency account is that it a ppears to accommodate all of the ordinary intuitions that lie behind the Liar argument, so it becomes pointless to search for a flaw in the Liar argument in order to avoid the inconsistency. Therefore, one should investigat e whether the inconsistency view of truth fares any better. There are different formulations of the incons istency views of truth. Charles Chihara [1979] characterizes an inconsistency view of truth as a view according to which a complete statement of what true means is inconsistent with all kno wn facts (in particular, it is inconsistent with a certain undeniably true sentence of reference). Tarski [1933] endorses the view that natural languages are inconsistent, which is in fact a corollary of an inconsistency view of truth. Finally, the inconsistency view can be understood as the view that the concept of truth is incoherent. [Ray 2002] These views should be distinguished from dialetheism, which in addition to the thesis that a statement of what true means is inconsistent with the facts holds that there could be true contradictions. I will mainly be concer ned with inconsistency views which are committed to the principle of non-contradiction. I argue that the inexpressibility view and the inconsistency view are not inconsistent with one another. Moreover, unless one adopts an extreme position, such as dialetheism, the inconsistency view entails inexpressibility. The in consistency view of truth has been frequently misunderstood. Thus, it has been argued that the id ea that languages can be inconsistent is of

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77 dubious coherence on the grounds that the semantic principles seem to be guaranteed to be true by fiat while the defenders of an inconsistency vi ew of truth, with the ex ception of dialetheists, think that some might fail to be true. Hans Herzberger [1967] and Scott Soames [1998] for instance, argue that the only plausible ways to articulate the idea of an inconsistent language suggest that inconsistent languages are impossible. I argue that the misunderstandings are due to th e failure to distinguish between intentional inconsistency and linguistic incons istency. Most of the attacks agai nst the inconsistency view of truth miss their target, because what they reject is the notion of li nguistic inconsistency (which is indeed incoherent), while the de fenders of an inconsistency view thought of it as a thesis of intentional inconsistency. Although the inconsistency view of truth is coherent, the arguments based on linguistic Liar paradoxes offered by Tarski and Chihara fail to establish that it is true. There are two ways to think of what it means for the meaning postulate s to be inconsistent, wh ich are quite different, but are normally confused with one another. What the Liar arguments can be used to show is that the rules which would be underwritt en by the intended meaning of t rue if true succeeded in expressing the concept of truth are inconsistent Thus, they do not establish that the intended meaning of true is inconsistent, but that it is inconsistent with the thesis that true succeeds in expressing its intended meaning. Even though the inconsistency view cannot be established by appealing to Liar sentences, it remains po ssible to prove it on independent grounds. Nevertheless, even if the inconsistency view hol ds, truth would be inexpressible. Therefore the inexpressibility remains the simplest acc ount of the linguistic Liar paradoxes. The Inconsistency View Many philosophers have argued th at the best way to account for the Liar paradox is to endorse an inconsistency view of truth. Recall that a Liar paradox consists in an argument that,

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78 starting from some principles that are prima faci e true, leads to a contradiction. The premises of the argument are semantic principles governing the meaning of truth (instances of the T-schema or some slightly different principles that are supposed to capture the same semantic intuitions) and certain (semantic or empirical ) reference facts about one or mo re Liar sentences. If such an argument is valid, then the set of premises is inconsistent. The difficulty stems from the fact that all semantic premises that are involved in the ar gument are prima facie tr ue (they appear to be derived from the meaning we associate with a seman tic term, in particular, with true), while the rules of inference are prima facie valid (they are commonly accepted as valid in every other area of human thought). Moreover, the reference fact is obviously true. Nevert heless, if the Liar argument is valid, unless one wants to reject the principle of non-cont radiction, the premises cannot all be true. Chihara [1979] draws a distincti on between two kinds of approa ch to the Liar paradoxes: the consistency and the inconsis tency views of truth. A consistency view of truth holds that an accurate statement of what true means will be logically consistent with all known facts, and in particular with all known facts of reference. [Chihara 1979: 607] Most of the accounts of the Liar paradoxes that have been offered are, according to Chihara, different versions of the consiste ncy view. Given that what a Liar argument seems to show is that some statements grounded in the meaning of the wo rd true are inconsiste nt with some facts of reference, either there is an il legitimate step in the argument or some of the semantic principles fail to correctly capture the mean ing of true. An account of the semantic paradoxes is supposed to point out either the ille gitimate rule of inference1 or the illegitimate semantic principle. On the other hand, an inconsistency view acknowledges the validity of the Liar argument and the 1 A good example of such a strategy is Skyrms attempt to show that the use of the intersubstitutivity principle is illegitimate in that context. See Skyrms paper in Martin [1984].

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79 legitimacy of all its semantic premises. The fact that a valid argument can be used to derive a contradiction is taken to show that a set of sentences is (derivably) inconsistent. More specifically, according to the inconsistency view, an accurate statement of what true means is logically inconsistent with the facts, in particul ar with some facts of reference. The semantic principle Chihara appeals to is an informal version of the T-schema: (Tr) A sentence is true if, and only if, what is sa id to be the case by the sentence is in fact the case. [Chihara 1979: 605] The Liar argument shows that (1) holds. (1) (Tr) together with a certain refe rence fact yields a contradiction. Sentence (1) together with (2) (2) (Tr) is part of an accurate statement of what true means. yields the inconsistency thesis: (3) An accurate statement of what true m eans is inconsistent with all known facts. One can see that the attitude of an advocate of the inconsistency view with respect to the semantic paradoxes is quite different from that of a defender of a consistency view. A Liar argument is valid and establishes that an accurate statement of what true means together with some facts of reference leads to a contradiction. No attempt is ma de to block the derivation of a contradiction. One rather tries to offer a diagnosis of this contradiction, which correctly captures our semantic intuitions. The Inconsistency of Natural Languages One version of the inconsistency view of truth has been articulated by Tarski in som e (but not all) of his writings. Even though he was mainly concerned with the notion of truth in a language and the languages he dea lt with are mainly formalized languages, some of his remarks about natural languages can be read as an endorsement of the inc onsistency view of truth. Tarski

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80 was engaged in the project of formulating a defi nition of the notion of a true sentence of a language. He introduced two requirements that need to be met by any candidate for a definition: it must be materially adequate and formally co rrect. The requirement of material adequacy is, roughly, the requirement that the definition capture the actual meaning of the ordinary notion of truth. In particular, the defin ition should entail the T-bicond itionals. Tarski does not say explicitely what the requirement of formal corre ctness amounts to, but it is reasonable to assume that it requires the truth definition to meet so me general constraints on definitions (such as consistency, non-circularity and conservativeness). Thus, an inconsistent definition would not be formally correct. Although there are languages for which the notion of a true sentence of that language can be defined, a definition of this so rt is not available for all languages. If the language has a certain degree of expressive power, then no definition of the required sort can be formulated in the language itself. More specif ically, if a language L has enough resources to allow one to construct a Liar argument for language L, then, in L, there can be no definition of a true sentence of L that is both materially adequate and formally corre ct. This is, roughly, the content of Tarskis indefinability theorem. Tarski distinguishes between semantically closed and semantica lly open languages. A language is semantically closed if in addition to its expressions, it contains the names of these expressions, and semantic terms such as true th at refer to sentences of the language and meet the requirement of material adequacy. If a langua ge is closed and has enough expressive power to enable one to construct a Liar argument, then there is no way to formulate a definition of a true sentence of that language, regardless of whet her one tries to do it in the language itself or in a metalanguage. On the other hand, for open languages a definition of truth can be done in a

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81 metalanguage. The metalanguage itself cannot contai n its own truth-predicate. One could define the notion of a true sentence of the metala nguage in a meta-metalanguage, and so on. Part of the indefinability argument is an argum ent that the principles that are required for material adequacy (the T-biconditionals), together with some facts of reference, lead to a contradiction. Therefore, Tarski takes it to be a corollary of the indefinability theorem that languages of a certain sort are inconsistent. In fact, his views about th e scope of his results changed. Thus, Tarski [1935] takes the inconsis tency result to apply to natural languages. English, for instance, cannot contai n a materially adequate and fo rmal correct definition of the notion of a true sentence of English. Moreover, since English contains a predicate, true, that meets the requirements for material adequacy, it follows that it is an inconsistent language. Tarski [1944] is more cautious and claims that the inconsistency theorem can only be established for languages with an exactly speci fied semantics. The issue of inconsistency occurs only in connection with languages with an exactly specified semantics. Since English does not have an exactly specified semantics, Tarski refrains from saying that it is an inconsistent language. However, languages that are close enough to English but have an exactly specified semantics are, even according to Tarski [1944], inconsistent. Mo reover, one can argue that the lack of an exactly specified semantics should not prevent Eng lish from being inconsistent. In fact, English still allows one to run the Liar argument, b ecause all the principles involved in it hold. Herzberger [1967] interprets Ta rskis view as entailing that English is inconsistent. Meaning Postulates Given that the Liar argu ment is accepted as valid and leads to a contradiction, a defender of the inconsistency view would have to either drop the principle of noncontradiction or accept that some semantic principles are not true. Both Tarski and Chihara are committed to the laws of classical logic and try to explain why it is plausibl e to think that some semantic principles are not

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82 true. This position is in conflict with the widely held view that definiti ons or meaning postulates are true by fiat. Therefore, a challenge for the de fenders of an inconsiste ncy view is to explain how it is possible for meaning postulates to fail to be true and still provid e a word with meaning. Moreover, one needs to explain how it is possibl e for expressions whose meanings are governed by inconsistent principles to be used in successful communication. Tarski thinks of the expressions of Englis h as having a meaning captured by a number of meaning postulates. These meaning postulates might happen to be inconsistent with one another or inconsistent with other true sentences. In pa rticular, the meaning postu lates that determine the meaning of true turn out to be inconsistent Since he is committed to the principle of noncontradiction, Tarski draws the conclusion that not all meaning postulates are true and some of them must be rejected. What is needed in order for a set of postulates to capture the meaning of a word is not the truth of those postulates but rath er the fact that those postulates are intended to govern the use of the word. Recall that unlike th e Expressive Adequacy requirement formulated in chapter three, Material Adequacy does not require that the T-biconditionals be true. Chihara also thinks of meanings in terms of meaning postulate s, and tries to explain how it is possible for some meaning postulates not to be true. He argues that in the case of truth there are some generally accepted conventi ons which give the meaning of true.2 A lesson that he wants to draw from the paradoxes is that these conventions or meaning postulates are inconsistent and some of them are not true.3 To show that the inconsistency of the meaning postulates does not preclude them from providing a word with a meaning, Chihara constructs the 2 See [Chihara 1979: 611]. 3 Matti Eklund [2002] also argues that the notion of an inconsistent language can be made intelligible by observing the distinction between semantic competence and semantic value.

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83 case of the Secretary Liberation club. The f ounding members of the Secretary Liberation club (Sec Lib) have decided that the eligibility to join Sec Lib is determined by the following rule: (R) A person is eligible to join this club if and only if, he (she) is secretary of a club which he (she) is not elig ible to join. [1979: 594] The rule is good enough to ensure the well-func tioning of the club. Problems occur only when Sec Lib hires Ms. Fineline, who happens to be secretary of no other club, as a secretary. According to (R), Ms. Fineline is eligible to join Sec Lib if and only of sh e is not eligible. To be more specific, (R) is inconsiste nt with the following two claims: (F1) Ms. Fineline is secretary of Sec Lib. (F2) Ms. Fineline is not secretary of any other club. Given that the truth of (F1) and (F2) can be established empirically, the only option is to deny the truth of (R). This shows that not all the meaning postu lates can be made true by fiat. (R) cannot be made true by fiat. This does not mean that they cannot provide the word with a meaning if these inconsistencies occur only in some isolated cases. Since inconsistenc ies only occur in some isolated cases (Ms.Fineline), (R) can still be us ed as an effective criterion of eligibility. Similarly, not all the T-biconditionals can be made true by fiat. They cannot all be made true because they are either inconsistent or inconsistent with some empirical facts. The Sec Lib case offers a nice model of how one should think of inconsistent meaning postulates, but there are additional worries that n eed to be addressed. On the face of it, it seems that a consequence of the inconsistency view of truth is that no sentence in which true functions as a predicate can make a statement. Chihara is aware of this difficulty which he formulates as follows: An explicating formula, determines whether a predicate applies to an object x only if there is no valid argument from true statements and the explicating formula itself to the

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84 conclusion both that the predicate does apply to x and that the predicate does not apply to x. [Chihara 1983: 225] Given that the explicating formul a (the T-schema) is inconsistent with some known facts, it follows that everything can be inferred from them This would mean that no sentences containing true as a predicate can be used to make a stat ement. Chihara thinks that the difficulty can be solved by saying that even though there is a way to derive a contradiction, it would not be reasonable to do so: in real life situations, one doesnt simply accept blindly the logical consequences of whatever one may initially have reason to believe [1983: 226]. It would also be wrong to think that an inconsistency view for ces one to endorse a c ontradiction. Although the meaning postulates enable one to infer that a Liar sentence is both true and not true, it is not reasonable to make this inference. Once one re alizes that the meaning postulates are inconsistent, one realizes that it does not make sense to ask wh ether the Liar is true or not4. Inconsistent Languages and the Inconsistency View of Truth There is a se nse in which the view that English is inconsistent and the inconsistency view of truth are different. The thesis that the language is inconsistent is, in fact, weaker than the thesis that truth is inconsistent with the facts. The inconsistency of the language might not have anything in particular to do with the concept of truth. It might be related with notions that occur in paradoxes other than the Liar paradox. Thus, the inconsistency might be caused by principles having to do with the notion of satisfaction, with the notion of reference, or it might be the case that no particular notion is respon sible for the inconsistency, but the language as a whole. In fact, 4 Gupta and Belnap raise an objection agai nst this answer provided by Chihara: By putting the entire burden of the theory on the notion of reasonable inference a notion of which no theoretical or even intuitive account is given it obscur es completely the contribution of the T-biconditionals to the meaning of true. [Gupta & Belnap 1993: 15] The problem, according to Gupta and Belnap, is that one needs to offer an acco unt of what it means for an inference to count as reasonable, but it is hard to see how such an account can be developed without losing the simplicity that was emphasized as one of the main reasons to pref er the inconsistency account over its competitors.

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85 Tarskis argument for the inconsistency of E nglish relies on a version of Grellings paradox rather than the Liar paradox. Nevertheless, he thought of the Grelling paradox as a paradox that is, just like the Liar paradox, based on the seman tic principles that govern the meaning of true. Thus, a defender of the inconsistency of English is normally also committed to an inconsistency view of truth. The Inconsistency of the Concept of Truth Tarski and Chihara talk about the m eaning of t rue and potential definitions of it, but they do not talk explicitly about the concept of truth. Nevertheless, the inconsistency story can be and has been extended to the level of concepts. Ray [fo rthcoming: 8] argues that in fact Tarski is best understood as having held an incohere nce view of the concept of truth. The Tbiconditionals (or, if one prefers, Chiharas (Tr) principle) are prin ciples that captu re part of the content (or maybe the entire conten t) of the concept of sentential truth. It makes sense to talk about inconsistent concepts if concepts are understood as coming supplied with explicit application rules associated with a predicate. Cert ainly, this is not the on ly way one can think of concepts, so one can reject the w hole strategy if one can argue that this is not the correct way to think of concepts5. Anyway, if concepts can be conceived of in this way, one can offer a pretty clear picture of what it means for a concept to be inconsistent. The concept is inconsistent if the application rules that are associated with it turn out to be inconsistent. These rules can offer full conceptual warrant to some se ntences in the language, but they cannot guarantee that those sentences are true. The idea of thinking of concep ts as coming supplied with application rules is also useful in characterizing the notion of an inconsistent language A language M is inconsistent 5 One can worry that concepts should not be tied up with a predicate, but they should rather be thought of in terms of conditions for objects to fall under it. This could still enable one to talk about inconsistent concepts. They would be concepts such that the conditions for objects to fall under them are inconsistent. This way one could talk about a concept as being inconsistent even if no predicate has ever been intended to express it.

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86 if a deductively inconsistent sentence is deriva ble by the rules of inference of M from the assertible sentences of M. [Ray 2002: 170] 6 A Priori or Empirical Inconsistency? The inconsistency view of truth does not de pend on any em pirical facts. The argument for inconsistency depends on a fact of reference wh ich could be either empi rical or non-empirical. An empirical Liar would appeal to a contingent identity sentences such as (4). (4) The first sentence on p. 6 in Quines fi rst published book is iden tical to The first sentence on p. 6 in Quines first published book is not true. The truth of the reference premise can be verified for instance, by a mere inspection of a page in a book. Tarskis argument for the inconsistency of the language consisted initially in a Liar argument based on an empirical fact of reference. Nevertheless, he argued th at the thesis that the language is inconsistent can also be derived without appealing to any empirical premise. One way to do this is to consider Grellings paradox instead of the Liar paradox. Tarski shows how one can run a Grelling argument th at involves only semantic premises and no empirical premise. Ray (forthcoming) argues that the version of the Grelling argument sketched by Tarski falls short of offering the desired result. Th is is because the argument assume s that the definition of selfapplicability is formally correct. This means th at the thesis that is delivered by the Grelling argument is not that truth is indefi nable, but that either truth or self-applicabil ity is an indefinable notion.7 Nonetheless, Ray shows how one can modify the Grelling argument so that it yields an indefinability theorem that depends neither on em pirical facts of refere nce nor on the formal correctness of some other notion, such as self-applicability. More over, Tarski could have made 6 The assertible sentences of the language are those that either have full conceptual warrant or have been confirmed empirically. 7 See [Ray forthcoming: 10].

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87 the same point by using a non-empirical Liar paradox instead of the Grelling paradox. Instead of using an empirical sentence such as (4), one could use instead (5) (5) (L) = (L) is not true. which is not a merely contingent truth, because the expressions on both sides of the identity sign are rigid designators. In the case of non-empirical Liar s all the reference facts involved in the argument are semantic facts. The corresponding identity premise is true by fiat (and, in addition, necessarily true), because Englis h is a flexible language that allows one to choose any name whatsoever for a certain sentence. The fact that natural languages have all the features needed for running a Liar argument for a non-empirical Liar shows that both theses can be formulated in the strong version as claims of inconsistency per se, not only inc onsistency with empirical facts. In what follows I will use the inconsistency thesis to refer to the strong version, unless I explicitly indicate that it is the weaker version that I have in mind. The fact that there is also a non-empirical way to support the incons istency view can be used to show that the incoherence of the concept of truth is a purely conceptual matter: it can be established a priori. In fact, to show that th e incoherence is a purely conceptual matter, one would not need to use a non-empi rical Liar: one could use a modified version of the empirical argument. Instead of arguing that there is a Liar sentence, one could argue that there could be a Liar sentence. Thus, (S) The first sentence on p. 6 in Quin es first published book is not true. could be used in a Liar argument even if the first sentence on p. 6 in Quines book is not (S) itself. It is enough that it c ould have been (S). Given that the T-biconditionals must hold necessarily, one can infer from them

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88 (S & ~S), from which one can derive (S & ~S) & ~ (S & ~S), which is an explicit contradiction. This means th at even for a language that does not have the resources to refer to the Liar sent ence via a rigid designator, one c ould still prove a priori that the concept of truth is incoherent. The Inconsistency View and Classical Logic Both Tarski and Chihara assum e that the la ws of classical logic hold for sentences of natural language. It is important to notice that the inconsistenc y view of truth does not actually depend on the assumption of classical logic. In chapter two I argued that one who rejects the principle of bivalence is normally also tempted to reject the pr inciple of the excluded middle or reductio as absurdum. This way one might hope th at although (Tr) together with the reference premise for (L) still allow one to deductively infer (C), (C) (L) is true iff (L) is not true. one no longer has resources to derive a contradiction of the form A & ~A. According to some very weak systems of logic, (C) would count as true if both sides of the biconditional lack a truth-value. If this is the case, then (Tr) is not inconsistent with all known facts. Other systems of logic, such as the system based on Kleenes three-va lued schema, take (C) to lack a truth value if both sides of the biconditional lack a truth-value. In this case (Tr) is still inconsistent with all known facts as long as inconsiste ncy is understood as a semantic notion. Nevertheless, they are not inconsistent in the sense th at one can derive a contradictio n. Chihara took the inconsistency to hold in both senses. If one takes (Tr) to be a complete statement of what true means8, then 8 Chihara seems to suggest that they indeed exhaust the meaning of true.

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89 by rejecting classical logic one can indeed weaken the strength of the inconsistency thesis; at least it would not have the worrisome consequen ce that the mere acceptance of the principles governing the meaning of true commits one to accepting the truth of a contradiction. Nevertheless, there are good reasons to think that (Tr) cannot exhaust the meaning that we associate with the predicate. In chapter two, I argu ed that there are other principles that capture other aspects of the meaning of true and these principles toge ther with (Tr) can be used to derive a contradiction on the basi s of some very weak principles of logic. For instance, TC* and FC*, repeated here, TC*: T(A) V ~T(A) [every sentence is either true or not true] FC*: F(A) V ~F(A) [every sentence is either false or not false] are principles that capture part of the meaning th at we associate with the predicates of truth and falsity. If TC* is added to th e list of meaning postulates, th en they can be proved to be inconsistent (in the sense that one can derive from them a contradi ction) from logical laws that are much weaker than classical logic. Thus, th e inconsistency view of truth does not depend on the assumption of classical logic.9 Skepticism with Respect to Inconsistency The inconsistency view has been treated by m any with skepticism. There are some prima facie reasons that have been offered to show that a view of this sort is incoherent. The consistency of natural languages, as well as the c onsistency of the meaning of true, seems to be a matter that is postulated rather than an issue that needs some investigation in order to be confirmed or infirmed. 9 Of course, the inconsistency view does not depend on the assumption of the principle of non-contradiction. Dialetheists have explored the possibility of solving the pa radoxes by denying the principle of non-contradiction, but I will not discuss this view in this study.

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90 It has been argued that there is no interesting way to articulate the notion of an inconsistent language. First of all, for uninterpreted languages the problem of consistency cannot make sense; there can be no sentences that ca n be evaluated as true or fals e and there are no axioms that would allow one to infer a contradiction.10 There is no set of sentences that could be tested for consistency. So one must assume that when the issue of consistency occurs, languages are understood as interpreted languages; when Tarski ta lks about inconsistent languages, he certainly has interpreted languages in mind. Scott Soames [1998] lists a number of possible ways to understand the notion of an inconsistent language, none of which are acceptable:11 A. A language is inconsistent iff there are inconsistent or contradictory sentences in it. B. A language is inconsistent iff some se ntence of the language and its negation are both true (iff at least one contradiction is true) C. A language is inconsistent iff there is so me theory, T, formulated in that language, such that T is inconsistent. The first proposal does not work, becaus e any language with negation contains inconsistent sentences and any language w ith both negation and conjunction contains contradictory sentences [Soame s 1998: 53]. The second characteri zation, attributed by Soames to Salmon, fares no better: as long as the principl e of non-contradiction is true, no language can be inconsistent in this sense. Finally, C cannot be the characteriz ation one is looking for, because the inconsistency of a language cannot be character ized as the inconsistency of any old theory 10 There might be inconsistent rules of syntax associated with a certain vocabulary, but it is hard to think of a language in the absence of a consistent system of rules of syntax. 11 Soames [1998: 53]

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91 formulated in that language. The inconsistency of a particular theory formulated in a certain language does not necessarily make the language itself inconsistent. Although C fails to provide the desired charac terization, it remains he lpful in that it indicates that a correct characteri zation should have to do not with some arbitrary theory that can be formulated in the language, but with a theory that has to do with the semantics of that language (formulated either in the language itse lf or in a metalanguage). One option that one might want to consider is the th eory that consists in all analytic sentences of that language, because the analytic sentences are those that are tr ue in virtue of the meanings of the words. An inconsistent language could then be characterized by the following clause: (*) A language is inconsistent iff its analytic sentences are inconsistent. Unfortunately, there cannot be incons istent languages in this sense.12 This is the conclusion of the following argument offered by Herzberger (f rom the Logical Consistency of Language): P1. The language L is not logically c onsistent in its analytic sentences. P2: L contains a nonempty set A of sentences such that: [from P1] P2a: Every sentence in A is analytic; and P2b: The set A is a logically inconsiste nt set of sentences. P3. Every sentence in A is true. [from P2a] P4. At least one sentence in A is not true. [from P2b] This means that the assumption that there is a language that is inconsistent in this sense leads to a contradiction. Thus, if one still wants to talk about inconsistent languages, one should characterize this notion in a different way. Soames believes that there is no acceptable characterization of the notion a nd rejects it as incoherent. 12 Priest [in Semantic Closure] argues that there are, but this is only because he denies LNC.

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92 The claim that the meaning of a term is inconsistent also faces prima facie difficulties. Traditionally, a definition has been taken as su ccessfully conferring meaning to a term only if it was consistent. Thus, a set of postulates used as a definition is either cons istent, in which case all postulates in the set are true by fiat, or it is inconsistent, in which case it fails to provide a definition. It is true that not all expressions in English have been introduced by a definition. However, there are semantic rules that govern th e use of these expressions. As long as these expressions are held to be meaningful those rules should be true by fiat. The notion of an inconsistent co ncept lends itself to similar criticism. In general, it is not clear whether and in what sense one can talk about inconsistent concepts. There is room for denying that there are inconsistent concepts. The notion of an inconsistent concept should be distinguished from the notion of a concept with an empty extensi on or with a necessarily empty extension. Normally, an expression like round squa re is taken to have an empty extension necessarily. One can find more or less reasonable grounds to deny that it actually expresses a concept. However, having a necessarily empt y extension does not offer enough grounds for denying that there is a concept expressed by the pred icate. It is hard to deny that prime number bigger than 23 but smaller than 29 expresses a concept, even t hough its extension (assuming that mathematical truths are necessary) is also necessa rily empty. In any case, inconsistent concepts are not merely empty concepts, concepts whose ex tension is (contingently or necessarily) empty. In their case there is no class or set that can be associated with th e concept. If the extension of a concept is understood as a class or set of objects to which the c oncept applies, then there are no inconsistent concepts, because th ey would have no extension. Th ere could still be objects to which the concept unproblematically applies or fails to apply; t hus, unlike the concept of a round square which applies to no object the concept of truth applie s without difficulties to many

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93 sentences, such as Snow is white, and it fails to apply to some other sentences. Nevertheless, this is not enough to determine an extension. Thus, one could argue against the idea that there are inconsistent concepts by saying that having an extension is a necessary condition for something to count as a concept. Intentional Inconsistency versus Linguistic Inconsistency The distinction between intended m eaning a nd linguistic meaning enables a better understanding of the inconsistency view and shows, at the same time, in what sense the view has been misunderstood by some of the opponents of the view. The inconsistency thesis, (3), can be read either as a thesis of intentional inconsistenc y or as a thesis of lingu istic inconsistency. I will argue that the thesis is properly understood as the thesis that the intended meaning of true is inconsistent. Tarskis thesis should be understood as the thesis that languages of a certain sort are intentionally inconsistent. Most of the objections that have been raised against inconsistency views are misguided because they stem from a mi sinterpretation of the inconsistency view as a thesis of linguistic inconsistency, which would indeed be an incoherent view. Nevertheless, I will argue that the arguments offered by Tarski, Chihara and others fail to establish that truth is intentionally inconsistent because there are two ways in which one could think of the meaning postulates which have not been properly distinguished from one another. Two Kinds of Inconsistency One can notice that Chih aras thesis of the inco nsistency of true can be read in two ways: as the thesis that a statement of what is the linguistic meaning of t rue is inconsistent, or as the thesis that the principles that characterize the intended meaning of true are inconsistent. Under the first reading, the inconsistenc y thesis requires that the rules that characterize the linguistic meaning of true (the T-biconditionals) are inconsistent. Under th e second reading, the inconsistency view is more modest. It only requires that the principles that are intended to govern

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94 the meaning of true are inconsistent; they do not ha ve to be true. It is obv ious that what Chihara is committed to is the view that the intended meaning of true is inconsistent. He allows the possibility that some of the m eaning postulates (some instances of the T-schema) are not true. Similarly, there are two ways in which a language can be said to be inconsistent. Tarskis indefinability theorem establishes that closed languages are inc onsistent. Nevertheless, he does not use this result to run a reduc tio argument to show that no language is in fact closed. He takes the inconsistency of a language to be a perfect ly acceptable result. There is no contradiction involved in claiming that a language is inconsistent. It on ly means that the set of rules that are intended to govern the use of the expre ssions of the language is inconsistent. On the other hand, Herzberger and others like him (Kripke, fo r instance) th ink that the indefinability theorem can be used to run a reductio argument to prove that there are no semantically closed languages. According to Herzberger, as long as one sticks with the Tbiconditionals, Tarskis definition [the definition of a semantically closed language] leaves too little room for any such languages [semantically closed languages] [H erzberger 1982: 481]. To support this idea he quotes th e following remark by Kripke: the result should rather be form ulated as such: no interpreted language in the ordinary firstorder predicate calculus containing number theory and so on, can be semantically closed; not that there are semantically closed ones but they are inconsistent. [Herzberger 1982: 481] It is clear that Tarski, on the one hand, and Herzberger and Kripke, on the other, do not talk about the same notion of language inconsistency. Thus, one can distinguish two senses in which a language can be said to be inconsistent13: 13 There is a parallel distinction between two senses in which a language can be closed: a language is linguistically closed if it has a predicate that ex presses the concept of truth as part of its linguistic meaning. -a language is intentionally closed if there is a predicate in the language that is intended to express the concept of truth.

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95 (Intentional Inconsistency) The principles that are intended to govern the use of the expressions of the langu age are inconsistent. (Linguistic Inconsistency) The principles underwritten by the linguistic meanings of the expressions of the langu age are inconsistent. Herzberger and Kripke (and Soam es as well) talk about linguis tic inconsistenc y and correctly argue that there cannot be language s of this sort. The thesis th at a language is linguistically inconsistent comes down to the thesis that its an alytic sentences are inconsistent, which is false because analytic sentences are true by definition. Nevertheless, what Tarski argues for is that natural languages are intentionally inconsistent. Herzberger and Soames miss the target, because they misinterpret Tarski as endorsing a thesis of linguistic inconsistency. The view that the linguistic meaning of true is inconsistent is indeed false. In order for an expression to have linguistic meaning a certain adequacy condition, Expressive Adequacy, must be met. This condition requires th at the principles that are inte nded to be true are true and determine the linguistic meaning of the expression. Therefore, the rules that determine the linguistic meaning of an expressi on cannot be inconsistent. In pa rticular, the linguistic meaning of true cannot be inconsistent. This means that the inconsistency view of truth is consistent only if it is understood as the th esis that the intended meaning of true is inconsistent. The objections raised by Herzberger and Soames ag ainst the inconsistency view miss the target, because they are objections ag ainst a view that nobody endorsed (the view that linguistic meaning is inconsistent). It is undeniable that there is a predicate in English that is intended to express the concept of truth. Tarskis thesis that English is inconsistent is supposed to be a consequence of the fact that English is intentionally closed. If one wants to reject the thesis that English is intentionally inconsistent, what one needs to show is that intentional closure does not in fact lead to a contradiction.

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96 I will argue that although the thesis that mean ings can be intentionally inconsistent is coherent, the paradox based arguments such as those offered by Tarski and Chihara fail to establish that the intended meani ng of truth is inconsistent or that English is inconsistent. Intentional Inconsistency I think that the thesis that the intended m eaning of a predicate is inconsistent is ambiguous. Suppose that one needs an expression to capture a certain content, which can be characterized by a certain set of application rules. When a predicate is assigned the role of capturing that content, then it is supposed to be used in accordan ce with a corresponding se t of principles. This corresponding set of principles could be identified with the application ru les of the concept (the intended meaning rules) or with the rules that would characteri ze the predicate in the hypothesis that it successfully expr esses the concept (the h ypothetical meaning rules14). In most cases, the hypothetical meaning rules are cons istent. Nevertheless, they coul d also be inconsistent. This does not necessarily mean that the intended mean ing is inconsistent. What it means is that if the predicate succeeds in playing the role that has be en assigned to it, then the intended meaning rules are inconsistent. There are two ways in which this situation could occur: either the intended meaning is inconsistent, or it is inconsistent w ith the hypothesis that the predicate successfully plays the role that has been assigned to it. Take, for instance, the predicate true. This predicates is intended to capture the concept of truth, which means that it is supposed to apply to a sentence just in case that sentence is true (the intended meaning rules). If the pred icate succeeds in playing this role, then its meaning is determined by the T-biconditionals (the hypothetical meaning rules), which can be proved to be inconsistent by using a Liar argument. This could be taken to mean that the intended meaning of true is in consistent or only that the intended meaning of 14 When one wants to express a certain concept in L, on is looking for a predicate of L which, if it is successful, would behave in the way prescribed by the concept.

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97 true is inconsistent with the hypothesis that t rue successfully plays the role that has been assigned to it. An inconsistenc y theorist chooses the former option, while a defender of the inexpressibility view chooses the latter. The di fference between the two options can be better emphasized by comparing the meanings of true and true*. Chihara [1979] introduces a new predicate, true*, which is implicitly defined by the following T-biconditionals: (T*) S is true* if and only if S. The meaning of true* is by definition exhausted by the above T*-biconditionals. Therefore, in this case the T*-biconditionals constitute the in tended meaning rules, not only the hypothetical meaning rules. The intended meaning of true* is characterized by the T*-biconditionals, which are inconsistent. On the other hand, true is not defined by the T-bi conditionals. It is a predicate intended to capture the concept of truth, which mean s that if it succeeds, then its meaning would be characterized by the T-bic onditionals. The T-biconditionals constitute the hypothetical meaning rules, not also the intended meaning rule s. Both true and tru e* fail to express the concepts that they are intended to express. The di fference is that the concept of a true* sentence of English is incoherent (the in tended meaning of true* is incons istent), while the concept of a true sentence of English could very well be consistent (the intende d meaning of true could very well be consistent) because Liar sentences do not offer any good reason to think that it is not consistent. Chiharas concept of an eligible memb er of Sec Lib would indeed be an incoherent concept (i.e., the intended meaning of eligible member of Sec Lib is inconsistent), but the case is more similar to the true* cas e rather than true, because the application rules of the concept are inconsistent regardless of wh ether the predicate succeeds in pl aying the role that has been assigned to it.

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98 What the above remarks show is not that the in consistency view is false. The thesis that some concepts are incoherent is indeed cohere nt. Nevertheless, they s how that the arguments offered by Tarski and Chihara fail to establish the inconsistency view, because they leave open the possibility that the intended meaning is consistent, but the predicate fails to express it. Moreover, it can be argued that the whether the concept of truth is cohe rent or not, it remains inexpressible. Therefore, the thesis that truth is inexpressible offers the best explanation of the Liar paradoxes and the thes is that the concept is in coherent is unjustified. Inconsistency Entails Inexpressibility The inexpressibility argum ent that I formulated in chapter three does not appeal to the assumption that truth is a consistent concept. Ther efore, the same argument can be used to argue that the concept is inexpressi ble even if the inconsistency view holds. However, one can establish a more general result: inconsistent concep ts are inexpressible. In order for a predicate to express a concept it must have a linguistic m eaning that would have to match its intended meaning. If one thinks of concepts, the way Ray suggested, in terms of application rules associated with a predicate, then the notion of a concept can be assimilated with the notion of intended meaning of a predicate: both are thought of in terms of some intended rules of application of a predicate. I already argued that if the hypothetical meaning rules are inconsistent, then they fail to provide the predic ate with linguistic meaning, because in order for there to be a match between the rules that characterize the linguistic meaning and the hypothetical meaning rules, the hypothetical meani ng rules must be consis tent. Nevertheless, if the intended meaning rules are inconsistent, then the hypothetical m eaning rules would be inconsistent as well. Thus, if the intended meaning rules are inconsistent, the predicate fails to have a linguistic meaning.

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99 Chihara recommends the following criteri on that can be used to choose between competitor theories: One consideration that moves reasonable peop le to prefer a theory to a competitor is whether the acceptance of the one would requir e less revision of our presently accepted scientific theories than would acceptan ce of the other. [Chihara 1979: 604] According to the inconsistency view, the Liar paradoxes are explained by assuming that the principles that characterize the intended meaning of true are inconsistent, a view that entails that the language cannot express the concept of truth. However, if this is how paradoxes are explained, one could also explai n them by just denying that the language can express the concept of truth without also having to assume that the co ncept is inconsistent. Thus, the inexpressibility account offers the simplest account of the semantic paradoxes.

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100 CHAPTER 5 NON-LINGUISTIC LIARS In chapter three I argued that the linguistic Liar paradoxe s (the paradoxes generated by Liar sen tences, Liar statements or Liar utterances) can be solved by saying that natural languages, appearances to the contrary notwithsta nding, are unfit to express the concept of truth. This account enables one to argu e that none of these Liar paradoxes give us good reason to think that the concept of truth is inc onsistent. However, there are non-li nguistic Liar paradoxes that are still unaccounted for which could provide such a reason. The inexpressibility view is compatible with the inconsistency view of trut h. In fact, one of the theses that I argued fo r in chapter four is that all inconsistent concepts are inexpressible. Nevertheless, the inexpressibility view would lose much of its appeal if inconsistencies can be proved to persist at the n on-linguistic level. In chapter four, I argued that the ine xpressibility view can be used to show that linguistic Liars do not suffice to show that the concept of truth is inconsistent. If it can be argued that an inconsistency survives at the le vel of non-linguistic Liars even if one drops the assumption that there is a predicate of English that expresses th e concept of truth, then the inexpressibility view would fail to provide a way to sa ve the consistency of truth. In this chapter I argue that all Liar argu ments generated by non-li nguistic Liars can be blocked. There are two kinds of non-linguistic Liar paradoxes that need to be accounted for, each of them correlated with a particular kind of trut h-bearer: propositions and mental representations. First, let us notice that non-linguis tic Liars that are still specified by means of a sentence do not raise special difficulties for the inexpressibility view. Consider, for instance, the following nonlinguistic Liar: (LP) The proposition expressed by (LP) is not true.

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101 It is clear that a defender of th e inexpressibility view would ha ve no difficulty blocking the Liar argument by denying that (LP) expresses a propos ition (because it fails to have a linguistic meaning). The difficult cases which I will focus on in this chapter are the potential Liars which are not specified via a sentence. I will first argue that no inconsiste ncy can be derived at the level of mental representations, and then s how that there are no Liar propositions. Mental Representations Consider first the possibility of Liar thoughts, understood as m ental representations. Could one think that ones thought is not true? On the face of it, it appears th at there should be Liar thoughts. Just like sentences, mental states can have the property of being about themselves. It is hard to deny that one could thi nk at t that the thought one has at t is not true. This thought is paradoxical, because both the assumption that it is true and the assumption that it is not true seem to lead to a contradiction. It seems also possible for there to be Pair Liar thoughts. If A thinks at t that Bs thought at t is not tr ue, and Bs thought at t is that As thought at t is true, then we seem to land again in paradox. If As thought is tr ue, then Bs thought is not true. Since what B thinks at t is that As thought is true, it woul d follow that As thought is not true. On the other hand, if As thought is not true, it follows that B s thought at t is not true. Since this is the content of As thought, it would follow that As thought at t is true after all. Again, both alternatives appear to le ad to a contradiction. I used thought in its ordinary sense. Thought s are very often understood in a very broad sense, to cover any mental state or psychological experience. Descartes, for instance, takes all intentional state (belief, desire, hope, etc.) to be a form of th inking. Nevertheless, one needs to distinguish the various senses in which one could talk about Liar thoughts, and see exactly which of them indeed generates a contradiction. Some of them do not. For instance, the Liar thought would be really unproblematic if it were understood as the mere act of entertaining a proposition.

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102 Frege distinguishes between merely entertaining a proposition and the act of judging, which is an act of assenting to that pr oposition or of holding-it-true.1 If my thought at t is the act of merely entertaining the proposition that my thought at t is not true, it would be a completely harmless Liar thought. The act of merely entertaining a co ntent is not the kind of entity that has a truthvalue. If at t I merely entert ain the proposition that my thought at t is not true, then my having that thought at t would not be enough to infer that my thought at t is not true. Likewise, no contradiction can be derived from my later judgment that the Liar thought is not tr ue. If at t I judge that my thought at t is not tr ue, then my judgment at t is true. However, this does not entail that the thought I have at t is true, becau se my belief at t and my thought at t differ in precisely that respect that enti tles the former but not the latter to have a truth-value: the psychological mode. Thoughts and Beliefs The Liar thought paradoxes becom e less trivia l when the structure of the mental state includes an assenting psychologica l mode in addition to the gras p of the propositional content. When one believes something, one does not merely en tertain a certain conten t but also assents to that propositional content. It is customary to distinguish betw een dispositional and occurrent beliefs. Normally, dispositional beliefs are considered to be those that I might not be actually aware of (I might have never had them present in my mind), but I have the disposition to assent to if asked. Occurrent beliefs are those that I am currently aware of. Accord ing to this picture, I have an indefinite number of dispositional beliefs, only a few of them (if any) being occurrent at a certain instant of time. Whether this is the right way to draw the distincti on is controversial. It 1 What exactly Frege meant by a judgment is controversial. Ricketts [1996], for instance, argues that the act of judging is to be understood as the act of acknowledging the truth of the content, and that Frege would want to say that one can judge only true propositions. I assumed here the more standard understanding, according to which the content that is judged does not have to be true. See also Kremer [2000] and Ricketts [1986].

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103 has been argued plausibly that the way we use b elief in English suggest s that it only applies to dispositional states2. If so, the proper distin ction should be between t acit and explicit beliefs. I will not engage in this controversy here, but from now on I will use belief to refer to dispositional states and thought to refer to what sometimes are called occurrent beliefs3. A complete account of the mental representation Liars would ha ve to deal with both Liar beliefs and Liar thoughts. I think that the notion of thought is more basic, because dispositional beliefs are characterized in terms of thoughts. Th erefore, I will first give an account of Liar thoughts and then I will apply the account to Liar beliefs. Liar Thoughts One way in which one could block the Liar thought argum ent is to deny that there are Liar thoughts. It is a fact that I have direct psychological evidence that I have a mental state at t, which I have good reasons to characterize as a thought. However, one co uld argue that although it might seem to me that I have a thought at t, what the Liar ar gument shows is that I actually do not have one. The difficulty with denying that there are Liar thoughts can be brought to light when considering empirical Liar thoughts. Ad apting a line of reasoning from Kripke, we can make the case as follows. It is implausible to argue that the mental states that are commonly described as empirical Liar t houghts are, in fact, not thoughts be cause they are paradoxical. In their case paradoxicality is an empirical matter, while being a thought is an intrinsic feature of the mental state. Had the empirical circumstances been different, then there would have been no problem calling that mental state a thought. This mean s that the mental state should also count as a thought under the current circumstances. Therefore, it would be a mistake to deny that there are 2 See Bach [1981] and Lycan [1986]. 3 I will distinguish later between thoughts and judgments and between acts of thinking and acts of judging. Not all thoughts constitute judgments.

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104 empirical Liar thoughts or Pair Li ar thoughts. Notice that this view does not assume an internalist view about thought content. It might seem that the externalist thesis that the content of a mental state might depend on external factors undermines the argument I used to support the thesis that the mental state I have at t is a thought. Nevert heless, even an externalist would have to admit that the mental states being a thought cannot depend on external fact ors. Therefore, it is safe to conclude that there c ould be Liar thoughts. Gappy Thoughts I will assume now that the m ental state I have at t is a Liar thought, T. If the Liar thought is true, it is hard to find a way to avoid a contradiction. The questi on is whether one can coherently maintain that the Liar thought is not true. The Liar argument s hows that there are prima facie reasons to think that one cannot. Su ppose that at t I think that th e Liar thought is not true, and suppose that the thought I have at t, T, is true. The problem is that the thought I have at t and the thought I have at t appear to have the same content. This would mean that in order to avoid the contradiction, one would have to deny the attr active principle that the truth-value of a thought is a function of its content and how the world is. I will argue that there are good reasons to think that the two thoughts, T and T, do not have the same content, but this will not be qui te enough to solve the problem. The challenge will be to argue that the difference in content is suffi cient to determine a difference in truth-value. The idea that the two thoughts ha ve the same content derives from the observation that in both cases, the same thing is thought about the sa me entity. Various traditional accounts of the relation between thought and thoug ht content indeed support the content identity thesis. One could take the two thoughts to be singular thoughts whose contents have the object of thought as a subpart. Since both thoughts are about the same object (the thought at t) and the other belief ingredients appear to be the sa me, then the beliefs would have to have the same content.

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105 Alternatively, one could also think that it is no t the object of thought that is part of the thought content, but a mode of presentati on that is insensitive to contextual features such as the thinker and the time at which the thought occurs. For in stance, the mode of presentation could be the content of a purely qualitative description. This view would also support the idea that T and T have the same content. Both Fregean and Russellian semantics have enough resources to support the idea that, in fact, T and T have different contents. The difference in cont ent can be explained if the contribution made by contextual factors such as the thinker and the time of the thinking event to the content of the thought is taken seriously4. Unfortunately, this is not yet enough to block the Liar Thought argument. What needs to be argued is that there is a differe nce in content that is sufficient to determine a difference in truth-value. It appears to be diffi cult to explain why two thoughts that differ only in that they occur at different instances of time differ in truth-value. For example, one could argue that my thought that Brutus killed Caesar and another persons thought that Brutus killed Caesar do not have exactly the same content, due to some contextual elements 4 For instance Traditional interpretations of Fregean semantics took modes of presentations to be contents of purely qualitative definite descriptions. However, it has been argued (Evans [1997] and Perry [1997]) that a proper understanding of the Fregean semantics is one that makes the mode of presentation sensitive to the thinker, the time of the thinking event and the object of thought. According to this picture, T and T would have to have different contents although they are thoughts of the same person and about the same object, because the mode of presentation of T at t is different from the mode of presentation of T at t. A similar story can be told within a non-Fregean semantic framework. Even if the possession of a thought such as T or T could be reported by using a proper name in the that-clause, as in (1) and (2), (1) At t I was thinking that T was not true. (2) At t I was thinking that T was not true. the contents of these thoughts are not singular propositions that contain T as a proper part. Leaving aside the fact that true fails to have a linguistic meaning, there are good reasons to think that the content of T or T is not identical with the content of the senten ce following the that-clause in the thought report, but rather with the content of a sentence that contains a description of the thought. Ludwig [1996] argues that if the description of the object is purely qualitative, then it might fail to pick out an object. If the universe we live in turns out to be a symmetrical or an infinitely repetitive universe, then the description would apply to more than one entity, so it would fail to pick out a particular object. Therefore, in order for a description to successfully pick out an object, it must be anchored in the self and the present time. This suggests that the content of a thought such as T or T should be identified with the content of a sentence containing a description that is anchor ed in the self and the present time. If this is the case, then it is clear that T and T will have different contents because they contain different descriptions, anchored in different times.

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106 (such as the thinker and the time when the thinking event occurs ) that are part of the thought content. Nevertheless, that diffe rence in content does not make for a difference in truth-value. The principle that requires that these thoughts have the same truth-value is not the thesis that sense (plus the world) determines reference, bu t a principle corresponding to the principle of intersubstitutivity at the level of thoughts. Although T and T have different contents, they involve different modes of presentations of the same entities, so the principle of intersubstitutivity would require that the m odes of presentations can be interchanged salva veritate In the next section I argue that there is more to the difference between the two thoughts than the differences in the description or the m ode of presentation of the object. One can explain why the two thoughts differ in truth-value without violating any intersubstit utivity principle. The difference in truth-value is not the result of a difference in propositional content. Intentional states and th eir propos itional content I will argue that the difference in content between the two thoughts is sufficient to determine a difference in truth-value. The stru cture of a propositional attitude consists in a propositional content an d a certain psychological mode that determines the type of the intentional state. One can disti nguish between thoughts, beliefs, de sires, commands or promises which are different attitudes one could have towa rds the same propositional content. Thus, it is clear that the propositional content is not enough to determine the semantic value of a thought. Moreover, I will argue that the conjunction of a propositional content and a psychological mode might also fail to determine a semantic value for the thought. It could happen for a certain thought to fail to be an act of judging although it is an act of thi nking. In particular, although it is a thought, the Liar thought fails to have what I call an assen tive content, so it fails to have a truth-value. When one has a t hought, one performs an action: the action of thinking a certain content. Therefore, one needs to draw a distinction between what one th inks and what one does

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107 in thinking it, which is parallel to the distinct ion drawn in speech act theory between what one says and what one does in saying it. I will argue that parallel to the way one can talk about infelicitous speech acts, one can al so talk about infelicitous thought acts. In particular, the Liar thought turns out to be an infelicitous thought ac t, and for this reason lacks a truth-value. This type of solution to the Liar thought paradox can be bett er explained by comparing it with the corresponding attempt to solve the Liar statement paradox. One strategy that has been proposed for solving the linguistic Li ar paradox is to say that truth and falsity apply to statements rather than sentences and deny that there is a Liar statement. Thus, Martinich [1983] proposes that Liar paradoxes should be treated not as semantic paradoxes but rather as speech act paradoxes. According to him, the paradox generate d by the Liar statement belongs to the same family of paradoxes as the paradoxes of comma nds, the paradoxes of promises and the paradoxes of bets. Utterances (3) (5)5 (3) I order you not to obey any of my orders. (4) I bet that I do not win this bet. (5) I promise you that I will not keep any promises. are paradoxical, according to him, in the same way as an utterance of (6) (6) This statement is not true. All these paradoxes can be solved, according to Martinich, by saying that the speech act is infelicitous. In particular, when one utters (6), one fails to make a statement (i.e., one fails to make an assertion), because there is an essent ial condition for a speech act to count as an assertion which is not met. Th is essential condition, pointed out by Searle, requires that the 5 See Martinich [1983]. There is a similar paradox of interrogative speech acts, proposed by Lappin [1982]: (Q) Is there a negative answer to this question? where a negative answer is any answer which answers a question, and which contains either negation of its main VP or negation of the main sentence [Lappin 1 982: 574]. For desires, a candidate would be: (D) I desire that my curren t desire remain unsatisfied.

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108 speaker intends that the audience will take his utterance as representing how things are6 [Martinich 1983: 64]. However, Martinich claims, a speaker cannot have th is intention if he utters (6) and knows what it mean s [1983: 64]. In the same way, u tterances of (3), (4) and (5) are infelicitous speech acts, because thei r corresponding essential condition is not met.7 In fact, I think that this stra tegy fails to offer a satisfactor y account of the linguistic Liar paradoxes. For one thing, the strategy cannot be extended to other linguistic Liar paradoxes. Martinich was content with the idea that his so lution does not apply to contingent Liars on the grounds that there is no good reason to expect that contingent a nd non-contingent Liar paradoxes receive the same kind of solution.8 Nevertheless, there are also non-contingent Liar paradoxes which cannot be solved in this way. In partic ular, the strategy does not apply to Pair Liar statements, because the essential condition could be met in their ca se, so one cannot deny that the two speech acts constitute statements. Moreover, the strategy fails to account for other paradoxical statements such as: (7) (7) is either not true or fails to be a statement. If one says that (7) fails to be a statement, a si mple rule of logic would force one to saying that (7) is either not true or fails to be a statement, which would m ean that (7) is true after all. The type of strategy proposed by Martinich is more appropriate to account for the Liar paradoxes that occur at the level of thought acts than for those at the level of speech acts. One can explain the Liar thought pa radoxes as thought act paradoxe s, and one can solve them by noticing that there is a certain infelicity that can be attributed to certain thoughts. 6 This might be spell out as the requirement that the mental state is identified as one w ith a mind-to-world direction of fit and with conditions of satisfaction. 7 Martinich [1983] characterizes his account as a pragmatic solution, but I think that it would be more appropriately called semantic. 8 Martinich [1983: 64-65].

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109 In particular, it can be argued that the Liar thought is in a cert ain sense infelicitous.9 It cannot be denied that when I think the Liar thought I am in a partic ular psychological mode. Nevertheless, the propositional content and the psychological mode fail to determine a truthvalue for the thought. Normally, the notion of conten t is thought of as determining the semanticvalue. Therefore, I will introduce a new notion, th e semantic content of a thought act, which is supposed to determine the truth-value of the thoug ht and consists in its propositional content and what I call its assentive content. The assent ive content is normally identified with the psychological mode of the thought, but sometimes as it happens for the Liar thought the two come apart. For Searle, in order for a mental state to c ount as a belief it must have a mind-to-world direction of fit and a sa tisfaction condition. It is normally t hought that these requirements are met if one is in a state with a certain type of ps ychological mode. Nevertheless, in some rare cases, such as the paradoxical thoughts, there is a conflict between the psychological mode and the propositional content. This c onflict precludes the t hought from having a direction of fit and satisfaction conditions. It would be helpful to draw a distinction between an act of thinking and an act of judging10. The former only requires the right psychological st ate. The latter requires that the act has an assentive content. The Liar thought fails to be a judgment. The Liar thought has in common with the mental state in which one merely entertains a proposition the fact that neither is an act of judging or of holding something true. They di ffer in that the former has a psychological 9 One could plausibly argue that there could be no Liar thought if at the time when the thought occurs I know what my Liar thought is about and I know what its propositional co ntent is. It was for this kind of reason that Martinich thought that there cannot be a Liar statement. Nevertheless, it is hard to deny that one could have a Liar thought if one does not know at the time when the thought occurs what the content of the thought he is thinking about is. 10 Act of judging is a technical term.

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110 component that the latter (the act of merely en tertaining a proposition) does not have. The Liar argument can be blocked by noticing that the thought I have at t does not determine a judgment. One might wonder why one should care about the semantic content and the assentive content of a thought. Why is it not enough to look at its propositional content and psychological mode? The answer is that th e notion of content is normally understood as determining satisfaction condition. This is how Searle thin ks of the content of an intentional state.11 The Liar Thought argument is run on the premise that psychological mode and th e propositional content (together with the facts in the world) are enough to determine the semantic value of the thought. This view can accommodate the in ternalist intuition that the cont ent of an intentional state is an intrinsic feature of it. Th is intuition would require that if two possible worlds differ only in what Bs thought at t is, then As thought that Bs thought at t is not true would have to have the same content in the two worlds. The view I articulated is compatible with saying that As thought would have the same propositional content. The propositional content of the intentional state could be the same regardless of how the wo rld is. However, the assentive content of the intentional state is not an intrinsic property of the state, and it could be different in different possible circumstances. There is a reason why the stra tegy described above is more successful for thought act Liar paradoxes. For both speech acts and thought acts one can distinguish between the propositional content and the type of the act (its force, to use Freges vocabulary). However, there is an important difference. Typically, the aspect of th e speech act that determines its type can be separated from the aspect that contributes the propositional content (the meaning of the sentence in the given context). Most fre quently there is a syntactical force indicator (the syntactical 11 See [Searle 2004: 169].

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111 structure of the sentence, the tense, the presence of some expressions such as promise or please, etc.), but these ar e not reliable indicators.12 A more reliable force indicator is how the speaker intends his act to be recognized by the hearer. Moreove r, the act succeeds, and the intention with which it is performed is fulfilled, if the audience recognizes that intention [Bach 1998]. One can easily imagine Liar statements that are not only intended by the speaker to be recognized as assertions, but are also recognized as such by the audience.13 One cannot deny that the speech act is infelicitous because there is a conflict between its force and its propositional content, because the felicity conditions are met. Therefore, in their case the strategy proposed by Martinich fails. On the other hand, a thought act is felicitous if it succeeds in be ing an act of holding something true. For propositional attitudes, the psychological state that determines the type of the state is inseparable from the attitude itself. In particular, the assentive part (the holdingsomething-true) cannot be separated from the t hought act. Therefore the assentive content of the thought cannot be determined independently of it s propositional content. This makes it possible to explain the infelicity of a thought act as a re sult of a conflict between the two aspects. Liar Beliefs I will now turn to th e case of Liar beliefs, where beliefs are understood as dispositional states. It is hard to imagine a non-contingent Liar belief that is not specified via a sentence or a thought. Nevertheless, there might we ll be contingent Liar beliefs or Pair Liar beliefs. Suppose that A believes that Bs favorite belief is not true and that B be lieves that As favorite belief is 12 To use an example offered by Bach [1998], section 1, by uttering I will call a lawyer one could perform speechacts of various types: predictions, promises or warnings. 13 This happens for Liar thoughts whose paradoxicality depends on some empirical facts that neither the speaker nor the hearer is aware of.

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112 true. Assume that it happens that the two are A s and Bs favorite beliefs. The two beliefs are paradoxical. The solution to this paradox will rely on the solution I proposed for the Liar thought paradoxes. Beliefs can be spelled out in terms of thoughts (or, if one prefer s, occurrent beliefs), so a Liar belief paradox comes down to some variation of a Liar thought paradox. Having a dispositional nature, beliefs cannot be said to be infelicitous. Only acts can be said to be infelicitous. Nonetheless, beliefs could display a dispositional infelicity if they could have thought occurrences which are infelicitous. Take, for instance, As belief that Bs favorite belief is not true. Saying that A has this belief amounts to saying that if asked, A would think that Bs favorite belief is not true. This thought of A would be infelicitous in the same sense as the Pair Liar thoughts.14 This can be seen from the following inferences: (8) If asked, As thought is true. (9) Bs favorite belief is not true. (10) Bs belief that As favorite belief is true is not true. (11) The thought B would have if asked (which is the thought that As favorite belief is true) is not true. (12) As favorite belief is not true. (13) If asked, As thought is not true. This shows that the Liar belief paradoxes ar e actually derived from a paradox at the level of thoughts. This paradox can be solved by sayi ng that As thought would lack a truth-value because it is infelicitous. Therefore, the two Liar beliefs also lack a truth-value, because they display a dispositional infelicity. 14 I take the notion of a true thought as basic, and treat the notion of a true belief as derivative.

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113 Liar Propositions Consider no w the case of propositions, understood in a broad enough sense to include both Russellian propositions and Fregean thoughts. If there is yet a Li ar argument at the level of propositions, then we still have a reason to think that there is something wrong with the concept of truth or with the property of truth. We might be forced to sayi ng that there is no property of truth, and so nothing to be expressed. I will argue that there are after all no Liar propositions that would force us to such a position. If there is a Liar proposition, it is a proposition that repres ents itself as not being true. Could there be a Liar proposition? Of course, the issue arises only if one grants that there are propositions, so I will assume that there are abstract entities of this sort. I will argue that none of our grounds for admitting propositions forces us to admit Liar propositions. Propositions are generally admitted in our ontology for two reasons: we need some entities that, first, play the role of truth bearers better than sentences and, second, serve the role of objects of our propositional attitudes. It is easy to see that the existence of a Liar proposit ion could not be motivated by the first of the above mentioned reasons, because no sentence of English could express a Liar proposition, for truth is inexpressi ble in English. It could be the case that no sentence in any language can express a Liar proposition. Could a Liar proposition be the propositional co ntent of a thought? Suppose that I think at t that the propositional content of my thought at t is not true.15 The propositional content of the thought I have at t appears to be paradoxical, because it can be used to run a Liar argument. If the propositional content is true, then it would also have to be not true. If it is not true, then it would follow that it is in fact true. However, there are ways to block the derivati on of this contradiction. 15 I could also consider the case in which I merely entertain at t the proposition that the proposition I am merely entertaining at t is not true.

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114 The Liar argument assumes that my thought at t has a propositional content, which means that it can be turned into a reductio argument whose co nclusion is that my thought at t in fact has no propositional content. This conclusion might strike one as im plausible on the grounds that if my thought at t is an empirical thought, then we would have had no reas on to deny that it had a propositional content had the circumstances been different. Ne vertheless, there is no good reason to believe that if a thought act has a propositional content it must have a propositional content in all counterfactual circumstances in which it is performed. A further co mplaint is that although one might deny that the thought has a propositional content, it nonetheless has a content which falls short of being a proposition. If this is gr anted, the challenge woul d be to account for the content of a slightly modified thought that I might have at t: I might think that the content16 of my thought at t is not true. If th is content is not a proposition, then it has to be neither true nor false. Therefore, at t I would be justified in thinking that the content of my thought at t is not true. I want to say that my thought at t has a propositional content, which is true. The difficulty is to explain why the thought at t has a proposit ional content while the th ought at t does not have one, although in both cases I seem to be thinking the same thing about the same entity. This might seem surprising, but it has to be accepted as a fact. Something can be called a proposition only in so far as it has a represen tational power. It turns out that th e content of the t hought at t is representational, while the cont ent of the thought at t is not. The difference between the two contents might be explained by the observation that in one case the content is part of what it is supposed to represent. 16 As opposed to the propositional content.

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115 What remains to be investigated is whether there could be a Liar proposition such that there could be no thought or sentence has it as its propositional content.17 Although none of the traditional grounds for admitting propositions would motivate the existence of a Liar thought of this sort, one who takes abstract entities such as propositions and their ingredients seriously might think that there are propos itions that are both inexpr essible and unthinkable. Although there could be propositions that are both inexpressible and unthi nkable, I think that there are good reasons to think that they cannot include Li ar propositions. The ingredients of propositions (concepts, senses of descriptions, etc.) coul d be recombined to obt ained new propositions. Nevertheless, one gets a proposition only if the new abstract entity is a representation. No arrangement of the ingredients would result in a Liar proposition. What is thought to be a Liar proposition cannot be a represen tation, therefore it is not a proposition. One might find it puzzling why some arrangements of the concept of truth and other i ngredients fail to constitute a proposition, but it is hard to see why the concept of truth could be made responsible for this. Therefore, there are no good Liar arguments at the level of propos itions that motivate the view that the concept of truth is inconsistent. Mentalese Liars I will end th is chapter with a brief discussi on of Mentalese Liar paradoxes. According to the representational view of thoughts and beliefs, having a thought or a belief is having a mental or physical representation with a certain propos itional content. What the Language of Thought Hypothesis adds to this picture is the idea that these representations constitute a mental language (Mentalese) that has a syntax and semantics just like natural language s. If this hypothesis is true, 17 One way to imagine a Liar proposition would be to think of some propositional ingredients that could be arranged in the appropriate order to generate the appropriate proposi tion. One ingredient is, certainly, the concept of truth. The nature of the remaining ingredients would depend on ones favorite view of semantics.

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116 then Mentalese appears to have all the ingredient s that are necessary to construct a Liar belief. The Liar belief would be a repr esentation that represents itself as not true. The question is whether the Mentalese Liar paradoxes should be accounted for the way linguistic Liar paradoxes have been accounted for, or like the non-linguistic Liar paradoxes. The fact that we are dealing with a language, albeit a langua ge of thought, might suggest th at Mentalese Liar paradoxes should be solved by applying the inexpressibility account. I will argue that this strategy would be unsuccessful, and that the para doxes should rather be treated like the other non-linguistic paradoxes. To apply the inexpressibility account to the Me ntalese Liars would involve saying that the concept of truth cannot be the content of any pr edicative part of a repr esentation, and that the proposition that the Mentalese Liar is not true cannot be the c ontent of any representation. The difficulty is that once one admits that one can think that the Mentalese Liar is not true, representationalism about thoughts fo rces one to accept that there is a representation of that thought in Mentalese. Moreover, to say that there is nothing that corresponds in Me ntalese to the concept of truth (i.e., the concept of truth is unthinkable) and that th e Mentalese Liar thought fails to have a content is highly implausible, b ecause it is hard to deny that one can grasp the concept of truth. Moreover, one cannot draw a distinction between an intended content and a linguistic content of a thought. Intentionality is an intrinsic feature of a thought, although it is not an intrinsic feature of sentences of English. Th erefore, the intended c ontent of a thought cannot fail to be its content. Notice th at this view does not involve an internalist assumption. Even for an externalist view about mental content, according to which the content of a belief is not an intrinsic feature of the belief, intentionality must be an intrinsic feature.

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117 Therefore, I think that the Mentalese Liar paradoxes shoul d be handled alongside other non-linguistic Liar paradoxes. What one should say is that the Me ntalese Liar is defective in some sense, because it fails to constitute a judgment. This would amount to saying that the thought fails to represent something as being th e case, because it has a self-undermining nature. The psychological evidence cannot guarantee th at the Mentalese Liar is a successful representation. The content of the thought cannot be reduced to the propositional content and the psychological mode.

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118 CHAPTER 6 AN EXTENSION OF THE ACCOUNT: SEMA NTIC VE RSUS LOGICAL PARADOXES I have already argued that the Liar paradoxes sh ow that truth is inex pressible in English. The purpose of this chapter is to investigate to what extent the inexpressibility account can be extended to other paradoxes. Ther e is a plethora of paradoxes that do not i nvolve the concept of truth, including semantic paradoxes, logical pa radoxes, paradoxes of vagueness, speech-act paradoxes, etc. I will only be concerned with thos e paradoxes that are simila r in some significant respect with the Liar paradoxes. If the following principle proposed by Graham Priest [1994: 32] is true (1) Same kind of paradox, same kind of solu tion. (The Principle of Uniform Solution) then a correct account for a particular paradox should be applic able to the paradoxes of the same kind. Priest uses this principle as a test of any proposed solution to a paradox: if a solution of the same type cannot be offered to the paradoxes of the same kind, then the proposed solution is unsuccessful. By all accounts, paradoxes involving semantic notions are similar enough to the Liar paradoxes to demand a similar kind of solution. In th e first part of this ch apter I argue that the inexpressibility account can be extended sm oothly to apply to all semantic paradoxes. Nevertheless, the account cannot be extended to the so -called logical paradoxes. It has been argued that semantic paradoxes and logical paradoxes are paradoxes of the same kind, so the Principle of Uniform Solution woul d require that they have the same kind of solution. If this is the case and the inexpressibility account indeed does not apply to logical paradoxes, then one has a reductio argument against the in expressibility account of the se mantic paradoxes. In the second part of the chapter I argue that semantic pa radoxes and logical paradoxes are not similar in a sense that would require that they have the same kind of solution. Therefore, the fact that the

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119 inexpressibility account cannot be extended to account for the logical paradoxes is not problematic. Semantic Paradoxes I will s tart with a brief survey of the ma in paradoxes that occur in connection with semantic notions other than truth, such as hetero logy, satisfaction, definability, etc. Thereafter, I argue that all these paradoxes can be solved by an inexpressibility account. Grellings Paradox One can distinguish between two categories of adjectiv es: those that are true of them selves and those that are not. By defin ition, those in the first category are called autological, while those in the latter category heterological. Thus, adjec tives such as short or polysyllabic are autological, while long, German, or monosy llabic are heterological. Grellings paradox arises with the question: is het erological heterological? If it is heterological, then it is not true of itself, so it is not heterological. On the other hand, if it is not heterologi cal, then it is true of itself, so it is heterological. The concept of satisfaction leads to a paradox in a way that rese mbles Grellings paradox. One can ask the question: does the expression x does not satisfy itself satisfy itself? If it satisfies itself, then it does not satisfy itself. If it does not satisfy itself, then it would follow that it satisfies itself. Paradoxes of Definability The following three paradoxes (Richards pa radox, Berrys paradox and Knigs paradox) are usually grouped together as paradoxes of definability. Richards paradox List in alphabetical order all pe rm utations of the twenty-six le tters of the English alphabet taken two at a time, followed by all permutations of the same letters taken three at a time, again

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120 in alphabetical order (repetitions of the same le tter are allowed), etc. Everything that can be named by finitely many words is named by some pe rmutation in the list. Eliminate from the list all those permutations that do not denote numbers. The set of numbers that are definable by finitely many words is the set containing the fi rst number defined by a permutation, the second number defined by a permutation, the third, a nd so on. Thus, the set of numbers that are definable by finitely many words, call it E, is a denumerably infinite set. The contradiction is obtained, using Richards own words, as follows: We can form a number not belonging to this set. Let p be the di git in the nth decimal place of the nth number of the set E; let us form a number having 0 for its integral part and, in its nth decimal place, p+1 if p is not 8 or 9, and 1 otherwise. This number N does not belong to the set E. If it were the nth number of the set E, the digit in its nth decimal place would be the same as the one in the nth decimal place of that number, which is not the case. I denote by G the collection of letters between quotation marks. The number N is defined by the words of the collection G, that is, by finitely many words; hence it should belong to the set E. But we have seen that it does not. Such is the contradiction. [Richard 1905: 143] Berrys paradox Berrys paradox was first published by Russell, w ho gives the credit for it to Mr. G. G. Berry of the Bodleian Library1. The number of syllables in English is finite. Therefore the number of descriptions or names under 19 syllab les is finite. Since there are infinitely many natural numbers, it follows that there are numbers that cannot be named or described in less than 19 syllables. The number two can be named, for in stance, by using the on e syllable expression two. Consider then the least integer not nameable in fewer than nineteen syllables It turns out that this number has just been named in 18 syllables. 1 Russell [1908: 153 fn. 3].

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121 Knigs paradox The reasoning behind Knigs para dox has initially been offered as part of a proof that the continuum is not well-ordered. Knig notices that only a denumerable set of real numbers is finitely definable. If the set of real numbers were well-ordere d, there would be a real number which is the least not finitely definable real number The problem is that that number has just been defined by the finite italic ized expression. Therefore, there would be a number that is both finitely definable and not finitely definable. K nigs conclusion was that the set of real numbers cannot be well-ordered2. Shortly after Knig made his proof public, due primarily to Zermelos proof that every set can be wellordered, but also to Richards pa radox that does not appeal to the notions of ordinal or well-ordening, Knigs conclusion had to be rejected. His argument towards a contradiction turned into a paradox wh ich is sometimes known as the Zermelo-Knig paradox. A Denotation Paradox There are other paradox es that belong to the paradoxes of definability. It is worth mentioning Simmons denotation paradox that resembles the pa radoxes above, but is mean t to make the selfreferential structure more explicit.3 Suppose the following expressions are written on the board in room 1014: The ratio of the circumference of a circle to its diameter. The positive square root of 36. The sum of the numbers denoted by expr essions on the board in room 101. 2 Knig [1905: 147-48]. 3 In addition to this, Simmons offers an example of a denotation paradox that does not have a self-referential structure. This paradox is supposed to stand to the self-referential denotation paradox as Yablos paradox stands to the Liar. See Simmons [2002] 4 Simmons [2002].

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122 The question is what is the denotation of the third definite description? The Inexpressibility of th e Semantic Concepts The above sem antic paradoxes can be solved in the same way as the Liar paradox. Consider, for instance, heterol ogy. Unlike the concept of truth, th e concept of a heterological term is not one that is used in everyday thought s. It is a concept that has been introduced by mathematicians and logicians for certain purpos es. However, heterology is normally thought of as a concept that is easily gr asped by ordinary speakers. Just as in the case of truth, the argument that leads to an inc onsistency involves the assumption that there is a predicate of English, H that expresses the co ncept of heterology. If H expresses the conc ept of heterology, then at least part of the concept of heterology should be captured by the following H-schema, (H-schema) P is H iff P is not P. where P is replaced by predicat es of English. Given that H is supposed to be a meaningful predicate of English, one can infer the following biconditional (2) H is H iff H is not H. which is an instance of the H-schema. Since H is neither vague, nor partially defined, (2) could also be turned into an explic it contradiction. Therefore, the a ssumption that H expresses the concept of truth leads to a contradiction and must be rejected. Moreover, no predicate of English can express the concept of heterology. Consequently, the predicate heterological fails to have a linguistic meaning. Since it lack s linguistic meaning, strictly speaking it does not apply to anything, not even to itself, therefore it is heterological. Nevertheless, the sentence Heterological is heterological is not true but rather lacks linguistic meaning. Of course, one needs again an answer to a poten tial self-defeat object ion, but the explanation would be the same as the one I offered in chapter three for truth.

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123 There is no need for thinking that the concept of heterology is incons istent, as there was no reason to say that the concept of truth is inconsistent. By anal ogy with truth and truth*, one can distinguish between heterology and heterology*. If one allows concep ts to be inconsistent, then heterology* would be an inconsistent concept and, at the same time, inexpressible (as it also is the case with truth*). The meaning of heterology* is implicitly defined by the following schema (H*-schema) P is heterological* iff P is not P where P is replaced by predicates of English. The same explanation can be extended smoothly to the groundedness, satisfaction and definability paradoxes as well as to any other semantic paradoxes, so there is no need to go over each particular inexpressibility argument. Noneth eless, some comments are worth being made in connection with the definability pa radoxes. In their case it is not obvious what semantic concepts are involved. This is due, at leas t in part, to the lack of a uni form terminology. There are various semantic expressions that are used in the formula tion of the definability paradoxes, most of them in a misleading way: definable, finitely defina ble, nameable, denote and can be written with finitely many words. Which of the corr esponding concepts is the one that should be claimed to be inexpressible? In order to be able to answer this question, one first needs to investigate which semantic concep t (or concepts, if there are more) is in fact intended to be captured by those rather misleading semantic expressi ons. Notice that the enti ties that are said to be definable are numbers. Moreover, a number is said to be definable in English if there is an expression of English that denotes or refers to that number. Definability is more commonly thought of as a property of expressions (or concepts): an expression of L is definable in L if and only if there is an expression of L that is synonymous w ith it. Thus, the paradoxes of

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124 definability are improperly called so. It w ould also be wrong to call them paradoxes of nameability, because they involve the notion of refe rence in a broader sense, to include definite descriptions. On the other hand, unrestricted re ference or denotability would not do, because demonstratives, or contingent definite descript ions such as Harrys favorite number, would trivialize the issue. The property that seems to be relevant to the de finability paradoxes is being denotable by some non-conti ngent singular referring term Let us call the property and the corresponding concept specifiability The definability paradoxes are solved by saying that the concept of specifiability cannot be expressed in English. Expre ssions such as the smallest number not specifiable in less than 19 syllables lack linguistic meaning and fail in fact to specify a number. Peanos idea that paradoxes su ch as Richards paradox belong to linguistics rather than mathematics is correct.5 Similarities between the Semantic and the Logical Paradoxes The inexpressibility account does not seem to be applic able to Russells paradox. The account would only enable one to deny that the c oncept of a class is e xpressible in English. However, even if the concept were not expressi ble in English, there woul d probably still be an extension of the concept. There w ould also have to be an extensi on of the concept of not being a member of itself. This would be enough to generate a paradox. Therefore, Russells paradox does not depend on any expressibility assumptions. This might appear to be in conflict with the fact that there are some similarities between the semantic paradoxes and the logical paradoxe s. Nevertheless, I will argue that these similarities are not as strong as to require that the two group s of paradoxes should receive the same kind of solution. I will start with a brief re view of Russell and Ramseys accounts of the 5 See also Ramsey [1990: 184].

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125 relation between the semantic and logical paradoxes. Thereafter, I will present Priests argument in favor of a uniform treatment of the paradoxe s, followed by my reject ion of the uniformity account. A Little Bit of History Russell believes that all paradoxes of se lf-reference share a common structure: In each con tradiction something is said about all cases of some kind, and from what is said a new case seems to be generated, which both is and is not of the same kind as the cases of which all were concerned in what was said. [Russell 1908a: 154] Consequently, he proposes that th ey should receive the same kind of solution, one that relies on his well-known vicious circle prin ciple. Russell introduces the viciou s circle principle in order to prevent the construction of expre ssions that involve vicious circ ularity, which leads him to the development of the ramified type theory. Ramsey argues that Russell is wrong in thinki ng that the two groups of paradoxes have the same structure, and claims that they belong, in fact, to two fundamenta lly distinct groups of paradoxes (the label that was current at that time is contradictions ) [Ramsey 1990: 183]: A. 1. The class of all classes whic h are not members of themselves. 2. The relation between two relations when one does not have itself to the other. 3. Burali Fortis contradiction of the greatest ordinal. B. 4. I am lying 5. The least integer not nameable in fewer than nineteen syllables. 6. The least indefinable ordinal. 7. Richards Contradiction. 8. Weyls contradicti on about heterologisch.6 6 It seems that Ramsey has misattributed Grellings paradox to Weyl.

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126 The first paradox in group A is Russells paradox, the well known paradox he communicated to Frege in 1903. The class of all classes which are no t members of themselves either is or is not a member of itself. However, if it is a member of itself, then it is not. If it is not a member of itself, then it is. Therefore, both options lead to a contradiction. The second paradox in group A is an extension of Russells paradox from sets to rela tions. As for the Burali-Forti paradox, it has been stated by Russell as follows: It can be shown that every well-ordered series has an ordinal number, that the series of ordinals up to and including any given ordinal exceeds the given ordinal by one, and (on certain very natural assumptions) that the series of all ordinals (in order of magnitude) is well-ordered. It follows that the series of all ordinals has an ordinal number, say. But in that case the series of all ordinals including has the ordinal number + 1, which must be greater that Hence is not the ordinal number of all ordinals. [Russell, 1908, p. 154] Paradoxes in group A are usually called logical pa radoxes (in fact, they would more properly be called mathematical or set-theoretical paradoxes), while those in group B semantic paradoxes, although this is not Ramseys terminology. The difference between them consists in the fact that [The contradictions of group A] involve only logical or mathematical terms such as class and number, and show that there must be so mething wrong with our logic or mathematics. But the contradictions of group B are not purel y logical, and cannot be stated in logical terms alone; for they all contain some refere nce to thought, language, or symbolism, which are not formal but empirical terms. So they ma y be due not to faulty logic or mathematics, but to faulty ideas concerning thoug ht and language. [Ramsey 1990: 183-4] Ramsey believes that the motivation Russell has for developing a ramified type theory is that he wants his theory to also provide a solution to the semantic para doxes. He argues that there is no point in trying to offer a unified account, because the two groups of paradoxes have a fundamentally different structure. Thus, the logical paradoxes should rather be solved by a simple type theory, while the semantic paradoxes w ould have to be accounted for in a totally different way. In any case, according to Ramsey, solving the semantic paradoxes should not be the business of mathematics, and should not require a solution to the logical paradoxes.

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127 Priests Uniformity Account Leaving aside the difficulties related to the notions of kind, both Russell and Ram sey appear to be committed to the Principle of Un iform Solution, but they disagree about whether paradoxes in the two classe s share the same underlying structure. Priest argues that Russell was right and Ramsey was wrong [Pri est 1994: 25]. Contrary to Rams eys suggestion that logical and semantic paradoxes are two different kinds of paradoxes, Graham Priest [1994; 2002] argues that they have in fact the same underlying structure. More specifi cally, he argues that (3) The Inclosure Schema is the structur e that underlies both logical and semantic paradoxes. Priest uses (3) together with the Pr inciple of Uniform Solution and (4), (4) Same underlying structure, same kind of paradox. which is supposed to be intuitivel y true, to show that the solutions that have been proposed in the past for both the semantic and the logical para doxes are wrong. He contends that since the orthodox solutions to the paradoxes of one kind are not applicable to the paradoxes of the other kind, the principles above are sufficient to sink virtually all orthodox solu tions to the paradoxes [Priest 1994: 33]. Since the inexpres sibility account is not applicab le to the logical paradoxes, it would follow that it is not even appropriate as a solution to the semantic paradoxes. Only Priests own dialetheist account is supposed to pass the test because it is the only one that applies equally well to both kinds of paradoxes. (3) is supposed to be estab lished by an examination of all major semantic and logical paradoxes and by noticing that they all share an underlying stru cture. Although paradoxes in group A involve the notion of a set, while those in group B involve a refe rence to thought and language, this is, accordi ng to Priest, only a superficial dist inction. Upon investigation, one can identify a common underlying structure, which is a slight modification of a schema discovered

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128 by Russell. Priest takes Russell to have already identified a structure that is common to all paradoxes in group A. They all display what Priest [2002: 129] calls Russells Schema: For given properties and function : 1. = y : (y) exists 2. if x is a subset of : a) (x) x and b) (x) Therefore, ( ) and ( ) The first clause is the Existence clause, and guara ntees that there is a totality of objects of a certain sort, while the second requires that Trans cendence (2a) and Closure (2b) hold for every subset of the totality guarantee d by the Existence clause. The function (x) is called a diagonaliser, because it is constructed in a way that resembles the technique of diagonalization made popular by Cantors theorem. Transcenden ce guarantees that the va lue the diagonaliser assigns to a subset, x, of does not belong to x, while Closure ensures that this value remains a member of the totality guranteed by the Existence clause. When the subset is itself, a contradiction can easily be derived. For Russells paradox, (x) is x x, is the Russell set, {y; y y}, and is the identity function. Priest [2002: 134] argues that although paradoxes in group B fail to satisfy Russells Schema, both para doxes of type A and B satisfy what he calls the Inclosure Schema (in [Priest 1994] it is called the Qualified Russells Schema): For given properties and and (possibly partial) function : 1. = x : (x) exists and ( ) 2. if x is a subset of su ch that (x): a) (x) x and b) (x) Therefore, ( ) and ( )

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129 The function has been added to the schema in order to ensure that Closure holds for paradoxes in group B. For semantic paradoxes, the mere fact that x is a subset of does not guarantee that (x) belongs to In particular, for Knigs paradox (x) is x is a definable ordinal, while (x) is the smallest ordinal that is not in x. If x is not definable, then there is no guarantee that (x) is a definable ordinal (i.e., ther e is no guarantee that Closure hol ds for x). Thus, the diagonaliser must be restricted in some way (to definable subsets of to sets that are definable in 19 words, etc.), which is why the additional function is needed. For the Liar paradox, (x) is x is true, (x) is x is definable, and is the set of true sentences. is a function constructed by diagonalization: (a) = were = < a> (the sentence (a) is supposed to say about itself that it is not in a). No similar restriction is needed for the logical paradoxes, but one can easily adjust them to fit the Inclosur e Schema. For Russells paradox is the universal property x = x. Priest also shows how various choices of and capture precisely the structures displayed by other paradoxes in groups A and B. It should be noticed that the need to add a restriction to definable sets for semantic paradoxes raises a worry with respect to Priest s uniform account: the universal property appears to be a mere ad-hoc addition that does not pl ay any role in Russell s paradox, while the definability requirement plays a genuine role in the semantic paradoxes. Priest takes this to be only an insignificant difference that has to do with the particular way in wh ich the schema of the paradox is obtained, rather than with the schema itself. There is only a difference in the class of objects the function ranges over. The fact that for seman tic paradoxes one has to restrict the function to a subset of the power set of is, in his view, only an accidental feature. In fact, according to Priest [2002], the In closure Schema does not only underlie the semantic and logical paradoxes. It is supposed to capture the structure of a much wider class of

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130 contradictions at the limits of thought that can be found in the history of Western philosophy (from Cratylus to Kant, Hegel, Russell, Heidegge r and Derrida), and in the Eastern tradition as well (N g rjuna). The Semantic Version of Russells Paradox Priest is not the only one w ho has m isgivings with respect to Ramseys position. Simmons [2002] also makes an attempt to blur the dis tinction between semantic and logical paradoxes. However, while Priest tries to reconstruct type B paradoxes in such a way that they reproduce the structure of the logical paradoxes, Simmons propos al is quite opposite. He tries to reconstruct Russells paradox in a manner that is slightly different from its or iginal form, and make it similar to the semantic paradoxes. Simm ons argues that Russells paradox is more appropriately treated as a semantic paradox. It is true that the pa radox involves in an esse ntial way the notion of extension, but extension should not be conceived of, according to Simmons, as reducible to the notions of a set or class. Extens ion should rather be re lated to the notion of predication. This is the reason why the paradox should count as a semantic paradox. For Simmons, Liar-like paradoxes, definability paradoxes and Russells paradoxes belong to the category of semantic paradoxes. The three categories are subcategorie s of the semantic paradoxes, and can be distinguished by the type of semantic value that is essentially involve d in the paradox: an extension, an object, or a truth value. Although there are important differences between them, they are subcategories of the same category of paradoxes, which suggests that they should be solved in the same way. Simmons believes that they all can be solved by a contextualist approach, in particular, his singu larity theory of the paradoxes. I think that Simmons view does not pose a serious threat to the inexpressibility account. The handling of the extensiona l version of Russells paradox w ould depend on how exactly the notion of extension is understood. If extensions are understood as totalities, albeit different from

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131 the traditional totalities of set-theory, then th e inexpressibility view w ould indeed not apply to Russells paradox, but there is no good reason to expect it to a pply, because Russells paradox would deal with totalities (and could be solved by denying that there are totalities of a certain sort), while the semantic paradoxes do not. On th e other hand, if the notio n of extension can be thought of in terms of predication only, independen tly of the notion of a totality, then there might be good reason to require a uniform solution, but the inexpressibility account would be applicable to Russells paradox. I th ink that Simmons is rather committed to the former view. He seems to think that extension should still be thought of as a totality (the collection of objects that satisfy a certain predicate), although this totality is no longer part of to the iterative/combinatorial universe of sets. Simmons own characteri zation of the alternative notion of extension is this: to any predicate that denotes a well-determi ned condition or concept (in Freges sense), such as the predicates abstrac t or set or class, there corresponds the collection of those things to which the predicate applies the collection of abstract th ings, the collection of sets, the collection of classes. Call this the predicative conception and call the collection of things to which a gi ven predicate applies the extension of the predicate. [Simmons, 2000, p. 116] Anyway, it is still assumed that for any meaningful predicate there is an entity (a collection of objects) that is the extension of that predicate. This is why Russe lls paradox survives even after the assumptions that there is a class of all classes, a set of all sets or of all self-membered sets is dropped. Russells paradox reoccurs as a paradox of extensions. Notice that the reason why the paradox survives is that one sti ll assumes that there is an entit y that is the extension of the predicate. The contradiction no longer occurs if the assumption that extension or selfmembered extension have an extension is droppe d. Of course, the fact th at some predicates fail to have an extension might still strike us as surprising. However, it is hard to think that it is part of the concept of a predicate that all predicates have an extension. If one thinks of predication in

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132 purely semantic terms, then one can avoid the contradiction. Thus, Simm ons idea that Russells paradox of extensions should count as a semantic paradox remains implausible. Even if one accepted Russells paradox of extension as a se mantic paradox, one should still count Russells paradox of sets or classes as a logical one. This suggests that some paradoxes might be amenable to two different formulations, such that there is both a semantic and a logical version of the paradox. Refuting the Uniformity Account The goal of this section is threefold. I first argu e that Priest fails to show that (3) provides a non-question-begging m ethod to sink virtually all orthodox solutions to the paradoxes. This is because he fails to prove that one could uncove r the structure underlying a certain paradox in the absence of a solution to it. Second, I argue that it is not the case that the Inclosure Schema is the schema that underlies the semantic paradoxes (in pa rticular, it is not the schema that underlies the Liar paradox). Third, I argue th at the temptation to treat sema ntic and logical paradoxes as sharing the same underlying structur e is, at least in part, due to the illegitimate identification of semantic paradoxes with their logi cal counterparts. I will first pres ent Priests arguments for the uniformity of semantic and logical paradoxes. An Objection from Circularity In this sec tion, I argue that Priest fa ils to show that (3) can be used to sink virtually all orthodox solutions to the paradoxes Unless one can provide a crit erion for establishing (3) independently of having found a solu tion to both paradoxes, the use of (3) to refute a certain type of solution to a paradox is question begging. Priest does not provide a crite rion of this sort. Smith [2000] argues that Priests argument that logical and semantic paradoxes should receive the same kind of soluti on fails because of an equivocation on the notion of kind. The difficulty stems from the fact that two object s can be of the same kind at some level of

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133 abstraction and of different kinds at another level of abstrac tion [Smith 2000: 118]. At some level of abstraction the logical paradoxes and the semantic para doxes are of different kinds: the former mention sets, the latter do not. At a different level of ab straction they are of the same kind, because both display the same Inclosure Sc hema. However, Priest describes the orthodox solutions [] at a far more concrete level than the level at which [] he describes paradoxes [Smith 2000: 118]. At the level of abstraction that corresponds to the Inclosure Schema the orthodox solutions can be said to be of the same kind: both are Inclosure Schema circumventers. Priest agrees that when one talks about para doxes being of the same kind, the notion of a kind is relative to a degree of abstraction [P riest 2000: 123]. However, he points out that when he talks about paradoxes being of the same kind, he has a certain level of abstraction in mind: the level of the underlying stru cture that generates and causes the contradictions [Priest 2000: 125]. Thus, the structure he is in terested in is not an arbitr ary structure, but the underlying structure of a paradox, i.e., the structure that generates or causes the paradox. As he further qualifies, causation should not be understood as physical causati on, but rather as explanation. Thus, paradoxes are said to be of the same kind in the sense that they have the same explanation. Priest acknowledges that it is not an easy task to characterize the noti on of explanation, but claims that an intuitive grasp of this noti on is enough for the points he wants to make. By choosing the structure to be not any old structure but an underlyi ng structure that generates and explains the paradox, Priest wants to make sure that paradoxes of the same kind indeed have the same kind of solution. The problem for Priests account of an underlying structure is that there is a sense of explanation that makes it the case that in orde r for one to know the explanation of a certain

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134 paradox, one would need to know its solution.7 Chihara, for instance, takes the problem of explaining a paradox to be nothi ng else than the problem of finding a diagnostic (solution): The problem of pinpointing that which is decei ving us and, if possible, explaining how and why the deception was produced is what I wish to call the dia gnostic problem of the paradox. [Chihara 1979: 590] Priest himself seems to think that having an explanation of a paradox presupposes having a solution to that paradox. Evidence for this is the fact that for him (1) is little more than a truism [Priest 2002: 287]. It is hard to see why the principl e would be trivial if having the same explanation does not entail having the same so lution. Another piece of evidence is Priests discussion of the following three paradoxes of infinity [Priest 2002: 288]: (i) If the world is infinite in time past th en the number of days before today is equal to the number of days before yesterday. But there are obviously fewer of these. (ii) If the world is infinite in time past th en the number of months before now must be twelve times the number of years before now, but this is already infinite, and so is as large as it can be. (iii) There are more natural numbers than even natural numbers; yet there must be the same number of each, since they can be put into one-to-one correspondence. Here is Priests comment on these three paradoxes: With the wisdom of hindsight, however, we can now see that they are the same kind of paradox. This is because they are all exampl es of a single phenomenon, namely, that an infinite set can have a proper subset that is the same cardinal size as itself. This fact also provides the solution to all the paradoxes. [Priest 2002: 289] Thus, for him, the explanation of the paradoxes o ffers at the same time a solution to them. It is easy to see why this notion of an explanation is problem atic for Priests claim that virtually all orthodox solutions to the paradoxes can be proved wrong by establishing (3). One could not establish that two paradoxes are of the same kind and require a uniform solution, 7 Solution here must be understood in a wide enough sense: it should cover not only accounts that indicate a way to block the argument toward contradiction, but also those accounts that take the derivation of a contradiction to be inescapable. I will keep using (following Priest) the word solution, although Chiharas notion of diagnosis would be more apt in this context.

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135 unless one already established that they have th e same kind of solution. Using (3) to reject a nonuniform solution to a certain paradox (i.e., a solu tion that cannot be exte nded to the paradoxes in the other category) would amount to begging the que stion against the propone nt of that solution, because one who entertains a non-uniform solu tion to a certain kind of paradoxes would not acknowledge that logical and semantic paradoxe s are both generated by the same underlying structure. This is actually a general objection th at applies to any schema that one might consider to be the structure of a paradox. In particular, in order for (3) to enable one to sink virtually all orthodox solutions it must be possible to establish that the Inclosure Schema underlies a certain paradox without having a solution to it. Priests explanati on of why the Inclosure Schema is the schema that explains and generates the logical and semantic paradoxes fails to meet this condition. The misleading suggestion that sharing the Inclosure Schema is enough to guarantee that th e solutions are of the same kind comes from reading too much into the comparison between the diagonaliser and a mechanism that generates an effect. Presumably, in order to prevent the effect from occurring, one would have to remove the mechanism that generates it. Priest draws heavily on this comparison to motivate (3). According to him, a ll inclosure contradictions are generated by the same underlying mechanism; also, the diagonaliser manages to lever itse lf out of a totality [Priest 2002: 289] or to operate on a totality of objects of a certain kind to produce a novel object of the same kind [Priest 2002: 136]. Neve rtheless, the comparison becomes illegitimate if one puts too much weight into the words gener ates or produces. The diagonaliser would be conferred the magic power of maki ng inconsistent principles that are otherwise innocuous and to bring novel objects into existence. Of course, none of these powers can be attributed to the diagonaliser. The fact that there is a technique that enables one to derive a contradiction from a

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136 number of intuitively true princi ples does not show that that technique is responsible for (or the cause of) the contradiction, but only shows that it can be used to bring into light the inconsistency of those principles One needs to provide additional reasons for thinking that the Inclosure Schema is the structure that generates the paradoxes. In the absence of these additional reasons, it might be the case that the technique is no more responsib le for the contradiction than a thermometer is responsible for a patients high fe ver. This provides us no reason to think that two paradoxes that share the Inclosure Schema could not be solved in different ways: one by rejecting Existence and the other by rejecting Closure for or, if one chooses to go that far, by embracing dialetheism. It is not my intention to deny that there ar e criteria for establishing that two paradoxes should have the same kind of solution in the absence of a solution to each. Th e point is that Priest fails to provide a non-question-begging criterion. Although it is not the pur pose of this paper to identify non-trivial criteria fo r paradoxes having solutions of the same kind, it seems more promising to relate them with the principles that appear to be inconsistent rather than with the mechanism that is used to br ing to light the inconsistency. The Liar and the Inclosure Schema In this sec tion I argue that the Inclosure Schema is not the structure th at underlies the Liar paradox and does not capture in a ny significant sense the essence of the phenomenon. Regardless of what the precise characteriza tion of an explanation is, it sh ould capture the essence of the phenomenon [Priest 2000: 24], not simple accidents. It turns out that the Inclosure Schema contains elements that are not essential to the Liar paradox and that ther e are essential elements of the Liar paradox that are not represented in the Inclosure Schema. This is enough to show that the Inclosure Schema fails to explain or to captu re in any significant sense the essence of the paradox. Moreover, it follows that the distinction between the two groups of paradoxes is not

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137 merely superficial, and that Ramsey was actually right in drawing a sh arp distinction between them. There is no good reason to expect them to receive the same kind of solution. In order to get clear on the struct ure of the Liar paradox, it is us eful to consider an informal version of it. Let (L) be the following sentence: (L) (L) is not true. The informal argument goes as follows. If (L) is true then it follows that (L) is not true, because (L) says that (L) is not true. Theref ore, one has to infer that (L) is not true. But, since this is what (L) says, (L) would have to be true after all. It can be noticed that this formulation makes no explicit appeal to there being a set of true senten ces. Nor does it appeal to the thesis that certain subsets of the set of true sent ences of English, including the set of true sentences of English itself, are definable in English (in the sense that there is a referring term or a predicate of English which refers to those subsets). This suggests that none of these theses should be made part of the explanation of the Liar paradox, because the Liar paradox survives even if one or both theses are rejected. On the other hand, other elements that are essential in the generation of any version of a Liar paradox, such as the T-schema and the assump tion that the Liar sentence says that the Liar is not true, do not show up in the Inclosure Schema. The Existence clause Part of the s tructure that generates the Liar paradox is, according to Priest, the Existence clause, namely, the assumption that there is a set of true sentences of E nglish. It is undeniable that the existence of a set of true sentences of English has no less prim a facie intuitive support than the thesis that there is a set of all sets that are not members of themselves. Nevertheless, there is an important difference: the Liar paradox survives afte r the Existence thesis is dropped, while the logical paradoxes do not. The orthodox solution to Ru ssells paradox is to deny that there is a set of all non-self-membered sets. Th e contradiction vanishes once the assumption is

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138 dropped. This does not mean that the orthodox soluti on is unproblematic, but at least in principle the argument is blocked if the set existence is denied. Unlike Russells paradox, the Liar paradox survives after the Existence thes is is dropped. The informal version of the Liar paradox, as well as the most familiar formal versions of it, do not actually make use of this assumption. As Tennant [1998: 28-29] notices, the Liar paradox appeals to a predicate, true, not to a set of true sentences. The fact that the contradiction is derived from weaker principles, and that the assumption of a set is not needed, is what ma kes semantic paradoxes particularly difficult to solve. Priest attempts to answer this objection by sayi ng that once one is willing to talk about true sentences of English, one is thereby committed to there being a set of true sentences of English. For him, there is a conceptual connection be tween satisfying a conditi on being true and being a member of a certain totality being one of the totality of true things [Priest 2002: 279]. Moreover, the reason that is co mmonly used to explain why th ere is no set of all non-selfmembered sets (the fact that the collection is to o big) cannot be applied to deny the existence of a set of true sentences of English, because there can only be countably many true sentences of English. Priests defense remains unsatisfactory for tw o reasons. First, the idea of a conceptual connection between satisfying a condition and being a member of a certain totality should not be taken as granted. Even if there could not be mo re than countably many objects that satisfy a certain condition (a meaningful predicate), there are ways in which the condition could fail to determine a totality. This might happ en, for instance, if the predicate turns out to be vague or if the meaning postulates associated with it are inconsiste nt with the facts. Second, even if one grants that there is a conceptual connection be tween a condition and ther e being a totality, it is

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139 not clear why the thesis that there is a totality of true sentences of English should be made part of the schema. In general, it does not seem to be the case that if A is part of the explanation of a certain paradox, and A entails B, then B is also part of the explanation of that same paradox. The diagonaliser and the Liar In this sec tion I will assume, for the sake of the argument, that there is a set of all true sentences of English and argue that the diagonaliser fa ils to play the role Prie st ascribes to it in the Liar paradox. Priest presents the Liar para dox as if there were a general mechanism (the diagonaliser) that for any definable subset of (the set of all true se ntences of English) makes both Transcendence and Closure true. When this general mechanism is applied to the set itself, a contradiction is obtained. Tennant [1998: 29] points out that the general mechanism is relevant only at the limit, i.e. at the level of Whether Transcendence and Closure obtain for other subsets of is completely irrelevant to the paradox. It is undeniable that the limit case is enough to obtain a contradiction. Neve rtheless, Priest counters Tenn ants objection by saying that by focusing only on the limit case one would miss complete ly an essential part of the story [Priest 2002: 279]. The reason is that the paradoxes are produced by a certain mechanism, and only when one understands this, one understands why contradictions of this kind arise [Priest 2002: 279]. In a previous section, I argued that it woul d be illegitimate to s upport the id ea that the Inclosure Schema generates the paradoxes by assi milating the role played by the diagonaliser with that of a causal mechanism. Priest tries to defend the idea that it is the Inclosure Schema that generates (and thus explains) the paradoxes by arguing that the diagonalisers are such that

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140 there is a genuine functiona l dependence of the value of the function on its argument: the argument is actually used in computing the va lue of the function [Priest 2002: 136, fn. 18].8 I will first argue that in the case of the Liar paradox the diagonaliser fails to provide the genuine functional dependency that would justify the thesis that the Inclosure Schema is the underlying structure that genera tes the Liar paradox. There is no good reason to make the requirement that Transcendence and Closure hold for subsets of the set of true sentences of English part of the underlying structure of the Liar paradox. Second, I will argue that the Inclosure Schema does not even hol d true for the Liar paradox. A closer look at the Liar paradox reveals that it is hard to identif y a genuine functional dependency that could be made responsible for the contradiction. U nderstood in a certain way, functional dependency is quite easy to get. Ande rson & al. [1992] take one-to-one functions to offer the paradigmatic case of genuine mathematical functional dependency. One-to-one functions are easy to construct, but it is obvious that more is needed in order to make the function in any sense responsible for generating the contradiction. In order to give credibility to the idea that the function is indeed responsible for generating the contra diction, one should take seriously Priests idea that the argument should actually be used in computing the value of the function.9 However, appearances to the contrary notwithstanding, the diagonaliser does not provide a genuine functional dependency, because the argument is not used in computing its value. Recall that in th e case of the Liar paradox is supposed to be the function (a) = were 8 Priest [2002: 136, fn. 18] refers to [Anderson et al. 1992] for more details about functional dependency. To his credit, he acknowledges that the problem of being able to tell when a functional dependency is genuine is a tricky and unresolved problem. 9 What Anderson et al. [1992] call syntactic dependency might come closer to what Priest has in mind. A function (formula) M depends syntactically on a variable x just in case M can be seen by syntactic inspection to be semantically strict in the variable x [Anderson et al. 1992 : 397]. A function (formula) M is semantically strict in the variable x just in case, if the value of x is undefined, so is the value of M [ibid. 397].

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141 = < a>. It should be noticed that Priests characterization of the function is ill-formed, because it involves a use-mention confusion: the fi rst occurrence of a is in the material mode, while the second is in the formal mode. Therefore, it is not clear which function Priest has in mind. One would have a genuine functional de pendency if the function were understood as ranging over expressions that refer to sets of true sentence s, rather than over sets of true sentences. However, in that case the structure of the Liar paradox would clearly be different from the Inclosure Schema, because the Inclosure Schema requires that range over a subset of P( ). The alternative is to interpret as a function that takes a set of true sentences, s, into a sentence that contains an expression that refers to s. That there is such a sentence for each set is guaranteed by the fact that the sets are restricted to definable sets of true sentences. The axiom of choice guarantees that there is a f unction that takes every definable se t of true sentences, s, into a sentence (s) = where = < a> and a is an expression that refers to the set s. would be in this case a function that ranges over a subset of P( ), the way Priest wants, but it can be noticed that the argument is not actually used in computing the value of the function. This shows that the diagonaliser cannot be made responsible for the contra diction; no good reason has been provided for thinking that the In closure Schema underlies in a ny significant sense the Liar paradox or that it captures an essential part of the story. The second point is that, far from being the underlying schema that generates the Liar paradox, the Inclosure Schema does not even hold true for the Liar paradox. Although the mere definability of a set is enough to guarantee Transcendence, it is not enough to guarantee Closure. It is hard to deny that Trans cendence and Closure hold for the set of true sentences itself, but this does not follow only from the fact that is definable. According to Priests own definition, something is definable iff there is a (non-indexical) noun-phrase that refers to it [1994: 28]. If

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142 refer is used in a wide enough sense, to also cover the relation betw een a predicate and its extension, then the definition is co mpatible with a set being definabl e if there is a predicate that has that set as its extension. Neve rtheless, Priest seems to think of a set to be definable only if there is a name or a definite desc ription in English that has that se t as its referent. Let s be a set of true sentences and (s) = where = < a> and a is an expres sion that refers to s. Transcendence would hold, because one can prove that (s) cannot belong to s. If it did, then it would be true, so it would follow th at it does not belong to the set a refers to, which is s. I will argue that Closure, namely the thesis that (s) is true, cannot be derived unless one makes some additional assumptions, such as the assumpti on of bivalence, or the assumption that (s) really says that (s) does not belong to s. One lesson that can be drawn from Kripke [ 1975] is that there can be languages that contain their own truth predicate. The price that must be paid for there being such a predicate is that some things cannot be said in that langu age, because some sentences would lack a truth value. There is no good reason to th ink that English could not contai n a similar predicate, true*, that represents the set of true sentences of Eng lish. If there is no such predicate in English, one could certainly add it to the langu age. In this case, the set of true* sentences would ensure the definability of the set of true sentences of English. Consider now the following sentence: (L*) (L*) does not belong to the set of true* sentences. (L*) would not belong to the set of true sentences of English (T ranscendence holds), but it would be inappropriate to infer that (L*) is true (Closure is not guaranteed). Thus, the definability of a subset of true sentences, a, is not enough to guarantee th at Closure holds (i.e., that (a) is true). What this shows is that the diagonalise r does not actually guarantee that both Transcendence and Closure hold for all definable subsets of which means that the Inclosure

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143 Schema is not an accurate representation of th e Liar paradox. In order to save the Inclosure Schema, one would have to further restrict to an even smaller subset of P( ). It might be held that if is restricted to sets that are definable by proper names, Closure could be saved on the grounds that expressions of the form a is no t a member of b, wher e a and b are proper names, could not lack a tr uth-value. The problem with this strategy is that the set of true sentences of English, as well as subsets of would have to be defined by proper names of English, but there is no intuitive support for this thesis. One could cert ainly add new names for these sets by stipulati on, but it is clear that the Liar pa radox should not be dependent on the existence of the stipulative names. A lternatively, one could try to restrict to those sets that guarantee that Transcendence and Closure hold, but this would be an explicit acknowledgement of the fact that it is not the general mechanism that should be made part of the structure of the Liar paradox, but only what makes Transcendence and Closure at the limit case possible. It can now be inferred that all that really ma tters for there to be a Liar paradox is that Transcendence and Closure hold for the set of true sentences of English. Moreover, definability alone is not enough to establish that the two c onditions hold at the limit case. Two additional elements emerge as essential aspects of the Li ar paradox: the T-schema which is needed to prove Transcendence, and the fact that the Liar (the sentence that says about itself that it is not a member of the set of true sentences of English) expresses in English the thought that the Liar is not true, which is needed (together with the Tschema) to establish Closure. These elements do not show up in the Inclosure Schema, but they are essential parts of the Li ar. Moreover, there are no similar elements in the structure of logical paradoxes.

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144 Semantic Liars and Logical Liars One source of the idea that sem antic and logical paradoxes have the same underlying structure is the fact that each semantic paradox can be associated and sometimes confused with a logical paradox. The Liar Paradox is a semantic pa radox because it deals with truth, which is a semantic notion. However, it is common practice to use the set-theoretical apparatus to analyze semantic notions. As Priest points out, this te ndency has been encouraged by the successes of Tarskis and Gdels work in semantics. The settheoretical apparatus enables one to reframe the Liar paradox as a paradox in set-theory. Instead of talking about the meaning of true or about the concept of truth, it is often more convenient to talk about the set of true sentences. On the one hand, it is intuitively true that there is a set of true sentences of English, which is represented in English by the predicate true se ntence of English. On the ot her hand, principles that are intuitively true enable one to pr ove that the set of true sentence s of English cannot be represented in English. This paradox, call it the Logical Liar, has indeed a structur e that is more similar to the structure of logical paradoxes. As such, it has all the appearances of a logical paradox, but it would be different from the original semantic Liar paradox. The temptation to identify the Liar paradox w ith the Logical Liar is encouraged by the tendency to take the advances in semantics to sh ow that there is no clear borderline between settheory and semantics. Priest argues that although at the time when Ramsey wrote his paper it was commonly held that the vocabularies of mathematic s and of semantics are indeed different, in the meantime things have changed: In particular, both syntactic and semantic lingui stic notions became quite integral parts of mathematics. Indeed, in a sense, the work of Gdel and Tarski showed how these notions could be reduced to other parts of mathematics (number theory and set theory, respectively) [Priest 1994: 26].

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145 Nevertheless, although some aspects of the semantic notions can be properly captured in set-theoretic terms, I think that the analogies shou ld not be pushed too far. Tarski showed indeed how the concept of truth for formalized langua ges can be extensionally characterized in mathematical terms, but this does not mean that the concept can be reduced to mathematical notions. Moreover, it is the concept of a true sent ence of English that gives rise to a paradox, not that of a true sentence of a certain formalized language, and Tarski did no t actually claim that the definition of truth can be ex tended to natural languages. One way to see that the two Liars should be sh arply distinguished from one another is to notice that the intuitions that lay behind the Logi cal Liar are parasitic upon those that lay behind the Liar paradox. Absent the intuitions that lead to the Liar paradox, the Logical Liar could be solved by denying that there is a set of true sentences of Englis h (the way one can block other logical paradoxes), or by denying that th e set is representable in English. Thus, the Liar paradox should not be identified with the logical Liar. For the same reasons, the structure of the Liar paradox, as well as the structure of other semantic paradoxes, is different from the structure of logical paradoxes. Consequen tly, it is perfectly reasonable to expect that the two kinds of paradoxes have different kinds of so lutions. The structural differences between the two kinds of paradoxes indicate that the lessons that should be drawn from them might be quite different. At the end of the day, logical paradoxe s suggest that there are fewer sets in the universe than one normally is tempted to allow (which is the idea behi nd the orthodox solutions to the logical paradoxes based on the axiomatization of set-theory ), while the semantic paradoxes suggest that fewer concepts are expressible in English th an one normally believes.

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146 LIST OF REFERENCES Anderson, Alan Ross, Belnap, Nuel D. and Dunn, J. Michael 1992. Enta ilment: the Logic of Relevance and Necessity, vol. 2 Princeton University Press. Bach, Kent 1981. An Analysis of Self-Deception, Philosophy and Phenomenological Research 41.3: 351-370. Bach, Kent 1998. Speech Acts, in Routledge Encyclopedia of Philosophy ed. E. Craig. London: Routledge. Badici, Emil 2005. The Simple Liar Takes Revenge, in The Logica Yearbook ed. M. Bilkova & O. Tomala, Prague: Filosofia. Beall, JC. and Otavio Bueno 2002. The Simple Liar without Bivalence?, Analysis 62.1: 26-34. Beth, E. W. 1965. Mathematical Thought Dordrecht: Reidel. Burge, Tyler 1984. Semantical Paradox, in Recent Essays on Truth and the Liar Paradox, ed. R. L. Martin, Oxford: Clarendon Press: 83-119. Chihara, Charles 1973. Ontology and the Vicious-Circle Principle Cornell University Press. Chihara, Charles 1979. The Semantic Pa radoxes: A Diagnostic Investigation, The Philosophical Review LXXXVIII.4: 590-618. Chihara, Charles 1984. The Semantic Paradoxes: Some Second Thoughts, Philosophical Studies 45: 223-29. Eklund, Matti 2002. Inconsistent Languages, Philosophy and Phenomenological Research 64.2: 251-75. Evans, Gareth 1997. Understanding Demonstratives, in Readings in the Philosophy of Language ed. P. Ludlow, The MIT Press: 717-44. Fitch, Friedrich 1964. Universal Metalanguages for Philosophy, Review of Metaphysics 17: 396402. Frege, Gottlob 1997. Begriffsschrift, in The Frege Reader ed. M. Beaney, Blackwell Publishers: 47-78. Frege, Gottlob 1997. Thought, in The Frege Reader ed. M. Beaney, Blackwell Publishers: 32545. Gupta, Anil and Nuel Belnap 1993. The Revision Theory of Truth Cambridge: MIT Press. Herzberger, Hans 1967. The Truth-Conditional Consistency of Natural Languages, Journal of Philosophy 64.2: 29-35.

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147 Herzberger, Hans 1970. Paradoxes of Grounding in Semantics, Journal of Philosophy 67: 14567. Herzberger, Hans 1982. Nave Sema ntics and the Liar Paradox, Journal of Philosophy 79.9: 47997. Hofweber, T. 2006. Inexpressible Properties and Propositions, in Oxford Studies in Metaphysics vol. 2, ed. D. Zimmerman, Oxford University Press. Kleene, S.C. 1952. Introduction to Metamathematics New York: Van Nostrand. Knig, J. 1967 (1905). On the Foundations of Se t Theory and the Continuum Problem, in From Frege to Gdel, ed. van Heijenoort, Cambridge MA : Harvard University Press. Kremer, M. 2000. Judgment and Truth in Frege, Journal of the History of Philosophy 38.4: 54981. Kripke, Saul 1975. Outline of a Theory of Truth, Journal of Philosophy 72: 690-716. Lappin, S. 1982. On the Pragmatics of Mood, Linguistics and Philosophy 4: 559-78. Ludwig, Kirk 1996. Singular Thought a nd the Cartesian Theory of Mind, Nous 30: 434-60. Lycan, William 1986. Tacit Belief, in Belief: Form, Content, and Function ed. R. Bogdan, Oxford: Clarendon Press: 61-82. Martin, R. L., ed. 1984. Recent Essays on Truth and the Liar Paradox Oxford: Clarendon Press. Martin, R. L. 1976. Are Natural Languages Universal?, Synthese 32: 271-91. Martinich, A. P. 1983. A Pragmatic Solution to the Liar Paradox, Philosophical Studies 43: 63-7. McGee, Vann 1990. Truth, Vagueness, and Paradox, Indianapolis: Hack ett Publishing Company. Parsons, Terence 1984. Assertion, Denial, and the Liar Paradox, Journal of Philosophical Logic 13:137-52. Perry, John 1997. Frege on Demonstratives, in Readings in the Philosophy of Language ed. P. Ludlow, The MIT Press: 693-716. Priest, Graham 1994. The Structure of the Paradoxes of Self-Reference, Mind 103.409: 25-34. Priest, Graham 1998. Whats so Bad about Contradictions?, Journal of Philosophy 96: 410-26. Priest, Graham 2000. On the Principle of Uniform Solution: A Reply to Smith, Mind 109.433: 123-6. Priest, Graham 2002. Beyond the Limits of Thought, Oxford University Press.

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148 Quine, W. V. 1969. Set Theory and Its Logic Cambridge MA: Harvard University Press. Ramsey, Frank Plumpton 1990. The Foundations of Mathematics, in Philosophical Papers Cambridge University Press. Ray, Greg 2002. Truth, the Liar, and Tarskian Truth Definition, in A Companion to Philosophical Logic, ed. D. Jacquette, Blackwell Publishers: 164-76. Ray, Greg 2005. On the Matter of Essential Richness, Journal of Philosophical Logic 34: 43357. Ray, Greg (forthcoming) Tarskis Grelling and the T-Strategy, in Truth and Probability: Essays in Honor of Hughes LeBlanc. Richard, J. 1967 (1905). The Principles of Mathematics and the Problem of Sets, in From Frege to Gdel ed. van Heijenoort, Cambridge MA: Harvard University Press. Ricketts, T. 1986. Objectivity and Objecthood : Freges Metaphysics of Judgment, in Frege Synthesized eds. Haaparanta and Hintikka, Dordrecht: Reidel: 65-95. Ricketts, T. 1996. Logic and Truth in Frege, Proceedings of the Aristotelian Society Supp. 70: 121-40. Russell, Bertrand 1967 (1905). Mathematical Logi c as Based on the Theory of Types, in From Frege to Gdel, ed. van Heijenoort, Cambridge MA : Harvard University Press. Searle, John 1969. Speech Acts: An Essay in the Philosophy of Language London: Cambridge University Press. Searle, John 2004. Mind. A Brief Introduction Oxford University Press. Simmons, Keith 1993. Universality and The Liar, Cambridge University Press. Simmons, Keith 2002. Semantical and Logical Paradox, in A Companion to Philosophical Logic, ed. D. Jacquette, Oxford: Blackwell Publishers: 115-30. Skyrms, Brian 1984. Intensional Aspects of Seman tical Self-Reference, in ed. R. L. Martin, Oxford: Clarendon Press: 121-26. Smith, Nicholas J.J. 2000. The Principle of Uniform Solution (of the Paradoxes of SelfReference), Mind 109: 117-22. Soames, Scott 1998. Understanding Truth Oxford University Press. Tarski, Alfred 1956 (1933). The Concept of Truth in Formalized Languages, in Logic, Semantics, Metamathematics, Oxford: Clarendon Press. Tarski, Alfred 1944. The Semantic Conception of Truth and the Foundations of Semantics. Philosophy and Phenomenological Research 4: 341-75.

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149 Tennant, Neil 1998. Critical Notice of Beyond the Limits of Thought, Philosophical Books 39: 20-38. van Fraassen, Baas 1968. Presupposition, Im plication, and Self-Reference, The Journal of Philosophy 65.5: 136-52. van Fraassen, Baas 1978. Truth and Paradoxical Consequences, in The Paradox of the Liar ed. R. L. Martin, Ridgeview Pub Co. van Fraassen, Baas 1970. Infere nce and Self-Reference, Synthese 21: 425-38. van Fraassen, Baas 1969. Presuppositions, Supervaluations, and Free Logic, in The Logical Way of Doing Things, ed. K. Lambert, New Haven and London: Yale University Press. van Heijenoort, Jean, ed. 1967. From Frege to Gdel Cambridge MA: Harvard University Press. Yablo, Steven 1993. Paradox without Self-Reference, Analysis 53: 251-52.

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150 BIOGRAPHICAL SKETCH I com pleted my undergraduate studies in Roma nia, where I received a BS in Mathematics and a BA in Philosophy from Bucharest University. I moved to Florida for the graduate school in August 1999, and I received an MA in philosophy form the Universi ty of Florida in December 2002. After completing and defending the dissertati on, I will receive my PhD in philosophy from the same university, in August 2007. My research has mainly been focused on some is sues in philosophical lo gic. In particular, I spent a fair amount of energy trying to find a solution to the semantic paradoxes and to better understand the nature of truth and the role it pl ays in human inquiry. The study of paradoxes is even more frustrating than the study of the origin s of the universe because it is very easy to get seduced by a path which turns out to be a dead end. I found consolation in the thought that many beautiful minds of the past have been tricked by the paradoxes in a similar way. Truth is an inexhaustible philosophical topic, so I plan on continuing my research on the Liar paradox and on some implications of the inexpressibility view and I also intend to sp end more time reflecting on the debate between deflationists and the defe nders of a more robust notion of truth. In addition to philosophical logic, I have strong interests in other areas of analytic philosophy, such as logic, metaphysics, ph ilosophy of language and the philosophy of mathematics. Although I am trying to keep up with the current philosophical literature, I have a vicious inclination towa rd outdated views.


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