liar paradox

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pdf version of the entry Liar Paradox http://plato.stanford.edu/archives/win2012/entries/liar-paradox/ from the Winter 2012 Edition of the Stanford Encyclopedia of Philosophy Edward N. Zalta Uri Nodelman Colin Allen John Perry Principal Editor Senior Editor Associate Editor Faculty Sponsor Editorial Board http://plato.stanford.edu/board.html Library of Congress Catalog Data ISSN: 1095-5054 Notice: This PDF version was distributed by request to mem- bers of the Friends of the SEP Society and by courtesy to SEP content contributors. It is solely for their fair use. Unauthorized distribution is prohibited. To learn how to join the Friends of the SEP Society and obtain authorized PDF versions of SEP entries, please visit https://leibniz.stanford.edu/friends/ . Stanford Encyclopedia of Philosophy Copyright c 2012 by the publisher The Metaphysics Research Lab Center for the Study of Language and Information Stanford University, Stanford, CA 94305 Liar Paradox Copyright c 2012 by the authors JC Beall and Michael Glanzberg All rights reserved. Copyright policy: https://leibniz.stanford.edu/friends/info/copyright/

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Page 1: liar paradox

pdf version of the entry

Liar Paradoxhttp://plato.stanford.edu/archives/win2012/entries/liar-paradox/

from the Winter 2012 Edition of the

Stanford Encyclopedia

of Philosophy

Edward N. Zalta Uri Nodelman Colin Allen John Perry

Principal Editor Senior Editor Associate Editor Faculty Sponsor

Editorial Board

http://plato.stanford.edu/board.html

Library of Congress Catalog Data

ISSN: 1095-5054

Notice: This PDF version was distributed by request to mem-

bers of the Friends of the SEP Society and by courtesy to SEP

content contributors. It is solely for their fair use. Unauthorized

distribution is prohibited. To learn how to join the Friends of the

SEP Society and obtain authorized PDF versions of SEP entries,

please visit https://leibniz.stanford.edu/friends/ .

Stanford Encyclopedia of Philosophy

Copyright c© 2012 by the publisher

The Metaphysics Research Lab

Center for the Study of Language and Information

Stanford University, Stanford, CA 94305

Liar Paradox

Copyright c© 2012 by the authors

JC Beall and Michael Glanzberg

All rights reserved.

Copyright policy: https://leibniz.stanford.edu/friends/info/copyright/

Page 2: liar paradox

Liar ParadoxFirst published Thu Jan 20, 2011

The first sentence in this essay is a lie. There is something odd aboutsaying so, as has been known since ancient times. To see why, rememberthat all lies are untrue. Is the first sentence true? If it is, then it is a lie,and so it is not true. Conversely, suppose that it is not true. As we (viz.,the authors) have said it, presumably with the intention of you believing itwhen it is not true, it is a lie. But then it is true!

That there is some sort of puzzle to be found with sentences like the firstone of this essay has been noted frequently throughout the history ofphilosophy. It was discussed in classical times, notably by the Megarians,but it was also mentioned by Aristotle and by Cicero. As one of theinsolubilia, it was the subject of extensive investigation by medievallogicians such as Buridan. More recently, work on this problem has beenan integral part of the development of modern mathematical logic, and ithas become a subject of extensive research in its own right. The paradoxis sometimes called the ‘Epimenides paradox’ as the tradition attributes asentence like the first one in this essay to Epimenides of Crete, who isreputed to have said that all Cretans are always liars. That some Cretanhas said so winds up in no less a source than New Testament![1]

Lying is a complicated matter, but what's puzzling about sentences likethe first one of this essay isn't essentially tied to intentions, social norms,or anything like that. Rather, it seems to have something to do with truth,or at least, some semantic notion related to truth. The puzzle is usuallynamed ‘the Liar paradox’, though this really names a family of paradoxesthat are associated with our type of puzzling sentence. The family is aptlynamed one of paradoxes, as they seem to lead to incoherent conclusions,such as: “everything is true”. Indeed, the Liar seems to allow us to reach

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such conclusions on the basis of logic, plus some very obvious principlesthat have sometimes been counted as principles of logic. Thus, we havethe rather surprising situation of something near or like logic aloneleading us to incoherence. This is perhaps the most virulent strain ofparadox, and dealing with it has been an important task in logic for aboutas long as there has been logic.

In this essay, we will review the important members of the family of Liarparadoxes, and some of the important ideas about how these paradoxesmight be resolved. The past few thousand years have yielded a greatnumber of proposals, and we will not be able to examine all of them;instead, we will focus on a few that have, in recent discussions, proved tobe important.

1. The Paradox and the Broader Phenomenon1.1 Simple-falsity Liar1.2 Simple-untruth Liar1.3 Liar cycles1.4 Boolean compounds1.5 Infinite sequences

2. Basic Ingredients2.1 Truth predicate2.2 Principles of truth2.3 The Liar in short

3. Significance4. Some Families of Solutions

4.1 Non-classical logics4.2 Classical logic4.3 Contextualist approaches4.4 The revision theory4.5 Inconsistency views

5. Concluding Remarks

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BibliographyAcademic ToolsOther Internet ResourcesRelated Entries

1. The Paradox and the Broader Phenomenon

1.1 Simple-falsity Liar

Consider a sentence named ‘FLiar’, which says of itself (i.e., says ofFLiar) that it is false.

This seems to lead to contradiction as follows. If the sentence ‘FLiar isfalse’ is true, then FLiar is false. But if FLiar is false, then the sentence‘FLiar is false’ is true. Since FLiar just is the sentence ‘FLiar is false’, wehave it that FLiar is false if and only if FLiar is true. But, now, if everysentence is true or false, FLiar itself is either true or false, in which case—given our reasoning above—it is both true and false. This is acontradiction. Contradictions, according to many logical theories (e.g.,classical logic, intuitionistic logic, and much more) imply absurdity—triviality, that is, that every sentence is true.

An obvious response is to deny that every sentence is true or false, i.e. todeny the principle of bivalence. As we will discuss in §4, somedescendants of this idea remain important in current work on the Liar.Even so, a simple variant Liar sentence shows that this immediate answeris not all there is to the story.

1.2 Simple-untruth Liar

FLiar: FLiar is false.

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Rather than work with falsehood, we can construct a Liar sentence withthe complex predicate ‘not true’.[2] Consider a sentence named ‘ULiar’(for ‘un-true’), which says of itself that it is not true.

The argument towards contradiction is similar to the FLiar case. In short:if ULiar is true, then it is not true; and if not true, then true. But, now, ifevery sentence is true or not true, ULiar itself is true or not true, in whichcase it is both true and not true. This is a contradiction. According tomany logical theories, a contradiction implies absurdity—triviality.

The two forms of the Liar paradox we have so far reviewed rely on someexplicit self-reference—sentences talking directly about themselves. Suchexplicit self-reference can be avoided, as is shown by our next family ofLiar paradoxes.

1.3 Liar cycles

Consider a very concise (viz., one-sentence-each) dialog between siblingsMax and Agnes.

What Max said is true if and only if what Agnes said is true. But whatAgnes said (viz., ‘Max's claim is not true’) is true if and only if what Maxsaid is not true. Hence, what Max said is true if and only if what Maxsaid is not true. But, now, if what Max said is true or not true, then it isboth true and not true. And this, as in the FLiar and ULiar cases, is acontradiction, implying, according to many logical theories, absurdity.

Liar paradoxes can also be formed using more complex sentence

ULiar: ULiar is not true.

Max: Agnes' claim is true.Agnes: Max's claim is not true.

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structure, rather than complex modes of reference. One that has beenimportant involves Boolean compounds.

1.4 Boolean compounds

Boolean compounds can enter into Liar sentences in many ways. Onerelatively simple one is as follows. Consider the following sentencenamed ‘DLiar’ (for ‘Disjunctive’).

Since 1≠0, we conclude (via disjunctive syllogism, which sanctionsinferring sentence B from the sentences A ∨ B and ¬A) that DLiar is nottrue. But a disjunction is true if and only if at least one disjunct is true.Hence, since ‘DLiar is not true’ is true, at least one of DLiar's disjuncts istrue (viz., the left one). Hence, DLiar is true. Hence, DLiar, like itscousins (FLiar, ULiar, etc), is true if and only if DLiar is not true. So, ifDLiar is true or not true, it is both true and not true. This is acontradiction, which, according to many logical theories, impliesabsurdity.

We pause to mention DLiar as it is connected with another importantparadox: Curry's paradox, which involves conditionals that say ofthemselves only that if they (the conditional itself) are true, so too is someabsurdity (e.g., ‘if this sentence is true, then 1=0’ or ‘if this sentence istrue, everything is true’ or so on). At least in languages where theconditional is the material conditional, and so A ⊃ B is equivalent to ¬A ∨B, DLiar is equivalent to the Curry sentence ‘DLiar is true ⊃ 1 = 0’.Though this may set up some relations between the Liar and Curry'sparadox, we pause to note an important difference. For the Curry paradoxis most important where the conditional is more than the materialconditional (or some modalized variant of it). In such settings, the Curryparadox does not wear negation on its sleeve, as DLiar does. For more

DLiar: Either DLiar is not true or 1=0.

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information, consult the entry on Curry's paradox.

1.5 Infinite sequences

The question of whether the Liar paradox really requires some sort ofcircularity has been the subject of extensive debate. Liar cycles (e.g., theMax–Agnes dialog) show that explicit self-reference is not necessary, butit is clear that such cycles themselves involve circular reference. Yablo(1993b) has argued that a more complicated kind of multi-sentenceparadox produces a Liar without circularity.

Yablo's paradox relies on an infinite sequence of claims A0,A1,A2,…,where each Ai says that all of the ‘greater’ Ak (i.e., the Ak such that k > i)are untrue. (In other words, each claim says of the rest that they're alluntrue.) Since we have an infinite sequence, this version of the Liarparadox appears to avoid the sort of circularity apparent in the previousexamples; however, contradiction still seems to emerge. If A0 is true, thenall of the ‘greater’ Ak are untrue, and a fortiori A1 is untrue. But, then,there is at least one true Ak greater than A1 (i.e., some Ak such that k > 1),which contradicts A0. Conversely, if A0 is untrue, then there's at least onetrue Ak greater than A0. Letting Am be such a one (i.e., a truth greater thanA0), we have it that Am+1 is untrue, in which case there's some truthgreater than Am+1. But this contradicts Am. What we have, then, is that ifA0 (the first claim in the infinite sequence) is true or untrue, then it isboth. And this, as in the other cases, is a contradiction.

Whether Yablo's paradox really avoids self-reference is much-debated.See, for instance, Beall (2001), Priest (1997), Sorensen (1998), and alsoCook (2006) and references therein.

2. Basic Ingredients

We have already seen a kind of characteristic reasoning that goes with the

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Liar. We have also seen some common structure across all our exampleLiar paradoxes, such as the presence of truth predicates, and somethinglike negation. We pause here to discuss these ingredients of the paradox,focusing on the basic Liars. Just what creates the Liar paradox, and justwhich of the puzzles we just surveyed is ‘basic’, is a contentious matter;different approaches to solving the Liar view these matters differently.Hence, our goal is merely to illuminate some common themes acrossdifferent Liars, not to offer a full diagnosis of the source of the paradox.

We highlight three aspects of the Liar: the role of truth predicates, thekinds of principles for reasoning about truth that are needed, and the waythat a paradox can be derived given these resources.

2.1 Truth predicate

The first ingredient in building a Liar is a truth predicate, which we writehere as Tr. We follow the usual custom in logic of treating this as apredicate of sentences. However, especially as we come to consider someways of resolving the Liar, it should be remembered that this treatmentcan be seen as more for convenience in exposition than a seriouscommitment to what truth bearers are.

We assume that we have, along with the truth predicate, appropriatenames of sentences. For a given sentence A, suppose that ⌜A⌝ is a namefor it. A predication of truth to A then looks like Tr(⌜A⌝).

We shall say that a predicate Tr(x) is a truth predicate for language L ifTr(⌜A⌝) is well-formed for every sentence A of L. We typically expect Trto obey some principles governing its behavior on sentences of a givenlanguage. It is to those we now turn.[3]

2.2 Principles of truth

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The tradition, going back to Tarski (1935), is that the behavior of the truthpredicate Tr is described by the following biconditional.

Indeed, Tarski took the biconditional here to be the material biconditionalof classical logic. This is usually called the T-schema. For more on the T-schema, and Tarski's views of truth, see the entries on Alfred Tarski andTarski's truth definitions.

The Liar paradox has been a locus of thinking about non-classical logics(as we already saw a taste of, for instance, in the idea that bivalencemight be rejected as part of a solution to the Liar). Thus, we should stopto consider what principles should govern the truth predicate Tr ifclassical logic is not to hold.

The leading idea for what might replace the T-schema points to two sortsof ‘rules’ (e.g., two sorts of ‘inference rules’ in some sense) or principlesthat are characteristic of the truth predicate. If you have a sentence A, youcan infer Tr(⌜A⌝), that is, you can ‘capture’ A with the truth predicate.Conversely, if you have Tr(⌜A⌝), you can infer A, that is, you can‘release’ A from the truth predicate. In some logics, capture and releasewind up being equivalent to the T-schema, but it is often helpful to breakthese up:

Implies here is a logical notion, though just which one, and what theoptions are, depends on what background logic is assumed. For ourdiscussion, we think of it in so-called rule form: that the argument from Ato B is valid, which we record (as above) via the turnstile. In some logicalsettings (e.g., classical logic, in which a certain so-called deduction

Tr(⌜A⌝) ↔ A.

Capture A implies Tr(⌜A⌝). (We also write this as A ⊢ Tr(⌜A⌝).)Release Tr(⌜A⌝) implies A. (We also write this as Tr(⌜A⌝⊢ A.)

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theorem holds), this is equivalent to the provability of a conditional, but insome settings, it is not. Either way, capture and release jointly make Aand Tr(⌜A⌝) logically equivalent in the sense of being inter-derivable.Thus, ⊢ is being used here as a schematic placeholder for a range ofdifferent logical notions, each of which will provide some notion of validinference in some logical theory.

(There are a number of logical subtleties here that we will not pursue,especially about how to formulate rules, and which rules are consistent.Different formulations of rules vary significantly in logical strength aswell.[4] See the entry on axiomatic theories of truth for more on howconsistent forms of capture and release can be formulated in classicallogic. In the terminology of Friedman and Sheard (1987), the rule formsof capture and release are called ‘T-Intro’ and ‘T-Elim’, and theconditional forms ‘T-In’ and ‘T-Out’. We prefer the broader terminology,since it highlights a general form of behavior common to a great varietyof predicates and operators, e.g., knowledge releases but doesn't capture;possibility captures but doesn't release; and so on; and truth is special indoing both.)

2.3 The Liar in short

The Liar paradox begins with a language containing a truth predicate,which obeys some form of capture and release. We now explore morecarefully how a paradox results from these assumptions.

2.3.1 Existence of Liar-like sentences

Putting aside Yablo-type paradoxes, the Liar relies on some form of self-reference, either direct, as in in the simple Liars above, or indirect, as inLiar cycles. Most natural languages have little trouble generating self-reference. The first sentence of this essay is one example. Self-referencecan be accidental, as in the case where someone writes ‘The only sentence

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on the blackboard in room 101 is not true’, by chance writing this in room101 itself (as C. Parsons (1974) noted).

In formal languages, self-reference is also very easy to come by. Anylanguage capable of expressing some basic syntax can generate self-referential sentences via so-called diagonalization (or more properly, anylanguage together with an appropriate theory of syntax or arithmetic).[5]

A language containing a truth predicate and this basic syntax will thushave a sentence L such that L implies ¬Tr(⌜L⌝) and vice versa:

This is a ‘fixed point’ of (the compound predicate) ¬Tr, and is, in effect,our simple-untruth Liar.

(It may help some readers, here and throughout, to think of such a Liarsentence L arising from a name c that denotes the sentence ¬Tr(c). In thisway, we can think of the existence of the Liar as being reflected in theidentity c = ⌜¬Tr(c)⌝.)

2.3.2 Other logical ‘laws’

Other conspicuous ingredients in common Liar paradoxes concern logicalbehavior of basic connectives or features of implication. A few of therelevant principles are:

Excluded middle (LEM): ⊢ A ∨¬A.Explosion (EFQ):[6] A,¬A ⊢ B.Disjunction principle (DP):[7] If A ⊢ C and B ⊢ C then A ∨ B ⊢ C.Adjunction: If A ⊢ B and A ⊢ C then A ⊢ B ∧ C.

(This is not to suggest that these are the only logical features involved incommon Liar paradoxes, but they're arguably the most important of the

L ⊣⊢¬Tr(⌜L⌝).

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salient ones.)

2.3.3 The Liar in abstract

Given the foregoing ingredients, we can now give a slightly more abstractform of the paradox. (Our hope is to use this abstract form to highlightdifferent responses to the paradox.) We suppose that we have a languageL with a truth predicate Tr, and that L allows enough syntax to constructa sentence L such that L ⊣⊢¬Tr(⌜L⌝). We also suppose that the logic ofL enjoys LEM and EFQ and satisfies DP and adjunction.

The argument that our Liar sentence L implies a contradiction runs asfollows.

1. Tr(⌜L⌝) ∨¬Tr(⌜L⌝) [LEM]2. Case One:

a. Tr(⌜L⌝)b. L [2a: release]c. ¬Tr(⌜L⌝) [2b: definition of L]d. ¬Tr(⌜L⌝) ∧ Tr(⌜L⌝) [2a, 2c: adjunction]

3. Case Two:a. ¬Tr(⌜L⌝)b. L [3a: definition of L]c. Tr(⌜L)⌝) [3b: capture]d. ¬Tr(⌜L⌝) ∧ Tr(⌜L⌝) [3a, 3c: adjunction]

4. ¬Tr(⌜L⌝) ∧ Tr(⌜L⌝) [1–3: DP]

This version of the Liar is one of many. With a little more complexity, forinstance, either capture or release can be avoided in favor of some otherbackground assumptions. Intuitionistic variants of the Liar are alsoavailable, though we shall not explore intuitionistic logic here.[8]

We have so far shown that with the given ingredients our Liar sentence L

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implies a contradiction (thus formalizing the reasoning in ULiar). Fromhere, it is one short step to all-out absurdity—if the lone contradictionweren't already absurd enough. We invoke EFQ to finish the proof. (Well,we also assume that A ∧ B implies A and B, i.e., that simplification isvalid in L.)

5. B [4: EFQ]

B, here, may be any—every—sentence that you like (or don't like, as thecase may be)! EFQ is the principle that every sentence follows from acontradiction; it sanctions the step from a single contradiction to outrighttriviality of logic.

In the face of such absurdity (triviality), we conclude that something iswrong in the foregoing Liar reasoning. The question is: what? This, in theend, is the question that the Liar paradox raises.

3. Significance

We have now seen that with some elementary assumptions about truthand logic, a logical disaster ensues. What is the wider significance of sucha result?

From time to time, the Liar has been argued to show us something far-reaching about philosophy. For instance, Grim (1991) has argued that itshows the world to be essentially ‘incomplete’ in some sense, and thatthere can be no omniscient being. McGee (1991) and others suggest thatthe Liar shows the notion of truth to be a vague notion. Glanzberg (2001)holds that the Liar shows us something important about the nature ofcontext dependence in language, while Eklund (2002) holds that it showsus something important about the nature of semantic competence and thelanguages we speak. Gupta and Belnap (1993) claim that it revealsimportant properties of the general notion of definition. And there are

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other lessons, and variations on such lessons, that have been drawn.

Of more immediate concern, at least for our purposes here, is what theLiar shows us about the basic principles governing truth, and about logic.In a skeptical vein, Tarski himself (1935, 1944) seems to have thought theLiar shows the ordinary notion of truth to be incoherent, and in need ofreplacement with a more scientifically respectable one. (For more onTarski, see the entries on Tarski and Tarski's truth definitions. For moreon Tarski's aims and purposes, see Heck (1997).) More common, andperhaps the dominant thread in the solutions to the Liar, is the idea thatthe basic principles governing truth are more subtle than the T-schemareflects.

The Liar has also formed the core of arguments against classical logic, asit is some key features of classical logic that allow capture and release toresult in absurdity. Most notable are the arguments for logics that areparacomplete (e.g., Kripke (1975), Field (2008), and others) andparaconsistent (e.g., Asenjo (1966), Priest (1984, 2006), and others).

In many cases inspired by wider views of the significance of the paradox,there have been a number of attempts to one way or another resolve theparadox. It is to these proposed solutions that we now turn.

4. Some Families of Solutions

In this section, we briefly survey some approaches to resolving the Liarparadox. We group proposed solutions into families, and try to explainthe basic ideas behind them. In many cases, a full exposition wouldinvolve a great deal of technical material, that we will not go into here.Interested readers are encouraged to follow the references we provide foreach basic idea.

4.1 Non-classical logics

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One of the leading ideas for how to resolve the Liar paradox is that itshows us something about logic, in fact, something far-reaching aboutlogic. The main idea is that the principles of capture and release are thefundamental conceptual principles governing truth, and cannot bemodified. Instead, basic logic must be non-classical, to avoid a logicaldisaster of the kind we reviewed in §2.

One important way to motivate non-classical solutions is to appeal to aform of deflationism about truth. Such views take something akin to theT-schema to be the defining characteristic of truth, and as such, not opento modification. (See, e.g., Horwich (1990).) Most strictly, so-calledtransparency or ‘see-through’ or ‘pure disquotational’ conceptions of truth(e.g., Field (1994, 2008) and Beall (2005, 2009)) take the definingproperty of truth to be intersubstitutability of A and Tr(⌜A⌝) in all non-opaque contexts. This makes capture and release, in unrestricted formapplying to all sentences of a language, a requirement for truth (at leastwhere we have A ⊢ A or, more strongly, ⊢ A → A).[9]

Holding capture and release fixed, and applying it to all sentences withoutrestriction, yields triviality unless the logic is non-classical. There are twomain sub-families of non-classical (transparency) truth theories:paracomplete and paraconsistent. We sketch the main ideas of each.

4.1.1 Paracomplete

According to paracomplete approaches to the Liar, the main lesson of theLiar is that LEM ‘fails’ in some sense. In other words: the Liar teaches usthat some sentences (notably, Liars!) ‘neither hold nor do not hold’ (insome sense), and so are neither true nor false. As a result, the logic oftruth is non-classical.

This idea is perhaps most natural in response to the simple-falsity Liar.

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There, it is tempting to say that there is some status other than truth andfalsity, and the Liar sentence L has it. But this will not suffice, forinstance, for the simple-untruth Liar. This says nothing about falsity.Rather, in some way the basic reasoning reviewed in §2.3 must fail, andthe culprit, in the paracomplete view, is LEM. Liar-instances of LEM‘fail’ (in some sense) according to the paracomplete approach; suchsentences fall into the ‘gap’ between truth and falsity (to use a commonmetaphor).

There have been many proposals for using such non-classical logics toaddress the Liar. An early example is van Fraassen (1968, 1970). ButKripke's work has been the most influential in recent times, not only toapproaches to the Liar based on non-classical logic, but a range of otherapproaches we will survey in §4.2 as well. Thus, we pause to describe atleast a little of Kripke's framework.

Kripke's theory

Logics where LEM fails are not themselves hard to come by. Amongmany such logics are a number of three-valued logics that allow sentencesto take a third value over and above true and false. Sentences like Liarsentences take the third value. One of the most commonly applied logicsis the Strong Kleene logic K3. We do not go into the details of K3 here,but only note the properties of K3 we need. (For more details, see theentry on many-valued logic, or Beall and van Fraassen (2003), or Priest(2008).) First and foremost, we have:

LEM fails. In fact, there are no logical truths (or valid sentences)according to K3. (We return to this on the topic of a ‘suitable conditional’below.)

⊬K3A ∨¬A.

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The challenge to using K3 to flesh out a paracomplete theory is to explainhow anything like (even rule-form) capture and release hold, and if youfollow the deflationist line, how full unrestricted capture and release hold.One way of understanding the important work of Kripke (1975) (andrelated work of Martin and Woodruff (1975) ) is as a way of achievingjust that.

Kripke begins with a fully classical language L0 containing no truthpredicate (or more generally, no semantic terms). (Recall, we areassuming a language comes equipped with a valuation scheme. For L0 itis classical.) He then considers extending it to a language L whichcontains a truth predicate Tr. The predicate Tr is taken to apply to everysentence of the expanded language L , including those of the original L0.Thus, it is a self-applicative truth predicate (as the deflationist-inspiredpicture we mentioned must require), even though we begin with alanguage without a truth predicate.

We can think of L0 as interpreted by a classical model M0. Kripke showsus how to build an interpretation M for the expanded language. Themain innovation is to see the truth predicate as partial. Rather than simplyhaving an extension, it has an extension (set of things of which it is true),and an anti-extension (set of things of which it is false). The extensionand anti-extension are mutually exclusive, but they need not jointlyexhaust the domain of M0. Pathological sentences like L fall in neither inthe extension or the anti-extension of Tr.

Falling into neither the extension or the anti-extension of Tr acts likehaving a third value, and we can interpret L as acting like a languagewith a K3 valuation scheme. Treating the language this way, Kripkeshows up to build up a very plausible extension and anti-extension for Tr,typically written E and A. The important property of the new extendedmodel 〈M0,〈E,A〉〉 is that the truth value of any sentence A and Tr(⌜A⌝)

+0

+0

+0

+0

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model 〈M0,〈E,A〉〉 is that the truth value of any sentence and ( )are exactly the same. A is true, false, or neither, just in case Tr(⌜A⌝) is.Furthermore, interpreting the expanded language L as a K3 language, wehave for K3 consequence A ⊣⊢ Tr(⌜A⌝), just as we desired.

Kripke shows how to build up E and A by an inductive process. Onestarts with an ‘approximation’ of the extension and anti-extension of Tr,and successively improves it until the improvement process ceases to beproductive (it reaches a ‘fixed point’). In fact, for the K3-based solution,the natural thing to do is start with an empty extension and anti-extension,and throw in sentences that are true at successive stages of the process.

Kripke's construction can be applied to a number of different logics,including other many-valued logics such as the ‘Weak Kleene’ logic, andsupervaluation logics. See, for instance, Burgess (1986) and McGee(1991) for discussion. Kripke-style constructions engage a fair bit ofmathematical subtlety. For an accessible overview of more of the details,see Soames (1999). For a more mathematically rich exposition, seeMcGee (1991).

Suitable conditionals

Logics like K3 suffer from the lack of a natural or ‘suitable’ conditional(in particular, one that satisfies A,A → B ⊢ B and ⊢ A → A). This revealsa limitation of the Kripkean approach to the Liar. The language L cannotreport the capture and release properties of truth itself in conditional form(i.e., T-biconditionals): Tr is transparent on this picture, and so Tr(⌜A⌝)and A are fully intersubstitutable. We don't have ¬A ∨ A true for allsentences A in this theory, and hence don't have ¬Tr(⌜A⌝) ∨ A for all A.But ¬Tr(⌜A⌝) ∨ A is equivalent to Tr(⌜A⌝) → A in the theory, since (inthe theory) → is just the material conditional. The Kripke construction athand, then, thus fails to enjoy all T-biconditionals—the natural candidatesfor expressing in the theory the basic capture and release features of truth.

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A recent, major step towards supplementing Kripke's framework with asuitable conditional is that of Field (2008). Field's theory is a majoradvance, but complex enough to be beyond the scope of this (very basic)introduction. Readers should consult Field's own discussion for a taste ofhow such a modification might proceed. See Field (2008), and furtherdiscussion in Beall (2009).

4.1.2 Paraconsistent

Here, the basic idea is to allow the contradiction (e.g., up to and includingstep 4 of the derivation in §2.3.3), but alter the logic by rejecting EFQ—and, hence, avoid the absurdity involved in step 5.

Like the paracomplete approach we just surveyed, paraconsistentapproaches to the Liar find easy, natural motivation in transparency orotherwise suitably ‘minimalist’ views of truth that require fullintersubstitutability of A and Tr(⌜A⌝), and thus cannot restrict capture andrelease. But paraconsistent approaches have also found motivation in aDummett-inspired anti-deflationist view, which takes the role of truth asthe aim of assertion seriously (cf. Dummett (1959)). Indeed, Priest (2006)argues that this (non-transparency) view of truth motivates both the T-schema and LEM, and that this implies that the Liar sentence L is bothtrue and not true. Hence, according to any such dialetheic line (accordingto which at least one sentence is both true and not true), the only option isto reject EFQ.

Dialetheism

Priest (1984, 2006) has been one of the leading voices in advocating aparaconsistent approach to solving the Liar paradox. He has proposed aparaconsistent (and non-paracomplete) logic now known as LP (for Logic

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of Paradox), which retains LEM, but not EFQ.[10] It has the distinctivefeature of allowing true contradictions. This is what Priest calls thedialetheic approach to truth. (See the entry on dialetheism for a moreextensive discussion.)

Formally, LP can be seen as a three-valued logic; but where K3 has truth-value gaps, LP has truth-value gluts. Thus, sentences in LP can be bothtrue and false.

Likewise, Kripke-style techniques can be applied to produce aninterpretation for a truth predicate, starting with a classical language L0not containing a truth predicate. Again, an extension and anti-extensionare assigned to Tr. Whereas Kripke's original construction had theextension and anti-extension disjoint but not exhausting the domain, inthis case we allow the extension and anti-extension to overlap, butsuppose that the two together exhaust the domain of the model. Relatedtechniques to Kripke's can then be used to build an extension and anti-extension for Tr. The result is again an interpretation where A andTr(⌜A⌝) get the same truth value in the model.

This construction was not given by Kripke himself, but variants have beenpursued by a number of authors, including Dowden (1984), Priest (1984,2006), Visser (1984), and Woodruff (1984). A recent discussion of directrelevance is in Beall (2009).

Combining paracompleteness and paraconsistency

Though we have identified paracomplete and paraconsistent approaches tothe Liar as two distinct options, they are not incompatible. Indeed, seen astheories of negation (if one wants), one might think that negation isneither exhaustive nor ‘explosive’ – i.e., satisfies neither LEM nor EFQ.An approach like this is the FDE-based (transparent) truth theory

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discussed in Dunn (1969, see Other Internet Resources), Gupta andBelnap (1993), Visser (1984), Woodruff (1984), Yablo (1993a), and—ineffect—Brady (1989).

(The LP-based theories and K3-based theories are—at least on one(standard-first-order) level—simply strengthened logics of the broaderFDE logic. For general discussion of such frameworks, see, e.g., Bealland van Fraassen (2003) and Priest (2008).)

4.1.3 Expressive power and ‘revenge’

Working in classical logic, Tarski (1935) famously concluded from theLiar paradox that a language cannot define its own truth predicate. Moregenerally, he took the lesson of the Liar to be that languages cannotexpress the full range of semantic concepts that describe their ownworkings. One of the main goals of the non-classical approaches to theLiar we have surveyed here is to avoid this conclusion, which many haveseen as far too drastic. However, how successful these approaches havebeen in this regard remains a highly contentious issue.

In one sense, both the paracomplete and paraconsistent approachesachieve the desired result: they present languages which contain truthpredicates which apply to sentences of that very language, and have thefeature that A and Tr(⌜A⌝) have the same truth value. In this respect, theyboth present languages which contain their own truth predicate.

In the paracomplete case, the issue of whether this suffices has been muchdebated. The paracomplete view holds that the Liar sentence L is neithertrue nor false, and this is key to retaining consistency. But note, theparacomplete approach we discussed above cannot state this fact, as itcannot come out true that ¬Tr(⌜L⌝).

Just what to make of this has been debated. It is certainly the case that the

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set of true sentences in the kind of model Kripke constructs does notinclude ¬Tr(⌜L⌝). Because of this, some authors, such as McGee (1991),T. Parsons (1984), and Soames (1999) have in effect maintained that thisis a further fact that is goes beyond what the truth predicate needs toexpress, and so is immaterial to the success of the solution to the Liar.(Actually, McGee's view has another aspect, which we discuss in §4.2.3.)

But nonetheless, it does appear that there is an important semantic factabout truth in the paracomplete language, closely related to if notidentical to a fact about truth per se, which the language cannot express. Itthus has been argued to fail to achieve a fully adequate theory of truth.Kripke himself notes that there are some semantic concepts that cannot beexpressed, and the argument has been pressed by Glanzberg (2004c) andC. Parsons (1974).

One way of spelling out what is missing in the paracomplete language isto introduce a new notion of determinateness, so that the status of the Liaris that of not being determinately true. If so, then the Kripke paracompletelanguage cannot express this concept of determinateness. Someapproaches taking paracomplete ideas on board have sought tosupplement the Kripke approach by adding notions of determinate truth.McGee (1991) does so in a basically classical setting. In a non-classical,paracomplete setting, Field (2008) supplements the basic paracompleteapproach with infinitely many different ‘determinately’ operators, eachdefined in terms of Field's ‘suitable conditional’, and each giving adifferent (stronger) notion of ‘truth’. (See also some of the papers in Beall(2008).)

It is often argued in favor of paraconsistent approaches that they have notrouble ‘characterizing’ the status of Liars: they're true and false (i.e., trueand have true negation). LP theories can state this. On the other hand,some such as Littmann and Simmons (2004) and S. Shapiro (2004), have

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thought that there is a dual problem: namely, ‘characterizing normalsentences’. (Some put this alleged problem as the problem ofcharacterizing being just true.) Whether this is a problem is something weleave open. (For some discussion, see Beall (2009), Field (2008), andPriest (2006).)

One other issue that arises here is that of so-called ‘revenge paradoxes’.We can illustrate this with the simple-falsity Liar. Suppose one starts withthis as the bench-mark Liar paradox, and proposes a simple solution thatrejects bivalence. In response one is shown the simple-untruth Liar, whichundercuts the simple solution. This is the pattern of ‘revenge’, where asolution to the paradox is rejected on the basis of what might be taken tobe a slightly modified form of the paradox. Revenge paradoxes forparacomplete solutions are often proposed: many points where theparacomplete language fails to express some semantic concept offer waysto construct a revenge problems. For instance, if the determinatenessavenue is taken, then one can construct a revenge problem via a sentencewhich says of itself that it is not determinately true.

We have seen at least some approaches (e.g. McGee (1991) in somerespects, T. Parsons (1984), and Soames (1999)) reject the revengeproblem, while some seek to solve it by additional apparatus (e.g. Field(2008)). As we discuss further in §4.3, contextualist views such as thoseof Burge (1979), Glanzberg (2001, 2004a), and C. Parsons (1974) tend tosee revenge not as a separate problem, but as the core Liar phenomenon.For more discussion about revenge and its nature, see the papers in Beall(2008), and L. Shapiro (2006).

4.2 Classical logic

We have now seen a range of options for responding to the Liar paradoxby reconsidering basic logic. There are also a number of approaches thatleave classical logic fixed, and try to find other ways of defusing the

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paradox.

One hallmark of most of these approaches is a willingness to somehowrestrict the range of application of capture and release, to block theparadoxical reasoning. This is antithetical to the kind of deflationist viewof truth we discussed in §4.1, but it is consistent with another view oftruth. This other view takes the main feature of truth to be that it reports anon-trivial semantic property of sentences (e.g. corresponding with a factin the world, or having a value in a model). Many approaches withinclassical logic embody the idea that a proper understanding of this featureallows for restricted forms of capture and release, and this in turn allowsthe paradox to be blocked, without any departure from classical logic.

We will consider a number of important approaches to the paradox withinclassical logic, most of which embody this idea in some form or another.

4.2.1 Tarski's hierarchy of languages

Traditionally, the main avenue for resolving the paradox within classicallogic is Tarski's hierarchy of languages and metalanguages. Tarskiconcluded from the paradox that no language could contain its own truthpredicate (in his terminology, no language can be ‘semantically closed’).

Instead, Tarski proposed that the truth predicate for a language is to befound only in an expanded metalanguage. For instance, one starts with aninterpreted language L0 that contains no truth predicate. One then ‘stepsup’ to an expanded language L1, which contains a truth predicate, but onethat only applies to sentences of L0. With this restriction, it is easyenough to define a truth predicate which completely accurately states thetruth values of every sentence in L0, obeys capture and release, and yieldsno paradox. Of course, this process does not stop. If we want to describetruth in L1, we need to step up to L2 to get a truth predicate for L1. Andso on. The process goes on indefinitely. At each stage, a new classical

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interpreted language is produced, which expresses truth for languagesbelow it. (For more on the mathematics of this sort of hierarchy oflanguages, see Halbach (1997).)

Why is there no Liar paradox in this sort of hierarchy of languages?Because the restriction that no truth predicate can apply to sentences of itsown language is enforced as a syntactic one. Any sentence L equivalent to¬Tr(⌜L⌝) is not syntactically well-formed. There is no Liar paradoxbecause there is no Liar sentence. See the entries on Tarski and Tarski'struth definitions for more on Tarski's views of truth.

Tarski's hierarchical approach has been subject to a number of criticisms.One is that in light of naturally occurring cases of self-reference, hisruling Liar sentences syntactically not well-formed seems overly drastic.Though Tarski himself was more concerned to resolve the Liar for formallanguages, his solution seems implausible as applied to many naturallyoccurring uses of ‘true’. Another important problem was highlighted byKripke (1975). As Kripke notes, any syntactically fixed set of levels willmake it extremely hard, if not impossible, to place various non-paradoxical claims within the hierarchy. For instance, if Jc says thateverything Michael says is true, the claim has to be made from a level ofthe hierarchy higher than everything Michael says. But if among thethings Micheal says is that everything Jc says is true, Michael's claimsmust be at a higher level than all of Jc's claims. Thus, some of Michael'sclaims must be higher than some of Jc's, and vice-versa. This isimpossible.

In light of these sorts of problems, many have concluded that Tarski'shierarchy of languages and metalanguages buys a solution to the Liarparadox at the cost of implausible restrictiveness.

4.2.2 The closed-off Kripke construction

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In light of these sorts of criticisms of Tarski's theory, a number ofapproaches to the Liar have sought to retain classical logic, but have somedegree of self-applicability for the truth predicate. We know from thereasoning in §2.3 that some restrictions on capture and release will thenbe required. One goal has been to work out which ones are well-motivated, and how to implement them.

One way to do this was suggested by Kripke himself. Rather than see theKripke apparatus we reviewed briefly in §4.1.1 as part of a non-classicallogical approach, one can see it as an intermediate step towards building aclassical interpretation of a self-applicative Tr.

Recall that the Kripke construction starts with a classical language L0with no truth predicate. It passes to an expanded language L , but unlikea Tarskian metalanguage, this language contains a truth predicate Tr thatapplies to all of L . Kripke shows how to build a partial interpretation ofTr, providing an extension E and an anti-extension A. But one can thensimply consider the classical model 〈M0,E〉, using only the extension.This it the ‘closed-off’ construction, as the gap between extension andanti-extension is closed off by throwing everything in the gap into thefalse category of a classical model.

We know this interpretation cannot make true all of capture and release.But it does make a restricted form true. The following holds in theclosed-off model:

This tells us that capture and release (in the form of the T-schema) holdsfor sentences that are well-behaved, in the sense of satisfying Tr(⌜A⌝) ∨Tr(⌜¬A⌝).

What happens to the Liar sentence on this approach? As in the three-

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valued case, the Liar is interpreted as falling within the gap. L is neitherin E nor A. L thus falls outside of the domain where Tr is interpreted aswell-behaved. Because the situation is classical, and ⌜L⌝∉E, we knowthat ¬Tr(⌜L⌝) is true in the closed-off model; likewise, so is ¬Tr(⌜¬L⌝).

On well-behaved sentences, we have the fixed point property that A andTr(⌜A⌝) have the same truth value, and so the semantics of L and thesemantics it assigns to Tr correspond exactly. On pathological sentenceslike L, they do not, and indeed, cannot, on pain of triviality.

In a point related to the closed-off construction, it was observed byFeferman (1984) that if we are careful about negation, we can dispensewith A altogether in the Kripke construction. Thus, the construction canbe done without any implicit appeal to many-valued logic. Related waysof thinking about Kripke's construction are discussed McGee (1991), andare applied in Glanzberg (2004a).

4.2.3 Determinateness revisited

In §4.1.3 we noted that paracomplete approaches to the paradox can bevulnerable to ‘revenge paradoxes’ based on some idea of indeterminatetruth or lacking a truth value. Related issue bear in the classical case. Wewill discuss a few in turn.

Grounding

The closed-off Kripke construction can help fill in the idea of adeterminately operator discussed in §4.1.3. Instead of an operator, itallows us to define a predicate D(⌜A⌝) by Tr(⌜A⌝) ∨ Tr(⌜¬A⌝). Drepresents ‘determinately’ in the sense of applying to sentences that havea truth value according to Tr, as it were, ‘determined’ by the modelproduced by the Kripke construction. It also, as we observed, applies to

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all the sentences which are well-behaved in the sense of obeying the T-schema (or capture and release).

Formally, the sentences to which D applies in the model generated by theKripke construction are those which fall in E or have their negations fallin E (equivalently fall in A). Kripke labeled this being grounded.[11]

It has often been noted that there is also a more informal notion ofdeterminateness or grounding, to which the formal notion expressed by Dat least roughly corresponds (cf. Herzberger (1970)). The idea is that thedeterminate sentences are the ones with well-defined semantic properties.Where we have no such well-defined semantic properties, we should notexpect the truth predicate to report anything well-behaved, nor should weexpect properties like capture and release to hold. Kripke's constructionbuilds up E in stages, starting with sentence with no semantic terms, andadding semantic complexity at each stage. One reaches E at the limit ofthis process, which allows us to think of E as indicating the limit of wheresemantic values are assigned by a well-defined process. Thus, the formalnotion of grounding provided by D is sometimes suggested to reflect theextent to which sentences have well-defined semantic properties.

McGee on truth and definite truth

Another view which makes use of a form of determinateness is advocatedby McGee (1991). McGee's theory, like many we have surveyed, here, isrich in complexity to which we cannot do justice. The theory has manycomponents, including a mathematically sophisticated approaches to truthrelated to the Kripkean ideas we have been discussing, in a setting whichholds to classical logic.

McGee relies on two notions: truth and definite truth. Definite truth is aform of the idea we glossed as determinateness. Formally, for McGee,

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definite truth is identified with provability in a partially interpretedlanguage, using an extension of classical logic which takes in facts aboutthe partial interpretation known as A-logic. It is thus different from thegrounding notion we just discussed. McGee treats definitely as apredicate, on par with the truth predicate, and not as an operator onsentences as some developments do. With the right notion of definitetruth, McGee shows that a partially interpreted language containing itsown truth predicate can meet restricted forms of capture and release put interms of definite truth. Where Def is the definiteness predicate, McGeeshow how to link truth and definite truth, by showing how to validate:

Indeed, McGee shows that these conditions can be met within a theory ofboth truth and definite truth, where truth meets appropriate forms ofcapture and release, and also where a formal statement of bivalence fortruth comes out definitely true. McGee thus provides a theory which hasstrongly self-applicative truth and definite truth, within a classical setting.

Though truth may satisfy the formal property of bivalence, it is crucial toMcGee's approach that definite truth is an open-ended notion, which maybe strengthened (formally, by strengthening a partially interpretedlanguage). Thus, definite truth meets weaker forms of capture and releasethan truth itself. (Some instances of Def(⌜A⌝) → A fail to be definitelytrue, according to McGee.) Furthermore, McGee suggests that thisbehavior of truth and definite truth makes truth a vague predicate. Itremains disputed whether McGee's theory avoids the kind of revengeproblems that plague other Kripkean approaches. An argument that it isvulnerable to forms of revenge is given in Glanzberg (2004c).

4.2.4 Other classical approaches

Def(⌜A⌝) iff Def(⌜Tr(⌜A⌝)⌝)Def(⌜¬A⌝) iff Def(⌜¬Tr(⌜A⌝)⌝)

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We have now surveyed some important representatives of approaches toresolving the Liar within classical logic. There are a number of others,may of them involving some complex mathematics. We will pause tomention a few of the more important of these, though given themathematical complexity, we will only gesture towards them.

Axiomatic theories of truth

There is an important strand of work in proof theory, which has sought todevelop axiomatic theories of self-applicative truth in classical logic,including work of Feferman (1984, 1991), Cantini (1996), and Friedmanand Sheard (1987). The idea is to find ways of expressing rules likecapture and release that retain consistency. Options include more careabout how proof-theoretic rules of inference are formulated, and morecare about formulating restricted rules. The main ideas are discussed inthe entry on axiomatic theories of truth, to which we will leave thedetails.

Truth and inductive definitions

Kripke's work on truth was developed in conjunction with some importantideas about inductive definitions (as we see, for instance, in the later partsof Kripke (1975)). These connections are explored further in work ofBurgess (1986) and McGee (1991). We also pause to mention work ofAczel (1980) combining ideas about inductive definitions and the lambdacalculus.

4.3 Contextualist approaches

Another family of proposed solutions to the Liar are contextualistsolutions. These also make use of classical logic, but base their solutions

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primarily on some ideas from the philosophy of language. They take thebasic lesson of the Liar to be that truth predicates show some form ofcontext dependence, even in otherwise non-context-dependent fragmentsof a language. They seek to explain how this can be so, and rely on it toresolve the problems faced by the Liar.

Contextualist theories share with a number of approaches we have alreadyseen the idea that there is something indeterminate or semantically notwell-formed about our Liar sentence L. But, contextualist views give aspecial role to issues of ‘revenge’ and lack of expressive power.

4.3.1 Instability and revenge

One way of thinking about why the truth predicate is not well-behaved onthe Liar sentence is that there is not really a well-defined truth bearerprovided by the Liar sentence. To make this vivid (as discussed by C.Parsons (1974) and subsequently Glanzberg (2001)), suppose that truthbearers are propositions expressed by sentences in contexts, and that theLiar sentence fails to express a proposition. This is the beginnings of anaccount of how the Liar winds up ungrounded or in some senseindeterminate. At least, we should not expect Tr to be well-behavedwhere sentences fail to express propositions.

But, it is an unstable proposal. We can reason that if the Liar sentencefails to express a proposition, it fails to express a true proposition. In themanner of a revenge paradox, if our Liar sentence had originally said ‘thissentence does not express a true proposition’, then we would have ourLiar sentence back. And, we have shown that this sentence sayssomething true, and so expresses a true proposition. Thus, from theassumption that the Liar sentence is indeterminate or lacks semanticstatus, we reason that it must have proper semantic status, and indeed saysomething true. We are hence back in paradox.

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Contextualists do not see this as a new ‘revenge’ paradox, but the basicproblem posed by the Liar. First of all, in a setting where sentences arecontext dependent, the natural formulation of a truth claim is always interms of expressing a true proposition, or some related semanticallycareful application of the truth predicate. But more importantly, to thecontextualist, the main issue behind the Liar is embodied in the reasoningon display here. It involves two key steps. First, assigning the Liarsemantically defective status—failing to express a proposition or beingsomehow indeterminate. Second, concluding from the first step that theLiar must be true—and so not indeterminate or failing to express aproposition—after all. Both steps appear to be the result of soundreasoning, and so both must be true. The main problem of the Liar,according to a contextualist, is to explain how this can be, and how thesecond step can be non-paradoxical. (Such reasoning is explored byGlanzberg (2001, 2004c) and C. Parsons (1974). For a critical discussion,see Gauker (2006).)

Thus, contextualists seek to explain how the Liar sentence can haveunstable semantic status, switching from defective to non-defective in thecourse of this sort of inference. They do so by appealing to the role ofcontext in fixing the semantic status of sentences. Sentences can havedifferent semantic status in different contexts. Thus, to contextualists,there must be some non-trivial effect of context involved in the Liarsentence, and more generally, in predication of truth.

4.3.2 Contextual parameters on truth predicates

One prominent contextualist approach, advocated by Burge (1979) anddeveloped by Koons (1992) and Simmons (1993), starts with the idea thatthe Tarskian hierarchy itself offers a way to see the truth predicate ascontext dependent. Tarski's hierarchy postulates a hierarchy of truthpredicates Tri. What if i is not merely a marker of level in a hierarchy, but

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a genuine contextual parameter? If so, then the Liar sentence is in factcontext-dependent: it has the form ¬Tri(⌜L⌝), where i is set by context.Context then sets the level of the truth predicate.

This idea can be seen as an improvement on the original Tarskianapproach in several respects. First, once we have a contextual parameter,the need to insist that Liar sentences are never well-formed disappears.Hence, we can think of each Tri as including some limited range ofapplicability to sentences of its own language. Using the Kripkeantechniques likes the closed-off construction we reviewed above,predicates like Tri can be constructed which have as much self-applicability as Kripke's own. (Burge (1979) and the postscript to C.Parsons (1974) consider briefly how Kripke techniques could be appliedin this setting. Though he works in a very different setting, ideas ofGaifman (1988, 1992) can be construed as showing how even more subtleways of interpreting a context-dependent truth predicate can bedeveloped.)

With suitable care, other problems for the Tarskian hierarchy can beavoided as well. Burge proposes that the parameter i in Tri is set by aGricean pragmatic process. In effect, speakers implicate that i is to be setto a level for which the discourse they are in can be coherently interpreted(with a maximal coherent extension for Tri). Thus, truth does indeed findits own level, and so Kripke's objection about how to fix levels for non-paradoxical sentences may be countered.

This approach gives substance to the idea that the Liar sentence is contextdependent. Any sentence containing Tri will be context dependent,inheriting a contextual parameter along the way. This offers a way tomake sense of the arguments for the instability of the semantic status of Lthat motivated contextualism. In an initial context, we fix some level i.This is the level at which L is interpreted. Call this interpretation Li. Li

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This is the level at which is interpreted. Call this interpretation i. isays ¬Tri(⌜Li

⌝). By the usual Liar reasoning, we show that Li must lackdeterminate semantic status—or fail to express a proposition. As wediscussed, we then reason that L must come out true. According to thecontextualist view at hand, this is the claim that Li is true according tosome other context, where a wider truth predicate is in play. This amountsto being true at some higher level of the hierarchy. We can conclude, forinstance, that the Liar sentence as it was used at level i is true accordingto a wider level k > i. Hence, Trk(⌜Li

⌝), where k > i.

This form of contextualism thus maintains that once we see the context-dependent behavior of Tri, we can make good sense of the instability of L.This can be seen as an improvement on both the Tarskian view, andembodying some of the techniques of classical logic we reviewed in §4.2.Depending how the Burge view is spelled out technically, it will eitherhave full capture and release at each level, or capture and release with thesame restrictions as the closed-off Kripke construction.

The view that posits contextual parameters on the truth predicate doesface a number of questions. For instance, it is fair to ask why we think thetruth predicate really has a contextual parameter, especially if we mean atruth predicate like the one we use in natural language. Merely noting thatsuch a parameter would avoid paradox does not show that it is present innatural language. Furthermore, whether it is acceptable to see truth ascoming in levels at all, context-based or not, remains disputed. (Not allthose who advocate contextual parameters on the truth predicate agreeabout the role of hierarchy. In particular, Simmons (1993) advocates aview he labels the ‘singularity theory’ which he proposes avoids outrighthierarchical structures.) Finally, the Burgean appeal to Griceanmechanisms to set levels of truth has been challenged. (For instance,Gaifman (1992) asks if the Gricean process does any substantial work inBurge's account.)

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Contextualist approaches come in many varieties, each of which makesuse of slightly different apparatus. With contextualist theories the choiceoften turns on issues in philosophy of language as well as logic. Wealready noted a different way of developing contextualist ideas fromGaifman (1988, 1992). We will now briefly review a few morealternatives.

4.3.3 Contextual effects on quantifier domains

Another contextualist approach, stemming from work of C. Parsons(1974) and developed by Glanzberg (2001, 2004a), seeks to build up thecontext dependence of the Liar sentence, and ultimately the contextdependence of the truth predicate, from more basic components. The keyis to see the context dependence of the Liar sentence as derived from thecontext dependence of quantifier domains.

Quantification enters the picture when we think about how to account forpredication of truth when sentences display context dependence. In suchan environment, it does not make good sense to predicate truth ofsentences directly. Not all sentences will have the right kind ofdeterminate semantic properties to be truth bearers; or, as we have beenputting it, not all sentences will express propositions. But then, to say thata sentence S is true in context c is to say that there is a proposition pexpressed by S in c, and that proposition p is true.

The current contextualist proposal starts with the observation thatquantifiers in natural language typically have context-dependent domainsof quantification. When we say ‘Everyone is here’, we do not meaneveryone in the world, but everyone in some contextually providedsubdomain. Context dependence enters the Liar, according to thiscontextualist view, in the contextual effects on the domain of thepropositional quantifier ∃p.

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In particular, this domain must expand in the course of the reasoningabout the semantic status of the Liar. In the initial context, ∃p must rangeover a small enough domain that there is no proposition for L to express.In the subsequent context, the domain expands to allow L to express sometrue proposition. Proposals for how this expansion happens, and how tomodel the truth predicate and the relation of expressing a proposition inthe presence of the Liar, have been explored by Glanzberg (2001, 2004a),building on work of C. Parsons (1974). Defenders of this approach arguethat it does better in locating the locus of context dependence than theparameters on truth predicates view.

4.3.4 Situation theory

Another variant on the contextualist strategy for resolving the Liar,developed by Barwise and Etchemendy (1987) and Groeneveld (1994),relies on situation theory rather than quantifier domains to provide thelocus of context dependence. Situation theory is a highly developed partof philosophy of language, so we shall again give only the roughestsketch of how their view works.

A situation is a partial state the world might be in: something like a beingF. Situations are classified by what are called situation types. Aproposition involves classifying a situation as being of a situation type.Thus, a proposition {s; [σ]} tell us that situation s is of type σ. Thesituation s here plays a number of roles, including that of providing acontext.

When it comes to the Liar, Barwise and Etchemendy construe Liarpropositions as having the form fs = {s; [Tr,fs; 0]}, relative to an initialsituation s. This is a proposition fs which says of itself that its falsity is afact that holds in s. There is a sense in which this proposition cannot beexpressed. In particular, the state of affairs 〈Tr,fs; 0〉 cannot be in s.

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(Actually, Barwise and Etchemendy say that the proposition isexpressible, but give up on what they call the F-closure of s. But there is acore observation in common between these two points, and the details donot matter for our purposes here.) There is then a distinct situation sʹ′ = s∪ {〈Tr,fs; 0〉}, and the proposition {sʹ′; [Tr,fs; 0]} relative to this newsituation—this new ‘context’—is true.

This idea clearly has a lot in common with the restriction on quantifierdomains view. In particular, both approaches seek to show how thedomain of contents expressible in contexts can expand, to account for theinstability of the Liar sentence. For discussion of relations between thesituation-theoretic and quantifier domain approaches, see Glanzberg(2004a). Barwise and Etchemendy discuss relations between theirsituation-based and a more traditional approach in (1987, Ch. 11). For adetailed match-up between the Barwise and Etchemendy framework and aBurgean framework of indexed truth predicates, see Koons (1992).

4.3.5 Issues for contextualism

It is a key challenge to contextualists to provide a full and well-motivatedaccount of the source and nature of the shift in context involved in theLiar, though of course, many contextualists believe they have met thischallenge. In favor of the contextualist approach is that it takes therevenge phenomenon to be the basic problem, and so is largely immuneto the kinds of revenge issues that affect other approaches we haveconsidered. But, it may be that there is another form of revenge whichmight be applied. To retain consistency, contextualists must applyrestrictions on quantifiers to such quantifiers as ‘all contexts’. To achievethis, it must presumably be denied that there are any absolutelyunrestricted quantifiers. Glanzberg (2004b, 2006) argues this is thecorrect conclusion, but it is highly controversial. For a survey of thinkingabout this, see the papers in Rayo and Uzquiano (2006).

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4.4 The revision theory

Another approach to the Liar, advocated by Gupta (1982), Herzberger(1982), Gupta and Belnap (1993), and a number of others, is the revisiontheory of truth. This approach shares some features with the views wesurveyed in §4.2, in that it takes classical logic for granted. We alsobelieve it has an affinity with the views discussed in §4.3, as it rethinkssome basic aspects of semantics. But it is a distinctive approach. We willsketch some of the fundamentals of this view. For a discussion of thefoundations of the revision theory, and its relations to contextualism, seeL. Shapiro (2006). More details, and more references, may be found inthe entry on the revision theory of truth.

The revision theory of truth starts with the idea that we may take the T-schema at face value. Indeed, Gupta and Belnap (1993) take up asuggestion from Tarski (1944), that the instances of the T-schema can beseen as partial definitions of truth; presumably with all the instancestogether, for the right language or family of languages constituting acomplete definition. At the same time, the revision theory holds fast toclassical logic. Thus, we already know, we have the Liar paradox for anylanguage with enough expressive resources to produce Liar sentences.

In response, the revision theory proposes a different way of approachingthe semantic properties of the truth predicate. In keeping with ourpractices here, we may begin with a classical model M0 for a languageL0 without a truth predicate, and consider what happens when we add atruth predicate Tr to form the extended language L . This language has afull self-applicative truth predicate, and so can generate the Liar sentenceL.

To build a classical model for L , we need an extension for Tr. Let uspick a set: call it H for a hypothesis about what the extension of Tr might

+0

+0

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be. H may be ∅, it may be the entire domain of M0, or it may beanything else. It need not be a particularly good approximation of thesemantic properties of Tr.

Even if it is not, 〈M0,H〉 still provides a classical model, in which we caninterpret L . With that, we can in effect apply the T-schema, relative toour hypothesis H, and see what we get. More precisely, we can let τ(H) ={⌜A⌝ | A is true in 〈M0,H〉}. τ(H) is generally a better hypothesis aboutwhat is true in our language than H might have been. At least, clearly, ifH made foolish guesses about the truth of sentence of the truth-freefragment L0, they are corrected in τ(H), which contains everything fromL0 true in M0. Thus, 〈M0,τ(H)〉 is generally a better model of L then〈M0,H〉.

Better in many respects. But when it comes to paradoxical sentences likeL, we see something different. As a starting hypothesis, let us consider H= ∅. Consider what happens to the truth of L as we apply τ:

The Liar sentence never stabilizes under this process. We reach analternation of truth values which will go on for ever.

In the terminology of the revision theory, τ is a revision rule. It takes usfrom one hypothesis about the interpretation of Tr to another. Sequences

+0

+0

n truth value of L in 〈M0,τn(∅)〉

0 true

1 false

2 true

3 false

4 true

⋮ ⋮

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of values we generate by such revision rules, starting with a given initialhypothesis, are revision sequences. We leave to a more full presentationthe important issue of the right way to define transfinite revisionsequences. (See the entry on revision theories of truth.)

The characteristic property of paradoxical sentences like the Liar sentenceis that they are unstable in revision sequences: there is no point in thesequence at which they reach a stable truth value. This classifiessentences as stably true, stably false, and unstable. The revision theorydevelops notions of consequence based on these, and related notions. Seethe entry on revision theories of truth for further exposition of this richtheory.

4.5 Inconsistency views

In §2.3.3 we saw that the Liar paradox, in the presence of unrestrictedcapture and release and classical logic, leads to contradiction. So long aswe have EFQ (as classical logic does), this results in triviality. Most ofthe proposed solutions we have considered (with the exception of therevision theory) try to avoid this result somehow, either by restrictingcapture and release or departing from classical logic. But there is anotheridea that has occasionally been argued, that the Liar paradox simplyshows that the kinds of languages we speak, which contain their own truthpredicates, are inconsistent.

This is not an easy view to formulate. Though Tarski himself seemed tosuggest something along these lines (for natural languages, specifically),it was argued by Herzberger (1967) that it is impossible to have aninconsistent language.

In contrast, Eklund (1902) takes seriously the idea that our semanticintuitions, expressed, for instance, by unrestricted capture and release,really are inconsistent. Eklund grants that this does not make sense if

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these intuitions have their source simply in our grasp of the truthconditions of sentences. But he suggests an alternative picture of semanticcompetence which does make sense of it (closely related to conceptualrole views of meaning). He suggests that we think of semanticcompetence in terms of a range of principles speakers are disposed toaccept in virtue of knowing a language. Those principles may beinconsistent. But even so, they determine semantic values. Semanticvalues will be whatever comes closest to satisfying the principles—whatever makes them maximally correct—even if nothing can satisfy allof them due to an underlying inconsistency.

Eklund thus supports an idea suggested by Chihara (1979). Chihara's mainaim is to provide what he calls a diagnosis of the paradox, which shouldexplain why the paradox arises and why it appears compelling. But alongthe way, he suggests that the source of the paradox is our acceptance ofthe T-schema (by convention, he suggests), in spite of its inconsistency.

A related, though distinct, view is defended by Patterson (2007, 2009).Patterson argues that competence with a language puts one in a cognitivestate relating to an inconsistent theory—one including the unrestricted T-schema and governed by classical logic. He goes on to explore how such acognitive state could allow us to successfully communicate, in spite ofrelating us to a false theory.

5. Concluding Remarks

There is much more to say about the Liar paradox than we have coveredhere: there are more approaches to the Liar variants we have mentioned,and more related paradoxes like those of denotation, properties, etc. Thereare also more important technical results, and more importantphilosophical implications and applications. Our goal here has been to bemore suggestive than exhaustive, and we hope to have given the reader an

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indication of what the Liar paradox is, and what its consequences mightbe.

Bibliography

Aczel, Peter, 1980, “Frege structures and the notions of proposition, truthand set”, in The Kleene Symposium, J. Barwise, H. J. Keisler, and K.Kunen, eds., Amsterdam: North-Holland, 31–59.

Anderson, Alan Ross, 1970, “St. Paul’s epistle to Titus”, in The Paradoxof the Liar, Robert L. Martin, ed., Atascadero: Ridgeview, 1–11.

Asenjo, F. G., 1966, “A calculus of antinomies”, Notre Dame Journal ofFormal Logic, 16: 103–105.

Barwise, Jon and Etchemendy, John, 1987, The Liar, Oxford: OxfordUniversity Press.

Beall, Jc, 2001, “Is Yablo’s paradox non-circular?”, Analysis, 61: 176–187.

Beall, Jc, 2005, “Transparent disquotationalism”, in Deflationism andParadox, Jc Beall and B. Armour-Garb, eds., Oxford: OxfordUniversity Press, 7–22.

Beall, Jc (ed.), 2008, Revenge of the Liar, Oxford: Oxford UniversityPress.

Beall, Jc, 2009, Spandrels of Truth, Oxford: Oxford University Press.Beall, Jc and Glanzberg, Michael, 2008, “Where the paths meet: Remarks

on truth and paradox”, in Midwest Studies in Philosophy VolumeXXXII: Truth and its Deformities, P. A. French and H. K. Wettstein,eds., Boston: Wiley-Blackwell.

Beall, Jc and van Fraassen, Bas C., 2003, Possibilities and Paradox,Oxford: Oxford University Press.

Brady, Ross T., 1989, “The non-triviality of dialectical set theory”, inParaconsistent Logic: Essays on the Inconsistent, G. Priest, R.Routley, and J. Norman, eds., Munich: Philosophia Verlag, 437–470.

Burge, Tyler, 1979, “Semantical paradox”, Journal of Philosophy, 76:

JC Beall and Michael Glanzberg

Winter 2012 Edition 41

Page 43: liar paradox

169–198. Reprinted in Martin (1984).Burgess, John P., 1986, “The truth is never simple”, Journal of Symbolic

Logic, 51: 663–681.Cantini, Andrea, 1996, Logical Frameworks for Truth and Abstraction:

An Axiomatic Study, Amsterdam: Elsevier.Chihara, Charles, 1979, “The semantic paradoxes: A diagnostic

investigation”, Philosophical Review, 88: 590–618.Cook, Roy, 2006, “There are non-circular paradoxes (but Yablo’s isn’t

one of them)”, The Monist, 89: 118–149.Dowden, Bradley H., 1984, “Accepting inconsistencies from the

paradoxes”, Journal of Philosophical Logic, 13: 125–130.Dummett, Michael, 1959, “Truth”, Proceedings of the Aristotelian

Society, 59: 141–162. Reprinted in Dummett (1978).Dummett, Michael, 1978, Truth and Other Enigmas, Cambridge: Harvard

University Press.Eklund, Matti, 2002, “Inconsistent languages”, Philosophy and

Phenomenological Research, 64: 251–275.Feferman, Solomon, 1984, “Toward useful type-free theories, I”, Journal

of Symbolic Logic, 49: 75–111. Reprinted in Martin (1984).Feferman, Solomon, 1991, “Reflecting on incompleteness”, Journal of

Symbolic Logic, 56: 1–49.Field, Hartry, 1994, “Deflationist views of meaning and content”, Mind,

103: 249–285.Field, Hartry, 2008, Saving Truth from Paradox, Oxford: Oxford

University Press.Friedman, Harvey and Sheard, Michael, 1987, “An axiomatic approach to

self-referential truth”, Annals of Pure and Applied Logic, 33: 1–21.Gaifman, Haim, 1988, “Operational pointer semantics: Solution to self-

referential puzzles I”, in Proceedings of the Second Conference onTheoretical Aspects of Reasoning about Knowledge, M. Y. Vardi,ed., Los Altos: Morgan Kaufmann, 43–59.

Gaifman, Haim, 1992, “Pointers to truth”, Journal of Philosophy, 89:

Liar Paradox

42 Stanford Encyclopedia of Philosophy

Page 44: liar paradox

223–261.Gauker, Christopher, 2006, “Against stepping back: A critique of

contextualist approaches to the semantic paradoxes”, Journal ofPhilosophical Logic, 35: 393–422.

Glanzberg, Michael, 2001, “The Liar in context”, Philosophical Studies,103: 217–251.

Glanzberg, Michael, 2004a, “A contextual-hierarchical approach to truthand the Liar paradox”, Journal of Philosophical Logic, 33: 27–88.

Glanzberg, Michael, 2004b, “Quantification and realism”, Philosophy andPhenomenological Research, 69: 541–572.

Glanzberg, Michael, 2004c, “Truth, reflection, and hierarchies”, Synthese,142: 289–315.

Glanzberg, Michael, 2006, “Context and unrestricted quantification”, inAbsolute Generality, A. Rayo and G. Uzquiano, eds., Oxford: OxfordUniversity Press, 45–74.

Grim, Patrick, 1991, The Incomplete Universe, Cambridge: MIT Press.Groeneveld, Willem, 1994, “Dynamic semantics and circular

propositions”, Journal of Philosophical Logic, 23: 267–306.Gupta, Anil, 1982, “Truth and paradox”, Journal of Philosophical Logic,

11: 1–60. Reprinted in Martin (1984).Gupta, Anil and Belnap, Nuel, 1993, The Revision Theory of Truth,

Cambridge: MIT Press.Halbach, Volker, 1997, “Tarskian and Kripean truth”, Journal of

Philosophical Logic, 26: 69–80.Heck, Richard, 1997, “Tarski, truth, and semantics”, Philosophical

Review, 106: 533–554.Herzberger, Hans G., 1967, “The truth-conditional consistency of natural

language”, Journal of Philosophy, 64: 29–35.Herzberger, Hans G., 1970, “Paradoxes of grounding in semantics”,

Journal of Philosophy, 67: 146–167.Herzberger, Hans G., 1982, “Notes on naive semantics”, Journal of

JC Beall and Michael Glanzberg

Winter 2012 Edition 43

Page 45: liar paradox

Philosophical Logic, 11: 61–102. Reprinted in Martin (1984).Horwich, Paul, 1990, Truth, Oxford: Basil Blackwell.Koons, Robert C., 1992, Paradoxes of Belief and Strategic Rationality,

Cambridge: Cambridge University Press.Kripke, Saul, 1975, “Outline of a theory of truth”, Journal of Philosophy,

72: 690–716. Reprinted in Martin (1984).Littmann, Greg and Simmons, Keith, 2004, “A critique of dialetheism”, in

The Law of Non-Contradiction, G. Priest, Jc Beall, and B. Armour-Garb, eds., Oxford: Oxford University Press, 314–335.

Martin, Robert L. (ed.), 1984, Recent Essays on Truth and the LiarParadox, Oxford: Oxford University Press.

Martin, Robert L. and Woodruff, Peter W., 1975, “On representing ‘true-in-L’ in L”, Philosophica, 5: 217–221. Reprinted in Martin (1984).

McGee, Vann, 1991, Truth, Vagueness, and Paradox, Indianapolis:Hackett.

Parsons, Charles, 1974, “The Liar paradox”, Journal of PhilosophicalLogic, 3: 381–412. Reprinted in Parsons (1983).

Parsons, Charles, 1983, Mathematics in Philosophy, Ithaca: CornellUniversity Press.

Parsons, Terence, 1984, “Assertion, denial, and the Liar paradox”,Journal of Philosophical Logic, 13: 137–152.

Patterson, Douglas, 2007, “Understanding the Liar”, in The Revenge ofthe Liar, Jc Beall, ed., Oxford: Oxford University Press, 197–224.

Patterson, Douglas, 2009, “Inconsistency theories of semantic paradox”,Philosophy and Phenomenological Research, 79: 387–422.

Priest, Graham, 1984, “Logic of paradox revisited”, Journal ofPhilosophical Logic, 13: 153–179.

Priest, Graham, 1997, “Yablo’s paradox”, Analysis, 57: 236–242.Priest, Graham, 2006, In Contradiction, Oxford: Oxford University Press,

second ed.Priest, Graham, 2008, An Introduction to Non-Classical Logic,

Cambridge: Cambridge University Press, second ed.

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Page 46: liar paradox

Rahman, Shahid, Tulenheimo, Tero, and Genot, Emmanuel (eds.), 2008,Unity, Truth and the Liar: The Modern Relevance of MedievalSolutions to the Liar Paradox, Berlin: Springer Verlag.

Rayo, Agustín and Uzquiano, Gabriel (eds.), 2006, Absolute Generality,Oxford: Oxford University Press.

Read, Stephen, 2002, “The Liar paradox from John Buridan back toThomas Bradwardine”, Vivarium, 40: 189–218.

Read, Stephen, 2006, “Symmetry and paradox”, History and Philosophyof Logic, 27: 307–318.

Restall, Greg, 2008, “Modal models for Bradwardine’s theory of truth”,Review of Symbolic Logic, 1: 225–240.

Shapiro, Lionel, 2006, “The rationale behind the revision theory”,Philosophical Studies, 129: 477–515.

Shapiro, Lionel, forthcoming, “Expressibility and the Liar’s revenge”,Australasian Journal of Philosophy.

Shapiro, Stewart, 2004, “Simple truth, contradiction, and consistency”, inThe Law of Non-Contradiction, G. Priest, Jc Beall, and B. Armour-Garb, eds., Oxford: Oxford University Press, 336–354.

Simmons, Keith, 1993, Universality and the Liar, Cambridge: CambridgeUniversity Press.

Soames, Scott, 1999, Understanding Truth, Oxford: Oxford UniversityPress.

Sorensen, Roy, 1998, “Yablo’s paradox and kindred infinite Liars”, Mind,107: 137–155.

Sorensen, Roy, 2003, A Brief History of Paradox, Oxford: OxfordUniversity Press.

Tarski, Alfred, 1935, “Der Wahrheitsbegriff in den formaliziertenSprachen”, Studia Philosophica, 1: 261–405. References are to thetranslation by J. H. Woodger as “The concept of truth in formalizedlanguages” in Tarski (1983).

Tarski, Alfred, 1944, “The semantic conception of truth”, Philosophy and

JC Beall and Michael Glanzberg

Winter 2012 Edition 45

Page 47: liar paradox

Phenomenological Research, 4: 341–375.Tarski, Alfred, 1983, Logic, Semantics, Metamathematics, Indianapolis:

Hackett, second ed. Edited by J. Corcoran with translations by J. H.Woodger.

van Fraassen, Bas C., 1968, “Presupposition, implication, and self-reference”, Journal of Philosophy, 65: 136–152.

van Fraassen, Bas C., 1970, “Truth and paradoxical consequence”, inParadox of the Liar, R. L. Martin, ed., Atascadero: Ridgeview, 13–23.

Visser, Albert, 1984, “Four valued semantics and the Liar”, Journal ofPhilosophical Logic, 13: 181–212.

Woodruff, Peter W., 1984, “Paradox, truth, and logic part 1: Paradox andtruth”, Journal of Philosophical Logic, 13: 213–232.

Yablo, Stephen, 1993a, “Hop, skip, and jump: The agnostic conception oftruth”, Philosophical Perspectives, 7: 371–396.

Yablo, Stephen, 1993b, “Paradox without self-reference”, Analysis, 53:251–252.

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Related Entries

Curry's paradox | dialetheism | Gödel, Kurt | Gödel, Kurt: incompletenesstheorem | logic: many-valued | logic: provability | Tarski, Alfred | Tarski,Alfred: truth definitions | truth | truth: axiomatic theories of | truth:revision theory of | vagueness

Notes to Liar Paradox

1. Thus, a paradox nearly occurs in the New Testament. For a delightfuldiscussion, see Anderson (1970). For a more thorough discussion of the

Preview the PDF version of this entry at the Friends of the SEPSociety.Look up this entry topic at the Indiana Philosophy OntologyProject (InPhO).Enhanced bibliography for this entry at PhilPapers, with linksto its database.

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history of the Liar, see Sorensen (2003). There has been some importantrecent work on medieval theories of the Liar, and their relevance tocurrent approaches. See the papers in Rahman, Tulenheimo, and Genot(2008), and, for instance, Read (2002, 2006), Restall (2008), andSimmons (1993).

2. Terminology here is not uniform. Van Fraassen (1968) introduced theterm ‘strengthened Liar’ to name what we are calling the simple-untruthLiar. Van Fraassen's term, however, has been more often used for a‘revenge-like’ paradox in the manner of C. Parsons (1974), as we discussin §4.1.3.

3. A note on terminology: when talking about a language, we mean aninterpreted language, including a syntax, an interpretation or model(containing a domain of objects and interpretations for non-logicalvocabulary), and a valuation scheme which determines truth in the modelfor complex formulas. Simplifying for exposition, we shall often speakloosely and blur the distinction between a logic and a language (thinking,in a not strictly accurate fashion, of the valuation scheme as the relevantbit of ‘logical consequence’ for the language). Moreover, at some points,it is a theory rather than a language that is at issue. We note when suchdistinctions become important. Given the wide range of ideas in logic thatwe survey in rather brief form in this essay, we leave many suchsubtleties to more leisurely presentations.

4. Hence, our terminology of ‘rule forms’ may not be ideal. For ourlimited purposes here, were use it merely to mark the difference betweena valid argument, recorded as a ‘rule form’, and the provability of aconditional, reported as a ‘conditional form’. Though this is not the onlylogical distinction to be drawn, it is the one which will be of mostimportance as our discussion progresses.

5. The situation with formal languages is actually somewhat subtler than

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our brief discussion indicates. In most cases, corner quotes really indicateformal terms for Gödel numbers of sentences, and are not genuinequotation marks in the usual sense (e.g., denoting the expression ‘inside’them). Hence, the sense in which such languages have reference tosentences is delicate. Yet with very minimal resources, syntax can berepresented and diagonal sentences constructed. Hence, there is a sense,albeit subtle, in which such languages can express self-reference. See theentry on provability logic and Gödel (see the section on theincompleteness theorems).

6.. It is this principle or rule that we repeated above by saying that by thelights of many logical theories an arbitrary contradiction implies absurdityor triviality, in the sense of implying all sentences. The principle is oftenlabeled by its classical title ‘ex falso quodlibet’, and hence its customaryabbreviation is ‘EFQ’ in spite of the name ‘explosion’.

7. This is often called ‘∨-Out’ or ‘∨-Elim’ or, more suggestively,‘reasoning by cases’.

8. Again for the classical case, Friedman and Sheard (1987) provide anexhaustive list of inconsistent theories relative to a fairly weak basetheory. They show that either of the classical conditional forms of captureor release are inconsistent by themselves, over the base theorysupplemented by the right choice of principles providing for completenessor consistency of truth.

9. We have argued for a close connection between general views on thenature of truth and available avenues for resolving the paradox in our(2008).

10. The logic was first advanced by Asenjo (1966), though has come tobe popularized by Priest’s work. The name ‘LP’ is due to Priest, while theterm ‘dialetheism’ was coined by Priest and Routley.

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11. A technical note: Kripke defined being grounded as appearing in theminimal fixed point. We have not discussed other fixed points here, butthere are many, and a sentence can appear in them without appearing inthe minimal one.

Copyright © 2012 by the authors JC Beall and Michael Glanzberg

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