In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance, declaring that "I am lying". If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied. In "this sentence is a lie" the paradox is strengthened in order to make it amenable to more rigorous logical analysis. It is still generally called the "liar paradox" although abstraction is made precisely from the liar making the statement. Trying to assign to this statement, the strengthened liar, a classical binary truth value leads to a contradiction.

If "this sentence is false" is true, then it is false, but the sentence states that it is false, and if it is false, then it must be true, and so on.

## History

The Epimenides paradox (circa 600 BC) has been suggested as an example of the liar paradox, but they are not logically equivalent. The semi-mythical seer Epimenides, a Cretan, reportedly stated that "All Cretans are liars."[1] However, Epimenides' statement that all Cretans are liars can be resolved as false, given that he knows of at least one other Cretan who does not lie (alternatively, it can be taken as merely a statement that all Cretans tell lies, not that they tell only lies).

The paradox's name translates as pseudómenos lógos (ψευδόμενος λόγος) in Ancient Greek. One version of the liar paradox is attributed to the Greek philosopher Eubulides of Miletus, who lived in the 4th century BC. Eubulides reportedly asked, "A man says that he is lying. Is what he says true or false?"[2]

The paradox was once discussed by St. Jerome in a sermon:

"I said in my alarm, Every man is a liar!" Is David telling the truth or is he lying? If it is true that every man is a liar, and David's statement, "Every man is a liar" is true, then David also is lying; he, too, is a man. But if he, too, is lying, his statement that "Every man is a liar", consequently is not true. Whatever way you turn the proposition, the conclusion is a contradiction. Since David himself is a man, it follows that he also is lying; but if he is lying because every man is a liar, his lying is of a different sort.[3]

The Indian grammarian-philosopher Bhartrhari (late fifth century AD) was well aware of a liar paradox which he formulated as "everything I am saying is false" (sarvam mithyā bravīmi). He analyzes this statement together with the paradox of "unsignifiability" and explores the boundary between statements that are unproblematic in daily life and paradoxes.[4][5]

There was discussion of the liar paradox in early Islamic tradition for at least five centuries, starting from late 9th century, and apparently without being influenced by any other tradition. Naṣīr al-Dīn al-Ṭūsī could have been the first logician to identify the liar paradox as self-referential.[6]

## Explanation and variants

The problem of the liar paradox is that it seems to show that common beliefs about truth and falsity actually lead to a contradiction. Sentences can be constructed that cannot consistently be assigned a truth value even though they are completely in accord with grammar and semantic rules.

The simplest version of the paradox is the sentence:

A: This statement (A) is false.

If (A) is true, then "This statement is false" is true. Therefore, (A) must be false. The hypothesis that (A) is true leads to the conclusion that (A) is false, a contradiction.

If (A) is false, then "This statement is false" is false. Therefore, (A) must be true. The hypothesis that (A) is false leads to the conclusion that (A) is true, another contradiction. Either way, (A) is both true and false, which is a paradox.

However, that the liar sentence can be shown to be true if it is false and false if it is true has led some to conclude that it is "neither true nor false".[7] This response to the paradox is, in effect, the rejection of the claim that every statement has to be either true or false, also known as the principle of bivalence, a concept related to the law of the excluded middle.

The proposal that the statement is neither true nor false has given rise to the following, strengthened version of the paradox:

This statement is not true. (B)

If (B) is neither true nor false, then it must be not true. Since this is what (B) itself states, it means that (B) must be true. Since initially (B) was not true and is now true, another paradox arises.

Another reaction to the paradox of (A) is to posit, as Graham Priest has, that the statement is both true and false. Nevertheless, even Priest's analysis is susceptible to the following version of the liar:

This statement is only false. (C)

If (C) is both true and false, then (C) is only false. But then, it is not true. Since initially (C) was true and is now not true, it is a paradox. However, it has been argued that by adopting a two-valued relational semantics (as opposed to functional semantics), the dialetheic approach can overcome this version of the Liar.[8]

There are also multi-sentence versions of the liar paradox. The following is the two-sentence version:

The following statement is true. (D1)
The preceding statement is false. (D2)

Assume (D1) is true. Then (D2) is true. This would mean that (D1) is false. Therefore, (D1) is both true and false.

Assume (D1) is false. Then (D2) is false. This would mean that (D1) is true. Thus (D1) is both true and false. Either way, (D1) is both true and false – the same paradox as (A) above.

The multi-sentence version of the liar paradox generalizes to any circular sequence of such statements (wherein the last statement asserts the truth/falsity of the first statement), provided there are an odd number of statements asserting the falsity of their successor; the following is a three-sentence version, with each statement asserting the falsity of its successor:

E2 is false. (E1)
E3 is false. (E2)
E1 is false. (E3)

Assume (E1) is true. Then (E2) is false, which means (E3) is true, and hence (E1) is false, leading to a contradiction.

Assume (E1) is false. Then (E2) is true, which means (E3) is false, and hence (E1) is true. Either way, (E1) is both true and false – the same paradox as with (A) and (D1).

There are many other variants, and many complements, possible. In normal sentence construction, the simplest version of the complement is the sentence:

This statement is true. (F)

If F is assumed to bear a truth value, then it presents the problem of determining the object of that value. But, a simpler version is possible, by assuming that the single word 'true' bears a truth value. The analogue to the paradox is to assume that the single word 'false' likewise bears a truth value, namely that it is false. This reveals that the paradox can be reduced to the mental act of assuming that the very idea of fallacy bears a truth value, namely that the very idea of fallacy is false: an act of misrepresentation. So, the symmetrical version of the paradox would be:

The following statement is false. (G1)
The preceding statement is false. (G2)

## Possible resolutions

### Fuzzy logic

In fuzzy logic, the truth value of a statement can be any real number between 0 and 1 both inclusive, as opposed to boolean logic, where the truth values may only be the integer values 0 or 1. In this system, the statement "This statement is false" is no longer paradoxical as it can be assigned a truth value of 0.5,[9][10] making it precisely half true and half false. A simplified explanation is shown below.

Let's denote the truth value of the statement "This statement is false" by x. The statement becomes

${\displaystyle x=NOT(x)}$

by generalizing the NOT operator to the equivalent Zadeh operator from fuzzy logic, the statement becomes

${\displaystyle x=1-x}$

from which it follows that

${\displaystyle x=0.5}$

### Alfred Tarski

Alfred Tarski diagnosed the paradox as arising only in languages that are "semantically closed", by which he meant a language in which it is possible for one sentence to predicate truth (or falsehood) of another sentence in the same language (or even of itself). To avoid self-contradiction, it is necessary when discussing truth values to envision levels of languages, each of which can predicate truth (or falsehood) only of languages at a lower level. So, when one sentence refers to the truth-value of another, it is semantically higher. The sentence referred to is part of the "object language", while the referring sentence is considered to be a part of a "meta-language" with respect to the object language. It is legitimate for sentences in "languages" higher on the semantic hierarchy to refer to sentences lower in the "language" hierarchy, but not the other way around. This prevents a system from becoming self-referential.

However, this system is incomplete. One would like to be able to make statements such as "For every statement in level α of the hierarchy, there is a statement at level α+1 which asserts that the first statement is false." This is a true, meaningful statement about the hierarchy that Tarski defines, but it refers to statements at every level of the hierarchy, so it must be above every level of the hierarchy, and is therefore not possible within the hierarchy (although bounded versions of the sentence are possible).[11][12] Saul Kripke is credited with identifying this incompleteness in Tarski's hierarchy in his highly cited paper "Outline of a theory of truth,"[12] and it is recognized as a general problem in hierarchical languages.[13][12]

### Arthur Prior

Arthur Prior asserts that there is nothing paradoxical about the liar paradox. His claim (which he attributes to Charles Sanders Peirce and John Buridan) is that every statement includes an implicit assertion of its own truth. Thus, for example, the statement "It is true that two plus two equals four" contains no more information than the statement "two plus two equals four", because the phrase "it is true that..." is always implicitly there. And in the self-referential spirit of the Liar Paradox, the phrase "it is true that..." is equivalent to "this whole statement is true and ...".

Thus the following two statements are equivalent:

This statement is false.
This statement is true and this statement is false.

The latter is a simple contradiction of the form "A and not A", and hence is false. There is therefore no paradox because the claim that this two-conjunct Liar is false does not lead to a contradiction. Eugene Mills[14] presents a similar answer.

### Saul Kripke

Saul Kripke argued that whether a sentence is paradoxical or not can depend upon contingent facts.[11]: 6  If the only thing Smith says about Jones is

A majority of what Jones says about me is false.

and Jones says only these three things about Smith:

Smith is a big spender.
Smith is soft on crime.
Everything Smith says about me is true.

If Smith really is a big spender but is not soft on crime, then both Smith's remark about Jones and Jones's last remark about Smith are paradoxical.

Kripke proposes a solution in the following manner. If a statement's truth value is ultimately tied up in some evaluable fact about the world, that statement is "grounded". If not, that statement is "ungrounded". Ungrounded statements do not have a truth value. Liar statements and liar-like statements are ungrounded, and therefore have no truth value.

### Jon Barwise and John Etchemendy

Jon Barwise and John Etchemendy propose that the liar sentence (which they interpret as synonymous with the Strengthened Liar) is ambiguous. They base this conclusion on a distinction they make between a "denial" and a "negation". If the liar means, "It is not the case that this statement is true", then it is denying itself. If it means, "This statement is not true", then it is negating itself. They go on to argue, based on situation semantics, that the "denial liar" can be true without contradiction while the "negation liar" can be false without contradiction. Their 1987 book makes heavy use of non-well-founded set theory.[15]

### Dialetheism

Graham Priest and other logicians, including J. C. Beall and Bradley Armour-Garb, have proposed that the liar sentence should be considered to be both true and false, a point of view known as dialetheism. Dialetheism is the view that there are true contradictions. Dialetheism raises its own problems. Chief among these is that since dialetheism recognizes the liar paradox, an intrinsic contradiction, as being true, it must discard the long-recognized principle of explosion, which asserts that any proposition can be deduced from a contradiction, unless the dialetheist is willing to accept trivialism – the view that all propositions are true. Since trivialism is an intuitively false view, dialetheists nearly always reject the explosion principle. Logics that reject it are called paraconsistent.

### Non-cognitivism

Andrew Irvine has argued in favour of a non-cognitivist solution to the paradox, suggesting that some apparently well-formed sentences will turn out to be neither true nor false and that "formal criteria alone will inevitably prove insufficient" for resolving the paradox.[7]

## Logical structure

For a better understanding of the liar paradox, it is useful to write it down in a more formal way. If "this statement is false" is denoted by A and its truth value is being sought, it is necessary to find a condition that restricts the choice of possible truth values of A. Because A is self-referential it is possible to give the condition by an equation.

If some statement, B, is assumed to be false, one writes, "B = false". The statement (C) that the statement B is false would be written as "C = 'B = false'". Now, the liar paradox can be expressed as the statement A, that A is false:

A = "A = false"

This is an equation from which the truth value of A = "this statement is false" could hopefully be obtained. In the boolean domain "A = false" is equivalent to "not A" and therefore the equation is not solvable. This is the motivation for reinterpretation of A. The simplest logical approach to make the equation solvable is the dialetheistic approach, in which case the solution is A being both "true" and "false". Other resolutions mostly include some modifications of the equation; Arthur Prior claims that the equation should be "A = 'A = false and A = true'" and therefore A is false. In computational verb logic, the liar paradox is extended to statements like, "I hear what he says; he says what I don't hear", where verb logic must be used to resolve the paradox.[18]

## Applications

### Gödel's first incompleteness theorem

Gödel's incompleteness theorems are two fundamental theorems of mathematical logic which state inherent limitations of sufficiently powerful axiomatic systems for mathematics. The theorems were proven by Kurt Gödel in 1931, and are important in the philosophy of mathematics. Roughly speaking, in proving the first incompleteness theorem, Gödel used a modified version of the liar paradox, replacing "this sentence is false" with "this sentence is not provable", called the "Gödel sentence G". His proof showed that for any sufficiently powerful theory T, G is true, but not provable in T. The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence.[19]

To prove the first incompleteness theorem, Gödel represented statements by numbers. Then the theory at hand, which is assumed to prove certain facts about numbers, also proves facts about its own statements. Questions about the provability of statements are represented as questions about the properties of numbers, which would be decidable by the theory if it were complete. In these terms, the Gödel sentence states that no natural number exists with a certain, strange property. A number with this property would encode a proof of the inconsistency of the theory. If there were such a number then the theory would be inconsistent, contrary to the consistency hypothesis. So, under the assumption that the theory is consistent, there is no such number.

It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "Q is the Gödel number of a false formula" cannot be represented as a formula of arithmetic. This result, known as Tarski's undefinability theorem, was discovered independently by Gödel (when he was working on the proof of the incompleteness theorem) and by Alfred Tarski.

George Boolos has since sketched an alternative proof of the first incompleteness theorem that uses Berry's paradox rather than the liar paradox to construct a true but unprovable formula.

## In popular culture

The liar paradox is occasionally used in fiction to shut down artificial intelligences, who are presented as being unable to process the sentence. In Star Trek: The Original Series episode "I, Mudd", the liar paradox is used by Captain Kirk and Harry Mudd to confuse and ultimately disable an android holding them captive. In the 1973 Doctor Who serial The Green Death, the Doctor temporarily stumps the insane computer BOSS by asking it "If I were to tell you that the next thing I say would be true, but that the last thing I said was a lie, would you believe me?" BOSS tries to figure it out but cannot and eventually decides the question is irrelevant and summons security.

In the 2011 video game Portal 2, artificial intelligence GLaDOS attempts to use the "this sentence is false" paradox to kill another artificial intelligence, Wheatley. However, lacking the intelligence to realize the statement is a paradox, he simply responds, "Um, true. I'll go with true. There, that was easy." and is unaffected. Humorously, all other AIs present barring GLaDOS, all of which are significantly less sentient and lucid than both her and Wheatley, are still killed from hearing the paradox. However, GLaDOS later notes that she almost killed herself from her own attempt to kill Wheatley.

The Devo song, Enough Said, includes the lyrics The next thing I say to you will be true / The last thing I said was false.

In the seventh episode of Minecraft: Story Mode titled "Access Denied" the main character Jesse and his friends are captured by a supercomputer named PAMA. After PAMA controls two of Jesse's friends, Jesse learns that PAMA stalls when processing and uses a paradox to confuse him and escape with his last friend. One of the paradoxes the player can make him say is the liar paradox.

In Douglas Adams The Hitchhiker's Guide to the Galaxy, chapter 21 he describes a solitary old man inhabiting a small asteroid in the spatial coordinates where it should have been a whole planet dedicated to Biro life forms. This old man repeatedly claimed that nothing was true, though he was later discovered to be lying.[20]

Rollins Band's 1994 song "Liar" alluded to the paradox when the narrator ends the song by stating "I'll lie again and again and I'll keep lying, I promise".

Robert Earl Keen's song "The Road Goes On and On" alludes to the paradox. The song is widely believed to be written as part of Keen's feud with Toby Keith, who is presumably the "liar" Keen refers to.[21]

## Notes

1. ^ Epimenides paradox has "All Cretans are liars." Titus 1:12
2. ^ Andrea Borghini. "Paradoxes of Eubulides". About.com (New York Times). Archived from the original on 2012-11-11. Retrieved 2012-09-04.
3. ^ St. Jerome, Homily on Psalm 115 (116B), translated by Sr. Marie Liguori Ewald, IHM, in The Homilies of Saint Jerome, Volume I (1-59 On the Psalms), The Fathers of the Church 48 (Washington, D.C.: The Catholic University of America Press, 1964), 294
4. ^ Jan E.M. Houben (1995). "Bhartrhari's solution to the Liar and some other paradoxes". Journal of Indian Philosophy. 23 (4): 381–401. doi:10.1007/bf01880219. JSTOR 23447805. S2CID 170337976.
5. ^ a b Jan E.M. Houben (2001). "Paradoxe et perspectivisme dans la philosophie de langage de Bhartrhari: langage, pensée et réalité" [Paradox and Perspectivism in Bhartrhari's Language Philosophy: Language, Thought and Reality]. Bulletin d'Études Indiennes (in French) (19): 173–199. Archived from the original on 2022-05-15. Retrieved 2018-08-04.
6. ^ Ahmed Alwishah and David Sanson (2009). "The Early Arabic Liar:The Liar Paradox in the Islamic World from the Mid-Ninth to the Mid-Thirteenth Centuries CE" (PDF). p. 1. Archived from the original (PDF) on August 16, 2011.
7. ^ a b Andrew Irvine, "Gaps, Gluts, and Paradox", Canadian Journal of Philosophy, supplementary vol. 18 [Return of the A priori] (1992), 273–299
8. ^ Zach Weber, Guillermo Badia and Patrick Girard (2015). "What is an Inconsistent Truth Table?". Australasian Journal of Philosophy. 94 (3): 7. doi:10.1080/00048402.2015.1093010. S2CID 170137819.
9. ^ Hájek, P.; Paris, J.; Shepherdson, J. (Mar 2000). "The Liar Paradox and Fuzzy Logic". The Journal of Symbolic Logic. 61 (1): 339–346. doi:10.2307/2586541. JSTOR 2586541. S2CID 6865763.
10. ^ Kehagias, Athanasios; Vezerides, K. (Aug 2006). "Computation of fuzzy truth values for the liar and related self-referential systems" (PDF). Journal of Multiple-Valued Logic and Soft Computing. 12 (5–6): 539–559. Archived (PDF) from the original on 2021-07-08. Retrieved 2021-02-17.
11. ^ a b Kripke, Saul (1975-11-06). Outline of a theory of truth. Seventy-Second Annual Meeting American Philosophical Association, Eastern Division. Vol. 72. Journal of Philosophy. pp. 690–716. doi:10.2307/2024634. JSTOR 2024634.
12. ^ a b c Beall, Jc; Glanzberg, Michael; Ripley, David (2016-12-12) [January 20, 2011]. "Liar Paradox: Section 4.3.1 Tarski's hierarchy of languages". Archived from the original on 2021-01-12. Retrieved 2021-01-16.
13. ^ Glanzberg, Michael (2015-06-17). "Complexity and hierarchy in truth predicates". Unifying the Philosophy of Truth. Logic, Epistemology, and the Unity of Science. Logic, Epistemology, and the Unity of Science. Springer, Dordrecht. 36: 211–243. doi:10.1007/978-94-017-9673-6_10. ISBN 978-94-017-9672-9.
14. ^ Mills, Eugene (1998). "A simple solution to the Liar". Philosophical Studies. 89 (2/3): 197–212. doi:10.1023/a:1004232928938. S2CID 169981769.
15. ^ a b Jon Barwise; John Etchemendy (1989). The Liar: An Essay on Truth and Circularity. New York: Oxford University Press. pp. 6, 188. ISBN 9780195059441. LCCN 86031260. Archived from the original on 2020-04-22. Retrieved 2016-02-22.
16. ^ Jan E. M. Houben, "Bhartrhari's Perspectivism (1)" in Beyond Orientalism ed. by Eli Franco and Karin Preisendanz, Amsterdam – Atlanta: Rodopi, 1997; Madeleine Biardeau recognized that Bhartrhari "wants to rise at once above all controversies by showing the conditions of possibility of any system of interpretation, rather than to prove the truth of a certain particular system" (Théorie de la connaissance et philosophie de la parole dans le brahmanisme classique, Paris – La Haye: Mouton, 1964, p. 263)
17. ^ Roberts, Julian. 1992. The Logic of Reflection. German Philosophy in the Twentieth Century. New Haven and London: Yale University Press. p. 43.
18. ^ Yang, T. (Sep 2001). "Computational verb systems: The paradox of the liar". International Journal of Intelligent Systems. 16 (9): 1053–1067. doi:10.1002/int.1049. S2CID 41448750.
19. ^ Crossley, J.N.; Ash, C.J.; Brickhill, C.J.; Stillwell, J.C.; Williams, N.H. (1972). What is mathematical logic?. London-Oxford-New York: Oxford University Press. pp. 52–53. ISBN 978-0-19-888087-5. Zbl 0251.02001.
20. ^ Adams, Douglas. "Chapter 21". The Hitchhiker's Guide to the Galaxy.
21. ^ Cohen, Jason (January 25, 2012). "Fightin' Words: Robert Earl Keen v. Toby Keith". Texas Monthly. Archived from the original on 2015-10-03. Retrieved July 12, 2021.