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philosophy Philosophy ('love of wisdom' in Ancient Greek) is a systematic study of general and fundamental questions concerning topics like existence, reason, knowledge, Value (ethics and social sciences), value, mind, and language. It is a rational an ...
and
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
, 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 A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true or apparently true premises, leads to a seemingly self-contradictor ...
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 In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Truth values are used in ...
leads to a
contradiction In traditional logic, a contradiction involves a proposition conflicting either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
. Assume that "this sentence is false" is true, then we can trust its content, which states the opposite and thus causes a contradition. Similarly, we get a contradiction when we assume the opposite.


History

The Epimenides paradox (c. 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 Crete ( ; , Modern Greek, Modern: , Ancient Greek, Ancient: ) is the largest and most populous of the Greek islands, the List of islands by area, 88th largest island in the world and the List of islands in the Mediterranean#By area, fifth la ...
, reportedly stated that "All Cretans are liars."Epimenides paradox has "All Cretans are liars." 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 Ancient Greek (, ; ) includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), 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?" The paradox was once discussed by Jerome of Stridon in a sermon: 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. 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 Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields. In natural language, natural or formal languages, ...
.


Explanation and variants

The problem of the liar paradox is that it seems to show that common beliefs about
truth Truth or verity is the Property (philosophy), property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth, 2005 In everyday language, it is typically ascribed to things that aim to represent reality or otherwise cor ...
and falsity actually lead to a
contradiction In traditional logic, a contradiction involves a proposition conflicting either with itself or established fact. It is often used as a tool to detect disingenuous beliefs and bias. Illustrating a general tendency in applied logic, Aristotle's ...
. Sentences can be constructed that cannot consistently be assigned a truth value even though they are completely in accord with
grammar In linguistics, grammar is the set of rules for how a natural language is structured, as demonstrated by its speakers or writers. Grammar rules may concern the use of clauses, phrases, and words. The term may also refer to the study of such rul ...
and
semantic Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
rules. The simplest version of the paradox is the sentence: 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".Andrew Irvine, "Gaps, Gluts, and Paradox", ''Canadian Journal of Philosophy'', supplementary vol. 18 'Return of the A priori''(1992), 273–299 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: 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 Graham Priest (born 1948) is a philosopher and logician who is distinguished professor of philosophy at the CUNY Graduate Center, as well as a regular visitor at the University of Melbourne, where he was Boyce Gibson Professor of Philosophy an ...
has, that the statement is both true and false. Nevertheless, even Priest's analysis is susceptible to the following version of the liar: 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. There are also multi-sentence versions of the liar paradox. The following is the two-sentence version: 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: 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: 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: There's also a version related to the problem of future contingents which cannot be answered without a contradiction arising: If the answer is 'yes', then the answer to the question is 'no', and if the answer is 'no', then the answer to the question is 'yes'.


Possible resolutions


Fuzzy logic

In
fuzzy logic Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
, the truth value of a statement can be any real number between 0 and 1 both inclusive, as opposed to
Boolean logic In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
, 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, making it precisely half true and half false. A simplified explanation is shown below. Let the truth value of the statement "This statement is false" be denoted by x. The statement becomes : x = NOT(x) by generalizing the NOT operator to the equivalent Zadeh operator from
fuzzy logic Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completely ...
, the statement becomes : x = 1 - x from which it follows that : x = 0.5


Alfred Tarski

Alfred Tarski Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...
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).
Saul Kripke Saul Aaron Kripke (; November 13, 1940 – September 15, 2022) was an American analytic philosophy, analytic philosopher and logician. He was Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emer ...
is credited with identifying this incompleteness in Tarski's hierarchy in his highly cited paper "Outline of a theory of truth," and it is recognized as a general problem in hierarchical languages.


Arthur Prior

Arthur Prior asserts that there is nothing paradoxical about the liar paradox. His claim (which he attributes to
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American scientist, mathematician, logician, and philosopher who is sometimes known as "the father of pragmatism". According to philosopher Paul Weiss (philosopher), Paul ...
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: The latter is a simple contradiction of the form "A and not A", and hence is false. Therefore, there is no paradox, because the claim that this two-conjunct Liar is false does not lead to a contradiction. Eugene Mills presents a similar answer.


Saul Kripke

Saul Kripke Saul Aaron Kripke (; November 13, 1940 – September 15, 2022) was an American analytic philosophy, analytic philosopher and logician. He was Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emer ...
argued that whether a sentence is paradoxical or not can depend upon contingent facts. If the only thing Smith says about Jones is and Jones says only these three things about Smith: 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 Kenneth Jon Barwise (; June 29, 1942 – March 5, 2000) was an American mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used. Education and career He was born in Indepen ...
and
John Etchemendy John W. Etchemendy (born 1952) is an American logician and philosopher who served as Stanford University's twelfth Provost (education), Provost. He succeeded John L. Hennessy to the post on September 1, 2000 and stepped down on January 31, 2017 ...
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.


Dialetheism

Graham Priest Graham Priest (born 1948) is a philosopher and logician who is distinguished professor of philosophy at the CUNY Graduate Center, as well as a regular visitor at the University of Melbourne, where he was Boyce Gibson Professor of Philosophy an ...
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 In classical logic, intuitionistic logic, and similar logical systems, the principle of explosion is the law according to which any statement can be proven from a contradiction. That is, from a contradiction, any proposition (including its n ...
, 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.


Bhartrhari's perspectivism

The Indian grammarian-philosopher Bhartrhari (late fifth century AD) dealt with paradoxes such as the liar in a section of one of the chapters of his magnum opus the Vākyapadīya. Bhartrhari's solution fits into his general approach to language, thought and reality, which has been characterized by some as "relativistic", "non-committal" or "perspectivistic". With regard to the liar paradox (''sarvam mithyā bravīmi'' "everything I am saying is false") Bhartrhari identifies a hidden parameter that can change unproblematic situations in daily communication into a stubborn paradox. Bhartrhari's solution can be understood in terms of the solution proposed in 1992 by Julian Roberts: "Paradoxes consume themselves. But we can keep apart the warring sides of the contradiction by the simple expedient of temporal contextualisation: what is 'true' with respect to one point in time need not be so in another ... The overall force of the 'Austinian' argument is not merely that 'things change', but that rationality is essentially temporal in that we need time in order to reconcile and manage what would otherwise be mutually destructive states." According to Robert's suggestion, it is the factor "time" which allows us to reconcile the separated "parts of the world" that play a crucial role in the solution of Barwise and Etchemendy. The capacity of time to prevent a direct confrontation of the two "parts of the world" is here external to the "liar". In the light of Bhartrhari's analysis, however, the extension in time that separates two perspectives on the world or two "parts of the world" – the part before and the part after the function accomplishes its task – is inherent in any "function": also the function to signify which underlies each statement, including the "liar". The unsolvable paradox – a situation in which we have either contradiction (''virodha'') or infinite regress (''anavasthā'') – arises, in case of the liar and other paradoxes such as the unsignifiability paradox ( Bhartrhari's paradox), when abstraction is made from this function (''vyāpāra'') and its extension in time, by accepting a simultaneous, opposite function (''apara vyāpāra'') undoing the previous one.


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 Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields. In natural language, natural or formal languages, ...
, 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: This is an equation from which the truth value of A = "this statement is false" could hopefully be obtained. In the
Boolean domain In mathematics and abstract algebra, a Boolean domain is a set consisting of exactly two elements whose interpretations include ''false'' and ''true''. In logic, mathematics and theoretical computer science, a Boolean domain is usually written ...
, "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.


Applications


Gödel's first incompleteness theorem

Gödel's incompleteness theorems Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the phi ...
are two fundamental theorems of
mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
which state inherent limitations of sufficiently powerful axiomatic systems for mathematics. The theorems were proven by
Kurt Gödel Kurt Friedrich Gödel ( ; ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly ...
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. 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 Alfred Tarski (; ; born Alfred Teitelbaum;School of Mathematics and Statistics, University of St Andrews ''School of Mathematics and Statistics, University of St Andrews''. January 14, 1901 – October 26, 1983) was a Polish-American logician ...
. George Boolos has since sketched an alternative proof of the first incompleteness theorem that uses
Berry's paradox The Berry paradox is a self-referential paradox arising from an expression like "The smallest positive integer not definable in under sixty letters" (a phrase with fifty-seven letters). Bertrand Russell, the first to discuss the paradox in print, ...
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 the '' Star Trek: The Original Series'' episode " I, Mudd", the liar paradox is used by
Captain Kirk James Tiberius Kirk, often known as Captain Kirk, is a fictional character in the ''Star Trek'' media franchise. Originally played by Canadian actor William Shatner, Kirk first appeared in ''Star Trek'' serving aboard the starship USS ''Enterp ...
and Harry Mudd to confuse and ultimately disable an android holding them captive. In the 1973 ''
Doctor Who ''Doctor Who'' is a British science fiction television series broadcast by the BBC since 1963. The series, created by Sydney Newman, C. E. Webber and Donald Wilson (writer and producer), Donald Wilson, depicts the adventures of an extraterre ...
'' 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 1967 film '' Bedazzled'', the Devil says to his subject, Stanley Moon, that "Everything he says is a lie, including this.". The Thin Lizzy song "Don't Believe a Word" contains the lyrics "Not a word of this is true." 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 Devo is an American new wave band from Akron, Ohio, formed in 1973. Their classic line-up consisted of two sets of brothers, the Mothersbaughs ( Mark and Bob) and the Casales (Gerald and Bob), along with Alan Myers. The band had a No. 14 ...
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 their 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 their last friend. One of the paradoxes the player can make Jesse say is the liar paradox. Robert Earl Keen's song "The Road Goes On and On" mentions the paradox. The song is part of Keen's feud with Toby Keith, who is presumably the "liar" Keen refers to.


See also

* Knights and Knaves * Performative contradiction *
Self-reference Self-reference is a concept that involves referring to oneself or one's own attributes, characteristics, or actions. It can occur in language, logic, mathematics, philosophy, and other fields. In natural or formal languages, self-reference ...
* List of self–referential paradoxes


Notes


References

* Greenough, P. M. (2001). " Free Assumptions and the Liar Paradox," ''American Philosophical Quarterly 38/2, pp. 115-135.'': * Hughes, G. E. (1992). ''John Buridan on Self-Reference : Chapter Eight of Buridan's Sophismata, with a Translation, and Introduction, and a Philosophical Commentary'', Cambridge Univ. Press, . Buridan's detailed solution to a number of such paradoxes. * Kirkham, Richard (1992). ''Theories of Truth''. MIT Press. Especially chapter 9. ISBN 0262611082, ISBN 978-0262611084 * * A. N. Prior (1976). ''Papers in Logic and Ethics''. Duckworth. * Smullyan, Raymond (1986). ''What Is the Name of This Book?'' . A collection of logic puzzles exploring this theme.


External links

* * {{DEFAULTSORT:Liar Paradox Communication of falsehoods Lying Self-referential paradoxes Ancient Greek logic