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In philosophy and logic, the liar paradox encompasses paradoxical statements such as "This sentence is false." or "The next sentence is false. The previous sentence is true." These statements are paradoxical because there is no way to assign them a consistent truth value. Consider that if "This statement is false" is true, then what it says is the case; but what it says is that it is false, hence it is false. On the other hand, if it is false, then what it says is not the case; thus, since it says that it is false, it must be true. Image File history File links Broom_icon. ... For other uses, see Philosophy (disambiguation). ... Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ... Look up paradox in Wiktionary, the free dictionary. ...

This is to be distinguished from the common colloquial expression "I tell a lie." when the speaker has realized that he has just accidentally told an untruth.

History

Epimenides and Eubulides

In the sixth century BC the philosopher-poet Epimenides, himself a Cretan, reportedly wrote: (2nd millennium BC - 1st millennium BC - 1st millennium) The 6th century BC started on January 1, 600 BC and ended on December 31, 501 BC. // Monument 1, an Olmec colossal head at La Venta The 5th and 6th centuries BC were a time of empires, but more importantly, a time... A philosopher is a person who thinks deeply regarding people, society, the world, and/or the universe. ... This article is about the art form. ... Epimenides of Knossos Epimenides of Knossos (Crete) (Greek: Επιμενίδης) was a semi-mythical 6th century BC Greek seer and philosopher-poet, who is said to have fallen asleep for fifty-seven years in a Cretian cave sacred to Zeus, after which he reportedly awoke with the gift of prophecy. ... For other uses, see Crete (disambiguation). ...

The Cretans are always liars.

The Epimenides paradox is often considered equivalent or interchangeable with the "liar paradox", but they are not the same. The liar paradox is a statement that cannot consistently be true or false, while Epimenides' statement is simply false, as long as there exists at least one Cretan who sometimes tells the truth. The Epimenides paradox is a problem in logic. ...

It is unlikely that Epimenides intended his words to be understood as a kind of liar paradox, and they were probably only understood as such much later in history. The oldest known version of the liar paradox is instead attributed to the Greek philosopher Eubulides of Miletus who lived in the fourth century BC. It is very unlikely that he knew of Epimenides's words, even if they were intended as a paradox. Eubulides reportedly said: Ancient Greece is the term used to describe the Greek-speaking world in ancient times. ... Eubulides of Miletus was a Greek philosopher who formulated the liar paradox in the 4th century BC. He was the successor of Euclid of Megara, the founder of the Megarian school of philosophy. ... The 4th century BC started the first day of 400 BC and ended the last day of 301 BC. It is considered part of the Classical era, epoch, or historical period. ...

A man says that he is lying. Is what he says true or false?

Variants of the paradox

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. Time Saving Truth from Falsehood and Envy, François Lemoyne, 1737 For other uses, see Truth (disambiguation). ... Falsity is a perversion of truth originating in the deceitfulness of one party, and culminating in the damage of another party. ... Broadly speaking, a contradiction is an incompatibility between two or more statements, ideas, or actions. ... For the rules of English grammar, see English grammar and Disputes in English grammar. ... In general, semantics (from the Greek semantikos, or significant meaning, derived from sema, sign) is the study of meaning, in some sense of that term. ...

Consider the simplest version of the paradox, the sentence:

This statement is false. (A)

If we suppose that the statement is true, everything asserted in it must be true. However, because the statement asserts that it is itself false, it must be false. So the hypothesis that it is true leads to the contradiction that it is false. Yet we cannot conclude that the sentence is false for that hypothesis also leads to contradiction. If the statement is false, then what it says about itself is not true. It says that it is false, so that must not be true. Hence, it is true. Under either hypothesis, we end up concluding that the statement is both true and false. But it has to be either true or false or so our common intuitions lead us to think, hence there seems to be a contradiction at the heart of our beliefs about truth and falsity.

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. This response to the paradox is, in effect, to reject the common beliefs about truth and falsity: the claim that every statement has to abide by the principle of bivalence, a concept related to the law of the excluded middle. In logic, the principle of bivalence states that for any proposition P, either P is true or P is false. ... The law of excluded middle (tertium non datur in Latin) states that for any proposition P, it is true that (P or ~P). ...

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 and so one is led to another paradox.

Another reaction to the paradox of (A) is to posit, as Graham Priest has, that the statement follows paraconsistent logic and is both true and false. Nevertheless, even Priest's analysis is susceptible to the following version of the liar: Graham (Grammy) Priest (born 1948) is Boyce Gibson Professor of Philosophy at the University of Melbourne and a regular visitor at St. ... A paraconsistent logic is a logical system that attempts to deal nontrivially with contradictions. ...

This statement is only false. (C)

If (C) is both true and false then it must be true. This means that (C) is only false, since that's what it says, but then it can't be true, and so one is led to another paradox.

Non-paradoxes

The statement "I always lie" is often considered to be a version of the liar paradox, but is not actually paradoxical. It could be the case that the statement itself is a lie, because the speaker sometimes tells the truth, and this interpretation does not lead to a contradiction. The belief that this is a paradox results from a false dichotomy - that either the speaker always lies, or always tells the truth - when it is possible that the speaker occasionally does both. The logical fallacy of false dilemma, also known as fallacy of the excluded middle, false dichotomy, either/or dilemma or bifurcation, is to set up two alternative points of view as if they were the only options, when they are not. ...

Possible resolutions

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 falsity) 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 falsity) 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. // Alfred Tarski (January 14, 1902, Warsaw, Russian-ruled Poland – October 26, 1983, Berkeley, California) was a logician and mathematician who spent four decades as a professor of mathematics at the University of California, Berkeley. ...

A.N. Prior

A. N. Prior asserts that there is nothing paradoxical about the liar paradox. His claim (which he attributes to Charles S. 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 is 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 ...". Arthur (A.N.) Prior (1914-1969) was one of the foremost logicians of the twentieth century. ... Charles Sanders Peirce Charles Sanders Peirce (September 10, 1839 – April 19, 1914) was an American logician, philosopher, scientist, and mathematician. ... Jean Buridan, in Latin Joannes Buridanus, (1300 - 1358) was a French philosopher who sowed the seeds of religious scepticism in Europe. ...

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^[1] and Neil Lefebvre and Melissa Schelein^[2] present similar answers.

But Prior never made clear how his approach would apply to the more complex versions of the paradox, such as the two sentence version: "The next sentence is false", "The preceding sentence is true". Moreover, if all sentences are really hidden conjunctions, then some rules of propositional logic, such as the rule that one can derive any conjunct immediately and the rule that from any two propositions one can immediately derive their conjunction, are called into question. If we can derive this statement is false from This statement is true and this statement is false, then the paradox is back. And if we are not allowed to make such a derivation, then Prior has, in effect, invented a new kind of conjunction whose truth value characteristics are so mysterious, we cannot really say with any confidence that the paradox has been dissolved.

It has also been argued that the interpretation of "this statement is false" should formally be expressed as an equation of the form A = { A = false } rather than a conjunction. In that case the paradox remains.^{[attribution needed]}

Saul Kripke

Saul Kripke points out that whether a sentence is paradoxical or not can depend upon contingent facts. Suppose that the only thing Smith says about Jones is Saul Aaron Kripke (born in November 13, 1940 in Bay Shore, New York) is an American philosopher and logician now emeritus from Princeton and teaches as distinguished professor of philosophy at CUNY Graduate Center. ...

A majority of what Jones says about me is false.

Now suppose that 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 the empirical facts are that Smith is a big spender but he 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, call that statement "grounded". If not, call that statement "ungrounded". Ungrounded statements do not have a truth value. Liar statements and liar-like statements are ungrounded, and therefore have no truth value.

Barwise and 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 their theory of "situational semantics", that the "denial liar" can be true without contradiction while the "negation liar" can be false without contradiction. Kenneth Jon Barwise (June 29, 1942 - March 5, 2000) was a US mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used. ... John W. Etchemendy is Stanford Universitys twelfth and current Provost. ...

Gödel's theorem

The proof of Gödel's incompleteness theorem uses self-referential statements that are similar to the statements at work in the Liar paradox. In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ...

In the context of a sufficiently strong axiomatic system A of arithmetic: In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. ...

This statement is not provable in A. (1)

The statement (1) does not mention truth at all (only provability) but the parallel is clear. Suppose (1) is provable, then what it says of itself, that it is not provable, is not true. But this conclusion is contrary to our supposition, so our supposition that (1) is provable must be false. Suppose the contrary that (1) is not provable, then what it says of itself is true, although we cannot prove it. Therefore, there is no proof that (1) is provable, and there is also no proof that its negation is provable (i.e., there is no proof that it is also unprovable). Whence, A is incomplete because it cannot prove all truths, namely, (1) and its negation. Statements like (1) are called undecidable. We take for granted that all the provable statements of logic and arithmetic are true; Gödel showed that the converse, that all the true statements of a system are provable in that system, is not the case. (This does not mean that all true statements are not provable in some system or other. Additionally, there are systems, such as first-order logic, in which all true statements of the system are provable.) In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer. ... First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. ...

Tarski's indefinability theorem, closely related to Gödel's Theorem, is a more direct application of the Liar Paradox, though there is no actual paradox involved; instead, the "paradox" simply demonstrates that all the true sentences of arithmetic are not arithmetically definable (or that arithmetic cannot define its own truth predicate; or that arithmetic is not "semantically closed"). In mathematical logic, Tarskis Indefinability Theorem is a theorem due to Alfred Tarski concerning the foundations of mathematics. ...

Dialetheism

Graham Priest and other logicians have proposed that the liar sentence should be considered to be both true and false, a point of view known as dialetheism. In a dialetheic logic, all statements must be either true, or false, or both. 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 ex falso quodlibet, which asserts that any sentence whatsoever can be deduced from a true contradiction. Thus, dialetheism only makes sense in systems that reject ex falso quodlibet. Such logics are called paraconsistent. Graham (Grammy) Priest (born 1948) is Boyce Gibson Professor of Philosophy at the University of Melbourne and a regular visitor at St. ... Dialetheism is a paraconsistent logic typified by its tolerance of at least some contradictions. ... The principle of explosion (also known as ex falso quodlibet, ex falso sequitur quodlibet (EFSQ for short), ex contradictione (sequitur) quodlibet (ECQ for short), and ex falso/contradictione (sequitur) aliquot) is the law of classical logic and a few other systems, for example, intuitionistic logic, according to which anything follows... A paraconsistent logic is a logical system that attempts to deal nontrivially with contradictions. ...

See also

• Quine's paradox
• List of paradoxes

Quines paradox is a paradox concerning truth values, attributed to W. V. O. Quine. ... This is a list of paradoxes, grouped thematically. ...

Notes

1. ^ Mills, Eugene (1998) ‘A simple solution to the Liar’, Philosophical Studies 89: 197-212.
2. ^ Lefebvre, N. and Schelein, M., "The Liar Lied," in Philosophy Now issue 51

Philosophy Now is a philosophy magazine, published every two months and sold from news-stands and bookstores in the USA, UK, Australia and Canada. ...

References

• Jon Barwise and John Etchemendy (1987) The Liar. Oxford University Press.
• Greenough, P.M., (2001) " ," American Philosophical Quarterly 38:
• 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, ISBN 0-521-28864-9. Buridan's detailed solution to a number of such paradoxes.
• Kirkham, Richard (1992) Theories of Truth. MIT Press. Especially chapter 9.
• Saul Kripke (1975) "An Outline of a Theory of Truth," Journal of Philosophy 72: 690-716.
• Lefebvre, Neil, and Schelein, Melissa (2005) "The Liar Lied," Philosophy Now issue 51.
• Graham Priest (1984) "The Logic of Paradox Revisited," Journal of Philosophical Logic 13: 153-179.
• A. N. Prior (1976) Papers in Logic and Ethics. Duckworth.
• Smullyan, Raymond (19nn) What is the Name of this Book?. ISBN 0-671-62832-1. A collection of logic puzzles exploring this theme.

Kenneth Jon Barwise (June 29, 1942 - March 5, 2000) was a US mathematician, philosopher and logician who proposed some fundamental revisions to the way that logic is understood and used. ... John W. Etchemendy is Stanford Universitys twelfth and current Provost. ... Saul Aaron Kripke (born in November 13, 1940 in Bay Shore, New York) is an American philosopher and logician now emeritus from Princeton and teaches as distinguished professor of philosophy at CUNY Graduate Center. ... Graham (Grammy) Priest (born 1948) is Boyce Gibson Professor of Philosophy at the University of Melbourne and a regular visitor at St. ... Arthur (A.N.) Prior (1914-1969) was one of the foremost logicians of the twentieth century. ... Raymond Merrill Smullyan (born 1919) is a mathematician, logician, philosopher, and magician. ...

External links

• Internet Encyclopedia of Philosophy: "Liar Paradox" -- by Bradley Dowden.