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If "This sentence is false" is true, then the sentence is false, which would in turn mean that it is actually true, but this would mean that it is false, and so on ad infinitum.

Similarly, if "This sentence is false" is false, then the sentence is true, which would in turn mean that it is actually false, but this would mean that it is true, and so on ad infinitum.

History

The Epimenides paradox (circa 600 BC) has been suggested as an example of the liar paradox, but they are not logically equivalent. The fictional speaker Epimenides, a Cretan, reportedly stated that "The Cretans are always liars."[citation needed] 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.

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.[citation needed]

The oldest known version of the actual liar paradox is attributed to the Greek philosopher Eubulides of Milet...

If "This sentence is false" is true, then the sentence is false, which would in turn mean that it is actually true, but this would mean that it is false, and so on ad infinitum.

Similarly, if "This sentence is false" is false, then the sentence is true, which would in turn mean that it is actually false, but this would mean that it is true, and so on ad infinitum.

History

The Epimenides paradox (circa 600 BC) has been suggested as an example of the liar paradox, but they are not logically equivalent. The fictional speaker Epimenides, a Cretan, reportedly stated that "The Cretans are always liars."[citation needed] 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.

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.[citation needed]

The oldest known version of the actual liar paradox is attributed to the Greek philosopher Eubulides of Miletus who lived in the 4th century BC. It is very unlikely that he knew of Epimenides's words, even if they were intended as a paradox.[citation needed] Eubulides reportedly asked, "A man says that he is lying. Is what he says true or false?"

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

"I said in my alarm, 'Every man is a liar!' (Psalm. 116:11) 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: "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."[1]

In early Islamic tradition liar paradox was discussed for at least 5 centuries starting from late 9th century 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. [2]

Explanation of the paradox 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:

This statement is false. (A)

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". 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.

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).

Non-paradoxes

Not all seemingly self-contradictory statements are liar 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 (and in reality almost certainly is) that the statement itself is a lie, because the speaker sometimes tells the truth, and this interpretation does not lead to a contradiction (unlike, for example, "I am lying"). 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.[citation needed]

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 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.

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[3] and Neil Lefebvre and Melissa Schelein[4] present similar answers.

But the claim that every statement is really a conjunction in which the first conjunct says "this statement is true" seems to run afoul of standard rules of propositional logic, especially the rule, sometimes called Conjunction Elimination, that from a conjunction any of the conjuncts can be derived. Thus, from, "This statement is true and this statement is false", it follows that "this statement is false" and so we have, once again, a paradoxical (and non-conjunctive) statement. It seems then that Prior's attempt at resolution requires either a whole new propositional logic or else the postulation that the "and" in, "This statement is true and this statement is false", is a special type of conjunctive for which Conjunction Elimination does not apply. But then we need, at least, an expansion of standard propositional logic to account for this new kind of "and".[5]

Saul Kripke

Saul Kripke argued that whether a sentence is paradoxical or not can depend upon contingent facts.[citation needed] 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.

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 situation semantics, that the "denial liar" can be true without contradiction while the "negation liar" can be false without contradiction.

Dialetheism

Graham Priest and other logicians[who?] 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 ex falso quodlibet, which asserts that any proposition can be deduced from a true 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 "ex falso quodlibet". Logics that reject "ex falso quodlibet" are called paraconsistent.

Chris Langan

Chris Langan in his work The Theory of Theories states:

... Consider the statement “this sentence is false.” It is easy to dress this statement up as a logical formula. Aside from being true or false, what else could such a formula say about itself? Could it pronounce itself, say, unprovable? Let’s try it: "This formula is unprovable". If the given formula is in fact unprovable, then it is true and therefore a theorem. Unfortunately, the axiomatic method cannot recognize it as such without a proof. On the other hand, suppose it is provable. Then it is self-apparently false (because its provability belies what it says of itself) and yet true (because provable without respect to content)! It seems that we still have the makings of a paradox…a statement that is "unprovably provable" and therefore absurd.

But what if we now introduce a distinction between levels of proof? For example, what if we define a metalanguage as a language used to talk about, analyze or prove things regarding statements in a lower-level object language, and call the base level of Gödel’s formula the "object" level and the higher (proof) level the "metalanguage" level? Now we have one of two things: a statement that can be metalinguistically proven to be linguistically unprovable, and thus recognized as a theorem conveying valuable information about the limitations of the object language, or a statement that cannot be metalinguistically proven to be linguistically unprovable, which, though uninformative, is at least no paradox. Voilà: self-reference without paradox! It turns out that "this formula is unprovable" can be translated into a generic example of an undecidable mathematical truth.

Logical structure of the liar paradox

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 statement like, "I hear what he says; he says what I don't hear", where verb logic must be used to resolve the paradox.[6]

Applications

Gödel's First Incompleteness Theorem

Gödel's incompleteness theorems are two fundamental theorems of mathematical logic which state inherent limitations of all but the most trivial 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 slightly modified version of the liar paradox, replacing "this sentence is false" with "this sentence is not provable", called the "Gödel sentence G". Thus for a 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.

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.

See http://en.wikipedia.org/wiki/... for Notes and References.

- if the Creten's statement is true then he must be lying because all Cretens are lairs.

- but if he is lying then his statement cannot be true

- at least some (or not all) Cretens are not lairs

- and there is a paradox

- if, however, the Creten's statement is a lie then at least some Cretens are not liers

- this does not present a paradox

- it is possible for him to lie and some Cretens to be truthful, just not him

- there are no other possible non-paradoxical answers, so that must be the right one