Liar paradox

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 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 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.[14] 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. There is therefore no paradox because the claim that this two-conjunct Liar is false does not lead to a contradiction. Eugene Mills[15] presents a similar answer.

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

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 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.[16]

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.

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]

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".[17] 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."[18] 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.[16]^: 188 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".[5] 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.

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.[20]

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.