Liar Paradox

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It is well known that Pinocchio's nose grows precisely when he is lying. But what happens when he says “My nose is growing right now”?

A liar's paradox is a paradox in philosophy or logic that arises when a sentence asserts its own falsehood (or untruth). If the proposition is true, it follows from its self-reference that it is false, and vice versa.

formulation

The simplest form of the liar's paradox is the following self-referential sentence:

"This sentence is wrong."

The paradox of this proposition is that it cannot reasonably be said to be true or false. Assuming it were wrong: Then what the sentence itself claims would apply and it should therefore be true. But if we assume that it is true, then what the sentence claims does not apply - which means that it is false.

This type of paradox is often referred to as semantic paradox in the philosophical discussion . It is made possible by the fact that the truth conditions of a sentence are specified in it (directly or indirectly) - but in a way that at least apparently no longer allows any meaningful attribution of truth or falsehood.

The name “Liar's Paradox” goes back to the fact that the paradox can also be formulated using the term lie , e.g. B. as follows:

(a person claims :) "I'm lying."

The person who claims this is claiming that his statement is a lie, so that it is not true. Ultimately, however, the same paradox arises as above.

In the paradox of Epimenides the sentence “All Cretans are liars” is used to represent the paradox. This sentence is asserted by Epimenides, who is himself a Cretan. But this is not a paradox in the full sense of the word, since the negation of the sentence, ie from “Some Cretans are not liars”, does not necessarily mean that Epimenides is telling the truth.

Extensions and related paradoxes

The paradoxical mechanism in the classic liar paradox is similar to that in other semantic paradoxes. A variant that already points more clearly to the problematic for logic is Curry's paradox . If the truth conditions of the logical subjunction are assumed for the following conditional, then it can be represented as follows, for example:

"If this sentence is true, then the moon is made of green cheese."

The truth value “false” cannot be consistently assigned to this sentence , because then the antecedent of the conditional would be false, which according to the assumed logical understanding would make the entire conditional true. The truth value “true”, on the other hand, can already be ascribed to the sentence; However, it must be assumed that the suffix “The moon is made of green cheese” is also true - otherwise the antecedent of the conditional would be true, but the suffix would be false, and the whole sentence would be wrong. If this sentence had to be assigned a truth value, then it would be an absurd “proof” that the moon is made of green cheese.

The proposed solution of countering the liar by rejecting the two-valued logic is countered by modified versions of the liar's paradox. The best known is the reinforced liar :

"This sentence is not true."

This paradox remains even if it is allowed that paradoxical sentences can neither be true nor false (so-called truth value “gaps”). However, it can still be avoided with a three-valued logic , which understands the third value as “both true and false” (so-called “Gluts”, e.g. represented by Graham Priest ). However, a variant of the reinforced liar can be cited:

"This sentence is not entirely true."

Paradoxes of the liar type can also be generated with several sentences, for example with the following two:

"The next sentence is wrong."
"The previous sentence is true."

This variant (suggested by Philip Jourdain , also known as the card problem ) avoids direct self-reference , but nevertheless creates exactly the same paradox as the classic liar. However, there is still an indirect self-reference, as there is a circle of references between the two sentences (similar to variants with a larger number of sentences).

According to its own claim, Yablo's paradox manages without self- referentiality . It consists of an infinite series of sentences, each of which claims that all of the following sentences are not true. Here, too, none of the sentences can be assigned a truth value without contradiction, because contradicting conditions would have to be placed on the series of the following sentences. If this paradox actually manages without self-referentiality (which is, however, sometimes denied in the philosophical discussion), then it shows that it is not self-referentiality that makes the paradox possible, but our handling of the terms “true” and “false”.

A proposition that instead of its falsehood asserts its own undecidability creates a related paradox.

history

Aristotle already discussed the liar's paradox in his sophistic refutations , albeit without a quote or author's name. Late antique sources name his contemporaries Eubulides as the speaker of the liar's paradox. Since the works of Eubulides are lost, his argumentation is only from the oldest quotations in Cicero et al. a. reconstructable; it could have had the following form of dialogue:

"If I lie to say that I am lying, am I lying or am I telling the truth?"
"You are telling the truth."
"If I say the truth and say that I am lying, I am lying."
"You are obviously lying."

This dialogue derives from the antinomy provoked by the paradoxical partial statement “I say that I am lying” .

Variants of this liar antinomy have been discussed throughout the history of logic. In modern mathematical logic it gained new importance through Bertrand Russell . He followed up on the paradox of Epimenides "Epimenides the Cretans said: All Cretans are liars"; this probably older, weaker pre-form of the liar's paradox does not yet produce an antinomy; he therefore tightened it up to the genuinely paradoxical sentence that generates the antinomy:

A man says: I am lying. - A man says: I'm lying.

Problems and solutions

Type-theoretic solution

To solve the paradox, Russell demanded a type theory with a hierarchy of statements and a hierarchy of truth predicates, namely statements of order n and truth predicates of order n (for n = 0, 1, 2, ...). A truth predicate of order n may only be made from a statement with an order less than n . So he solved the liar's paradox by syntactically excluding self-referential statements.

Separation of object and metalanguage

The liar's paradox has been seen as a significant problem for a philosophical theory of truth since the 20th century . Alfred Tarski formulates the problem in his influential essay The Concept of Truth in Formal Languages as follows: Colloquial language is "universalistic", i. that is, it absorbs all semantic expressions. However:

“Following this universalistic tendency of colloquial language with regard to semantic investigations, we must consequently [...] include such semantic expressions as“ true statement ”,“ name ”,“ designate ”etc. On the other hand, it is precisely this universalism of colloquial language in the field of semantics that is probably the essential source of all so-called semantic antinomies, such as the antinomies of the liar or the heterological words; these antinomies seem simply to be proof that a contradiction must arise on the basis of every language which would be universal in the above sense and for which the normal laws of logic should apply. "

Tarski shows below that such paradoxes do not arise for artificial languages ​​in which a separation of object language and metalanguage is consistently carried out. An essential characteristic of this separation is that no statements about this language can be made within the object language - that is reserved for the metalanguage for this language. For statements about the metalanguage, however, a metalanguage is then required for this metalanguage, so that a so-called “Tarski hierarchy” results. A reference to sentences in this language is therefore always excluded within a language.

Soundness

An alternative to the Tarski hierarchy that is supposed to provide a model of natural language is based on Saul Kripke's concept of soundness. Kripke provides a semantic truth theory in which statements about the truth of other sentences can also be assigned a truth value, as long as they are "well-founded". “Unfounded” statements are not recognized as propositions that are true or false; According to Kripke, however, they are not pointless insofar as they express possible propositions according to their form and could still be treated with the help of a three-valued logic .

The basic idea of ​​Kripke's theory of truth is as follows: In a first step, all statements that do not depend on the truth value of other statements (i.e., e.g. do not claim from another sentence that this is true) are assigned a truth value - simply by comparison with the reality. In a second step, all statements about the truth value of other statements are now considered. If a value can be assigned to these statements based on the truth values ​​distributed up to this point, this also happens. This second step is repeated until no new truth values ​​have been distributed in a repetition of this step. Sentences that have no truth value at this "smallest fixed point" are considered unfounded.

Kripke thinks that with his theory of truth he has evaded the usual formulations of the liar. He evades the versions of the reinforced liar by stating that "unfounded" is not a third truth value, and he emphasizes that the classical logic remains valid for the area of ​​propositions. However, new paradoxes can still be formulated with the aid of the concept of unfoundedness (which are often referred to in the literature as “the liar's revenge”). Kripke sees this and does not claim to have given a universal semantics of the concept of truth. Ultimately, he admits to Tarski the need for a metalanguage for terms like “unfounded” or “paradoxical”. He only wanted to give a model for the everyday language of non-philosophical speakers, but more refined terms could not handle this.

General formalization

A formalization of the argument solves the paradox even without syntactic restrictions. Classical propositional logic with additional predicates " X is lying" (in the sense of " X is lying") and " X says that A " and two syllogisms are sufficient as a calculus :

(1)    X lies and X says that A → not- A    (momentary lying)
(2)    X doesn't lie and X says that AA    (momentary truth-telling)

This calculation is free of contradictions : Both syllogisms apply at least in a world in which nobody says anything (there their second premise is wrong). So no antinomy can be derived in the calculus. The liar's paradox here is a syntactically correct self-reference in variable form:

(3)    X says X is lying   (general liar paradox)

In the calculus, the following theorem, based on Arthur Prior , applies :

The Liar's Paradox (3) is refutable; the negation of (3) applies.

Indirect proof: assumption (3). First case: X is lying; then it follows from assumption (3) with (1) and the modus ponens : X does not lie. So this case is contradicting itself. In the other case: X does not lie; but then it follows from the assumption (3) with (2) per modus ponens: X lies. Both possible cases are thus contradictory and (3) is refuted.

The proof specifies Eubulides' argumentation, but at the same time emphasizes his hidden assumption: the liar's paradox, without which the argumentation would not work. It turns out to be a sophism that is not relatively consistent with the declared calculus and therefore ruled out as a logical argument. The proof is independent of the definition of the predicates in special models, because the formalization is a general axiomatization that allows different models.

Impersonal model

Since the formalization leaves the assignment of the variables open, X can be a statement that also says something and is wrong if it lies; this can be summed up by two definitions:

(4)    X says that A   XA  
(5)    X is wrong   X is lying   

These definitions generate from (3) the impersonal liar paradox “ XX is wrong”, for which Prior's theorem applies as well and is originally formulated. The model leaves open how the predicate “ X lies” is defined. It can happen on an extended semantic level of language, which is why the liar is considered a semantic paradox. However, this is not mandatory, as the following model shows.

Propositional model

The impersonal liar model becomes a propositional model in that the predicate lying is not shifted to a higher semantic language level, but is also defined as a statement:

(6)    X does     n't lie- X

With the definitions (4) and (6) the syllogisms (1) and (2) become provable, and the liar paradox (3) becomes equivalent to the self-reference " X ↔ not- X ", which is known to be wrong.

Popular culture

Embellished variants of the liar's paradox are circulating in popular culture: A frequent motif among science fiction authors is overcoming an overpowering artificial intelligence by confronting the paradox, which is supposed to lead to an infinite computation loop.

literature

  • JC Beall: Spandrels of Truth. Oxford 2009.
  • Elke Brendel: The truth about the liar. A philosophical and logical analysis of the liar's antinomy. Berlin 1992.
  • Tyler Burge : Semantical Paradox. In: Journal of Philosophy. 76, 1979, pp. 169-198.
  • Hartry Field , Saving Truth from Paradox. New York 2008.
  • William Kneale : Russell's Paradox and Some Others. In: The British Journal for the Philosophy of Science. Vol. 22, 4, 1971, pp. 321-338.
  • Saul Kripke : Outline of a Theory of Truth. In: Journal of Philosophy. 72, 1975, pp. 690-716; also reprinted in: ders .: Philosophical Troubles. (= Collected Papers. Volume I). Oxford 2011, pp. 75-98.
  • Wolfgang Künne : Epimenides and other liars. Frankfurt am Main 2013.
  • Graham Priest , The Logic of Paradox. In: Journal of Philosophical Logic. 8, 1979, pp. 219-241.
  • Alexander Riistow : The liar. Theory, History and Resolution . Leipzig 1910. (Reprints New York 1987 and Cologne 1994)
  • Alfred Tarski : The Concept of Truth in Formalized Languages . In: Studia Philosophica. [Lemberg] 1, 1936, pp. 261-405.
  • Stephen Yablo: Paradox without Self-Reference . In: Analysis. 53, 1993, pp. 251f.

Web links

Individual evidence

  1. see e.g. B. Béla Juhos : Elements of the New Logic , 1954, p. 222 (oldest source of this version to date)
  2. So z. B. by Tyler Burge: Semantical Paradox. In: Journal of Philosophy. 76, 1979, pp. 169-198.
  3. See also Michael Clark: Paradoxien von A bis Z. Stuttgart 2012, pp. 66–68.
  4. Yablo 1993, pp. 251f.
  5. Michael Clark: Paradoxien von A bis Z. Stuttgart 2012, pp. 292–294.
  6. ^ Aristotle: Sophistic refutations (= Topic. IX). 25, 180b2-7
  7. Diogenes Laertios: About the life and teachings of famous philosophers. II 108
  8. Alexander Riistow: The liar. P. 40. There list of quotations, Rustow's reconstruction in Greek, part translated above, as far as it is based on the oldest Cicero quotations. (PDF; online)
  9. Elke Brendel: The truth about the liar: a philosophical-logical analysis of the antinomy of the liar. Part II: The story of the liar. Berlin / New York 1993, pp. 19-40. (on-line)
  10. ^ A b Bertrand Russell: Mathematical logic as based on the theory of types. (PDF; 1.9 MB). In: American Journal of Mathematics. 30, 1908, p. 222 (1): “The oldest contradiction of the kind in question is the Epimenides. Epimenides the Cretan said that all Cretans were liars, and all other statements made by Cretans were certainly lies. What is this a lie? The simplest form of this contradiction is afforded by the man who says "I am lying"; if he is lying, he is speaking the truth, and vice versa. "
  11. ^ A b Whitehead Russell: Principia Mathematica . 1910, p. 63 (1)
  12. ^ Whitehead Russell: Principia Mathematica. 1910, p. 65 (1). (on-line)
  13. Tarski 1935, p. 278.
  14. Kripke 1975, pp. 699f.
  15. Kripke 1975, pp. 702-705.
  16. Kripke 1975, p. 700, fn. 8.
  17. So z. B. in Rudolf Schüßler : Offspring for the Liar. From liar and fortified liar to super liar. In: Knowledge. 24, 1986, pp. 219-234.
  18. Kripke 1975, p. 715.
  19. Kripke 1975, p. 714, fn. 34.
  20. a b Arthur Prior: Epimenides the Cretan. In: Journal of Symbolic Logic. 23, 1958, pp. 261-266; there impersonal version "This sentence is false"
  21. ^ A b Prior presented in a modern way in: Andras Kornai: Mathematical Linguistics. Springer, 2007, p. 143, Theorem 6.1 (online)
  22. Logic Bomb on TVTropes.org