Liar paradox

A Liar Paradox is a paradox that arises when a sentence asserts its own falsity (or untruth ). If the sentence is true, then it follows by its self-reference, that he is wrong, and vice versa.

Formulation

The most popular form of the liar is the following self-referential sentence:

The paradox of this sentence is that he could not reasonably claim is true or false. Suppose it would be wrong: Then that would be true, which asserts the sentence itself, and he would actually rather be true. But we assume that it is true, then it seems that what the sentence asserts, no longer apply - which means that the set actually but again would be wrong.

This kind of paradox is often referred to in the philosophical debate as semantic paradox. It is possible because the truth conditions of a sentence in this even be specified (directly or indirectly ) - but in a way that at least seems to be no sensible ascription of truth or falsity longer permits.

The name " Liar Paradox" follows on from that the paradox can be formulated also using the concept of a lie, for example, as follows:

The Cretan who says so, so also claims, among other things, that the above statement is a lie, so therefore does not reflect the truth. Thus there arises ultimately the same paradox as above.

In the paradox of Epimenides the sentence " All Cretans are liars. " Used to represent the paradox. Here it would be necessary that we additionally assume that with " liar " only persons are meant who say only the truth. There is a liar someone meant the occasional untruth says, then goes out the paradox.

Extensions and related paradoxes

The paradoxical mechanism in the classical liar paradox is similar to that in other semantic paradoxes. A variant which is already clearly points out the problem for the logic curries paradox. If the truth conditions of the logical subjunction be assumed for the following conditional, then it may for example be reproduced:

This set may be the truth value "false " is not attributed consistent, because the antecedent of the conditional tense then would be wrong, which would make true the whole conditional on the assumed logical understanding. The truth value "true", however, can already be attributed to the set; However, it must be assumed that also the consequent " The moon is made of green cheese. " is true - otherwise the antecedent of the conditional tense of postscript, however, the whole sentence would be true, false, and so wrong again. If this set so a truth value should be attributed to, he would be an absurd "proof" that the moon is made ​​of green cheese.

The proposed solution to meet the liar by a rejection of classical logic, modified versions of the Liar paradox be pleaded. The best known is the increased liars:

This paradox remains active even exist if it is allowed that paradoxical sentences can be neither true nor false (so-called truth - value "gaps "). You can, however, still avoid a three-valued logic, which conceives the third value as " both true and false" (so-called " Gluts ", for example, represented by Graham Priest ). On the hand, a variant of the amplified liar cite:

Paradoxes of the liar - type can also generate multiple sets, such as the following two:

This variant avoids the direct self-reference, but is still exactly the same as the classic paradox ago liar. An indirect self-reference is still available because a circle of references of the two sets consists among each other (similar in models with a larger number of sets ).

Its own claim by without self-reference comes from Yablo 's paradox. It consists of an infinite series of sentences, each of which maintains that all now following sentences are not true. Again, none of the sets can without contradiction a truth value attributed to, because each conflicting conditions would be imposed on the number of subsequent blocks. If this paradox actually works without self-reference (which is, however, occasionally contested in the philosophical discussion), then it shows that not self-reference allows the paradox, but our use of the terms " true" and "false".

A sentence which asserts its own undecidability instead of his falsehood, generates a related paradox.

History

Already Aristotle discussed in his Sophistic refutations the liar paradox, but without citation and author is acknowledged. Late Ancient sources mention his contemporaries Eubulides as speakers of the Liar paradox. Since the works of Eubulides are lost, his argument is only of the oldest citations in Cicero, among others reconstructed; they could have had the following dialogue form:

This dialog manages the paradoxical part by the statement " I am saying that I am lying " provoked antinomy from.

Variants of this Liar antinomy were discussed by the whole logic throughout history. In modern mathematical logic, she won on new importance by Bertrand Russell. He followed up on the paradox of Epimenides " Epimenides the Cretan said: All Cretans are liars "; this probably older, weaker preform of the liar paradox produces no antinomy; he tightened it therefore for real paradoxical statement that generates the antinomy:

Problems and solutions

Type Theoretical solution

Russell called for the solution of the paradox a type theory with a hierarchy of statements and a hierarchy of truth predicates, namely n statements of order and truth predicates of order n ( for n = 0, 1, 2, ...). A truth predicate of order n can be predicated only of a statement with an order less than n. So he solved the liar paradox by excluded self-referential statements syntactically.

Separation of object and metalanguage

The Liar paradox has been considered from the 20th century as a significant problem for a philosophical theory of truth. Alfred Tarski formulated the problem in his influential essay The concept of truth in formal languages ​​as: the colloquial, " universalist ", ie they take all the semantic expressions into itself. However:

Tarski below shows that such paradoxes do not arise for artificial languages ​​, in which a separation of the object language and metalanguage is applied consistently. The essential characteristic of this separation is that within the object language no statements can be made about this language - the remains of the metalanguage for that language reserved. Then, however, a meta-language for this meta-language is required for statements about the meta-language, so that a so-called " Tarski hierarchy " results. Within a language, a reference to sentences of this language is thus always excluded.

Soundness

An alternative to Tarski hierarchy, which is to provide a model of natural language, based on Saul Kripke's concept of soundness. Kripke provides a semantic theory of truth, in the statements about the truth of other sentences a truth value can be assigned, as long as they " sound " are. " Unfounded " statements are not recognized as propositions that are true or false; they are according to Kripke yet not meaningless, insofar as they express the mold after possible propositions and using a three-valued logic could still be treated.

The basic idea of Kripke's theory of truth is the following: In a first step, all statements that do not depend on the truth value of other statements ( not claim say for example from another set, this is true ) is assigned a truth value - simply by comparison with reality. In a second step, all statements about the truth value of other statements now be considered. Unless the basis of the distributed thus far truth values ​​such statements, a value can be assigned this happens. This second step will be repeated until a repetition of this step no new truth values ​​more were distributed. Sentences in this "least fixed point " has no truth- value are regarded as unfounded.

Kripke said to have escaped the usual formulations of the liar with his theory of truth. The enhanced versions of the liar he evades by stating that it is " unfounded " is no third truth value in, he also stressed that the classical logic would continue to be valid for the range of propositions. However, can nevertheless be using the concept of Unfundiertheit formulate new paradoxes (which are often referred to in the literature as " revenge Liar "). Kripke sees this and does not claim to have given a universal semantics of the concept of truth. Ultimately, he admits Tarski the need for metalanguage for terms such as " unfounded " or " paradoxical " to. He wanted to give just a model of the everyday language is not philosophical spokesman, however, more sophisticated concepts can not absorb this.

General formalization

A formalization of reasoning resolves the paradox without syntactic restrictions. As calculus classical propositional logic with additional predicates is sufficient " X lies " ( " X is lying straight" within the meaning of ) and "X says that A" and two syllogisms:

This calculus is consistent: both syllogisms are, at least in a world where no one says anything ( there is their second premise wrong). So no antinomy is derivable in the calculus. The Liar paradox here is a syntactically correct self-reference in variable form:

In calculus, the following applies to Arthur Prior returning theorem:

Indirect proof: assumption ( 3). First case: X is lying; then follows from the assumption (3 ) with ( 1) and the modus ponens: X does not lie. This case is so contradictory. In the other case: X is not lying; but then follows from the assumption (3 ) with ( 2) by Modus Ponens: X is lying. Thus both possible cases are contradictory and ( 3) is refuted.

The proof clarifies the reasoning of Eubulides but also stresses the hidden assumption: the liar paradox, without which the argument does not work. It proves to be a sophism, which is not quite consistent with the declared calculus and therefore ceases to be a logical argument. The proof is independent of the definition of the predicates in specific models, since the formalization is a general axiomatization that allows for different models.

Impersonal model

Since the formalization leaves open the assignment of variables, X may be a statement that indeed says something is wrong and if she's lying; Can you believe two definitions:

These definitions generate from (3) the impersonal liar paradox " X ↔ X is wrong " is true for the prior's theorem and is just as originally formulated. The model leaves open is defined as the predicate " X is lying ." It can be done on an extended semantic level of language, which is why the liar is considered as semantic paradox. But this is not mandatory, as the following model shows.

Statements Logical model

The impersonal Liars model for propositional model by lying predicate is not shifted to a higher semantic level of language, but is also defined as a statement:

With the definitions (4) and ( 6) the syllogisms ( 1) and (2) provable, and the liar paradox (3) is equivalent to the self-reference " X ↔ non- X ", which is known to be wrong.

Popular culture

In popular culture circulate embellished versions of the liar paradox: a common motif in science fiction authors is to overcome an overpowering artificial intelligence by the confrontation with the paradox, which should lead to an infinite loop calculation.