Joe Lau

Tarski and the Liar Paradox

Readings

  • [Required] Alfred Tarski (1969) Truth and Proof. Scientific American, June Issue.
  • Alfred Tarski (1956) The concept of truth in formalized languages. In Logic, Semantics, Metamathematics. Oxford University Press.

Tarski on the liar

  1. The classical / semantic conception of truth - implies every instance of : "p" is true if and only if p.
  2. Suppose sentence S = "S is not true"
  3. "S is not true" is true if and only if S is not true.
  4. S is true if and only if S is not true.
  5. S is both true and not true.

Tarski's explanation of why the paradox comes about : There are two assumptions.

  1. Natural language is "semantically closed". (a) The language contains both expressions and names of these expressions. (b) The language has a truth-predicate that applies to sentences of the same language.
  2. The laws of classical logic.
  • With (1), it is possible to write down a sentence which says of itself that it is false (or not true). With (2), we can derive a contradiction from the sentence.
  • Note: Tarski gives up (1). TV gaps/gluts give up (2).

@We shall try to find a solution that will keep the classical concept of truth essentially intact. The applicability of the notion of truth will have to undergo some restrictions, but the notion will remain available at least for the purpose of scholarly discourse.@

More details

  • No liar paradox will arise in a formalized language L as long as L does not contain its own truth predicate.
    • A formalized language is a language with a clearly defined vocabulary and syntax, and which does not contain any indexical expressions, expressions whose referents depend on the context, e.g. "he", "it", etc..
    • A truth predicate for a language L is a predicate (e.g. "is true") that applies to all and only the true sentences of L.
  • Given a formalized language L, we can define the truth predicate for L in a larger language L' that contains L, but the truth-predicate is not part of the vocabulary of L. L is called the object-language and L' the meta-language.
    • The proof appears in Tarski (1956).
    • No paradox will arise in a hierarchy of languages where a higher language can only speak of the truth or falsity of sentences of a lower language. (But see Yablo)

Evaluation

  • Kripke's objections - See LiarKripke
  • A consistent universal language is not possible?

@There is, however, no need to use universal languages in all possible situations. In particular, such languages are in general not needed for the purposes of science (and by science I mean here the whole realm of intellectual inquiry)@

Category.LogicAndMaths