Joe Lau

Kripke on the liar paradox

Readings

  • Saul Kripke. 1975. Outline of a Theory of Truth. Journal of Philosophy, 72: 690 - 716. doi:10.2307/2024634

Kripke on Tarski's solution

@(a) Dean: All of Nixon's statements about Watergate are false.
(b) Nixon: Everything Dean says about Watergate is false.@

  • Ordinary speakers use 'true' and not 'true1', 'true2', etc.
  • Even if indices are to be used, speakers would not know which indices to assign. The index associated with (a) depends on the level of other statements.
  • The hierarchical view does not allow for the situation that corresponds to (a) and (b) where each index is supposed to be higher than the other one.
  • The concept of truth is risky in that the semantic and syntactic features of sentence (a) cannot help us determine whether the sentence itself is paradoxical. It depends on the empirical facts.

Kripke's approach

Warning: Informal and grossly misleadingly presentation.

  • A predicate is defined by an extension and an anti-extension <E,A>, e.g. "red" defined by <{red things},{non-red things}>
  • Define a truth predicate by defining <{true sentences},{false sentences}>, leaving out the ungrounded sentences such as liar sentences and the truth-teller.
  • Kripke shows how to define such a predicate by adding true sentences to the extension and false sentences to the anti-extension until a fixed-point is reached.
  • Sentence L: "L is not true" is neither a member of the extension nor the anti-extension of the truth predicate, and the truth predicate applies to sentences in the same language as L.

Limitations

@there are assertions we can make about the object language which we cannot make in the object language, in the sense that the inductive process never makes them true; but we are precluded from saying this in the object language by our interpretation of negation and the truth predicate. ... The necessity to ascend to a metalanguage may be one of the weaknesses of the present theory. The ghost of the Tarski hierarchy is still with us.@

  • (K) K is not* true.
  • (K) K is false or undefined.

Category.LogicAndMaths