A fixed point theorem for the weak Kleene valuation scheme
Tarski's theorem shows that no language with a classical interpretation can contain its own truth predicate. \textit{S. Kripke} [''Outline of a theory of truth'', J. Philos. 72, 690-716 (1975)] and others, have shown that if its interpretation allows for truth-value gaps, a language may contain a predicate whose extension is exactly the set of sentences true in the interpretation. This paper shows that if, in addition, the interpretation is a weak Kleene interpretation (distinguished, essentially, by the condition that a formula is truth-valueless in an interpretation iff some variant interpretation makes a sub-formula truth- valueless) then the language may also contain a predicate whose extension is exactly the set of sentences that are neither true nor false. However, even for such interpretations there will be important notions not, in general, definable in the language. For example, if there were a formula of one free variable, \(\phi\) (x), whose extension is exactly the complement of the set of true sentences, then, given sufficient diagonalising machinery, we could construct a sentence \(\phi\) (a), denoted by ''a''. The usual liar-paradox contradiction would then be forthcoming.
scientific article; zbMATH DE number 3863165