A revenge-immune solution to the semantic paradoxes
This paper proposes an extension of (the strong Kleene version of) \textit{S. Kripke}'s theory of truth [``Outline of a theory of truth'', J. Philos. 72, 690-716 (1975; Zbl 0952.03513)] to include a conditional \(\to\) such that both \(A\leftrightarrow A\) and \(\text{True}(\langle A\rangle)\leftrightarrow A\) are theorems. The resulting logic LCC (for ``logic of circularly defined concepts'') is non-classical: Explosion \((A\wedge\neg A\vdash B)\) holds, but Importation, Contraction, \(\to\)-Introduction, and Permutation fail, as does the Law of Excluded Middle. Nevertheless, it can be described in a sublanguage of the object language with \(\to\), and \(\to\) coincides with \(\supset\) in contexts, like ordinary arithmetic, where the Law of Excluded Middle holds. For any theory \(T\) which includes the arithmetic required for formal syntax, and any classical model \(M\) of \(T\) that is standard with respect to arithmetic, one can define a new non-classical model for the logic LCC which satisfies naive truth theory. The domain of the new model is that of the old, and all the predicates of \(T\) have the same extension as on the old, so that the new model is standard with respect to arithmetic. The procedure accords with Tarski's theorem that ``true'' (as against ``true in a model'') cannot be defined. Field argues that the system is immune to any introduction of hyper-versions of the paradox.
scientific article; zbMATH DE number 1924906
scientific article