Saving the truth schema from paradox
In view of the semantic paradoxes, we cannot have both classical truth theory and classical logic. \textit{S. Kripke} [``Outline of a theory of truth'', J. Philos. 72, 690-716 (1975; Zbl 0952.03513)] showed that we can preserve intersubstitutivity of \(A\) and \(\text{Tr}\langle A\rangle\) in all contexts, but the resulting logic will not validate \(\langle\text{If }A\) then \(A\rangle\). Field adds to Kleene's strong three-valued logic an extra conditional \(\to\) for which contraction does not hold. \((A\wedge A\to C\) yields \(A \to C\), but \(A\to(B\to C)\) is not equivalent to \(A\wedge B\to C.)\) (We retain the effect of classical logic in the field to which the logic is to be applied.) This is like the addition of the conditional of Łukasiewicz's continuum-valued semantics (the conditional of fuzzy logic), which was shown [\textit{G. Restall}, ``Arithmetic and truth in Łukasiewicz's infinitely valued logic'', Log. Anal., Nouv. Sér. 35, No. 139-140, 303-312 (1992; Zbl 0832.03010)] not to handle more complex paradoxical sentences. Field's conditional is fully embeddable, and the logic is not merely consistent but \(\omega\)-consistent, and includes T (i.e., \(\text{Tr}(\langle A\rangle) \leftrightarrow A\), for the new conditional) and full inter-substitutivity of \(\text{Tr} (\langle A\rangle)\) with \(A\) in all contexts, even within the scope of \(\to\). Where \(Q\) asserts its own untruth, we cannot assert \(Q\vee\neg Q\) nor \(Q\wedge\neg Q\), though we have many special cases of excluded middle, notably \((A\to B)\vee \neg(A\to B)\) for the new conditional.
scientific article; zbMATH DE number 1784287
scientific article