Saving the truth schema from paradox
type: Item
instance of: scholarly article
title: Saving the truth schema from paradox
zbMATH Open document ID: 1008.03004
DOI: 10.1023/A:1015063620612
author: Hartry Field
published in: Journal of Philosophical Logic
publication date: 2002-08-20T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03A05
Mathematics Subject Classification ID: 03B50
zbMATH DE Number: 1784287
zbMATH Keywords: conditional
zbMATH Keywords: law of excluded middle
zbMATH Keywords: semantic paradoxes
zbMATH Keywords: truth
reviewed by: Jim Mackenzie
MaRDI profile type: Publication
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full work available at URL: a:1015063620612
OpenAlex ID: W1996660455
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description: scientific article; zbMATH DE number 1784287