A revenge-immune solution to the semantic paradoxes
type: Item
instance of: scholarly article
title: A revenge-immune solution to the semantic paradoxes
zbMATH Open document ID: 1026.03009
DOI: 10.1023/A:1023027808400
author: Hartry Field
published in: Journal of Philosophical Logic
publication date: 2003-06-09T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03A05
zbMATH DE Number: 1924906
zbMATH Keywords: conditionals
zbMATH Keywords: Curry paradox
zbMATH Keywords: determinateness
zbMATH Keywords: liar paradox
zbMATH Keywords: truth
reviewed by: Jim Mackenzie
MaRDI profile type: Publication
cites work: Q4397069
cites work: Q3032252
cites work: Saving the truth schema from paradox
cites work: Q4845457
cites work: The liar paradox and fuzzy logic
cites work: Outline of a Theory of Truth
cites work: Q4856958
cites work: Q5573961
Recommended article: Q3299146
Recommended article: SUPER LIARS
Recommended article: Theories of truth for countable languages which conform to classical logic
Recommended article: A mathematically derived definitional/semantical theory of truth
Recommended article: CONSISTENCY AND THE THEORY OF TRUTH
Recommended article: Supervaluation-style truth without supervaluations
Recommended article: Q3058982
Recommended article: Indicative conditionals, restricted quantification, and naive truth
Recommended article: Post-Tarskian truth
Recommended article: Q2939751
description: scientific article; zbMATH DE number 1924906