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Truth, Disjunction, and Induction

Journal article
Authors Ali Enayat
Fedor Pakhomov
Published in Archive for mathematical logic
Volume 58
Issue 5-6
Pages 753–766
ISSN 0933-5846
Publication year 2019
Published at Department of Philosophy, Linguistics and Theory of Science
Pages 753–766
Language en
Links https://doi.org/10.1007/s00153-018-...
Keywords Axiomatic truth, Compositional theory of truth, Conservativity
Subject categories Algebra and Logic, Logic

Abstract

By a well-known result of Kotlarski et al. (1981), first-order Peano arithmetic PA can be conservatively extended to the theory CT−[PA] of a truth predicate satisfying compositional axioms, i.e., axioms stating that the truth predicate is correct on atomic formulae and commutes with all the propositional connectives and quantifiers. This result motivates the general question of determining natural axioms concerning the truth predicate that can be added to CT−[PA] while maintaining conservativity over PA . Our main result shows that conservativity fails even for the extension of CT−[PA] obtained by the seemingly weak axiom of disjunctive correctness DC that asserts that the truth predicate commutes with disjunctions of arbitrary finite size. In particular, CT−[PA]+DC implies Con(PA) . Our main result states that the theory CT−[PA]+DC coincides with the theory CT0[PA] obtained by adding Δ0 -induction in the language with the truth predicate. This result strengthens earlier work by Kotlarski (1986) and Cieśliński (2010). For our proof we develop a new general form of Visser’s theorem on non-existence of infinite descending chains of truth definitions and prove it by reduction to (Löb’s version of) Gödel’s second incompleteness theorem, rather than by using the Visser–Yablo paradox, as in Visser’s original proof (1989).

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