Journal of Symbolic Logic

A proof-theoretic study of the correspondence of classical logic and modal logic

Abstract

It is well known that the modal logic S5 can be embedded in the classical predicate logic by interpreting the modal operator in terms of a quantifier. Wajsberg proved this fact in a syntactic way. Mints extended this result to the quantified version of S5; using a purely proof-theoretic method he showed that the quantified S5 corresponds to the classical predicate logic with one-sorted variable. In this paper we extend Mints’ result to the basic modal logic S4; we investigate the correspondence between the quantified versions of S4 (with and without the Barcan formula) and the classical predicate logic (with one-sorted variable). We present a purely proof-theoretic proof-transformation method, reducing an LK-proof of an interpreted formula to a modal proof.

Article information

Source
J. Symbolic Logic, Volume 68, Issue 4 (2003), 1403-1414.

Dates
First available in Project Euclid: 31 October 2003

https://projecteuclid.org/euclid.jsl/1067620195

Digital Object Identifier
doi:10.2178/jsl/1067620195

Mathematical Reviews number (MathSciNet)
MR2017363

Zentralblatt MATH identifier
1056.03009

Citation

Kushida, H.; Okada, M. A proof-theoretic study of the correspondence of classical logic and modal logic. J. Symbolic Logic 68 (2003), no. 4, 1403--1414. doi:10.2178/jsl/1067620195. https://projecteuclid.org/euclid.jsl/1067620195

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