## 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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