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

Published online by Cambridge University Press:  12 March 2014

H. Kushida  and
M. Okada
H. Kushida
Affiliation:
Department of Philosophy, Keio University, 2-15-45 Mita, Minato-Ku, Tokyo 108-8345, Japan, E-mail: mitsu@abelard.flet.keio.ac.jp
M. Okada
Affiliation:
Department of Philosophy, Keio University, 2-15-45 Mita, Minato-Ku, Tokyo 108-8345, Japan, E-mail: kushida@abelard.flet.keio.ac.jp
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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 [10] proved this fact in a syntactic way. Mints [7] 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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 4 , December 2003 , pp. 1403 - 1414
Copyright
Copyright © Association for Symbolic Logic 2003

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References

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