Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-14T08:52:21.307Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Buss, S., Introduction to proof theory, Handbook of proof theory (Buss, S., editor), 1998, pp. 1–78.Google Scholar
[2] Dyckhoff, R. and Pinto, L., Permutability of proofs in intuitionistic sequent calculi, Theoretical Computer Science, vol. 212 (1999), pp. 141–155.CrossRefGoogle Scholar
[3] Feys, R., Modèles à variables de différentes sortes pour les logiques modales M” ou S5, Synthése, vol. 12 (1960), pp. 182–196.Google Scholar
[4] Gentzen, G., Untersuchungen über das logische Schließen, Mathematische Zeitschrift, vol. 39 (1935), pp. 176–210, 405–431.CrossRefGoogle Scholar
[5] Hughes, G. and Cresswell, M., A new introduction to modal logic, Routledge, London and New York, 1996.CrossRefGoogle Scholar
[6] Kleene, S. C., Permutability of inferences in Gentzen's calculi LK and LJ, Memoirs of the American Mathematical Society, vol. 10 (1952), pp. 1–26.Google Scholar
[7] Mints, G. E., On some calculi of modal logic, Proceedings of the Steklov Institute of Mathematics, 1968, pp. 97–124.Google Scholar
[8] Takeuti, G., Proof theory, second ed., North-Holland, Amsterdam, 1987.Google Scholar
[9] van Benthem, J., Modal logic and classical logic, Bibliopolis, Naples, 1985.Google Scholar
[10] Wajsberg, M., Ein erweiterter Klassenkalkül, Monatshefte für mathematische Physik, vol. 40 (1933), pp. 113–126.CrossRefGoogle Scholar