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A cut-free Gentzen-type system for the modal logic S51

Published online by Cambridge University Press:  12 March 2014

Masahiko Sato
Masahiko Sato*
Affiliation:
University of Tokyo, College Of General Education, Tokyo 153, Japan
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Extract

The modal logic S5 has been formulated in Gentzen-style by several authors such as Ohnishi and Matsumoto [4], Kanger [2], Mints [3] and Sato [5]. The system by Ohnishi and Matsumoto is natural, but the cut-elimination theorem in it fails to hold. Kanger's system enjoys cut-elimination theorem, but, strictly speaking, it is not a Gentzen-type system since each formula in a sequent is indexed by a natural number. The system S5+ of Mints is also cut-free, and its cut-elimination theorem is proved constructively via the cut-elimination theorem of Gentzen's LK. However, one of his rules does not have the so-called subformula property, which is desirable from the proof-theoretic point of view. Our system in [5] also enjoys the cut-elimination theorem. However, it is also not a Gentzen-type system in the strict sense, since each sequent in this system consists of a pair of sequents in the usual sense.

In the present paper, we give a Gentzen-type system for S5 and prove the cut-elimination theorem in a constructive way. A decision procedure for S5 can be obtained as a by-product.

The author wishes to thank the referee for pointing out some errors in the first version of the paper as well as for his suggestions which improved the readability of the paper.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 45 , Issue 1 , March 1980 , pp. 67 - 84
Copyright
Copyright © Association for Symbolic Logic 1980

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Footnotes

1

This work was partially supported by the Sakkokai Foundation.

References

REFERENCES

[1]Gentzen, G., Untersuchungen über das logische Schliessen. I, II, Mathematische Zeitschrift, vol. 39 (1935), pp. 176–210, pp. 405–431; English translation in The collected papers of Gerhard Gentzen (M. E. Szabo, Editor), North-Holland, Amsterdam, 1969.CrossRefGoogle Scholar
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