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Subformula property in many-valued modal logics

Published online by Cambridge University Press:  12 March 2014

Mitio Takano*
Affiliation:
Department of Mathematics, Faculty of Education, Niigata University, Niigata 950-21, Japan

Extract

Fitting, in [1] and [2], investigated two families of many-valued modal logics. The first, which is somewhat familiar in the literature, is that of the logics characterized using a many-valued version of the Kripke model (binary modal model in his terminology) with a two-valued accessibility relation. On the other hand, those logics which are characterized using another many-valued version of the Kripke model (implicational modal model), with a many-valued accessibility relation, form the second family. Although he gave a sequent calculus for each of these logics, it is far from having the cut-elimination property (CEP) or the subformula property. So we will give a substitute for his system enjoying the subformula property, though it is not of ordinary sequent calculus but of the many-valued version of sequent calculus initiated by Takahashi [7] and Rousseau [3].

The author, unaware of the deduction systems with CEP, had given in [8] and [9], after Rousseau [4], the deduction systems for the intuitionistic many-valued logics which enjoy CEP only for a certain restricted class of proofs. Then in [10], he gave for three-valued modal logics the ones with CEP, but these systems have a rule of inference which is unnecessary if the Cut rule is present. Why are we particular about CEP? The author's answer is that a cut-free proof is easy to examine since it is composed solely of subformulas of the formulas which form its conclusion. In this direction, the author has given, for modal logics with the Brouwerian axiom [11], the ones without CEP which nevertheless enjoy the subformula property. This paper is a sequel to the study in [11].

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1994

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References

REFERENCES

[1]Fitting, M. C., Many-valued modal logics, Fundamenta Informaticae, vol. 15 (1991), pp. 235254.CrossRefGoogle Scholar
[2]Fitting, M. C., Many-valued modal logics II, Fundamenta Informaticae, vol. 17 (1992), pp. 5573.CrossRefGoogle Scholar
[3]Rousseau, G., Sequents in many-valued logic I, Fundamenta Mathematicae, vol. 60 (1967), pp. 2333.CrossRefGoogle Scholar
[4]Rousseau, G., Sequents in many-valued logic II, Fundamenta Mathematicae, vol. 67 (1970), pp. 125131.CrossRefGoogle Scholar
[5]Sohotch, P. K., Jensen, J. B., Larsen, P. F., and MacLellan, E. J., A note on three-valued modal logic, Notre Dame Journal of Formal Logic, vol. 19 (1978), pp. 6368.Google Scholar
[6]Segerberg, K., Some modal logics based on a three-valued logic, Theoria, vol. 33 (1967), pp. 5371.CrossRefGoogle Scholar
[7]Takahashi, M., Many-valued logics of extended Gentzen style I, Science Reports of Tokyo Kyoiku Daigaku, Section A, vol. 9 (1967), pp. 271292.Google Scholar
[8]Takano, M., Extending the family of intuitionistic many-valued logics introduced by Rousseau, Annals of the Japan Association for Philosophy of Science, vol. 7 (1986), no. 1, pp. 4756.CrossRefGoogle Scholar
[9]Takano, M., Cut-elimination in the intuitionistic many-valued logic based on a partial order, Annals of the Japan Association for Philosophy of Science, vol. 7 (1988), no. 3, pp. 117123.CrossRefGoogle Scholar
[10]Takano, M., Cut-free systems for three-valued modal logics, Notre Dame Journal of Formal Logic, vol. 33 (1992), pp. 359368.CrossRefGoogle Scholar
[11]Takano, M., Subformula property as a substitute for cut-elimination in modal propositional logics, Mathematica Japonica, vol. 37 (1992), pp. 11291145.Google Scholar
[12]Thomason, S. K., Possible worlds and many truth values, Studia Logica, vol. 37 (1978), pp. 195204.CrossRefGoogle Scholar