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Subformula property in many-valued modal logics. (English) Zbl 0813.03014

M. Fitting investigated [Ann. Soc. Math. Pol., Ser. IV, Fundam. Inf. 15, No. 3/4, 235-254 (1991; Zbl 0745.03018); ibid. 17, No. 1/2, 55- 73 (1992; Zbl 0772.03006)] 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 or the subformula formula. So we 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 M. Takahashi [Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 9, 271-292 (1968; Zbl 0172.008)] and G. Rousseau [Fundam. Math. 60, 23-33 (1967; Zbl 0154.255)].

MSC:

03B45 Modal logic (including the logic of norms)
03B50 Many-valued logic
03F05 Cut-elimination and normal-form theorems
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[2] Fundamenta Mathematicae 60 pp 23– (1967)
[3] Fundamenta Informaticae 17 pp 55– (1992)
[4] Fundamenta Informaticae 15 pp 235– (1991)
[5] DOI: 10.1007/BF02124804 · Zbl 0391.03014 · doi:10.1007/BF02124804
[6] DOI: 10.1305/ndjfl/1093634401 · Zbl 0778.03005 · doi:10.1305/ndjfl/1093634401
[7] Annals of the Japan Association for Philosophy of Science 7 pp 117– (1988) · Zbl 0648.03014 · doi:10.4288/jafpos1956.7.117
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[11] DOI: 10.1305/ndjfl/1093888207 · Zbl 0332.02018 · doi:10.1305/ndjfl/1093888207
[12] Mathematica Japonica 37 pp 1129– (1992)
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