The decidability of admissibility problems for modal logics S4.2 and S4.2Grz and superintuitionistic logic KC
type: Item
instance of: scholarly article
title: The decidability of admissibility problems for modal logics S4.2 and S4.2Grz and superintuitionistic logic KC
zbMATH Open document ID: 0795.03017
DOI: 10.1007/BF02259929
author: S. V. Babenyshev
published in: Algebra and Logic
publication date: 1994-04-12T00:00:00Z
review text: In studies dealing with intuitionistic logic the following question arises: Is the admissibility problem for the logic Int decidable? (the Kuznetsov-Friedman problem). In a paper of \textit{V. V. Rybakov} [Algebra Logika 20, 440-464 (1981; Zbl 0489.03005)], it was noted that a rule \(\alpha/\beta\) is admissible in a superintuitionistic logic \(\lambda\supseteq\text{Int}\) if and only if the rule \(T(\alpha)/T(\beta)\) is admissible in \(\sigma(\lambda)\) -- the greatest modal companion of \(\lambda\) -- where \(T(\alpha)\) is the Gödel translation of the formula \(\alpha\). In another paper of Rybakov [ibid. 23, No. 5, 546-572 (1984; Zbl 0598.03013)], the Kuznetsov-Friedman problem was solved positively, using this fact and the decidability of the admissibility problem for modal logic S4, the modal companion of Int. Of special interest among superintuitionistic logics is the logic KC with the Weak Law of Excluded Middle. The greatest modal companion of KC is the logic S4.2Grz. In the present paper we obtain algorithms to verify whether the inference rules in the modal logics S4.2Grz are admissible. As a consequence, it is proved that the admissibility problem for KC is decidable.
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03B55
zbMATH DE Number: 536667
zbMATH Keywords: admissibility problem
zbMATH Keywords: superintuitionistic logic
zbMATH Keywords: Kuznetsov-Friedman problem
zbMATH Keywords: decidability
zbMATH Keywords: Weak Law of Excluded Middle
zbMATH Keywords: modal companion
zbMATH Keywords: algorithms
MaRDI profile type: Publication
cites work: Admissible rules for pretable modal logics
cites work: A criterion for admissibility of rules in the modal system S4 and intuitionistic logic
cites work: A general filtration method for modal logics
cites work: Epistemic considerations in game theory
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description: scientific article; zbMATH DE number 536667