Canonical formulas for K4. part III: the finite model property

Zakharyaschev, Michael (1997) Canonical formulas for K4. part III: the finite model property. The journal of Symbolic Logic 62 (3), pp. 950-975. ISSN 0022-4812.

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Official URL: https://doi.org/10.2307/2275581

Abstract

This paper, a continuation of the series [22, 24], presents two methods for establishing the finite model property (FMP, for short) of normal modal logics containing K4. The methods are oriented mainly to logics represented by their canonical axioms and yield for such axiomatizations several sufficient conditions of FMP. We use them to obtain solutions to two well known open FMP problems. Namely, we prove that • every normal extension of K4 with modal reduction principles has FMP and • every normal extension of S4 with a formula of one variable has FMP. These results are interesting not only from the technical point of view. Actually, they reveal important properties of a quite natural family of modal logics—formulas of one variable and, in particular, modal reduction principles are typical axioms in modal logic. Unfortunately, the technical apparatus developed in this paper is applicable only to logics with transitive frames, and the situation with FMP of extensions of K by modal reduction principles, even by axioms of the form □np → □mp still remains unclear. I think at present this is one of the major challenges in completeness theory. The language of the canonical formulas, introduced in [22] (I'll refer to that paper as Part I), is a way of describing the “geometry and topology” of formulas' refutation (general) frames by means of some finite refutation patterns.

Metadata

Item Type:	Article
School:	Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences
Depositing User:	Sarah Hall
Date Deposited:	08 Nov 2021 19:25
Last Modified:	09 Aug 2023 12:52
URI:	https://eprints.bbk.ac.uk/id/eprint/46652

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