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Matching Topological and Frame Products of Modal Logics

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Abstract

The simplest combination of unimodal logics \({\mathrm{L}_1 \rm and \mathrm{L}_2}\) into a bimodal logic is their fusion, \({\mathrm{L}_1 \otimes \mathrm{L}_2}\), axiomatized by the theorems of \({\mathrm{L}_1 \rm for \square_1 \rm and of \mathrm{L}_2 \rm for \square_{2}}\). Shehtman introduced combinations that are not only bimodal, but two-dimensional: he defined 2-d Cartesian products of 1-d Kripke frames, using these Cartesian products to define the frame product \({\mathrm{L}_1 \times \mathrm{L}_2 \rm of \mathrm{L}_1 \rm and \mathrm{L}_2}\). Van Benthem, Bezhanishvili, ten Cate and Sarenac generalized Shehtman’s idea and introduced the topological product \({\mathrm{L}_1 \times_{t}\mathrm{L}_2}\), using Cartesian products of topological spaces rather than of Kripke frames. Frame products have been extensively studied, but much less is known about topological products. The goal of the current paper is to give necessary and sufficient conditions for the topological product to match the frame product, for Kripke complete extensions of \({\mathrm{S}4: \mathrm{L}_1 \times_t \mathrm{L}_2 = \mathrm{L}_1 \times \mathrm{L}_2 \rm iff \mathrm{L}_1 \supsetneq \mathrm{S}5 \rm or \mathrm{L}_2 \supsetneq \mathrm{S}5 \rm or \mathrm{L}_1, \mathrm{L}_2 = \mathrm{S}5}\).

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  1. Department of Philosophy, University of Toronto Scarborough, 1265 Military Trail, Toronto, ON, M1C 1A4, Canada

    Philip Kremer

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Kremer, P. Matching Topological and Frame Products of Modal Logics. Stud Logica 104, 487–502 (2016). https://doi.org/10.1007/s11225-015-9648-6

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