Topological-Frame Products of Modal Logics

Abstract

The simplest bimodal combination of unimodal logics $$\text {L} _1$$ and $$\text {L} _2$$ is their fusion, $$\text {L} _1 \otimes \text {L} _2$$ , axiomatized by the theorems of $$\text {L} _1$$ for $$\square _1$$ and of $$\text {L} _2$$ for $$\square _2$$ , and the rules of modus ponens, necessitation for $$\square _1$$ and for $$\square _2$$ , and substitution. Shehtman introduced the frame product $$\text {L} _1 \times \text {L} _2$$ , as the logic of the products of certain Kripke frames: these logics are two-dimensional as well as bimodal. Van Benthem, Bezhanishvili, ten Cate and Sarenac transposed Shehtman’s idea to the topological semantics and introduced the topological product $$\text {L} _1 \times _t \text {L} _2$$ , as the logic of the products of certain topological spaces. For almost all well-studies logics, we have $$\text {L} _1 \otimes \text {L} _2 \subsetneq \text {L} _1 \times \text {L} _2$$ , for example, $$\text {S4} \otimes \text {S4} \subsetneq \text {S4} \times \text {S4} $$ . Van Benthem et al. show, by contrast, that $$\text {S4} \times _t \text {S4} = \text {S4} \otimes \text {S4} $$ . It is straightforward to define the product of a topological space and a frame: the result is a topologized frame, i.e., a set together with a topology and a binary relation. In this paper, we introduce topological-frame products $$\text {L} _1 \times _ tf \text {L} _2$$ of modal logics, providing a complete axiomatization of $$\text {S4} \times _ tf \text {L} $$ , whenever $$\text {L} $$ is a Kripke complete Horn axiomatizable extension of the modal logic D: these extensions include $$\text {T} , \text {S4} $$ and $$\text {S5} $$ , but not $$\text {K} $$ or $$\text {K4} $$ . We leave open the problem of axiomatizing $$\text {S4} \times _ tf \text {K} $$ , $$\text {S4} \times _ tf \text {K4} $$ , and other related logics. When $$\text {L} = \text {S4} $$ , our result confirms a conjecture of van Benthem et al. concerning the logic of products of Alexandrov spaces with arbitrary topological spaces.