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.