A calculus for modal compact Hausdorff spaces
Nick Bezhanishvili, Luca Carai, Silvio Ghilardi, Zhiguang Zhao
TL;DR
The paper addresses axiomatizing modal compact Hausdorff spaces by leveraging de Vries duality and extends the strict symmetric implication calculus ${\mathsf{S^2IC}}$ to a modal system ${\mathsf{MS^2IC}}$ with a universal modality. It introduces ${\mathsf{UMS^2IC}}$ and proves it is strongly sound and complete for upper continuous modal de Vries algebras, aided by $\Pi_2$-rules expressing upper continuity. Through a relational semantics based on modal contact frames and regular stable p-morphisms, the authors establish Kripke completeness for ${\mathsf{MS^2IC}}$ and show the admissibility of key $\Pi_2$-rules, yielding the equivalence ${\mathsf{MS^2IC}} = {\mathsf{UMS^2IC}}$ and strong completeness for zero-dimensional/descriptive frameworks. The results provide a robust logical calculus for modal KHaus spaces, connecting algebraic, topological, and relational semantics and enabling canonical representations and completeness across multiple algebraic/space variants.
Abstract
The symmetric strict implication calculus $\mathsf{S^2IC}$ is a modal calculus for compact Hausdorff spaces. This is established through de Vries duality, linking compact Hausdorff spaces with de Vries algebras-complete Boolean algebras equipped with a special relation. Modal compact Hausdorff spaces are compact Hausdorff spaces enriched with a continuous relation. These spaces correspond, via modalized de Vries duality, to upper continuous modal de Vries algebras. In this paper we introduce the modal symmetric strict implication calculus $\mathsf{MS^2IC}$, which extends $\mathsf{S^2IC}$. We prove that $\mathsf{MS^2IC}$ is strongly sound and complete with respect to upper continuous modal de Vries algebras, thereby providing a logical calculus for modal compact Hausdorff spaces. We also develop a relational semantics for $\mathsf{MS^2IC}$ that we employ to show admissibility of various $Π_2$-rules in this system.