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arXiv · 2402.00528

A calculus for modal compact Hausdorff spaces

Nick Bezhanishvili·Luca Carai·Silvio Ghilardi·Zhiguang Zhao
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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 $\Pi_2$-rules in this system.

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Nick Bezhanishvili, Luca Carai, Silvio Ghilardi, Zhiguang Zhao. 2024-02-01. A calculus for modal compact Hausdorff spaces. https://arxiv.org/abs/2402.00528

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