A calculus for modal compact Hausdorff spaces
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866913683586678784 |
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| author | Bezhanishvili, Nick Carai, Luca Ghilardi, Silvio Zhao, Zhiguang |
| author_facet | Bezhanishvili, Nick Carai, Luca Ghilardi, Silvio Zhao, Zhiguang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_00528 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A calculus for modal compact Hausdorff spaces Bezhanishvili, Nick Carai, Luca Ghilardi, Silvio Zhao, Zhiguang Logic 03B45, 06E15, 54E05, 06E25 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. |
| title | A calculus for modal compact Hausdorff spaces |
| topic | Logic 03B45, 06E15, 54E05, 06E25 |
| url | https://arxiv.org/abs/2402.00528 |