Two-dimensional Kripke Semantics II: Stability and Completeness
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929634448244736 |
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| author | Kavvos, G. A. |
| author_facet | Kavvos, G. A. |
| contents | We revisit the duality between Kripke and algebraic semantics of intuitionistic and intuitionistic modal logic. We find that there is a certain mismatch between the two semantics, which means that not all algebraic models can be embedded into a Kripke model. This leads to an alternative proposal for a relational semantics, the stable semantics. Instead of an arbitrary partial order, the stable semantics requires a distributive lattice of worlds. We constructively show that the stable semantics is exactly as complete as the algebraic semantics. Categorifying these results leads to a 2-duality between two-dimensional stable semantics and categories of product-preserving presheaves, i.e. models of algebraic theories in the style of Lawvere. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_03578 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two-dimensional Kripke Semantics II: Stability and Completeness Kavvos, G. A. Logic in Computer Science Category Theory Logic 03B45 (Primary), 03B20, 03B70, 68Q55, 06D20, 06D22, 06D05, 06D50, 18A15, 18A40, 18F20, 18D60, 18C10 (Secondary) F.4.1; F.3.2 We revisit the duality between Kripke and algebraic semantics of intuitionistic and intuitionistic modal logic. We find that there is a certain mismatch between the two semantics, which means that not all algebraic models can be embedded into a Kripke model. This leads to an alternative proposal for a relational semantics, the stable semantics. Instead of an arbitrary partial order, the stable semantics requires a distributive lattice of worlds. We constructively show that the stable semantics is exactly as complete as the algebraic semantics. Categorifying these results leads to a 2-duality between two-dimensional stable semantics and categories of product-preserving presheaves, i.e. models of algebraic theories in the style of Lawvere. |
| title | Two-dimensional Kripke Semantics II: Stability and Completeness |
| topic | Logic in Computer Science Category Theory Logic 03B45 (Primary), 03B20, 03B70, 68Q55, 06D20, 06D22, 06D05, 06D50, 18A15, 18A40, 18F20, 18D60, 18C10 (Secondary) F.4.1; F.3.2 |
| url | https://arxiv.org/abs/2406.03578 |