Two-dimensional Kripke Semantics II: Stability and Completeness
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...
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| Published in: | Electronic Notes in Theoretical Informatics and Computer Science Vol. 4 - Proceedings of... |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
11.12.2024
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| ISSN: | 2969-2431, 2969-2431 |
| Online Access: | Get full text |
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| Summary: | 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.
Comment: Accepted at MFPS 2024 |
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| ISSN: | 2969-2431 2969-2431 |
| DOI: | 10.46298/entics.14767 |