Gödel-McKinsey-Tarski and (not quite) Blok-Esakia for Heyting-Lewis Implication
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866914356645593088 |
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| author | de Groot, Jim Litak, Tadeusz Pattinson, Dirk |
| author_facet | de Groot, Jim Litak, Tadeusz Pattinson, Dirk |
| contents | Heyting-Lewis Logic is the extension of intuitionistic propositional logic with a strict implication connective that satisfies the constructive counterparts of axioms for strict implication provable in classical modal logics. Variants of this logic are surprisingly widespread: they appear as Curry-Howard correspondents of (simple type theory extended with) Haskell-style arrows, in preservativity logic of Heyting arithmetic, in the proof theory of guarded (co)recursion, and in the generalization of intuitionistic epistemic logic.
Heyting-Lewis Logic can be interpreted in intuitionistic Kripke frames extended with a binary relation to account for strict implication. We use this semantics to define descriptive frames (generalisations of Esakia spaces), and establish a categorical duality between the algebraic interpretation and the frame semantics. We then adapt a transformation by Wolter and Zakharyaschev to translate Heyting-Lewis Logic to classical modal logic with two unary operators. This allows us to classical results to obtain the finite model property and decidability for a large family of Heyting-Lewis logics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_01873 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Gödel-McKinsey-Tarski and (not quite) Blok-Esakia for Heyting-Lewis Implication de Groot, Jim Litak, Tadeusz Pattinson, Dirk Logic Logic in Computer Science Heyting-Lewis Logic is the extension of intuitionistic propositional logic with a strict implication connective that satisfies the constructive counterparts of axioms for strict implication provable in classical modal logics. Variants of this logic are surprisingly widespread: they appear as Curry-Howard correspondents of (simple type theory extended with) Haskell-style arrows, in preservativity logic of Heyting arithmetic, in the proof theory of guarded (co)recursion, and in the generalization of intuitionistic epistemic logic. Heyting-Lewis Logic can be interpreted in intuitionistic Kripke frames extended with a binary relation to account for strict implication. We use this semantics to define descriptive frames (generalisations of Esakia spaces), and establish a categorical duality between the algebraic interpretation and the frame semantics. We then adapt a transformation by Wolter and Zakharyaschev to translate Heyting-Lewis Logic to classical modal logic with two unary operators. This allows us to classical results to obtain the finite model property and decidability for a large family of Heyting-Lewis logics. |
| title | Gödel-McKinsey-Tarski and (not quite) Blok-Esakia for Heyting-Lewis Implication |
| topic | Logic Logic in Computer Science |
| url | https://arxiv.org/abs/2105.01873 |