Correspondence Theory for Many-valued Modal Logic
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916788776730624 |
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| author | Britz, Cecelia Conradie, Willem Morton, Wilmari |
| author_facet | Britz, Cecelia Conradie, Willem Morton, Wilmari |
| contents | The aim of the present paper is to generalise Sahlqvist correspondence theory to the many-valued modal semantics defined by Fitting, assuming a perfect Heyting algebra as truth value space. We present the standard translations between many-valued modal languages and suitably defined first-order and second-order correspondence languages and prove their correctness. We introduce a notion of many-valued modal frame correspondence with a truth value parameter. Exploring the consequences of this definition, we define many-valued analogues of the syntactically specified classes of Sahlqvist and inductive formulas. We adapt the ALBA algorithm to effectively compute many-valued parameterized local frame correspondents for all many-valued Sahlqvist and inductive formulas. Lastly we prove that the many-valued frame correspondent (parameterized with any non-zero truth value) of every classical Sahlqvist formula is syntactically identical to its standard crisp frame correspondent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07894 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Correspondence Theory for Many-valued Modal Logic Britz, Cecelia Conradie, Willem Morton, Wilmari Logic 03B45, 06D50, 06D10, 03G10, 06E15 The aim of the present paper is to generalise Sahlqvist correspondence theory to the many-valued modal semantics defined by Fitting, assuming a perfect Heyting algebra as truth value space. We present the standard translations between many-valued modal languages and suitably defined first-order and second-order correspondence languages and prove their correctness. We introduce a notion of many-valued modal frame correspondence with a truth value parameter. Exploring the consequences of this definition, we define many-valued analogues of the syntactically specified classes of Sahlqvist and inductive formulas. We adapt the ALBA algorithm to effectively compute many-valued parameterized local frame correspondents for all many-valued Sahlqvist and inductive formulas. Lastly we prove that the many-valued frame correspondent (parameterized with any non-zero truth value) of every classical Sahlqvist formula is syntactically identical to its standard crisp frame correspondent. |
| title | Correspondence Theory for Many-valued Modal Logic |
| topic | Logic 03B45, 06D50, 06D10, 03G10, 06E15 |
| url | https://arxiv.org/abs/2401.07894 |