Blok-Esakia Theorems via Stable Canonical Rules
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866915463826505728 |
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| author | Bezhanishvili, Nick Cleani, Antonio Maria |
| author_facet | Bezhanishvili, Nick Cleani, Antonio Maria |
| contents | We present a new uniform method for studying modal companions of superintuitionistic rule systems and related notions, based on the machinery of stable canonical rules. Using this method, we obtain alternative proofs of the Blok-Esakia theorem and of the Dummett-Lemmon conjecture for rule systems. Since stable canonical rules may be developed for any rule system admitting filtration, our method generalizes smoothly to richer signatures. Using essentially the same argument, we obtain a proof of an analogue of the Blok-Esakia theorem for bi-superintuitionistic and tense rule systems, and of the Kuznetsov-Muravitsky isomorphism between rule systems extending the modal intuitionistic logic $\logic{KM}$ and modal rule systems extending the provability logic $\logic{GL}$. In addition, our proof of the Dummett-Lemmon conjecture also generalizes to the bi-superintuitionistic and tense cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_08863 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Blok-Esakia Theorems via Stable Canonical Rules Bezhanishvili, Nick Cleani, Antonio Maria Logic 03B45, 03B55, 03B60, 03B44, 03F45 We present a new uniform method for studying modal companions of superintuitionistic rule systems and related notions, based on the machinery of stable canonical rules. Using this method, we obtain alternative proofs of the Blok-Esakia theorem and of the Dummett-Lemmon conjecture for rule systems. Since stable canonical rules may be developed for any rule system admitting filtration, our method generalizes smoothly to richer signatures. Using essentially the same argument, we obtain a proof of an analogue of the Blok-Esakia theorem for bi-superintuitionistic and tense rule systems, and of the Kuznetsov-Muravitsky isomorphism between rule systems extending the modal intuitionistic logic $\logic{KM}$ and modal rule systems extending the provability logic $\logic{GL}$. In addition, our proof of the Dummett-Lemmon conjecture also generalizes to the bi-superintuitionistic and tense cases. |
| title | Blok-Esakia Theorems via Stable Canonical Rules |
| topic | Logic 03B45, 03B55, 03B60, 03B44, 03F45 |
| url | https://arxiv.org/abs/2206.08863 |