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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.07209 |
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| _version_ | 1866910817590444032 |
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| author | Bezhanishvili, Guram Bezhanishvili, Nick Moraschini, Tommaso |
| author_facet | Bezhanishvili, Guram Bezhanishvili, Nick Moraschini, Tommaso |
| contents | A classic result in modal logic, known as the Blok Dichotomy Theorem, states that the degree of incompleteness of a normal extension of the basic modal logic $\sf K$ is $1$ or $2^{\aleph_0}$. It is a long-standing open problem whether Blok Dichotomy holds for normal extensions of other prominent modal logics (such as $\sf S4$ or $\sf K4$) or for extensions of the intuitionistic propositional calculus $\mathsf{IPC}$. In this paper, we introduce the notion of the degree of finite model property (fmp), which is a natural variation of the degree of incompleteness. It is a consequence of Blok Dichotomy Theorem that the degree of fmp of a normal extension of $\sf K$ remains $1$ or $2^{\aleph_0}$. In contrast, our main result establishes the following Antidichotomy Theorem for the degree of fmp for extensions of $\mathsf{IPC}$: each nonzero cardinal $κ$ such that $κ\leq \aleph_0$ or $κ= 2^{\aleph_0}$ is realized as the degree of fmp of some extension of $\mathsf{IPC}$. We then use the Blok-Esakia theorem to establish the same Antidichotomy Theorem for normal extensions of $\sf S4$ and $\sf K4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_07209 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Degrees of the finite model property: the antidichotomy theorem Bezhanishvili, Guram Bezhanishvili, Nick Moraschini, Tommaso Logic A classic result in modal logic, known as the Blok Dichotomy Theorem, states that the degree of incompleteness of a normal extension of the basic modal logic $\sf K$ is $1$ or $2^{\aleph_0}$. It is a long-standing open problem whether Blok Dichotomy holds for normal extensions of other prominent modal logics (such as $\sf S4$ or $\sf K4$) or for extensions of the intuitionistic propositional calculus $\mathsf{IPC}$. In this paper, we introduce the notion of the degree of finite model property (fmp), which is a natural variation of the degree of incompleteness. It is a consequence of Blok Dichotomy Theorem that the degree of fmp of a normal extension of $\sf K$ remains $1$ or $2^{\aleph_0}$. In contrast, our main result establishes the following Antidichotomy Theorem for the degree of fmp for extensions of $\mathsf{IPC}$: each nonzero cardinal $κ$ such that $κ\leq \aleph_0$ or $κ= 2^{\aleph_0}$ is realized as the degree of fmp of some extension of $\mathsf{IPC}$. We then use the Blok-Esakia theorem to establish the same Antidichotomy Theorem for normal extensions of $\sf S4$ and $\sf K4$. |
| title | Degrees of the finite model property: the antidichotomy theorem |
| topic | Logic |
| url | https://arxiv.org/abs/2307.07209 |