Arithmetical completeness for some extensions of the pure logic of necessitation
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911277040795648 |
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| author | Kogure, Haruka |
| author_facet | Kogure, Haruka |
| contents | We investigate the arithmetical completeness theorems of some extensions of Fitting, Marek, and Truszczyński's pure logic of necessitation $\mathbf{N}$. For $m,n \in ω$, let $\mathbf{NA}_{m,n}$, which was introduced by Kurahashi and Sato, be the logic obtained from $\mathbf{N}$ by adding the axiom scheme $\Box^n A \to \Box^m A$. In this paper, among other things, we prove that for each $m,n \geq 1$, the logic $\mathbf{NA}_{m,n}$ becomes a provability logic, that is, there exists a provability predicate $\mathrm{Pr}_T(x)$ of $T$ whose $T$-verifiable modal principles are exactly the logic $\mathbf{NA}_{m,n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00938 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Arithmetical completeness for some extensions of the pure logic of necessitation Kogure, Haruka Logic We investigate the arithmetical completeness theorems of some extensions of Fitting, Marek, and Truszczyński's pure logic of necessitation $\mathbf{N}$. For $m,n \in ω$, let $\mathbf{NA}_{m,n}$, which was introduced by Kurahashi and Sato, be the logic obtained from $\mathbf{N}$ by adding the axiom scheme $\Box^n A \to \Box^m A$. In this paper, among other things, we prove that for each $m,n \geq 1$, the logic $\mathbf{NA}_{m,n}$ becomes a provability logic, that is, there exists a provability predicate $\mathrm{Pr}_T(x)$ of $T$ whose $T$-verifiable modal principles are exactly the logic $\mathbf{NA}_{m,n}$. |
| title | Arithmetical completeness for some extensions of the pure logic of necessitation |
| topic | Logic |
| url | https://arxiv.org/abs/2409.00938 |