On distance logics of Euclidean spaces
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908521538256896 |
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| author | Agnew, Gabriel Gutierrez-Hougardy, Uzias Harding, John Shapirovsky, Ilya West, Jackson |
| author_facet | Agnew, Gabriel Gutierrez-Hougardy, Uzias Harding, John Shapirovsky, Ilya West, Jackson |
| contents | We consider logics derived from Euclidean spaces $\mathbb{R}^n$. Each Euclidean space carries relations consisting of those pairs that are, respectively, distance more than 1 apart, distance less than 1 apart, and distance 1 apart. Each relation gives a uni-modal logic of $\mathbb{R}^n$ called the farness, nearness, and constant distance logics, respectively. These modalities are expressive enough to capture various aspects of the geometry of $\mathbb{R}^n$ related to bodies of constant width and packing problems. This allows us to show that the farness logics of the spaces $\mathbb{R}^n$ are all distinct, as are the nearness logics, and the constant distance logics. The farness and nearness logics of $\mathbb{R}$ are shown to strictly contain those of $\mathbb{Q}$, while their constant distance logics agree. It is shown that the farness logic of the reals is not finitely axiomatizable and does not have the finite model property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04884 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On distance logics of Euclidean spaces Agnew, Gabriel Gutierrez-Hougardy, Uzias Harding, John Shapirovsky, Ilya West, Jackson Logic 03B45, 06E25 F.4.1 We consider logics derived from Euclidean spaces $\mathbb{R}^n$. Each Euclidean space carries relations consisting of those pairs that are, respectively, distance more than 1 apart, distance less than 1 apart, and distance 1 apart. Each relation gives a uni-modal logic of $\mathbb{R}^n$ called the farness, nearness, and constant distance logics, respectively. These modalities are expressive enough to capture various aspects of the geometry of $\mathbb{R}^n$ related to bodies of constant width and packing problems. This allows us to show that the farness logics of the spaces $\mathbb{R}^n$ are all distinct, as are the nearness logics, and the constant distance logics. The farness and nearness logics of $\mathbb{R}$ are shown to strictly contain those of $\mathbb{Q}$, while their constant distance logics agree. It is shown that the farness logic of the reals is not finitely axiomatizable and does not have the finite model property. |
| title | On distance logics of Euclidean spaces |
| topic | Logic 03B45, 06E25 F.4.1 |
| url | https://arxiv.org/abs/2501.04884 |