Existential completions and Herbrand's theorem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911114545070080 |
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| author | Wrigley, Joshua L. |
| author_facet | Wrigley, Joshua L. |
| contents | Recently, Abbadini and Guffanti gave an algebraic proof of Herbrand's theorem using a completion for Lawvere doctrines that freely adds existential and universal quantifiers. A more direct argument can be given by only completing with respect to existential quantifiers. We construct the free existential completion on a presheaf of distributive lattices, and deduce Herbrand's theorem for coherent logic from the explicit description. We also discuss the cases involving presheaves of meet-semilattices, due to Trotta, and presheaves of frames. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15518 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existential completions and Herbrand's theorem Wrigley, Joshua L. Logic Category Theory 18C10 (Primary) 03G30, 08B20 (Secondary) Recently, Abbadini and Guffanti gave an algebraic proof of Herbrand's theorem using a completion for Lawvere doctrines that freely adds existential and universal quantifiers. A more direct argument can be given by only completing with respect to existential quantifiers. We construct the free existential completion on a presheaf of distributive lattices, and deduce Herbrand's theorem for coherent logic from the explicit description. We also discuss the cases involving presheaves of meet-semilattices, due to Trotta, and presheaves of frames. |
| title | Existential completions and Herbrand's theorem |
| topic | Logic Category Theory 18C10 (Primary) 03G30, 08B20 (Secondary) |
| url | https://arxiv.org/abs/2508.15518 |