Existential completions and Herbrand's theorem

Joshua L. Wrigley

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arXiv · 2508.15518

Existential completions and Herbrand's theorem

Joshua L. Wrigley

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Abstract

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.

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BibTeX RIS

Joshua L. Wrigley. 2025-08-21. Existential completions and Herbrand's theorem. https://arxiv.org/abs/2508.15518

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