Conceptual completeness for subgeometric logics

2026-07-02

AI Summary

This study investigates conceptual completeness for sub-geometric fragments of first-order logic, including coherent logic, regular logic, disjunctive logic, and essentially algebraic logic with falsum. By recasting conceptual completeness as a duality between theories and Grothendieck toposes, and integrating categorical and proof-theoretic methods, the work unifies semantic reconstruction with proof-theoretic perspectives. The main contributions are twofold: first, it establishes that all the aforementioned logical fragments enjoy conceptual completeness and admit conservative embeddings into full geometric logic; second, within this unified framework, it recovers and generalizes Makkai’s semantic reconstruction theorem, thereby revealing a deep correspondence between these logical fragments and their categories of models.

Abstract

We explore the notion of conceptual completeness for a fragment of geometric logic in the framework developed by the first and third author. Unlike its traditional interpretation as a reconstruction of syntax from semantics, in this paper we characterise conceptual completeness of a fixed fragment in terms of a duality between theories and topoi. We then show that conceptually complete fragments are conservatively embedded in full geometric logic, thus casting conceptual completeness in a new proof-theoretic light. We give a new proof of conceptual completeness for coherent logic, and we also show that regular, disjunctive, and essentially algebraic logic with falsum are conceptually complete. Finally, we show that our notion is equivalent to a traditional reconstruction result under the assumption of completeness with respect to set-based models: in the coherent case, we thus recover Makkai's original reconstruction theorem via ultracategories.

Problem

• conceptual completeness
• subgeometric logics
• duality
• reconstruction
• topoi

Innovation

• conceptual completeness
• geometric logic
• duality
• topos theory
• ultracategories