Categorical Proof-Theoretic Semantics

TL;DR

This paper explores proof-theoretic semantics for propositional logic, establishing completeness without models by connecting semantics to categorical proof theory and sheaf categories.

Contribution

It reconstructs Sandqvist's semantics categorically, demonstrating naturality and extending the understanding of proof-theoretic validity in intuitionistic logic.

Findings

Categorical reconstruction of proof-theoretic semantics

Demonstration of completeness without formal models

Connection of semantics to sheaf categories

Abstract

In proof-theoretic semantics, model-theoretic validity is replaced by proof-theoretic validity. Validity of formulae is defined inductively from a base giving the validity of atoms using inductive clauses derived from proof-theoretic rules. A key aim is to show completeness of the proof rules without any requirement for formal models. Establishing this for propositional intuitionistic logic (IPL) raises some technical and conceptual issues. We relate Sandqvist's (complete) base-extension semantics of intuitionistic propositional logic to categorical proof theory in presheaves, reconstructing categorically the soundness and completeness arguments, thereby demonstrating the naturality of Sandqvist's constructions. This naturality includes Sandqvist's treatment of disjunction that is based on its second-order or elimination-rule presentation. These constructions embody not just validity, but certain forms of objects of justifications. This analysis is taken a step further by showing that from the perspective of validity, Sandqvist's semantics can also be viewed as the natural disjunction in a category of sheaves.