The category of propositional deductive systems

TL;DR

This paper explores the categorical structure of propositional deductive systems, establishing their equivalence to certain propositional logics and analyzing coproducts and amalgamation properties within this framework.

Contribution

It introduces a categorical perspective on propositional deductive systems, proves the existence of coproducts and pushouts, and links these to logical coproducts and interpretations.

Findings

Coproducts and pushouts exist in the category of quantale modules.

The subcategory of propositional deductive systems is equivalent to a category of propositional logics.

The logical coproduct corresponds to the coproduct in the subcategory.

Abstract

We define the category $\mathcal{QM}$ of quantales and their modules and prove the existence of coproducts, and the one of pushout and amalgamated coproducts under certain conditions. Then we define the non-full subcategory $\mathcal{DS}_0$, of propositional deductive systems, show that it is equivalent to the one of ``real'' propositional logics whose morphisms are interpretations (modulo a language translation, when needed), and prove that the coproduct in $\mathcal{DS}_0$ is precisely the deductive system called ``logical coproduct'' in [Russo,2022]. Last, we discuss amalgamation in $\mathcal{DS}_0$.