The category of propositional deductive systems

Abstract

We define the category $\QM$ of quantales and their modules and prove the existence of coproducts, and the existence of pushout and amalgamated coproducts under certain conditions. Then we define the non-full subcategory $\DS_0$ 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 $\DS_0$ is precisely the deductive system called ``logical coproduct'' in \cite{ruslu}. Last, we discuss amalgamation in $\DS_0$.

To appear in The Review of Symbolic Logic