The category of propositional deductive systems

2025/08/26 by Ciro Russo, Russo, Ciro
Mathematics · #math.LO #math.CT

paper · pdf · doi:10.48550/arxiv.2508.19426

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 \DS₀ 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<sub>0</sub> is precisely the deductive system called ``logical coproduct'' in \citeruslu. Last, we discuss amalgamation in \DS₀.

Related

• Coproduct and amalgamation of deductive systems by means of ordered algebras
• Towards a good notion of categories of logics
• Representing Conjunctive Deductions by Disjunctive Deductions
• Propositional theories are strongly equivalent to logic programs
• General Logic-Systems that Determine Significant Collections of Consequence Operators