On a paraconsistentization functor in the category of consequence structures

TL;DR

This paper introduces a categorical functor to transform classical logic into a paraconsistent logic, enabling reasoning in the presence of contradictions without trivialization.

Contribution

It proposes a novel functor within category theory to systematically convert classical consequence structures into paraconsistent ones.

Findings

Defined a category of consequence structures

Constructed a functor for paraconsistentization

Applied the method to classical propositional logic

Abstract

This paper is an attempt to solve the following problem: given a logic, how to turn it into a paraconsistent one? In other words, given a logic in which \emph{ex falso quodlibet} holds, how to convert it into a logic not satisfying this principle? We use a framework provided by category theory in order to define a category of consequence structures. Then, we propose a functor to transform a logic not able to deal with contradictions into a paraconsistent one. Moreover, we study the case of paraconsistentization of propositional classical logic.