Toward a Stronger Constraint for Non-Trivial Inconsistent Theories

This article discusses the definition of paraconsistency understood as the property of a consequence relation that does not trivialize inconsistent theories. Some logicians have argued that standard paraconsistency, the requirement of a non-explosive consequence relation, is insufficient for that purpose. In this article, we have a twofold goal. First, we offer an exposition of some attempts to strengthen standard paraconsistency in the literature. After discussing the shortcomings of those attempts, we examine the concepts of triviality in relation to which those concepts of paraconsistency are defined. Then, as our second goal, we propose an alternative definition of paraconsistency that aims to avoid the trivialization of inconsistent theories in a stricter sense