Identifying roles of formulas in inconsistency under Priest's minimally inconsistent logic of paradox

Abstract

It has been increasingly recognized that identifying roles of formulas of a knowledge base in the inconsistency of that base can help us better look inside the inconsistency. However, there are few approaches to identifying such roles of formulas from a perspective of models in some paraconsistent logic, one of typical tools used to characterize inconsistency in semantics. In this paper, we characterize the role of each formula in the inconsistency arising in a knowledge base from informational as well as causal aspects in the framework of Priest's minimally inconsistent logic of paradox. At first, we identify the causal responsibility of a formula for the inconsistency based on the counterfactual dependence of the inconsistency on the formula under some contingency in semantics. Then we incorporate the change on semantic information in the framework of causal responsibility to develop the informational responsibility of a formula for the inconsistency to capture the contribution made by the formula for the inconsistent information. This incorporation makes the informational responsibility interpretable from the point of view of causality, and capable of catching the role of a formula in inconsistent information concisely. In addition, we propose notions of naive and quasi naive responsibilities as two auxiliaries to describe special relations between inconsistency and formulas in semantic sense. Some intuitive and interesting properties of the two kinds of responsibilities are also discussed.