Minimal-Inconsistency Tolerant Logics: A Quantitative Approach

In order to reason in a non-trivializing way with contradictions, paraconsistent logics reject some classically valid inferences. As a way to recover some of these inferences, Graham Priest proposed to nonmonotonically strengthen the Logic of Paradox by allowing the selection of “less inconsistent” models via a comparison of their respective inconsistent parts. This move recaptures a good portion of classical logic in that it does not block, e.g., disjunctive syllogism, unless it is applied to contradictory assumptions. In Priest’s approach the inconsistent parts of models are compared in an extensional way by consid- ering their inconsistent objects. This distinguishes his system from the standard format of (inconsistency-)adaptive logics pioneered by Diderik Batens, according to which (atomic) contradictions validated in models form the basis of their comparison. A well-known prob- lem for Priest’s extensional approach is its lack of the Strong Reassurance property, i.e., for specific settings there may be infinitely descending chains of less and less inconsistent models, thus never reaching a minimally inconsistent model. In the following paper, we show that Strong Reassurance holds for the extensional ap- proach under a cardinality-based comparison of the inconsistent parts of models. We develop and study the meta-theory of a class of nonmonotonic inconsistency-tolerant logics based on the extensional and the quantitative comparisons of their respective models, including important model-theoretic properties, such as the Löwenheim-Skolem theorems, as well as principles of nonmonotonic inference.