Paraconsistency Everywhere

Abstract

Paraconsistent logics are, by definition, inconsistency tolerant: In a paraconsistent logic, inconsistencies need not entail everything. However, there is more than one way a body of information can be inconsistent. In this paper I distinguish {contradictions} from {other inconsistencies}, and I show that several different logics are, in an important sense, "paraconsistent" in virtue of being inconsistency tolerant without thereby being contradiction tolerant. For example, even though no inconsistencies are tolerated by intuitionistic propositional logic, some inconsistencies are tolerated by intuitionistic predicate logic. In this way, intuitionistic predicate logic is, in a mild sense, paraconsistent. So too are orthologic and quantum propositional logic and other formal systems. Given this fact, a widespread view—that traditional paraconsistent logics are especially repugnant because they countenance inconsistencies—is undercut. Many well-understood nonclassical logics countenance inconsistencies as well.