Paraconsistency and Topological Semantics

The well-studied notion of deductive explosion describes the situation where any formula can be deduced from an inconsistent set of formulas. In other words, in deductively explosive logics, we have {φ,¬φ} ⊢ ψ for all formulas φ, ψ where ⊢ is a logical consequence relation. In this respect, both “classical” and intuitionistic logics are known to be deductively explosive. Paraconsistent logic, on the other hand, is the umbrella term for logical systems where the logical consequence relation is not explosive. Variety of philosophical and logical objections can be raised against paraconsistency, and almost all of these objections can be defended in a rigorous fashion. Here, we will not be concerned about the philosophical implications of it, yet we refer the reader to the following references for comprehensive defenses of paraconsistency with a variety of well-structured applications chosen from mathematics and philosophy with a rigorous history of the subject (?; Priest, 2002; Priest, 1998; ?).