Abductive Reasoning in a Paraconsistent Framework

We explore the problem of explaining observations starting from a classically inconsistent theory by adopting a paraconsistent framework. We consider two expansions of the well-known Belnap--Dunn paraconsistent four-valued logic mathsfBD: mathsfBD_circ introduces formulas of the form circφ (the information on φ is reliable), while mathsfBD_triangle augments the language with triangleφ's (there is information that φ is true). We define and motivate the notions of abduction problems and explanations in mathsfBD_circ and mathsfBD_triangle and show that they are not reducible to one another. We analyse the complexity of standard abductive reasoning tasks (solution recognition, solution existence, and relevance / necessity of hypotheses) in both logics. Finally, we show how to reduce abduction in mathsfBD_circ and mathsfBD_triangle to abduction in classical propositional logic, thereby enabling the reuse of existing abductive reasoning procedures.