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 BD: BD_ introduces formulas of the form (the information on is reliable), while BD_ augments the language with 's (there is information that is true). We define and motivate the notions of abduction problems and explanations in BD_ and BD_ 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 BD_ and BD_ to abduction in classical propositional logic, thereby enabling the reuse of existing abductive reasoning procedures.