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-circ introduces formulas of the form circ phi (‘the information about phi is reliable’), while BD-triangle augments the language with formulas triangle phi (‘there is information that phi is true’). We define and motivate the notions of abduction problems and explanations in BD-circ and BD-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 BD-circ and BD-triangle to abduction in classical propositional logic, thereby enabling the reuse of existing abductive reasoning procedures.