In this paper, we explore the problem of explaining observations starting from a classically inconsistent theory by adopting a paraconsistent framework. More precisely, we consider theories formulated in the well-known Belnap–Dunn paraconsistent four-valued logic BD and its implicative expansion BD⊃. Abductive solutions are then given in one of the two further expansions of BD: BD∘, which introduces formulas of the form ∘φ (‘the information on φ is reliable’), and BD△, which augments the language with formulas of the form △φ (‘there is information that φ is true’). We show that explanations in BD∘ and BD△ are not reducible to one another. We analyse the complexity of standard abductive reasoning tasks (solution recognition, solution existence, and relevance/necessity of hypotheses) depending on the language of the solution (BD∘ or BD△) and on the language of the theory (BD or BD⊃). In addition, we consider the complexity of abductive reasoning in the Horn fragment of BD⊃. By showing how to reduce abduction in BD and its expansions to abduction in classical propositional logic, we enable the reuse of existing abductive reasoning procedures.
Bienvenu et al. (Sun,) studied this question.