Abductive Reasoning in Expansions of Belnap-Dunn Logic

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.