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 $\mathsf{B}\mathsf{D}$: ${\mathsf{B}\mathsf{D}}_{\circ }$ introduces formulas of the form $\circ \phi$ (the information on $\phi$ is reliable), while ${\mathsf{B}\mathsf{D}}_{\mathrm{△}}$ augments the language with $\mathrm{△}\phi$’s (there is information that $\phi$ is true). We define and motivate the notions of abduction problems and explanations in ${\mathsf{B}\mathsf{D}}_{\circ }$ and ${\mathsf{B}\mathsf{D}}_{\mathrm{△}}$ 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 ${\mathsf{B}\mathsf{D}}_{\circ }$ and ${\mathsf{B}\mathsf{D}}_{\mathrm{△}}$ to abduction in classical propositional logic, thereby enabling the reuse of existing abductive reasoning procedures.