Abductive Reasoning in a Paraconsistent Framework

TL;DR

This paper investigates abductive reasoning within paraconsistent logics, specifically extensions of Belnap--Dunn logic, and demonstrates how to reduce such reasoning to classical propositional logic for practical application.

Contribution

It introduces two new paraconsistent frameworks for abductive reasoning and analyzes their complexity, providing a method to translate these problems into classical propositional logic.

Findings

Abduction problems in the new frameworks are not reducible to each other.

Complexity analysis of abductive reasoning tasks in both logics.

Reduction of abduction in paraconsistent logics to classical propositional logic.

Abstract

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