Reasoning with belief functions over Belnap–Dunn logic

Abstract

We design an expansion of Belnap–Dunn logic with belief and plausibility functions that allows non-trivial reasoning with contradictory and incomplete probabilistic information. We also formalise reasoning with non-standard probabilities and belief functions in two ways. First, using a calculus of linear inequalities, akin to the one presented in [23]. Second, as a two-layered modal logic wherein reasoning with evidence (the outer layer) utilises paraconsistent expansions of Łukasiewicz logic. The second approach is inspired by [3]. We prove completeness for both kinds of calculi and show their equivalence by establishing faithful translations in both directions.