Reasoning with belief functions over Belnap--Dunn logic

TL;DR

This paper extends Belnap--Dunn logic with belief and plausibility functions to enable reasoning with inconsistent and incomplete probabilistic information, introducing two formal frameworks and proving their completeness and equivalence.

Contribution

It introduces a novel expansion of Belnap--Dunn logic incorporating belief functions, formalizes reasoning with non-standard probabilities, and establishes completeness and translation between two logical approaches.

Findings

Formalization of belief functions over Belnap--Dunn logic

Development of two calculi for reasoning with belief functions

Proof of completeness and equivalence of the calculi

Abstract

We design an expansion of Belnap--Dunn logic with belief and plausibility functions that allow non-trivial reasoning with inconsistent 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~\cite{FaginHalpernMegiddo1990}. Second, as a two-layered modal logic wherein reasoning with evidence (the outer layer) utilises paraconsistent expansions of \L{}ukasiewicz logic. The second approach is inspired by~\cite{BaldiCintulaNoguera2020}. We prove completeness for both kinds of calculi and show their equivalence by establishing faithful translations in both directions.