Measuring evidence: a probabilistic approach to an extension of Belnap–Dunn logic

Abstract

This paper introduces the logic of evidence and truth \({ LET}_{F}\) as an extension of the Belnap–Dunn four-valued logic \({ FDE}\). \({ LET}_{F}\) is a slightly modified version of the logic \({ LET}_{J}\), presented in Carnielli and Rodrigues (Synthese 196:3789–3813, 2017). While \({ LET}_{J}\) is equipped only with a classicality operator \({\circ }\), \({ LET}_{F}\) is equipped with a non-classicality operator \({\bullet }\) as well, dual to \({\circ }\). Both \({ LET}_{F}\) and \({ LET}_{J}\) are logics of formal inconsistency and undeterminedness in which the operator \( \circ \) recovers classical logic for propositions in its scope. Evidence is a notion weaker than truth in the sense that there may be evidence for a proposition \( \alpha \) even if \( \alpha \) is not true. As well as \({ LET}_{J}\), \({ LET}_{F}\) is able to express preservation of evidence and preservation of truth. The primary aim of this paper is to propose a probabilistic semantics for \({ LET}_{F}\) where statements \(P(\alpha )\) and \(P({\circ }\alpha )\) express, respectively, the amount of evidence available for \(\alpha \) and the degree to which the evidence for \( \alpha \) is expected to behave classically—or non-classically for \( P({\bullet }\alpha ) \). A probabilistic scenario is paracomplete when \(P(\alpha ) + P(\lnot \alpha ) < 1\), and paraconsistent when \(P(\alpha ) + P(\lnot \alpha ) > 1\), and in both cases, \(P(\circ \alpha ) < 1\). If \(P(\circ \alpha ) = 1\), or \(P({\bullet }\alpha ) = 0\), classical probability is recovered for \( \alpha \). The proposition \( {\circ }\alpha \vee {\bullet }\alpha \), a theorem of \({ LET}_{F}\), partitions what we call the information space, and thus allows us to obtain some new versions of known results of standard probability theory.