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 F DE. LET F is a slightly modified version of the logic LET J , presented in Carnielli and Rodrigues (2017). While LET J is equipped only with a classicality operator ○, LET F is equipped with a non-classicality operator • as well, dual to ○. Both LET F and LET J are logics of formal inconsistency and undeterminedness in which the operator ○ 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 α even if α 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 (α) and P (○α) express, respectively, the amount of evidence available for α and the degree to which the evidence for α is expected to behave classically-or non-classically for P (•α). A probabilistic scenario is paracomplete when P (α) + P (¬α) < 1, and paraconsistent when P (α) + P (¬α) > 1, and in both cases, P (○α) < 1. If P (○α) = 1, or P (•α) = 0, classical probability is recovered for α. The proposition ○α ∨ •α, 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.

The evidence approach to paraconsistency versus the paraconsistent approach to evidence

“…What we find, however, are explicit claims that paraconsistent logics must be interpreted in terms of evidence. Besides the mentioned sources of the epistemic approach, we also find it explicitly in Rodrigues et al (2020) p. 11 the claim that evidence is "well-suited to a non-dialetheist reading of paraconsistency" (we take it that "reading" and "interpreting" do the same work here). Along with Carnielli and Rodrigues, Barrio and da Re (2018) also advance the claim that systems of logic must be interpreted in a philosophical way, although they also do not explicitly define what that would mean.…”

Probabilities with Gaps and Gluts

“…A similar project has been undertaken in parallel in [30], complementing our semantic, model based approach with a syntactic perspective focusing on deductive systems. In [30] the authors mainly work on a slight extension of Belnap-Dunn logic with two modalities for expressing which formulas adhere to the laws of classic logic and which to first degree entailment. These modalities are intended to track the amount of available information for a proposition.…”

“…Just as in the standard case, non-normal Bayes conditioning can be defined as extremal case of Jeffrey updates. In fact, non-normal Bayes conditioning has been studied independently, for instance in [23] and recently in [30]. The current framework generalizes these approach by also incorporating Jeffrey updating and by Fig.…”

“…The current framework generalizes these approach by also incorporating Jeffrey updating and by Fig. 2 Classic Conditioning identifying a number of different Bayes like updates, containing the ones put forward by [23] and [30].…”

“…They further substantiate the approach by providing a subjectivist interpretation of non-standard probabilities and the corresponding Dutch Book Argument. In a very recent paper, Rodrigues, Bueno-Soler and Carnieli [30] have independently presented a similar approach for single valued probability assignments to general substructural logics that is equivalent to that of [6] on Belnap-Dunn Logic.…”