Rodrigues, Abilio and Bueno-Soler, Juliana and Carnielli, Walter
(2020)
Measuring evidence: a probabilistic approach to an extension of Belnap-Dunn Logic.
[Preprint]
Abstract
This paper introduces the logic of evidence and truth LETF as an extension of the Belnap-Dunn four-valued logic F DE. LETF is a slightly modified version of the logic LETJ, presented in Carnielli and Rodrigues (2017). While LETJ is equipped only with a classicality operator ○, LETF is equipped with a non-classicality operator ● as well, dual to ○. Both LETF and LETJ 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 LETJ,
LETF is able to express preservation of evidence and preservation of truth. The primary aim of this paper is to propose a probabilistic semantics for LETF 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 LETF , partitions what we call the information space, and thus allows us to obtain some new versions of known results of standard probability theory
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