A capacity-based framework encompassing Belnap–Dunn logic for reasoning about multisource information

Abstract

Open Archive Toulouse Archive OuverteOATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible A capacity-based framework encompassing Belnap-Dunn logic for reasoning about multisource information ✩,✩✩ a b s t r a c t Keywords: Belnap-Dunn logic Inconsistency handling Kleene and Priest logics Capacities Possibility theory Modal logicBelnap-Dunn four-valued logic is one of the best known logics for handling elementary information items coming from several sources. More recently, a conceptually simple framework, namely a two-tiered propositional logic augmented with classical modal axioms (here called BC logic), was suggested by the second author and colleagues, for the handling of multisource information. It is a fragment of the non-normal modal logic EMN, whose semantics is expressed in terms of two-valued monotonic set functions called Boolean capacities. We show BC logic is more expressive than Belnap-Dunn logic by proposing a consequence-preserving translation of Belnap-Dunn logic in this setting. As special cases, we can recover already studied translations of three-valued logics such as Kleene and Priest logics. Moreover, BC logic is compared with the source-processor logic of Avron, Ben Naim and Konikowska. Our translation bridges the gap between Belnap-Dunn logic, epistemic logic, and theories of uncertainty like possibility theory or belief functions, and paves the way to a unified approach to various methods for handling inconsistency due to several conflicting sources of information. epistemic truth-values, two of which refer to ignorance and conflict. Truth tables for conjunction, disjunction, and negation are used to compute the epistemic status of other more complex formulas. This logic underlies both Kleene three-valued logic [40] (recovered when no conflict between sources is observed) and the three-valued Priest Logic of Paradox [46,47] (when sources are never ignorant and always assign truth-values to each elementary proposition). In the former logic, the truth-value referring to the idea of "contradiction" is eliminated, while in the latter, the truth-value referring to the idea of "ignorance" is eliminated. Belnap-Dunn logic uses all four values forming a bilattice structure, in which one partial ordering expresses relative strength of information (from no information to too much information), and the other partial ordering represents relative truth (from false to true).In recent works [20,21], we have focused attention on three-valued logics of uncertainty or inconsistency (paraconsistent logics). We were able to express them in a two-tiered propositional logic called MEL [8], which adopts the syntax of a fragment of epistemic logic, and borrows axioms of the KD logic. MEL can be viewed as the simplest logical framework for reasoning about incomplete information and at the semantic level it expresses the all-or-nothing version of possibility theory [28]. This kind of two-tiered logic dates backs to works by Hájek...

Qualitative capacities: Basic notions and potential applications

“…Theorem 1. γ is separable if and only if γ = ⊗E∈Fγ N E . 12 This result strikingly differs from the case of numerical separable belief functions, whose focal sets do not satisfy Eq. ( 17), but are closed under conjunction [67].…”

“…In [12], it is shown that Belnap logic can be encoded in a simplified modal logic where modalities are interpreted by means of Boolean capacities.…”