Probabilities with Gaps and Gluts

Abstract

Belnap-Dunn logic (BD), sometimes also known as First Degree Entailment, is a four-valued propositional logic that complements the classical truth values of True and False with two non-classical truth values Neither and Both. The latter two are to account for the possibility of the available information being incomplete or providing contradictory evidence. In this paper, we present a probabilistic extension of BD that permits agents to have probabilistic beliefs about the truth and falsity of a proposition. We provide a sound and complete axiomatization for the framework defined and also identify policies for conditionalization and aggregation. Concretely, we introduce four-valued equivalents of Bayes’ and Jeffrey updating and also suggest mechanisms for aggregating information from different sources.

How to reason with inconsistent probabilistic information?

“…This section contains the necessary technicalities on algebraic structures (namely certain product bilattices, residuated bilattices and MV algebras) and logics (namely BD logic and two logics derived from Lukasiewicz logic) used in the framework, and finally non-standard probabilities. For more details, the reader can refer to [18,15] for bilattices, [5,11] and [17] for BD logic and de Morgan algebras, [10] for Lukasiewicz logic and MV algebras, and [16] for non-standard probabilities.…”

“…In such frames, µ B interprets B as a non-standard probability (see Appendix 6.6). From [16,Theorem 4], we know that it is the induced non-standard probability function of exactly one mass function on the BD states, which in fact yields completeness w.r.t. the intended frames described above.…”

“…This assumption is naturally applicable in the probabilistic framework. We will distinguish positive probability p + of a claim ϕ representing (probabilistic) information supporting ϕ and negative probability of ϕ representing (probabilistic) information rejecting ϕ. Non-standard probabilities of this kind have been discussed in the literature, we introduce them in Section 2.3 following the presentation given in [16]. Two-layer modal logics.…”

Reasoning with belief functions over Belnap--Dunn logic

“…This is why one needs a probability theory that accounts for contradictory and incomplete information. Such generalisation of the classical probability theory was undertaken in [21], where non-classical probabilistic extensions of BD were presented. Furthermore, two versions of non-classical probability functions were given a complete axiomatisation.…”

Constraint tableaux for two-dimensional fuzzy logics

“…Non-standard probabilities [11,21] generalise the notion of independent positive and negative support of a statement in presence of uncertainty. They encode the positive and negative probabilistic information about a statement ϕ with a couple p(ϕ) = (p + (ϕ), p − (ϕ)).…”