Two-Layered Logics for Paraconsistent Probabilities

We discuss two-layered logics formalising reasoning with paraconsistent probabilities that combine the Lukasiewicz [0, 1]-valued logic with Baaz Delta operator and the Belnap-Dunn logic. The first logic Pr-Delta(L2) (introduced in [7]) formalises a 'two-valued' approach where each event phi has independent positive and negative measures that stand for, respectively, the likelihoods of phi and (sic)phi. The second logic 4Pr(L Delta) that we introduce here corresponds to 'four-valued' probabilities. There, phi is equipped with four measures standing for pure belief, pure disbelief, conflict and uncertainty of an agent in phi. We construct faithful embeddings of 4Pr(L Delta) and Pr-Delta(L2) into one another and axiomatise 4Pr(L Delta) using a Hilbert-style calculus. We also establish the decidability of both logics and provide complexity evaluations for them using an expansion of the constraint tableaux calculus for L.