Two-Layered Logics for Paraconsistent Probabilities

Abstrakt

We discuss two-layered logics formalising reasoning with paraconsistent probabilities that combine the Łukasiewicz [0,1]-valued logic with Baaz Delta operator and the Belnap–Dunn logic. The first logic formalises a ‘two-valued’ approach where each event ϕ has independent positive and negative measures that stand for, respectively, the likelihoods of ϕ and ¬ϕ. The second logic that we introduce here corresponds to ‘four-valued’ probabilities. There, ϕ is equipped with four measures standing for pure belief, pure disbelief, conflict and uncertainty of an agent in ϕ. We construct faithful embeddings of and into one another and axiomatise 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.