NP-containment for the coherence test of assessments of conditional probability: a fuzzy logical approach

In this paper we investigate the problem of testing the coherence of an assessment of conditional probability following a purely logical setting. In particular we will prove that the coherence of an assessment of conditional probability χ can be characterized by means of the logical consistency of a suitable theory Tχ defined on the modal-fuzzy logic FPk(R L∆) built up over the many-valued logic R L∆. Such modal-fuzzy logic was previously introduced in [11] in order to treat conditional probability by means of a list of simple probabilities following the well known (smart) ideas exposed in [18] by Halpern and in [7] by Coletti and Scozzafava. Roughly speaking, such logic is obtained by adding to the language of R L∆ a list of k modalities for “probably” and axioms reflecting the properties of simple probability measures. Moreover we proved that the satisfiability problem for modal formulas of FPk(R L∆) is NP-complete. Finally, as main result of this paper, we used FPk(R L∆) in order to prove that the problem of establishing the coherence of rational assessments of conditional probability is NP-complete.