Coherent Conditional Probability in a Fuzzy Logic Setting

Very recently, a (fuzzy modal) logic to reason about coherent conditional probability, in the sense of de Finetti, has been introduced by the authors. Under this approach, a conditional probability μ (· ∣ ·) is taken as a primitive notion defined over conditional events of the form “φ given ψ”, φ | ψ for short, where ψ is not the impossible event. The logic, called FCP(ŁΠ), exploits an idea already used by Hájek and colleagues to define a logic for (unconditional) probability in the framework of fuzzy logics. Namely, we take the probability of the conditional event “φ | ψ” as the truth-value of the (fuzzy) modal proposition P(φ ∣ ψ), read as “φ | ψ is probable”. The logic FCP(ŁΠ), which is built up over the many-valued logic ŁΠ $\frac{1}{2}$ (a logic which combines the well-known Łukasiewicz and Product fuzzy logics), was shown to be complete for modal theories with respect to the class of probabilistic Kripke structures induced by coherent conditional probabilities. Indeed, checking coherence of a (generalized) probability assessment to an arbitrary family of conditional events becomes tantamount to checking consistency of a suitably defined theory over the logic FCP(ŁΠ). In this paper we provide further results for the logic FCP(ŁΠ). In particular, we extend the previous completeness result by allowing the presence of non-modal formulas in the theories, which are used to describe logical relationships among events. This increases the knowledge modelling power of FCP(ŁΠ). Then, we improve the results concerning checking consistency of suitably defined theories in FCP(ŁΠ) to determine coherence by showing parallel results w.r.t. the notion of generalized coherence when dealing with imprecise assessments. Moreover we also show and discuss compactness results for our logic. Finally, FCP(ŁΠ) is shown to be a powerful tool for knowledge representation. Indeed, following ideas already investigated in the related literature, we show how FCP(ŁΠ) allows the definition of suitable notions of default rules which enjoy the core properties of nonmonotonic reasoning characterizing system P and R.