A Logic for Reasoning About Coherent Conditional Probability: A Modal Fuzzy Logic Approach

Abstract

In this paper we define a logic to reason about coherent conditional probability, in the sense of de Finetti. Under this view, a conditional probability μ(· | ·) is a primitive notion that applies over conditional events of the form “ϕgiven ψ”, where ψ is not the impossible event. Our approach exploits an idea already used by Hájek and colleagues to define a logic for (unconditional) probability in the frame of fuzzy logics. Namely, in our logic for each pair of classical propositions ϕ and ψ, we take the probability of the conditional event “ϕgiven ψ”, ϕ∣ψ for short, as the truth-value of the (fuzzy) modal proposition P(ϕ | ψ), read as “ϕ∣ψ is probable”. Based on this idea we define a fuzzy modal logic FCP(ŁΠ), built up over the many-valued logic Ł$\Pi\frac{1}{2}$ (a logic which combines the well-known Lukasiewicz and Product fuzzy logics), which is shown to be complete with respect to the class of probabilistic Kripke structures induced by coherent conditional probabilities. Finally, we show that checking coherence of a probability assessment to an arbitrary family of conditional events is tantamount to checking consistency of a suitable defined theory over the logic FCP(ŁΠ).