A first-order conditional probability logic

Abstract

In this article, we present the probability logic LFOCP which is suitable to formalize statements about conditional probabilities of first order formulas. The logical language contains formulas such as CP ≥s (φ,Θ) and CP≤s (φ,Θ) with the intended meaning 'the conditional probability of φ given Θ is at least s' and 'at most s', respectively, where φ and Θ are first-order formulas. We introduce a class of first order Kripke-like models that combine properties of the usual Kripke models and finitely additive probabilities. We propose an infinitary axiom system and prove that it is sound and strongly complete with respect to the considered class of models. In this article, the terms finitary and infinitary concern meta language only, i.e. the logical language is countable, formulas are finite, while only proofs are allowed to be infinite. We analyse decidability of LFOCP and provide a procedure which decides satisfiability of a given conditional probability formula, in the case when the underlying first order theory is decidable. Relationships to other systems and possible extensions of the presented approach are discussed.