Completeness Theorem for Biprobability Models

The aim of the paper is to prove the completeness theorem for biprobability models. This also solves Keisler's Problem 5.4 (see [4]). Let be a countable admissible set and ω ∈ . The logic is similar to the standard probability logic . The only difference is that two types of probability quantifiers and are allowed. A biprobability model is a structure ( , μ 1 , μ 2 ) where is a classical structure without operations and μ 1 , μ 2 are two types of probability measures such that μ 1 is absolutely continuous with respect to μ 2 , i.e. μ 1 ≪ μ 2 . The quantifiers are interpreted in the natural way, i.e. for i = 1, 2. (The measure is the restriction of the completion of to the σ -algebra generated by the measurable rectangles and the diagonal sets Axioms and rules of inference are those of , as listed in [2] with the axiom B 4 from [4], with the remark that both P 1 and P 2 can play the role of P , together with the following axioms: Axioms of continuity . 1) . 2) . Axiom of absolute continuity : where and Φ n = { φ ∈ Φ : φ has n free variables}.