A propositional logic is defined which in addition to propositional language contains a list of probabilistic operators of the form P ≥s (with the intended meaning ‘‘the probability is at least s’’). The axioms and rules syntactically determine that ranges of probabilities in the corresponding models are always finite. The completeness theorem is proved. It is shown that completeness cannot be generalized to arbitrary theories.

A topological class logic is an infinitary logic formed by combining a first-order logic with the quantifier symbols O and C. The meaning of a formula closed by quantifier O is that the set defined by the formula is open. Similarly, a formula closed by quantifier C means that the set is closed. The corresponding models are a topological class spaces introduced by Ćirić and Mijajlović (Math Bakanica 1990). The completeness theorem is proved.

This paper is devoted to fill the gap in studying logics for biprobability structures. We introduce the logic equation image with two conditional expectation operators and prove the completeness theorem. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.