Completeness theorem for propositional probabilistic models whose measures have only finite ranges

Abstract

A prepositional logic is defined which in addition to prepositional 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.