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

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.This research was supported by Ministarstvo za nauku, tehnologije i razvoj Republike Srbije, through Matematiki institut, under grant 1379Mathematics Subject Classification (2000): 03C70, 03B48