Completeness theorem for a logic with imprecise and conditional probabilities

Abstract

We present a propositional probability logic which allows making formulas that speak about imprecise and conditional probabilities. A class of Kripke-like probabilistic models is defined to give semantics to probabilistic formulas. Every possible world of such a model is equipped with a probability space. The corresponding probabilities may have nonstandard values. The proposition "the probability is close to r" means that there is an infinitesimal ε, such that the probability is equal to r-ε (or r+ε). We provide an infinitary axiomatization and prove the corresponding extended completeness theorem.