Completeness Theorem for a Logic With Imprecise and Conditional Probabilities

Zoran,Ognjanović ,Zoran,Marković ,Miodrag,Rašković

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 $\epsilon$, such that the probability is equal to $r-\epsilon$ (or $r+\epsilon$). We provide an infinitary axiomatization and prove the corresponding extended completeness theorem.