Conditional Probability Logic over Conditional Probability Spaces

Abstract

We present a logical system sound and complete with respect to the class of conditional probability spaces. The universal modality has an important role in our language: conditional probabilities have a peculiar behavior when conditionalized over empty events, so we use the universal modality to express nonemptiness of events. We add public announcement to our language in a way that we are able to express sentences like: “the agent believes the probability of p is zero and after the announcement of p he/she believes the probability of p is greater than zero.” Because of the Conditional Probability Language we are able to express reduction axioms for the probability Public Announcement Logic.