A logical and algebraic treatment of conditional probability

This paper is devoted to a logical and algebraic treatment of conditional probability. The main ideas are the use of non-standard probabilities and of some kind of standard part function in order to deal with the case where the conditioning event has probability zero, and the use of a many-valued modal logic in order to deal probability of an event phi as the truth value of the sentence phi is probable, along the lines of Hajek's book [H98] and of [EGH96]. To this purpose, we introduce a probabilistic many-valued logic, called FP(SLPi), which is sound and complete with respect a class of structures having a non-standard extension [0,1](star) of [0,1] as set of truth values. We also prove that the coherence of an assessment of conditional probabilities is equivalent to the coherence of a suitably defined theory over FP(SLPi) whose proper axioms reflect the assessment itself.