From conditional probability to the logic of doxastic actions

Abstract

We investigate the discrete (finite) case of the Popper-Renyi theory of conditional probability, introducing discrete conditional probabilistic for knowledge and conditional belief, and comparing them with the more standard plausibility models. We also consider a related notion, that of safe belief, which is a weak (nonnegatively introspective) type of knowledge. We develop a probabilistic version of this concept (degree of safety) and we analyze its role in games. We completely axiomatize the logic of conditional belief, knowledge and safe belief over conditional probabilistic models. We develop a theory of probabilistic dynamic belief revision, introducing action models and a notion of probabilistic update product, that comes together with appropriate reduction laws.