Dynamic Doxastic Probability Logic

In this paper we will propose a dynamic doxastic probability logic which is inspired by two already existing systems and tries to meet them somewhere in between. On the one hand you have the original simple Kooi system of public announcement logic with hard public information only, and on the other hand you have the very rich and powerful BGK system of van Benthem, Gerbandy and Kooi [5]. Public announcement logic is built on a static logic which models the knowledge of agents by their information states. This system is made dynamic by the public announcement operator. The idea is that only truthful information can be announced and that the announcement is public (reaches everyone of the group). Along the same line there is the dynamic doxastic logic, which except for knowledge, also models beliefs of agents. Beliefs are modeled by an ordering relation which specifies which world is more plausible than another world. The update modalities in this logic consist of updates in the plausibility order. Note that this logic can only express updates that make certain worlds top worlds, since the language of beliefs only allows one to speak about the top worlds. Changing the order among worlds without one of them becoming top worlds is not expressible here. Extensions of this kind of dynamic epistemic logics with probabilistic information have been investigated by several researchers ([3], [4] and [5]). In this paper we will focus on the work of van Benthem, Gerbrandy and Kooi [5], which is a subsuming of the other two systems. The BGK system is a complete probability update logic. As static language they take a simple instance of a system described by Halpern and Tutle [2] and further developed by Fagin and Halpern [1] (of which we will use a simple instance in this paper). Then they use probabilistic update models as update modality