Conditional Belief, Knowledge and Probability

A natural way to represent beliefs and the process of updating beliefs is presented by Bayesian probability theory, where belief of an agent a in P can be interpreted as a considering that P is more probable than not P. This paper attempts to get at the core logical notion underlying this. The paper presents a sound and complete neighbourhood logic for conditional belief and knowledge, and traces the connections with probabilistic logics of belief and knowledge. The key notion in this paper is that of an agent a believing P conditionally on having information Q, where it is assumed that Q is compatible with what a knows. Conditional neighbourhood logic can be viewed as a core system for reasoning about subjective plausibility that is not yet committed to an interpretation in terms of numerical probability. Indeed, every weighted Kripke model gives rise to a conditional neighbourhood model, but not vice versa. We show that our calculus for conditional neighbourhood logic is sound but not complete for weighted Kripke models. Next, we show how to extend the calculus to get completeness for the class of weighted Kripke models. Neighbourhood models for conditional belief are closed under model restriction (public announcement update), while earlier neighbourhood models for belief as `willingness to bet' were not. Therefore the logic we present improves on earlier neighbourhood logics for belief and knowledge. We present complete calculi for public announcement and for publicly revealing the truth value of propositions using reduction axioms. The reductions show that adding these announcement operators to the language does not increase expressive power.