  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.
