A simple modal logic for belief revision (Q813420)

From Bayes' rule for posterior probabilities one can extract a qualitative principle: when the support of a prior probability function is not disjoint from new evidence, then their intersection equals the support of the posterior probability function. Here the `support' of a probability function is the set of worlds (states) that receive a non-zero value under that function. The author calls this the `qualitative Bayes rule', and uses it as the guiding idea for constructing a modal logic for belief revision. This logic contains three modal operators: one for prior belief, one for posterior belief, and one for (all of) the input information that permits passage from the former to the latter. Postulates and possible-worlds semantics are given, and soundness/completeness theorems proven. Although the semantics for the information-input operator is rather unusual, the author suggests that the system has the advantage of demonstrating that ``three operators are sufficient to axiomatize the qualitative version of Bayes' rule. Previous modal axiomatizations of belief revision required an infinite number of modal operators''. Reviewer's comment: A limitation of the system, as one might expect from its starting point, is that it is able to consider only information inputs that are consistent with initial beliefs. Thus it covers situations that give rise to what in the AGM tradition is known as `belief expansion', constituting a limiting case of belief revision. The subject of the paper might thus be described as `a simple modal logic for belief expansion'. From the point of view of revision, the interesting cases are those in which the input contradicts initial beliefs.