On the Consistency between Belief Revision and Belief Update

Belief revision and belief update are two fundamental and well-studied processes of belief change. In the present article, we introduce a consistency principle which dictates that the revision and update policies employed by a rational agent are not independent, but ought to be related in a certain coherent way. We formalize our consistency principle both axiomatically and semantically, and we establish a representation result explicitly connecting the two formalizations. Furthermore, we show that two important concrete types of belief change, namely uniform belief change and parametrized-difference belief change, serve as proof-of-concept examples for the introduced consistency principle, as they fully comply with it. Additionally, we identify an intriguing property of uniform belief change in which revision and update become indistinguishable when an epistemic input contradicts the initial state of belief, as both processes produce identical outcomes. Lastly, it is shown that, unlike parametrized-difference belief change, uniform belief change is incompatible with Parikh’s notion of relevance. Consequently, building on previous results, it is demonstrated that parametrized-difference belief change is relevance-sensitive — indicating that the proposed principle of consistency is compatible with relevance — while uniform belief change is not.