Relevance in belief revision

Possible-world semantics are provided for Parikh's relevance-sensitive axiom for belief revision, known as axiom (P). Loosely speaking, axiom (P) states that if a belief set K can be divided into two disjoint compartments, and the new information ϕ relates only to the first compartment, then the second compartment should not be effected by the revision of K by ϕ. Using the well-known connection between AGM revision functions and preorders on possible worlds as our starting point, we formulate additional constraints on such preorders that characterise precisely Parikh's axiom (P). Interestingly, the additional constraints essentially generalise a criterion of plausibility between possible worlds that predates axiom (P). A by-product of our study is the identification of two possible readings of Parikh's axiom (P), which we call the strong and the weak versions of the axiom. Regarding specific operators, we show that Dalal's belief revision operator satisfies both weak and strong (P), and it is therefore relevance-sensitive.