Global Descriptor Revision

• Sven Ove Hansson
Chapter
Part of the Trends in Logic book series (TREN, volume 46)

Abstract

In local change, the operation $$\circ$$ is specific for the original belief set K. Formally it is a function that takes us from a descriptor $$\Psi$$ to an element $$K\circ \Psi$$ of the outcome set $$\mathbb {X}$$ (the set of belief sets that are potential outcomes of belief change). It only represents changes that have K as their starting-point. In this chapter the framework of descriptor revision is widened to global (iterated) belief change. This means that the operation $$\circ$$ can be applied to any potential belief set. Formally, it is a function that takes us from a pair consisting of a belief set K and a descriptor $$\Psi$$ to a new belief set $$K\circ \Psi$$. This makes it possible to cover successive changes, such as $$K\circ \mathfrak {B}p\circ \lnot \mathfrak {B} p$$. Several constructions of global descriptor revision are presented and axiomatically characterized. The most orderly of these constructions is based on pseudodistances (distance measures that allow the distance from X to Y to differ from the distance from Y to X). For any elements X and Y of the outcome set, i.e. the set of belief sets that are eligible as outcomes, there is a number $$\delta (X, Y)$$ denoting how far away Y is from X. When revising a belief set K by some descriptor $$\Psi$$, the outcome $$K\circ \Psi$$ is the belief set satisfying $$\Psi$$ that is closest to K, as measured with $$\delta$$. If we revise $$K\circ \Psi$$ by $$\Xi$$, then the outcome $$K\circ \Psi \circ \Xi$$ is the belief set $$\delta$$-closest to $$K\circ \Psi$$ that satisfies $$\Xi$$, etc. The chapter also provides a generalization of blockage revision to global operations.