A Conditional Perspective on the Logic of Iterated Belief Contraction

Abstract

Classical AGM belief change theory provides only limited information on how to deal with iterative belief revision, leading to the question of how appropriate iteration principles look alike. The contributions to the interconnection between iterative belief revision and conditional beliefs lead to fundamental insights on that matter. Besides of clarifying to what amount iterated revision should respect conditional beliefs, these investigations identified several principles for iterated revision that are today widely accepted by the belief change community and beyond. Specifically, due to the seminal work by Darwiche and Pearl, it is standard, that principles for iterated revision and change of conditional beliefs are considered as two sides of the same coin. However, for iterated contraction the situation is different, there are only a small amount of iteration principles present in the literature. Moreover, for none of these iterated principles for contraction, there is a known equivalent formulation in terms of change of conditional beliefs.In this article, we consider iteration principles for contraction, with the goal of identifying properties for contractions that respect conditional beliefs. Therefore, we investigate and evaluate four groups of iteration principles for contraction which consider the dynamics of conditional beliefs. For all these principles, we provide semantic characterization theorems and provide formulations by postulates which highlight how the change of beliefs and of conditional beliefs is constrained, whenever that is possible. The first group is similar to the syntactic Darwiche-Pearl postulates. As a second group, we consider semantic postulates for iteration of contraction by Chopra, Ghose, Meyer and Wong, and by Konieczny and Pino Pérez, respectively, and we provide novel syntactic counterparts. Third, we propose a contraction analogue of the independence condition by Jin and Thielscher. For the fourth group we consider natural and moderate contraction by Nayak. Methodically, we make use of conditionals for contractions, so-called contractionals and furthermore, we propose and employ the novel notion of α-equivalence for formulating some of the new postulates.