Elementary Iterated Revision and the Levi Identity

TL;DR

This paper explores extending the Levi Identity to iterated belief revision, providing new characterizations of classic revision operators and analyzing the implications of different extension proposals.

Contribution

It offers the first collective characterization of elementary iterated revision operators and compares two plausible extensions of the Levi Identity, linking them to existing postulates.

Findings

Nayak's proposal is equivalent to new postulates under mild assumptions.

Under certain conditions, both extensions imply that rational revision coincides with natural revision.

The paper discusses the potential for defining iterated revision solely in terms of iterated contraction.

Abstract

Recent work has considered the problem of extending to the case of iterated belief change the so-called `Harper Identity' (HI), which defines single-shot contraction in terms of single-shot revision. The present paper considers the prospects of providing a similar extension of the Levi Identity (LI), in which the direction of definition runs the other way. We restrict our attention here to the three classic iterated revision operators--natural, restrained and lexicographic, for which we provide here the first collective characterisation in the literature, under the appellation of `elementary' operators. We consider two prima facie plausible ways of extending (LI). The first proposal involves the use of the rational closure operator to offer a `reductive' account of iterated revision in terms of iterated contraction. The second, which doesn't commit to reductionism, was put forward some years ago by Nayak et al. We establish that, for elementary revision operators and under mild assumptions regarding contraction, Nayak's proposal is equivalent to a new set of postulates formalising the claim that contraction by $\neg A$ should be considered to be a kind of `mild' revision by $A$. We then show that these, in turn, under slightly weaker assumptions, jointly amount to the conjunction of a pair of constraints on the extension of (HI) that were recently proposed in the literature. Finally, we consider the consequences of endorsing both suggestions and show that this would yield an identification of rational revision with natural revision. We close the paper by discussing the general prospects for defining iterated revision in terms of iterated contraction.