On Strengthening the Logic of Iterated Belief Revision: Proper Ordinal Interval Operators

TL;DR

This paper introduces Proper Ordinal Interval (POI) operators for iterated belief revision, extending existing principles with new conditions that enable more nuanced belief state updates while satisfying key AGM postulates.

Contribution

It proposes a novel class of belief revision operators based on POI assignments, addressing limitations of previous reductionist approaches and satisfying iterated AGM postulates.

Findings

POI operators generalize lexicographic and restrained revision.

Operators satisfy iterated AGM postulates including Superexpansion.

New principles govern relations between different posterior belief sets.

Abstract

Darwiche and Pearl's seminal 1997 article outlined a number of baseline principles for a logic of iterated belief revision. These principles, the DP postulates, have been supplemented in a number of alternative ways. Most of the suggestions made have resulted in a form of `reductionism' that identifies belief states with orderings of worlds. However, this position has recently been criticised as being unacceptably strong. Other proposals, such as the popular principle (P), aka `Independence', characteristic of `admissible' revision operators, remain commendably more modest. In this paper, we supplement both the DP postulates and (P) with a number of novel conditions. While the DP postulates constrain the relation between a prior and a posterior conditional belief set, our new principles notably govern the relation between two posterior conditional belief sets obtained from a common prior by different revisions. We show that operators from the resulting family, which subsumes both lexicographic and restrained revision, can be represented as relating belief states that are associated with a `proper ordinal interval' (POI) assignment, a structure more fine-grained than a simple ordering of worlds. We close the paper by noting that these operators satisfy iterated versions of a large number of AGM era postulates, including Superexpansion, that are not sound for admissible operators in general.