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