Interval Orders, Biorders and Credibility-limited Belief Revision

Abstract

Rational belief revision is commonly viewed as being based on a preference order between possible worlds, with the resulting new belief set being those sentences true in all the most preferred models of the incoming new information. Usually, such a preference order is taken to be a total preorder. Nevertheless, there are other, more general classes of ordering that can also be employed. In this paper, we explore two such classes that have been studied within the theory of rational choice but have seen limited or no application in belief revision. We begin with interval orders, introduced by Fishburn in the '80s, which associate with each possible world a nonnegative `interval' of plausibility. We then move on to biorders, studied by Aleskerov, Bouyssou, and Monjardet, which generalise interval orders by allowing the intervals to have negative lengths, a feature that can be used to capture a notion of dissonance or instability. We provide axiomatic characterisations of these two resulting families of belief revision operators, as well as of two further families of interest that lie between interval orders and biorders. We show that while biorder-based revisions satisfy the Success postulate, they do not always yield consistent outputs. By modifying their definition to discard inputs that lead to inconsistency as `incredible', we derive new families of so-called non-prioritised revision that satisfy the Consistency postulate, but not the Success one. These families are linked to credibility-limited revision operators of Hansson et al., but for which the set of credible sentences does not satisfy the single-sentence closure condition. We argue that the biorder-based approach is well-suited for scenarios where an agent might initially reject new information, but may accept it when presented with additional explanation.