Revising Epistemic States by Conditional Beliefs

Abstract

Usually, the belief sets in AGM theory (cf. Section 2.2) are assumed to be deductively closed sets of propositional formulas, or to be represented by one single propositional formula, respectively, and the revising beliefs are taken to be propositional formulas. So the AGM postulates constrain revisions of the form ψ* A, the revision operator * connecting two propositional formulas ψ and A, where ψ represents the initial state of belief and A stands for the new information. A representation theorem (see [KM91a]) establishes a relationship between AGM revision operators and total pre-orders ⩽_ψ on the set of possible worlds, proving the revised belief set ψ* A to be satisfied precisely by all minimal A-worlds (see also Section 2.2).

Belief sets represent what is known for certain and are of specific interest. They are, however, only poor re ections of the complex attitudes an individual may hold. The limitation to propositional beliefs severely restricts the frame of AGM theory, in particular, when iterated revisions have to be performed. So belief revision should not only be concerned with the revision of propositional beliefs but also with the modification of revision strategies when new information arrives (cf. [DP97a, Bou93, BG93]). These revision strategies may be given implicitly by some kind of preference relation like a plausibility ordering or an epistemic entrenchment (cf. Section 2.4), or may be taken explicitly as conditional beliefs. Revisions of the complex structure of an epistemic state so as to allow iterated revisions are denoted as transmutations of knowledge systems in [Wil94]. As a counterpart to the paradigm of minimal propositional change guiding the AGM postulates, the new paradigm of preserving conditional beliefs, shortly referred to as conditional preservation, arises in the framework of revising epistemic states.

Darwiche and Pearl [DP97a] explicitly took conditional beliefs into account by revising epistemic states instead of belief sets, and they advanced four postulates in addition to the AGM axioms as an approach to describe conditional preservation under revision by propositional beliefs (cf. the DP-postulates on page 22 in Section 2.4).

In the sequel, we broaden the framework for revising epistemic states (as presented, for instance, in [DP97a, Bou94, Wil94]) so as to include also the