A General Family of Preferential Belief Removal Operators

Most belief change operators in the AGM tradition assume an underlying plausibility ordering over the possible worlds which is transitive and complete. A unifying structure for these operators, based on supplementing the plausibility ordering with a second, guiding, relation over the worlds was presented in [Booth et al., 2004]. However it is not always reasonable to assume completeness of the underlying ordering. In this paper we generalise the structure of [Booth et al., 2004] to allow incomparabilities between worlds. We axiomatise the resulting class of belief removal functions, and show that it includes an important family of removal functions based on finite prioritised belief bases. We also look at some alternative notions of epistemic entrenchment which become distinguishable once we allow incomparabilities.