Yet some more considerations on cumulativity in default logics

Abstract

From the time that [Makinson, 89] noticed the non-cumulativity of Reiter's and Łukaszewicz's default logics, many attempts have been made to introduce this property in full default logic. Following [Brewka, 91], many of these attempts are associated with the consistency of the set of justifications used for deriving an extension. But this modification of the original formalism of Reiter is not always suitable. Another point is that most of these attempts rely on a modification of the underlying langage and/or a weakening of cumulativity such as initially interpreted by [Makinson, 89]. In the present paper, a different approach to cumulativity in default logics is investigated. A (trivial) cumulative variant of default logic that does not require commitment to justifications is first introduced for skeptical reasoning in the sense of [Makinson, 89]. It can be noted that both Reiter's and Łukaszewicz's approaches are embedded in it, and that no extension of the underlying language is required. Then, using this framework and following a proposition of [Voorbraak, 93], a criterion is proposed to distinguish cumulative default theories in the sense of Łukaszewicz from non-cumulative ones.

Supported by a grant from the Natural Science and Engineering Research Council of Canada.