On cumulative defaults logics

Recently cumulative variants of Reiter's default logic have been proposed. All of them happen to be semimonotonic and to commit to assumptions, i.e. to require consistency of justifications. In this paper we define two new cumulative variants of default logic in the style of Brewka's CDL, which show that the above two properties are not required to achieve cumulativity. In fact, the first variant commits to assumptions, like Brewka's CDL, but it is not semimonotonic, and hence it allows priorities among defaults to be represented. The other cumulative variant does not even commit to assumptions, and thus it is very close to Reiter's default logic.