Open Default Theories over Closed Domains

In this paper we compare two approaches to the meaning of a free variable x in an open default α(x) : β1(x), . . . , βm(x) γ(x) . The first treats x as a metavariable for the ground terms of the underlying theory, whereas the second treats it as a “name” of arbitrary elements of the theory universe. We show that, for normal default theories, under the domain closure assumption, the two approaches are equivalent. In the general case, the approaches are equivalent in the presence of both the domain closure assumption and the unique name assumption.