Open Default Theories over Closed Domains

Abstract

We compare two approaches to the meaning of a free variable x in an open default $\frac{{\alpha (x):M\beta _1 (x),...,M\beta _m (x)}} {{\gamma (x)}}$ . The first treats x as a metavariable for the ground terms of the underlying theory, whereas the second threats 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.

In fact, Poole in [11] considers only normal defaults without prerequisites, i.e., defaults of the form $\frac{{:M\beta (x)}} {{\beta (x)}}$ .