Projection and scope-determined circumscription

Abstract

We develop a semantic framework that extends first-order logic by literal projection and a novel second semantically defined operator, raising, which is only slightly different from literal projection and can be used to define a generalization of parallel circumscription with varied predicates in a straightforward and compact way. We call this variant of circumscription scope-determined, since like literal projection and raising its effects are controlled by a so-called scope, that is, a set of literals, as parameter. We work out formally a toolkit of propositions about projection, raising and circumscription and their interaction. It reveals some refinements of and new views on previously known properties. In particular, we apply it to show that well-foundedness with respect to circumscription can be expressed in terms of projection, and that a characterization of the consequences of circumscribed propositional formulas in terms of literal projection can be generalized to first-order logic and expressed compactly in terms of new variants of the strongest necessary and weakest sufficient condition.