Description Logics with Contraries, Contradictories, and Subcontraries

Abstract

Several constructive description logics,12) in which classical negation was replaced by strong negation as a component to treat negative atomic information have been proposed as intuitionistic variants of description logics. For conceptual representation, strong negation alone and in a combination with classical negation seems to be useful and necessary due to their respective predicate denial (e.g., not happy) and predicate term negation (e.g., unhappy) properties. In this paper, we propose an alternative description logic \({\mathcal{ALC}}_{~}^{n}\) with classical negation and strong negation. We adhere in particular to the notions of contraries, contradictories, and subcontraries (as discussed in 6)), generated from conceivable statement types using predicate denial and predicate term negation. To capture these notions, our formalization includes a semantics that suitably interprets various combinations of classical negation and strong negation. We show that our semantics preserves contradictoriness and contrariness for \({\mathcal{ALC}}_{~}^{n}\)-concepts, but the semantics of constructive description logic \({\mathcal{CALC}}_{~}^{2}\) with Heyting negation and strong negation cannot preserve the property for \({\mathcal{CALC}}_{~}^{2}\)-concepts.