Negations in Description Logic – Contraries, Contradictories, and Subcontraries

In [10], several constructive description logics were proposed as intuitionistic variants of description logics, in which classical negation was replaced by strong negation as a component to treat negative atomic information. For conceptual representation, not only strong negation but also combining it with classical negation seems to be useful and necessary due to their respective features corresponding to predicate denial (e.g., not happy) and predicate term negation (e.g., unhappy). In this paper, we propose an alternative description logic ALC∼ with classical negation and strong negation. In particular, we adhere to the notions of contraries, contradictories, and subcontraries (in [5]), generated from conceivable statement types using predicate denial and predicate term negation. To capture these notions, our formalization provides an improved semantics that suitably interprets various combinations of classical negation and strong negation. We show that our semantics preserves contradictoriness and contrariness for ALC∼-concepts but the semantics of constructive description logic CALC∼ with Heyting negation and strong negation cannot preserve the property for CALC∼concepts.