Constructive Description Logics Hybrid-Style

Constructive modal logics come in several different flavours and constructive description logics, not surprisingly, do the same. We introduce an intuitionistic description logic, which we call iALC (for intuitionistic ALC, since ALC is the name of the canonical description logic system) and provide axioms, a Natural Deduction formulation and a sequent calculus for it. The system iALC is related to Simpson’s constructive modal logic IK the same way Mendler and Scheele’s cALC is related to constructive CK and in the same way classical multimodal K is related to ALC. In the system iALC, as well as in cALC, the classical principles of the excluded middle C ¬C = T, double negation ¬¬C = C and the definitions of the modalities ∃R.C = ¬∀R.¬C and ∀R.C = ¬∃R.¬C are no longer validities, but simply non-trivial TBox statements used to axiomatize specific application scenarios. Meanwhile in iALC, like in classical ALC, we have that the distribution of existential roles over disjunction i.e. ∃R.(C D) = ∃R.C ∃R.D and (the nullary case) ∃R.⊥ = ⊥ hold, which is not true for cALC. We intend to use iALC for modelling juridical Artificial Intelligence (AI) systems and we describe briefly how.