Description logics are among the most prominent formalisms for knowledge representation. Many expressive but still decidable description logics have been investigated and implemented. Although there is some work on description logics using non-monotonic, many-valued, or fuzzy-logic, see [1, Chapter 6], the underlying logic of almost all systems of description logic is classical. One of the rare exceptions is the work [2] by Patel-Schneider. The author considers four valued semantics for description logic, but he does not introduce non-classical connectives in the object language. In the present paper, we shall replace the classical base logic by a non-classical logic, namely Nelson's four-valued constructive and paraconsistent logic N4, see, for example, [3]. In particular, we shall replace material implication by intuitionistic implication and Boolean negation by constructive, strong negation.