  Non-normal modal logics, interpreted on neighbourhood models which generalise
the usual relational semantics, have found application in several areas, such
as epistemic, deontic, and coalitional reasoning. We present here preliminary
results on reasoning in a family of modal description logics obtained by
combining ALC with non-normal modal operators. First, we provide a framework of
terminating, correct, and complete tableau algorithms to check satisfiability
of formulas in such logics with the semantics based on varying domains. We then
investigate the satisfiability problems in fragments of these languages
obtained by restricting the application of modal operators to formulas only,
and interpreted on models with constant domains, providing tight complexity
results.
