  The study of Description Logics have been historically mostly focused on
features that can be translated to decidable fragments of first-order logic. In
this paper, we leave this restriction behind and look for useful and decidable
extensions outside first-order logic. We introduce universally quantified
concepts, which take the form of variables that can be replaced with arbitrary
concepts, and define two semantics of this extension. A schema semantics allows
replacements of concept variables only by concepts from a particular language,
giving us axiom schemata similar to modal logics. A second-order semantics
allows replacement of concept variables with arbitrary subsets of the domain,
which is similar to quantified predicates in second-order logic.
  To study the proposed semantics, we focus on the extension of the description
logic $\mathcal{EL}$. We show that for a useful fragment of the extension, the
conclusions entailed by the different semantics coincide, allowing us to use
classical $\mathcal{EL}$ reasoning algorithms even for the second-order
semantics. For a slightly smaller, but still useful, fragment, we were also
able to show polynomial decidability of the extension. This fragment, in
particular, can express a generalized form of role chain axioms, positive self
restrictions, and some forms of (local) role-value-maps from KL-ONE, without
requiring any additional constructors.

Logic in Computer Science;Artificial Intelligence