Fuzzy Description Logics (DLs) are a formalism for the representation of structured knowledge that is imprecise or vague by nature. In fuzzy DLs, restricting to a finite set of degrees of truth has proved to be useful, both for theoretical and practical reasons. In this paper, we propose finite fuzzy DLs as a generalization of existing approaches. We assume a finite totally ordered set of linguistic terms or labels, which is very useful in practice since expert knowledge is usually expressed using linguistic terms. Then, we consider fuzzy DLs based on any smooth t-norm defined over this set. Initially we focus on the finite fuzzy DL
, studying some logical properties, and showing the decidability of the logic by presenting a reasoning preserving reduction to the classical case. Finally, we extend our logic in two directions: by considering non-smooth t-norms and by considering additional DL constructors.