  The notions of formal contexts and concept lattices, although introduced by
Wille only ten years ago, already have proven to be of great utility in various
applications such as data analysis and knowledge representation. In this paper
we give arguments that Wille's original notion of formal context, although
quite appealing in its simplicity, now should be replaced by a more semantic
notion. This new notion of formal context entails a modified approach to
concept construction. We base our arguments for these new versions of formal
context and concept construction upon Wille's philosophical attitude with
reference to the intensional aspect of concepts. We give a brief development of
the relational theory of formal contexts and concept construction,
demonstrating the equivalence of "concept-lattice construction" of Wille with
the well-known "completion by cuts" of MacNeille. Generalization and
abstraction of these formal contexts offers a powerful approach to knowledge
representation.
