Conceptual Collectives
This is an incremental improvement in knowledge representation for data analysis and AI applications.
The paper argues that Wille's formal contexts should be replaced with a more semantic notion, leading to a modified concept construction approach, and demonstrates equivalence with MacNeille's completion by cuts.
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.