A classification model: Syntax and semantics for classification

We present here Semantic and Descriptive Models for Classification as components of our Classification Model (definition 17). We do so within a framework of a General Data Mining Model (definition 4) which is a model for Data Mining viewed as a generalization process and sets standards for defining syntax and semantics and its relationship for any Data Mining method. In particular, we define the notion of truthfulness, or a degree of truthfulness of syntactic descriptions obtained by any classification algorithm, represented within the Semantic Classification Model by a classification operator. We use our framework to prove (theorems 1 and 3) that for any classification operator (method, algorithm) the set of all discriminant rules that are fully true form semantically the lower approximation of the class they describe. The set of characteristic rules describes semantically its upper approximation. Similarly, the set of all discriminant rules for a given class that are partially true is semantically equivalent to approximate lower approximation of the class. The notion of the approximate lower approximation extends to any classification operator (method, algorithm) the ideas first expressed in 1986 by Wong, Ziarko, Ye [9], and in the VPRS model of Ziarko [10].