Theory of classification using the object-predicate table for our better understanding of intergroup relationship

In the Zadehian formalization, a grade function is defined with respect to a single object, and not to a group of objects. This paper proposes a formal framework which enable us to examine a complicated relationship among groups of objects, and to deal with loosely-defined concepts at the non-quantitative level. Assuming that each object is characterized by a set of predicates as a binary vector, whose components are 1 or 0 according to whether the object affirms a predicate or not, we first examine the relationship among objects in reference to the choice of predicates (or classifier) and formulate a formal theory of classification of objects by predicates. The formalization is perfectly symmetrical between the objects and the predicates, and it allows us to consider a classification of predicates by objects. Based on this reciprocal view of classification, we clarify a formal basis of equality between two groups of objects, propose a new approach to clustering and discuss its merits by showing examples