Object-predicate Reciprocity in the Nominal Measurement

Within a formal framework, where each object is characterized by a set of predicates as a binary sequence of 0 and 1, a nominal level of measurement is discussed. The nominal scale is reinterpreted as a set of predicates, and nominal measurement as an assignment of binary sequences to the objects. One remarkable character of this formalization is a reciprocal relationship between the objects and the predicates. Taking this reciprocity as a heuristic guiding principle, a new notion of generalized indiscernibility between two sets of objects is introduced, which enable us to examine various complicated relationship among subsets of objects. As an application to nominal, non-quantitative measurement, algorithm of clustering is proposed, and its merit is discussed in comparison with a method of pattern recognition based on the principle of minimum entropy. The result suggests the importance of structural inter-group relationship that cannot be reduced easily to simple numerical variables. The notion of object-predicate reciprocity will helps us to analyse complicated object-predicate relations reflected in the result of nominal measurement, and to enlarge the scope of the newly emerging measurement science.