Measuring the Empirical Properties of Sets

Abstract

An important problem which frequently arises is the problem of measuring the degree to which a collection of objects has the empirical properties of its elements. Examples of this are found in the question of whether a network of switches is on or off, the degree to which a group of factories pollutes the air, and even the probability of success defined on an empirical frequency distribution. While each of these questions may be addressed with an esoteric formalism, there exists no general methodology for addressing the general problem because the properties of a set's elements are not defined on the set itself. For example, the set of all tall buildings is neither tall nor a building. Rather, it is an abstraction and has only those properties defined on such abstractions, e.g., cardinality. The purpose of the present paper is to develop a broad new formalism for addressing the noted general problem in a very general way. To begin with, the property p defined on the object x is formalized as a function p(x) = [0,1] which may be ``Boolean'' (binary valued) or ``formal'' (many valued or fuzzy). A set P may likewise have such a property to the degreep(P). Such a set is referred to as a ``property set,'' and these are partitioned into two general categories: ``property systems'' and ``collectives.'' A property system is characterized by the fact that its elements are related through the operators AND or OR.