Abstract

Zermelo, Frege and Russell accepted a common theme regarding classes. Classes were determined by other entities—functions, concepts, properties or conditions—and a class was only acceptable in the theory if there was such a determining entity. Thus, the existence of a class was taken to be dependent on a concept, function, condition, etc. whose satisfaction or fulfillment by an element determined the element to be a member of the class. This feature was behind Russell’s “no class theory,” where a class was not only taken to be “determined by” a function but was considered to be “replaceable” by its determining function, as well as Frege’s problematic claim that every concept determined a class. Thus, Russell could identify the class of φs with, and Frege could take there to be a class of φs whose elements were all things falling under φ. Treating classes in such a manner does not provide for an adequate ontological assay of classes. If classes are entities, then whether or not there exists a class cannot depend on there being a property or function or concept or condition specifying membership in the class. Such a concept or property merely allows us to provide a definite description of the class. It is not the basis for the existence of the class. Russell’s “no class” theory of the first edition of Principia takes class abstracts and the classes they purportedly denote as “incomplete symbols.” As applied to the class abstract signs, the notion of an “incomplete symbol” means that such signs are only meaningful in contexts, in fact, are contextually defined. Thus, their meaning is not supplied by an entity such a sign purportedly denotes. As applied to a purported entity, to say of such a “thing” that it is an incomplete symbol is to say that it doesn’t exist. Thus, Russell classified the King of France in 1905, the king, as well as the definite description, as an incomplete symbol and called propositions “incomplete symbols” when he denied that there were such entities.