Russell's Paradox and the Theory of Classes in The Principles of Mathematics

Abstract

We can regard Russell's theory of classes in The Principles of Mathematics as a dual theory of classes based upon the distinction between a class as one and a class as many. In this paper, I shall show the following (A)-(D) from the viewpoint of how Russell dealt in the Principles with the paradox that bears his own name. (A) The dual theory of classes is one that deserves serious considerations. (B) Russell offered measures against the paradox other than the simple theory of types developed in an appendix to the Principles. (C) Russell avoided the paradox without forbidding self-memberships of classes in general. (D) Though introducing a hierarchy of types into pseudo-entities is a very important step to Russell's later theories (e.g., the substitutional theory and the ramified theory of types), we can find this strategy already in the Principles.