The 1910 *Principia*'s Theory of Functions and Classes and the Theory of Descriptions

Abstract

It is generally acknowledged that the 1910 Principia does not deny the existence of classes, but claims only that the theory it advances can be developed so that any apparent commitment to them is eliminable by the method of contextual analysis. The application of contextual analysis to ontological questions is widely viewed as the central philosophical innovation of Russell’s theory of descriptions. Principia’s “no-classes theory of classes” is a striking example of such an application. The present paper develops a reconstruction of Principia’s theory of functions and classes that is based on Russell’s epistemological applications of the method of contextual analysis. Such a reconstruction is not eliminativist—indeed, it explicitly assumes the existence of classes—and possesses certain advantages over the no–classes theory advocated by Whitehead and Russell.

Keywords

• Principia Mathematica
• contextual definition
• ramified type theory
• axiom of reducibility
• impredicative definition
• propositional function