The Origins of the Propositional Functions Version of Russell's Paradox

Abstract

Russell discovered the classes version of Russell's Paradox in spring 1901, and the predicates version near the same time. There is a problem, however, in dating the discovery of the propositional functions version. In 1906, Russell claimed he discovered it after May 1903, but this conflicts with the widespread belief that the functions version appears in The Principles of Mathematics, finished in late 1902. I argue that Russell's dating was accurate, and that the functions version does not appear in the Principles. I distinguish the functions and predicates versions, give a novel reading of the Principles, section 85, as a paradox dealing with what Russell calls assertions, and show that Russell's logical notation in 1902 had no way of even formulating the functions version. The propositional functions version had its origins in the summer of 1903, soon after Russell's notation had changed in such a way as to make a formulation possible.