Russell, His Paradoxes, and Cantor's Theorem: Part I

Sequel to Part I. In these articles, I describe Cantor’s power-class theorem, as well as a number of logical and philosophical paradoxes that stem from it, many of which were discovered or considered (implicitly or explicitly) in Bertrand Russell’s work. These include Russell’s paradox of the class of all classes not members of themselves, as well as others involving properties, propositions, descriptive senses, class-intensions and equivalence classes of coextensional properties. Part II addresses Russell’s own various attempts to solve these paradoxes, including strategies that he considered and rejected (limitation of size, the zigzag theory, etc.), as well as his own final views whereupon many purported entities that, if reified, lead to these contradictions, must not be genuine entities, but ‘logical fictions’ or ‘logical constructions’ instead. This article is a sequel to ‘Russell, His Paradoxes, and Cantor’s Theorem: Part I’, in which various Cantorian diagonal paradoxes either discovered or considered by Bertrand Russell were outlined. These include Russell’s famous class paradox involving the class of all classes not members of themselves, his predication paradox involving the property of non-self-instantiation, as well as similar paradoxes involving propositions, descriptive senses, class-intensions and equivalence classes of coextensive properties. Four different lines of solutions considered by Russell were also discussed: (i) the theory of limitation of size, (ii) the zigzag theory, (iii) logical types of things, and (iv) the ‘no classes (etc.)’ theory. (In what follows, it is assumed that the reader has read Part I.) In this sequel, we examine in further detail the impact of these paradoxes on Russell’s own philosophy, his consideration of possible solutions of all these kinds, and his reasons for, in the end, moving towards a rejection of robust metaphysical realism about many kinds of abstract objects in favor of viewing them as ‘logical fictions’, or mere facons de parler, so that such