Russell discovered the paradox which bears his name in May 1901, while working on his Principles of Mathematics (1903). The paradox arose in connection with the set of all sets which are not members of themselves. Such a set, if it exists, will be a member of itself if and only if it is not a member of itself. The significance of the paradox follows since, in classical logic, all sentences are entailed by a contradiction. In the eyes of many mathematicians (including David Hilbert and Luitzen Brouwer) it therefore appeared that no proof could be trusted once it was discovered that the logic apparently underlying all of mathematics was contradictory. A large amount of work throughout the early part of this century in logic, set theory, and the philosophy and foundations of mathematics was thus prompted.

Russell’s paradox arises as a result of naive set theory’s so-called unrestricted comprehension (or abstraction) axiom. Originally introduced by Georg Cantor, the axiom states that any predicate expression, P(x), which contains x as a free variable, will determine a set whose members are exactly those objects which satisfy P(x). The axiom gives form to the intuition that any coherent condition may be used to determine a set (or class). Most attempts at resolving Russell’s paradox have therefore concentrated on various ways of restricting or abandoning this axiom.

Russell’s own response to the paradox came with the introduction of his theory of types. His basic idea was that reference to troublesome sets (such as the set of all sets which are not members of themselves) could be avoided by arranging all sentences into a hierarchy (beginning with sentences about individuals at the lowest level, sentences about sets of individuals at the next lowest level, sentences about sets of sets of individuals at the next lowest level, etc.). Using the vicious circle principle also adopted by Henri Poincaré, together with his so-called “no class” theory of classes, Russell was then able to explain why the unrestricted comprehension axiom fails: propositional functions, such as the function “x is a set”, should not be applied to themselves since self-application would involve a vicious circle. On this view, it follows that it is possible to refer to a collection of objects for which a given condition (or predicate) holds only if they are all at the same level or of the same “type”.

Although first introduced by Russell in 1903 in the Principles, his theory of types finds its mature expression in his 1908 article Mathematical Logic as Based on the Theory of Types and in the monumental work he co-authored with Alfred North Whitehead, Principia Mathematica (1910, 1912, 1913). Thus, in its details, the theory admits of two versions, the “simple theory” and the “ramified theory”. Both versions of the theory later came under attack. For some, they were too weak since they failed to resolve all of the known paradoxes. For others, they were too strong since they disallowed many mathematical definitions which, although consistent, violated the vicious circle principle. Russell’s response to the second of these objections was to introduce, within the ramified theory, the axiom of reducibility. Although the axiom successfully lessened the vicious circle principle’s scope of application, many claimed that it was simply too ad hoc to be justified philosophically.

Of equal significance during this same period was Russell’s defence of logicism, the theory that mathematics was in some important sense reducible to logic. First defended in his Principles, and later in more detail in Principia Mathematica, Russell’s logicism consisted of two main theses. The first is that all mathematical truths can be translated into logical truths or, in other words, that the vocabulary of mathematics constitutes a proper subset of that of logic. The second is that all mathematical proofs can be recast as logical proofs or, in other words, that the theorems of mathematics constitute a proper subset of those of logic.

Like Gottlob Frege, Russell’s basic idea for defending logicism was that numbers may be identified with classes of classes and that number-theoretic statements may be explained in terms of quantifiers and identity. Thus the number 1 would be identified with the class of all unit classes, the number 2 with the class of all two-membered classes, and so on. Statements such as “there are two books” would be recast as “there is a book, x, and there is a book, y, and xis not identical to y“. It followed that number-theoretic operations could be explained in terms of set-theoretic operations such as intersection, union, and the like. In Principia Mathematica, Whitehead and Russell were able to provide detailed derivations of many major theorems in set theory, finite and transfinite arithmetic, and elementary measure theory. A fourth volume on geometry was planned but never completed.

In much the same way that Russell wanted to use logic to clarify issues in the foundations of mathematics, he also wanted to use logic to clarify issues in philosophy. As one of the founders of “analytic philosophy”, Russell is remembered for his work using first-order logic to show how a broad range of denoting phrases could be recast in terms of predicates and quantified variables. Thus, he is also remembered for his emphasis upon the importance of logical form for the resolution of many related philosophical problems. Here, as in mathematics, it was Russell’s hope that by applying logical machinery and insights one would be able to resolve otherwise intractable difficulties.