Classes and Relations

A class is determined by its members: the class of prime numbers less than ten is identical with the class {2, 3, 5, 7}, regardless of how we describe it. Russell argues that classes are not mysterious additional entities over and above their members: they are logical constructions defined by propositional functions — expressions with a free variable, like "x is a prime number". The class of prime numbers just is what we get when we ask "for which values of x is 'x is a prime number' true?" This deflationary attitude toward abstract objects — treating them as logical constructions rather than Platonic inhabitants of a third realm — is a hallmark of Russell's philosophical style.

Russell's treatment of relations was one of the most important contributions of the Principles. Earlier logicians, working within a subject-predicate framework, had struggled to give an adequate account of asymmetric relations: "John is taller than Mary" resists reduction to a predicate of John alone or of Mary alone. Russell insists that relations are genuine logical entities — that "x is to the left of y" expresses a fact irreducible to any combination of facts about x and y separately. This liberation of logic from the subject-predicate straitjacket made possible the logical analysis of order, series, and the continuum that underlies modern mathematical logic.

The deepest consequence of Russell's analysis is that mathematics is the science of abstract structure: of what is common to all systems that satisfy certain relational conditions. The number 2 is not a specific thing but a structural role — the role played by the second element in any simply infinite series. Two systems are mathematically equivalent if and only if there is a structure-preserving mapping between them. This structuralist insight, refined by later philosophers and mathematicians, became one of the central themes of the philosophy of mathematics in the twentieth century.

The analysis of classes and relations occupies much of Parts I–III of The Principles of Mathematics (1903). Russell's debt to Peano's logical notation and to Cantor's set theory is acknowledged throughout. The "no-class theory" — which treats class expressions as incomplete symbols to be eliminated in analysis — is developed more fully in Principia Mathematica.