The Logic of Relations

Traditional logic, from Aristotle through Leibniz and Kant, was primarily a logic of subject-predicate propositions: every proposition was thought to attribute a property to a subject. Relations between two or more terms — "Plato is earlier than Aristotle," "Paris is between London and Berlin" — were awkward for this framework, since they seem to resist reduction to a property of either term alone. Russell and Whitehead's formal logic treats polyadic relations as primitive logical entities alongside monadic properties, opening up the resources needed to formalise the logic of order, series, and structure that mathematics requires.

The formal vindication of external relations — relations that do not affect the intrinsic nature of their terms — was, for Russell, also a philosophical victory over the British Idealists, and over Bradley's argument in particular. Bradley had argued that external relations are incoherent: a relation must either be internal to its terms (modifying their nature) or it is a mere fiction incapable of genuinely connecting them. Russell's response was that Bradley's argument confuses the ontological question (what kind of thing is a relation?) with the semantic question (what is the logical form of a relational proposition?). A relational fact — aRb — is simply a fact, not a further item requiring cement.

The significance of the logic of relations for mathematics is enormous. Mathematical structures — the natural numbers in their order, the real numbers in their density, geometric spaces in their topology — are fundamentally relational: they are defined not by the intrinsic properties of their elements but by the relations between those elements. The logical analysis of relations in Principia Mathematica provides the tools needed to define these structures precisely and to derive their properties from logical principles alone, making good on the logicist promise that mathematical structure is logical structure.

The logic of relations is developed in detail in Part IV of Principia Mathematica, Vol. II. Russell's philosophical case for external relations against Bradley is made in "The Nature of Truth" (1906) and in the introduction to The Principles of Mathematics (1903).