The Contradiction (iii): A Ramified Solution

Abstract

Abstract Poincaré had recommended a ‘Vicious Circle Principle’, and Russell accepted this in his final ‘ramified’ theory of types. He claims both that this principle resolves a range of paradoxes (including those concerning propositions that were earlier unresolved), and that it has a certain ‘consonance with common sense’. The first claim, which deals with ‘self-reference’ and ‘self-quantification’, is acceptable, but the solution offered is somewhat extravagant. The second apparently requires a conceptualist approach to abstract objects. (Principia Mathematica seems to sidestep this requirement, but the alternative justification which it offers is clearly faulty.) The Vicious Circle Principle introduces many problems for Russell’s derivation of mathematics, which he overcomes only by introducing an axiom of reducibility. This, he thinks, gives him all the right results: it still blocks the so-called ‘semantic’ paradoxes, but also allows the deduction of mathematics.