Chapter

The Contradiction (iii): A Ramified Solution

David Bostock

in Russell's Logical Atomism

Published in print April 2012 | ISBN: 9780199651443
Published online September 2012 | e-ISBN: 9780191741197 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199651443.003.0005
The Contradiction (iii): A Ramified Solution

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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.

Keywords: Poincaré; vicious circle principle; ramified type theory; self-reference; self-quantification; conceptualism; axiom of reducibility; semantic paradoxes; Principia Mathematica

Chapter.  11986 words. 

Subjects: History of Western Philosophy

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