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Russell's Logical Atomism$
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David Bostock

Print publication date: 2012

Print ISBN-13: 9780199651443

Published to Oxford Scholarship Online: September 2012

DOI: 10.1093/acprof:oso/9780199651443.001.0001

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The Contradiction (iii): A Ramified Solution

The Contradiction (iii): A Ramified Solution

Chapter:
(p.74) 5 The Contradiction (iii): A Ramified Solution
Source:
Russell's Logical Atomism
Author(s):

David Bostock

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199651443.003.0005

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

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