Full Idea
The Axiom of Reducibility avoids impredicativity, by asserting that for any predicate of given arguments defined by quantifying over higher-order functions or classes, there is another co-extensive but predicative function of the same type of arguments.
Gist of Idea
Reducibility says any impredicative function has an appropriate predicative replacement
Source
Bernard Linsky (Russell's Metaphysical Logic [1999], 1)
Book Reference
Linsky,Bernard: 'Russell's Metaphysical Logic' [CSLI 1999], p.4
A Reaction
Eventually the axiom seemed too arbitrary, and was dropped. Linsky's book explores it.
Related Idea
Idea 21704 'Impredictative' definitions fix a class in terms of the greater class to which it belongs [Linsky,B]