Impredicative definition
Impredicative definition
Poincare in a proposed resolution (1906) of the paradoxes of Burali-Forti and Richard (see Paradoxes, logical), introduced the principle thnt, in making a definition of a particular member of any class, no reference should be allowed to the totality of members of that class. Definitions in violation of this principle were called impredicative (non predicatives) and were held to involve a vicious circle.
The prohibition against impredicative definition was incorporated by Russell into his ramified theory of types (1908) and is now usually identified with the restriction to the ramified theorv of types without the axiom of reducibility. (Poincare, however, never made his principle exact and may have intended, vaguely, a less severe restriction than this — as indeed some passages in later writings would indicate.) — A. C.
H. Poincare, Les mathematiques et la logique. Revue de Metaphysique et de Morale, vol. 14 (1906), pp. 294-317. R. Carnap, The Logical Syntax of Language, New York and London, 1937.