impredicative definition the definition of a concept in terms of the totality to which it belongs. Russell, in the second (1925) edition of Principia Mathematica, introduced the term ‘impredicative’, prohibiting this kind of definition from the conceptual foundations of mathematics, on the grounds that they imply formal logical paradoxes. The impredicative definition of the set R of all sets that are not members of themselves in Russell’s paradox leads to the self-contradictory conclusion that R is a member of itself if and only if it is not a member of itself. To avoid antinomies of this kind in the formalization of logic, Russell first implemented in his ramified type theory the vicious circle principle, that no whole may contain parts that are definable only in terms of that whole. The limitation of ramified type theory is that without use of impredicative definitions it is impossible to quantify over all mathematical objects, but only over all mathematical objects of a certain order or type. Without being able to quantify over all real numbers generally, many of the most important definitions and theorems of classical real number theory cannot be formulated. Russell for this reason later abandoned ramified in favor of simple type theory, which avoids the logical paradoxes without outlawing impredicative definition by forbidding the predication of terms of any type (object, property and relation, higher-order properties and relations of properties and relations, etc.) to terms of the same type. See also DEFINITION , PHILOSOPHY OF MATHEMATICS, QUANTIFICATION, SET-THEO- RETIC PARADOXES , TYPE THEOR. D.J.