Nonpredicative Definition

a definition by means of which there is created or introduced for consideration a subject that is one of the meanings of an indefinite noun (the “variable”) that is part of the defining expression.

The incorrectness of a nonpredicative definition lies in the fact that the subject introduced by means of the definition can alter by its introduction the meaning of the defining expression and, thus, the meaning of the very subject to be defined. When this possibility does not come about (as is the case if all entries of the aforementioned indefinite noun are unessential, that is, can be eliminated by logical means), the incorrectness of the non-predicative definition may be disregarded, but in such cases the problem of the nonpredicative definition also does not arise. If just one entry of an indefinite noun is irremovable, then the object created by the definition takes part in its own definition as one of the meanings of the noun, and the definition is faulty, insofar as it does not effect reduction of the defined object to previously known objects and concepts. From the point of view of the theory of definitions, such fallacious nonpredicative definitions should be considered just as impermissible as circular arguments.

H. Poincaré was the first to point out nonpredicative definitions in mathematical analysis. He also introduced the term “nonpredicative definition.” The best known examples of nonpredicative definitions are encountered in “naïve” classical attempts to substantiate axiomatic set theory. For example, the proof of the existence of the union (set-theoretic sum) of an arbitrary set of sets is nonpredicative (since in defining set the word “set” is found twice in the defining expression). Various means (modifications of naïve set theory), in particular the theory of types, have been proposed in order to avoid the difficulties associated with this problem.