The Principles of Mathematics (1903)

§ 110

Now this definition by abstraction, and generally the process employed in such definitions, suffers from an absolutely fatal formal defect: it does not show that only one object satisfies the definition[76]. Thus instead of obtaining one common property of similar classes, which is the number of the classes in question, we obtain a class of such properties, with no means of deciding how many terms this class contains. In order to make this point clear, let us examine what is meant, in the present instance, by a common property. What is meant is, that any class has to a certain entity, its number, a relation which it has to nothing else, but which all similar classes (and no other entities) have to the said number. That is, there is a many-one relation which every class has to its number and to nothing else. Thus, so far as the definition by abstraction can show, any set of entities to each of which some class has a certain many-one relation, and to one and only one of which any given class has this relation, and which are such that all classes similar to a given class have this relation to one and the same entity of the set, appear as the set of numbers, and any entity of this set of the number of some class. If, then, there are many such sets of entities--and it is easy to prove that there are an infinite number of them--every class will have many numbers, and the definition wholly fails to define the number of a class. This argument is perfectly general, and shows that definition by abstraction is never a logically valid process.(§ 110 ¶ 1)

§ 110 n. 1. On the necessity of this condition, cf. Padoa, loc. cit., p. 324. Padoa appears not to perceive, however, that all definitions define the single individual of a class; when what is defined is a class, this must be the only term of some class of classes.