# The Principles of Mathematics (1903)

## § 101

Let us leave these paradoxical consequences, and attempt the exact statement of the contradiction itself. We have first the statement in terms of predicates, which has been given already. If x be a predicate, x may or may not be predicable of itself. Let us assume that not-predicable of oneself is a predicate. Then to suppose either that this predicate is, or that it is not, predicable of itself, is self-contradictory. The conclusion, in this case, seems obvious: not-predicable of oneself is not a predicate.(§ 101 ¶ 1)

Let us now state the same contradiction in terms of class-concepts. A class-concept may or may not be a term of its own extension. Class-concept which is not a term of its own extension appears to be a class-concept. But if it is a term of its own extension, it is a class-concept which is not a term of its own extension, and vice versâ. Thus we must conclude, against appearances, that class-concept which is not a term of its own extension is not a class-concept.(§ 101 ¶ 2)

In terms of classes the contradiction appears even more extraordinary. A class as one may be a term of itself as many. Thus the class of all classes is a class; the class of all the terms that are not men is not a man, and so on. Do all the classes that have this property form a class? If so, is it as one a member of itself as many or not? If it is, then it is one of the classes which, as ones, are not members of themselves as many, and vice versâ. Thus we must conclude again that the classes which as ones are not members of themselves as many do not form a class--or rather, that they do not form a class as one, for the argument cannot show that they do not form a class as many.(§ 101 ¶ 3)