ABSTRACT
Let w be a class-concept which can be asserted of itself, i.e. such that “w is a w”. Instances are class-concept, and the negations of ordinary class-concepts, e.g. not-man. Then (α ) if w be contained in another class v, since w is a w, w is a v; consequently there is a term of v which is a class-concept that can be asserted of itself. Hence by contraposition, (β) if u be a class-concept none of whose members are class-concepts that can be asserted of themselves, no classconcept contained in u can be asserted of itself. Hence further, (γ) if u be any class-concept whatever, and u' the class-concept of those members of u which are not predicable of themselves, this class-concept is contained in itself, and none of its members are predicable of themselves; hence by (β) u' is not predicable of itself. Thus u' is not a u' , and is therefore not a u; for the terms of u that are not terms of u' are all predicable of themselves, which u' is not. Thus (δ) if u be any class-concept whatever, there is a class-concept contained in u which is not a member of u, and is also one of those class-concepts that are not predicable of themselves. So far, our deductions seem scarcely open to
question. But if we now take the last of them, and admit the class of those class-concepts that cannot be asserted of themselves, we find that this class must contain a class-concept not a member of itself and yet not belonging to the class in question.
