The Principles of Mathematics (1903)

§ 105

Other ways of evading the contradiction, which might be suggested, appear undesirable, on the ground that they destroy too many quite necessary kinds of propositions. It might be suggested that identity is introduced in x is not an x in a way which is not permissible. But it has been already shown that relations of terms to themselves are unavoidable, and it may be observed that suicides or self-made men or the heroes of Smiles's Self-Help are all defined by relations to themselves. And generally, identity enters in a very similar way into formal implication, so that it is quite impossible to reject it.(§ 105 ¶ 1)

A natural suggestion for escaping from the contradiction would be to demur to the notion of all terms or of all classes. It might be urged that no such sum-total is conceivable; and if all indicates a whole, our escape from the contradiction requires us to admit this. But we have already abundantly seen that if this view were maintained against any term, all formal truth would be impossible, and Mathematics, whose characteristic is the statement of truths concerning any term, would be abolished at one stroke. Thus the correct statement of formal truths requires the notion of any term or every term, but not the collective notion of all terms.(§ 105 ¶ 2)

It should be observed, finally, that no peculiar philosophy is involved in the above contradiction, which springs directly from common sense, and can only be solved by abandoning some common-sense assumption. Only the Hegelian philosophy, which nourishes itself on contradictions, can remain indifferent, because it finds similar problems everywhere. In any other doctrine, so direct a challenge demands an answer, on pain of a confession of impotence. Fortunately, no other similar difficulty, so far as I know, occurs in any other portion of the Principles of Mathematics.(§ 105 ¶ 3)