The Principles of Mathematics (1903)

§ 44

We can now see how, where the analysis in to subject and assertion is legitimate, to distinguish implications in which there is a term which can be varied from others in which this is not the case. Two ways of making the distinction may be suggested, and we shall have to decide between them. It may be said that there is a relation between the two assertions is a man and is a mortal, in virtue of which, when the one holds, so does the other. Or again, we may analyze the whole proposition Socrates is a man implies Socrates is a mortal into Socrates and an assertion about him, and say that the assertion in question holds of all terms. Neither of these theories replaces the above analysis of x is a man implies x is a mortal into a a class of material implications; but whichever of the two is true carries the analysis one step further. The first theory suffers from the difficulty that it is essential to the relation of assertions involved that both assertions should be made of the same subject, though it is otherwise irrelevant what subject we choose. The second theory appears objectionable on the ground that the suggested analysis of Socrates is a man implies Socrates is a mortal seems scarcely possible. The proposition in question consists of two terms and a relation, the terms being Socrates is a man and Socrates is a mortal; and it would seem that when a relational proposition is analyzed into a subject and an assertion, the subject must be one of the terms of the relation which is asserted. This objection seems graver than that against the former view; I shall therefore, at any rate for the present, adopt the former view, and regard formal implication as derivd from a relation between assertions.(§ 44 ¶ 1)

It is important to realize that, according to the above analysis of formal implication, the notion of every term is indefinable and ultimate. A formal implication is one which holds of every term, and therefore every cannot be explained by means of formal implication. If a and b be classes, we can explain every a is a b by means of x is an a implies x is a b; but the every which occurs here is a derivative and subsequent notion, presupposing the notion of every term. It seems to be the very essence of what may be called a formal truth, and of formal reasoning generally, that some assertion is affirmed to hold of every term; and unless the notion of every term is admitted, formal truths are impossible.(§ 44 ¶ 2)