The Principles of Mathematics (1903)

§ 77

A relation which, before Peano, was almost universally confounded with , is the relation of inclusion between classes, as e.g. between men and mortals. This is a time-honored relation, since it occurs in the traditional form of the syllogism: it has been a battle-ground between intension and extension, and has been so much discussed that it is astonishing how much remains to be said about it. Empiricists hold that such propositions mean an actual enumeration of the terms of the contained class. They must, it is to be inferred, regard it as doubtful whether all primes are integers, since they will scarcely have the face to say that they have examined all primes one by one. Their opponents have usually held, on the contrary, that what is meant is a relation of whole and part between the defining predicates, but turned in the opposite sense from the relation, between the classes: i.e. the defining predicate of the larger class is part of that of the smaller. This view seems far more defensible than the other; and wherever such a relation does hold between the defining predicates, the relation of inclusion follows. But two objections may be made, first, that in some cases of inclusions there is no such relation between the defining predicates, and secondly, that in any case what is meant is a relation between the classes, not a relation of their defining predicates. The first point may be easily established by instances. The concept even prime does not contain as a constituent the concept integer between 1 and 10; the concept English King whose head was cut off does not contain the concept people who died in 1649; and so on through innumerable obvious cases. This might be met by saying that, though the relation of the defining predicates is not one of whole and part, it is one more or less analogous to implication, and is always what is really meant by propositions of inclusion. Such a view represents, I think, what is said by the better advocates of intension, and I am not concerned to deny that a relation of the kind in question does always subsist between defining predicates of classes one of which is contained in the other. But the second of the above points remains valid as against any intensional interpretation. When we say that men are mortals, it is evident that we are saying something about men, not about the concept man or the predicate human. The question is, then, what exactly are we saying?(§ 77 ¶ 1)

Peano held, in earlier editions of his Formulaire, that what is asserted is the formal implication x is a man implies x is mortal. This is certainly implied, but I cannot persuade myself that it is the same proposition. For in this proposition, as we saw in Chapter III, it is essential that x should take all values, and not only such as are men. But when we say all men are mortals, it seems plain that we are only speaking of men, and not of all other imaginable terms. We may, if we wish for a genuine relation of classes, regard the assertion as one of whole and part between the two classes, each considered as a single term. Or we may give a still more purely extensional form to our proposition, by making it mean: Every (or any) man is a mortal. This proposition raises very interesting questions in the theory of denoting: for it appears to assert an identity, yet it is plain that what is denoted by every man is different from what is denoted by a mortal. These questions, however, interesting as they are, cannot be pursued here. It is only necessary to realize clearly what are the various equivalent propositions involved where one class is included in another. The form most relevant to Mathematics is certainly the one with formal implication, which will receive a fresh discussion in the following chapter.(§ 77 ¶ 2)

Finally, we must remember that classes are to be derived, by means of the notion of such that, from other sources than subject-predicate propositions and their equivalents. Any propositional function in which a fixed assertion is made of a variable term is to be regarded, as was explained in Chapter II, as giving rise to a class of values satisfying it. This topic requires a discussion of assertions; but one strange contradiction, which necessitates the care in discrimination aimed at in the present chapter, may be mentioned at once.(§ 77 ¶ 3)