# The Principles of Mathematics (1903)

## § 499

The first point may be illustrated by somewhat simpler ones. There are, we know, more classes than individuals; but predicates are individuals. Consequently not all classes have defining predicates. This result, which is also deducible from the Contradiction, shows how necessary it is to distinguish classes from predicates, and to adhere to the extensional view of classes. Similarly there are more ranges of couples than there are couples, and therefore more than there are individuals; but verbs, which express relations intensionally, are individuals. Consequently not every range of couples forms the extension of some verb, although every such range forms the extension of some propositional function containing two variables. Although, therefore, verbs are essential in the logical genesis of such propositional functions, the intensional standpoint is inadequate to give all the objects which Symbolic Logic regards as relations.(§ 499 ¶ 1)

In the case of propositions, it seems as though there were always an associated verbal noun which is an individual. We have x is identical with x and the self-identity of x, x differs from y, and the difference of x and y; and so on. The verbal noun, which is what we called the propositional concept, appears on inspection to be an individual; but this is impossible, for the self-identity of x has as many values as there are objects, and therefore more values than there are individuals. This results from the fact that there are propositions concerning every conceivable object, and the definition of identity shows (§ 26) that every object concerning which there are propositions, is identical with itself. The only method of evading this difficulty is to deny that propositional concepts are individuals; and this seems to be the course to which we are driven. In is undeniable, however, that a propositional concept and a colour are two objects; hence we shall have to admit that it is possible to form mixed ranges, whose members are not all of the same type, but such ranges will always be of a different type from what we may call pure ranges, i.e. such as have only members of one type. The propositional concept seems, in fact, to be nothing other than the proposition itself, the difference being merely the psychological one that we do not assert the proposition in one case, and do assert it in the other.(§ 499 ¶ 2)