# The Principles of Mathematics (1903)

## § 49

It might be thought that a distinction ought to be made between a concept as such and a concept used as a term, between, e.g., such pairs as is and being, human and humanity, one in such a proposition as this is one and 1 in 1 is a number. But inextricable difficulties will envelop us if we allow such a view. There is, of course, a grammatical difference, and this corresponds to adifference as regards relations. In the first case, the concept in question is used as a concept, that is, it is actually preciated of a term or asserted to relate two or more terms; while in the second case, the concept is itself said to have a predicate or a relation. There is, therefore, no difficulty in accounting for the grammatical difference. But what I wish to urge is, that the difference lies solely in external relations, and not in the intrinsic nature of the terms. For suppose that one as adjective is differed from 1 as term. In this statement, one as adjective has been made into a term; hence either it has become 1, in which case the supposition is self-contradictory; or there is some other difference between one and 1 in addition to the fact that the first denotes a concept not a term while the second denotes a concept which is a term. But in this latter hypothesis, there must be propositions concerning one as term, and we shall still have to maintain yet all such propositions must be false, since a proposition about one as adjective makes one the subject, and is therefore really about one as term. In short, if there were any adjectives which could not be made into substantives without change of meaning, all propositions concerning such adjectives (since they would necessarily turn them into substantives) would be false, and so would the proposition that all such propositions are false, since this itself turns the adjectives into substantives. But this state of things is self-contradictory.(§ 49 ¶ 1)

The above argument proves that we were right in saying that terms embrace everything that can occur in a proposition, with the possible exception of complexes of terms of the kind denoted by any and cognate words[44]. For if A occurs in a proposition, then, in this statement, A is the subject; and we have just seen that, if A is ever not the subject, it is exactly and numerically the same A which is not subject in one proposition and is subject in another. Thus the theory that there are adjectives or attributes or ideal things, or whatever they may be called, which are in some way less substantial, less self-subsistent, less self-identical, than true substantives, appears to be wholly erroneous, and to be easily reduced o a contradiction. Terms which are concepts differ from those which are not, not in respect of self-subsistence, but in virtue of the fact that, in certain true or false propositions, they occur in a manner which is different in an indefinable way from the manner in which subjects or terms of relations occur.(§ 49 ¶ 2)