The Principles of Mathematics (1903)

§ 58

There is, connected with every predicate, a great variety of closely allied concepts, which, in so far as they are distinct, it is important to distinguish. Starting, for example, with human, we have man, men, all men, every man, any man, the human race, of which all except the first are twofold, a denoting concept and an object denoted; we have also, less closely analogous, the notions a man and some man, which again denote objects[49] other than themselves. This vast apparatus connected with every predicate must be borne in mind, and an endeavour must be made to give an analysis of all the above notions. But for the present it is the property of denoting, rather than the various denoting concepts, that we are concerned with.(§ 58 ¶ 1)

The combination of concepts as such to form new concepts, of greater complexity than their constituents, is a subject upon which writers on logic have said many things. But the combination of terms as such, to form what by analogy may be called complex terms, is a subject upon which logicians, old and new, give us only the scantiest discussion. Nevertheless, the subject is of vital importance to the philosophy of mathematics, since the nature both of number and of the variable turns upon just this point. Six words, of constant occurrence in daily life, are also characteristic of mathematics: these are the words all, every, any, a, some, and the. For correctness of reasoning, it is essential that these words should be sharply distinguished from one another; but the subject bristles with difficulties, and is almost wholly neglected by logicians[50].(§ 58 ¶ 2)

It is plain, to begin with, that a phrase containing one of the above six words always denotes. It will be convenient, for the present discussion, to distinguish a class-concept from a predicate; I shall call human a predicate, and man a class-concept, though the distinction is perhaps only verbal. The characteristic of a class-concept, as distinguished from terms in general, is that x is a u is a propositional function when, and only when, u is a class-concept. It must be held that when u is not a class-concept, we do not have a false proposition, but simply no proposition at all, whatever value we may give to x. This enables us to distinguish a class-concept belonging to the null-class, for which all propositions of the above form are false, from a term which is not a class-concept at all, for which there are no propositions of the above form. Also it makes it plain that a class-concept is not a term in the proposition x is a u, for u has a restricted variability if the formula is to remain a proposition. A denoting phrase, we may now say, consists always of a class-concept preceded by one of the above six words or some synonym of them.(§ 58 ¶ 3)

§ 58 n. 1. I shall use the word object in a wider sense than term, to cover both singular and plural, and also cases of ambiguity, such as a man. The fact that a word can be framed with a wider meaning than term raises grave logical problems. Cf. § 47.

§ 58 n. 2. On the indefinite article, some good remarks are made by Meinong, Abstrahiren und Vergleichen, Zeitschrift für Psychologie und Physiologie der Sinnesorgane, Vol. XXIV, p. 63.