The Principles of Mathematics (1903)

§ 56

The notion of denoting, like most of the notions of logic, has been obscured hitherto by an undue admixture of psychology. There is a sense in which we denote, when we point or describe, or employ words as symbols for concepts; this, however, is not the sense that I wish to discuss. But the fact that description is possible--that we are able, by the employment of concepts, to designate a thing which is not a concept--is due to a logical relation between some concepts and some terms, in virtue of which such concepts inherently and logically denote such terms. It is this sense of denoting which is here in question. This notion lies at the bottom (I think) of all theories of substance, of the subject-predicate logic, and of the opposition between things and ideas, discursive thought and immediate perception. These various developments, in the main, appear to me mistaken, while the fundamental fact itself, out of which they have grown, is hardly ever discussed in its logical purity.(§ 56 ¶ 1)

A concept denotes when, if it occurs in a proposition, the proposition is not about the concept, but about a term connected in a certain peculiar way with the concept. If I say, I met a man, the proposition is not about a man: this is a concept which does not walk the streets, but lives in the shadowy limbo of the logic-books. What I met was a thing, not a concept, an actual man with a tailor and a bank-account or a public-house and a drunken wife. Again, the proposition any finite number is odd or even is plainly true; yet the concept any finite number is neither odd nor even. It is only particular numbers that are odd or even; there is not, in addition to these, another entity, any number, which is either odd or even, and if there were, it is plain that it could not be odd and could not be even. Of the concept any number, almost all the propositions that contain the phrase any number are false. If we wish to speak of the concept, we have to indicate the fact by italics or inverted commas. People often assert that man is mortal; but what is mortal will die, and yet we should be surprised to find in the Times a notice such as the following: Died at his residence of Camelot, Gladstone Road, Upper Tooting, on the 18th of June 19--, Man, eldest son of Death and Sin. Man, in fact, does not die; hence if man is mortal were, as it appears to be, a proposition about man, it would be simply false. The fact is, the proposition is about men; and here again, it is not about the concept men, but about what this concept denotes. The whole theory of definition, of identity, of classes, of symbolism, and of the variable is wrapped up in the theory of denoting. The notion is a fundamental notion of logic, and, in spite of its difficulties, it is quite essential to be as clear about it as possible.(§ 56 ¶ 2)