The Principles of Mathematics (1903)

§ 62

It appears from the above discussion that, whether there are different ways of denoting or not, the objects denoted by all men, every man, etc. are certainly distinct. It seems therefore legitimate to say that the whole difference lies in the objects, and that denoting itself is the same in all cases. There are, however, amny difficult problems connected with the subject, especially as regards the nature of the objects denoted. All men, which I shall identify with the class of men, seems to be an unambiguous object, although grammatically it is in the plural. but in the other cases the question is not so simple: we may doubt whether an ambiguous object is unambiguously denoted, or a definite object ambiguously denoted. Consider again the proposition I met a man. It is quite certain, and is implied by this proposition, that what I met was an unambiguous perfectly definite man: in the technical language which is here adopted, the proposition is expressed by I met some man. But the actual man whom I met forms no part of the proposition in question,and is not specially denoted by some man. Thus the concrete event which happened is not asserted in the proposition. What is asserted is merely that some one of a class of concrete events took place. The whole human race is involved in my assertion: if any man who ever existed or will exist had not existed or been going to exist, the purport of my proposition would have been different. Or, to put the same point in more intensional language, if I substitute for man any of the other class-concepts applicable to the individual whom I had the honour to meet, my proposition is changed, although the individual in question is just as much denoted as before. What this proves is, that some man must not be regarded as actually denoting Smith and actually denoting Brown, and so on: the whole procession of human beings throughout the ages is always relevant to every proposition in which some man occurs, and what is denoted is essentially not each separate man, but a kind of combination of all men. This is more evident in the case of every, any, and a. There is, then, a definite something, different in each of the five cases, which must, in a sense, be an object, but is characterized as a set of terms combined in a certain way, which something is denoted by all men, every man, any man, a man, or some man; and it is with this very paradoxical object that propositions are concerned in which the corresponding concept is used as denoting.(§ 62 ¶ 1)