# The Principles of Mathematics (1903)

## § 60

When a class-concept a is given, it must be held that the various terms belonging to the class are also given. That is to say, any term being proposed, it can be decided whether or not it belongs to the class. In this way, a collection of terms can be given otherwise than by enumeration. Whether a collection can be given otherwise than by enumeration or by a class-concept, is a question which, for the present, I leave undetermined. But the possibility of giving a collection by a class concept is highly important, since it enables us to deal with infinite collections, as we shall see in Part V. For the present, I wish to examine the meaning of such phrases as all a's, every a, any a, an a, and some a. All a's, to begin with, denotes a numerical conjunction; it is definite as soon as a is given. The concept all a's is a perfectly definite single concept, which denotes the terms of a taken all together. The terms so taken have a number, which may thus be regarded, if we choose, as a property of the class-concept, since it is determinate for any given class-concept. Every a, on the contrary, though it still denotes all the a's, denotes them in a different way, i.e. severally instead of collectively. Any a denotes only one a, but it is wholly irrelevant which it denotes and what is said will be equally true whichever it may be. Moreover, any a denotes a variable a, that is, whatever particular a we may fasten upon, it is certain that any a does not denote that one; and yet of that one any proposition is true which is true of any a. An a denotes a variable disjunction: that is to say, a proposition which holds of an a may be false concerning each particular a, so that it is not reducible to a disjunction of propositions. For example, a point lies between any point and any other point; but it would not be true of any one particular point that it lay between any point and any other point, since there would be many pairs of points between which it did not lie. This brings us finally to some a, the constant disjunction. This denotes just one term of the class a, but the term it denotes may be any term of the class. Thus some moment does not follow any moment would mean that there was a first moment in time, while a moment precedes any moment means the exact opposite, namely, that every moment has predecessors.(§ 60 ¶ 1)