Introduction to Mathematical Philosophy

proposition we shall have an attribute or relation holding of or between one, two or more "subjects" in the extended sense. (A relation may have more than two terms: e.g. " gives to " is a relation of three terms.) Now it often happens that, on a closer scrutiny, the apparent subjects are found to be not really subjects, but to be capable of analysis; the only result of this, however, is that new subjects take their places. It also happens that the verb may grammatically be made subject: e.g. we may say, "Killing is a relation which holds between Brutus and Cæsar." But in such cases the grammar is misleading, and in a straightforward statement, following the rules that should guide philosophical grammar, Brutus and Cæsar will appear as the subjects and killing as the verb. We are thus led to the conception of terms which, when they occur in propositions, can only occur as subjects, and never in any other way. This is part of the old scholastic definition of substance; but persistence through time, which belonged to that notion, forms no part of the notion with which we are concerned. We shall define "proper names" as those terms which can only occur as subjects in propositions (using "subject" in the extended sense just explained). We shall further define "individuals" or "particulars" as the objects that can be named by proper names. (It would be better to define them directly, rather than by means of the kind of symbols by which they are symbolised; but in order to do that we should have to plunge deeper into metaphysics than is desirable here.) It is, of course, possible that there is an endless regress: that whatever appears as a particular is really, on closer scrutiny, a class or some kind of complex. If this be the case, the axiom of infinity must of course be true. But if it be not the case, it must be theoretically possible for analysis to reach ultimate subjects, and it is these that give the meaning of "particulars" or "individuals." It is to the number of these that the axiom of infinity is assumed to apply. If it is true of them, it is true [Pg 142] of classes of them, and classes of classes of them, and so on; similarly if it is false of them, it is false throughout this hierarchy. Hence it is natural to enunciate the axiom concerning them rather than concerning any other stage in the hierarchy. But whether the axiom is true or false, there seems no known method of discovering. [Pg 143]