# The Principles of Mathematics (1903)

## § 48

Among terms, it is possible to distinguish two kinds, which I shall call respectively things and concepts. The former are the terms indicated by proper names, the latter those indicated by all other words. Here proper names are to be understood in a somewhat wider sense than is usual, and things also are to be understood as embracing all particular points and instants, and many other entities not commonly called things. Among concepts, again, two kinds at least must be distinguished, namely those indicated by adjectives and those indicated by verbs. The former kind will often be called predicates or class-concepts; the latter are always or almost always relations. (In intransitive verbs, the notion expressed by the verb is complex, and usually associates a definite relation to an indefinate relatum, as in Smith breathes.)(§ 48 ¶ 1)

In a large class of propositions, we agreed, it is possible, in one or more ways, to distinguish a subject and an assertion about the subject. The assertion must always contain a verb, but except in this respect, assertions appear to have no universal properties. In a relational proposition, say A is greater than B, we may regard A as the subject and is greater than B as the assertion, or B as the subject and A is greater than as the assertion. There are thus, in the case proposed, two ways of analyzing the proposition into subject and assertion. Where a relation has more than two terms, as in A is here now,[42] there will be more than two ways of making the analysis. But in some propositions, there is only a single way: these are the subject-predicate propositions, such as Socrates is human. The proposition humanity belongs to Socrates, which is equivalent to Socrates is human, is an assertion about humanity; but it is a distinct proposition. In Socrates is human, the notion expressed by human occurs in a different way from that in which it occurs when it is called humanity, the difference being that in the latter case, but not in the former, the proposition is about this notion. This indicates that humanity is a concept, not a thing. I shall speak of the terms of a proposition as those terms, however numerous, which occur in a proposition and may be regarded as subjects about which the proposition is. It is characteristic of the terms of a proposition that any one of them may be replaced by any other entity without our ceasing to have a proposition. Thus we shall say that Socrates is a human is a proposition having only one term; of the remaining components of the proposition, one is the verb, the other is a predicate. With the sense which is has in this proposition, we no longer have a proposition at all if we replace human by something other than a predicate. Predicates, then, are concepts, other than verbs, which occur in propositions having only one term or subject. Socrates is a thing, because Socrates can never occur otherwise than as term in a proposition: Socrates is not capable of that curious twofold use which is involved in human and humanity. Points, instants, bits of matter, particular states of mind, and particular existents generally, are things in the above sense, and so are many terms which do not exist, for example, the points in non-Euclidean space and the pseudo-existents of a novel. All classes, it would seem, as numbers, men, spaces, etc., when taken as single terms, are things; but this is a point for Chapter VI.(§ 48 ¶ 2)

Predicates are distinguished from other terms by a number of very interesting properties, chief among which is their connection with what I shall call denoting. One predicate always gives rise to a host of cognate notions: thus in addition to human and humanity, which only differ grammatically, we have man, a man, some man, any man, every man, all men[43], all of which appear to be genuinely distinct one from another. The study of these various notions is absolutely vital to any philosophy of mathematics; and it is on account of them that the theory of predicates is important.(§ 48 ¶ 3)

§ 48 n. 1. This proposition means A is in this place at this time. It will be shown in Part VII that the relation expressed is not reducible to a two-term relation.

§ 48 n. 2. I use all men as collective, i.e. as nearly synonymous with the human race, but differing therefrom by being many and not one. I shall always use all collectively, confining myself to every for the distributive sense. Thus I shall say every man is mortal, not all men are mortal.