and depth, for extension and comprehension respectively, and shall employ them in different senses, which I shall distinguish by different adjectives.
By the informed breadth of a term, I shall mean all the real things of which it is predicable, with logical truth on the whole in a supposed state of information. By the phrase “on the whole” I mean to indicate that all the information at hand must be taken into account, and that those things of which there is not on the whole reason to believe that a term is truly predicable are not to be reckoned as part of its breadth.
If T be a term which is predicable only of S′, S″, and S‴, then the S′s, the S″‘s, and the S‴’s will constitute the informed breadth of T. If at the same time, S′ and S″ are the subjects of which alone another term T can be predicated, and if it is not known that all S‴’s are either S′ or S″, then T is said to have a greater informed breadth than T. If the S‴s are known not to be all among the S′’s and S″’s, this excess of breadth may be termed certain, and, if this is not known, it may be termed doubtful. If there are known to be S‴’s, not known to be S′’s or S″’s, T is said to have a greater actual breadth than T’; but if no S″’s are known except such are known to be S′’s, and S″’s (though there may be others), T is said to have a greater potential breadth than T′. If T and T′ are conceptions in different minds, or in different states of the same mind, and it is known to the mind which conceives T that every S‴ is either S″ or S′, then T is said to be more extensively distinct than T′19
By the informed depth of a term, I mean all the real characters (in contradistinction to mere names) which can be predicated of it20 (with logical truth, on the whole) in a supposed state of information; no character being counted twice over knowingly in the supposed state of information. The depth, like the breadth, may be certain or doubtful, actual or potential, and there is a comprehensive distinctness corresponding to extensive distinctness.
The informed breadth and depth suppose a state of information which lies somewhere between two imaginary extremes. These are, first, the state in which no fact would be known, but only the meaning of terms; and, second, the state in which the information would amount to an absolute intuition of all there is, so that the things we should know would be the very substances themselves, and the qualities we should know would be the very concrete forms themselves. This suggests two other sorts of breadth and depth corresponding to these two states of information, and which I shall term respectively the essential and the substantial breadth and depth.
By the essential depth of a term, then, I mean the really conceivable qualities predicated of it in its definition.
The defined term will not perhaps be applicable to any real objects whatever. Let, for example, the definition of the term T be this,
Any T is both P′ and P″ and P‴,
then this sums up its whole meaning; and, as it may not be known that there is any such thing as P′, the meaning of T does not imply that it exists. On the other hand, we know that neither P′, P″, nor P‴ is coextensive with the whole sphere of being. For they are determinate qualities, and it is the very meaning of being that it is indeterminate, that is, is more extensive than any determinate term. In fact, P′, for example, is a real notion which we never could have except by means of its contrast to something else. Hence we must know that
Whatever is not-P′ is not-T,
Whatever is not-P″ is not-T,
and Whatever is not-P‴ is not-T.
Thus if we define the essential breadth of a term as those real things of which, according to its very meaning, a term is predicable, not-T has an essential breadth. We may therefore divide all terms into two classes, the essentially affirmative or positive and the essentially negative; of which the former have essential depth, but no essential breadth, and the latter essential breadth, but no essential depth. It must be noted, however, that this division is not the same as the similar one which language makes. For example, being, according to this, is an essentially negative term, in as much as it means that which can be predicated of whatever you please, and so has an essential breadth; while nothing is an essentially positive term, in as much as it means that of which you are at liberty to predicate what you please, and therefore has an essential depth. The essential subjects of being cannot be enumerated, nor the essential predicates of nothing.
In essential breadth or depth, no two terms can be equal; for, were that the case, the two terms would have the same meaning, and therefore, for logical purposes, would be the same term. Two terms may have unknown relations in these quantities, on account of one or other of them not being distinctly conceived.
Substantial breadth is the aggregate of real substances of which alone a term is predicable with absolute truth. Substantial depth is the real concrete form which belongs to everything of which a term is predicable with absolute truth.
General terms denote several things. Each of these things has in itself no qualities, but only a certain concrete form which belongs to itself alone. This was one of the points brought out in the controversy in reference to the nature of universals.21 As Sir William Hamilton says, not even the humanity of Leibniz belongs to Newton, but a different humanity. It is only by abstraction, by an oversight, that two things can be said to have common characters. Hence, a general term has no substantial depth. On the other hand, particular terms, while they have substantial depth, in as much as each of the things, one or other of which are predicated of them, has a concrete form, yet have no substantial breadth, in as much as there is no aggregate of things to which alone they are applicable. In order to place this matter in a clearer light, I must remark, that I, in common with most logicians, take the copula in the sense of a sign of attribution, and not, like Hamilton, in the sense of a sign of equality in extension or comprehension. He exposes the proposition, “man is an animal,” thus:—
| The extension of man | Subject. |
| equals | Copula. |
| a part or all of the extension of animal | Predicate. |
And thus he makes the predicate particular. Others interpret it thus:—
| Every man | Subject. |
| has all the attributes common to | Copula. |
| every animal | Predicate. |
It is in this latter sense that the copula is considered in this paper. Now, a particular is, as has been said, an alternative subject. Thus, “Some S is M” means, if S′, S″, and S‴ are the singular S′s, that “either S′, or else S″, or else S‴, has all the attributes belonging to M.” A particular term, then, has a substantial depth, because it may have a predicate which is absolutely concrete, as in the proposition, “Some man is Napoleon.” But if we put the particular into the predicate we have such a proposition as this: “M has all the attributes belonging to S′, or else all those belonging to S″, or else all those belonging to S″.” And this can never be true unless M is a single individual. Now a single individual substance is, I will not say an atom, but the smallest part of an atom, that is, nothing at all. So that a particular can have no substantial breadth. Now take the universal term “S.” We can say, “Any S is M,” but not if M is a real concrete quality. We cannot say, for instance, “Any man is Napoleon.” On the other hand, we can say “Any