Two terms may be equal in their substantial breadth and depth, and differ in their essential breadth and depth. But two terms cannot have relations of substantial breadth and depth which are unknown in the state of information supposed, because in that state of information everything is known.
In informed breadth and depth, two terms may be equal, and may have unknown relations. Any term, affirmative or negative, universal or particular, may have informed breadth or depth.
§6. The Conceptions of Quality, Relation, and Representation, applied to this Subject
In a paper presented to the Academy last May, I endeavored to show that the three conceptions of reference to a ground, reference to a correlate, and references to an interpretant, are those of which logic must principally make use. I there also introduced the term “symbol,” to include both concept and word. Logic treats of the reference of symbols in general to their objects. A symbol, in its reference to its object, has a triple reference:—
1st, Its direct reference to its object, or the real things which it represents;
2d, Its reference to its ground through its object, or the common characters of those objects;
3d, Its reference to its interpretant through its object, or all the facts known about its object.
What are thus referred to, so far as they are known, are:—
1st, The informed breadth of the symbol;
2d, The informed depth of the symbol;
3d, The sum of synthetical propositions in which the symbol is subject or predicate, or the information concerning the symbol.
By breadth and depth, without an adjective, I shall hereafter mean the informed breadth and depth.
It is plain that the breadth and depth of a symbol, so far as they are not essential, measure the information concerning it, that is, the synthetical propositions of which it is subject or predicate. This follows directly from the definitions of breadth, depth, and information. Hence it follows:—
1st, That, as long as the information remains constant, the greater the breadth, the less the depth;
2d, That every increase of information is accompanied by an increase in depth or breadth, independent of the other quantity;
3d, That, when there is no information, there is either no depth or no breadth, and conversely.
These are the true and obvious relations of breadth and depth. They will be naturally suggested if we term the information the area, and write—
Breadth x Depth = Area.
If we learn that S is P, then, as a general rule, the depth of S is increased without any decrease of breadth, and the breadth of P is increased without any decrease of depth. Either increase may be certain or doubtful.
It may be the case that either or both of these increases does not take place. If P is a negative term, it may have no depth, and therefore adds nothing to the depth of S. If S is a particular term, it may have no breadth, and then adds nothing to the breadth of P. This latter case often occurs in metaphysics, and, on account of not-P as well as P being predicated of S, gives rise to an appearance of contradiction where there really is none; for, as a contradiction consists in giving to contradictory terms some breadth in common, it follows that, if the common subject of which they are predicated has no real breadth, there is only a verbal, and not a real contradiction. It is not really contradictory, for example, to say that a boundary is both within and without what it bounds. There is also another important case in which we may learn that “S is P,” without thereby adding to the depth of S or the breadth of P. This is when, in the very same act by which we learn that S is P, we also learn that P was covertly contained in the previous depth of S, and that consequently S was a part of the previous breadth of P. In this case, P gains in extensive distinctness and S in comprehensive distinctness.
We are now in condition to examine Vorländer’s objection to the inverse proportionality of extension and comprehension. He requires us to think away from an object all its qualities, but not, of course, by thinking it to be without those qualities, that is, by denying those qualities of it in thought. How then? Only by supposing ourselves to be ignorant whether it has qualities or not, that is, by diminishing the supposed information; in which case, as we have seen, the depth can be diminished without increasing the breadth. In the same manner we can suppose ourselves to be ignorant whether any American but one exists, and so diminish the breadth without increasing the depth.
It is only by confusing a movement which is accompanied with a change of information with one which is not so, that people can confound generalization, induction, and abstraction. Generalization is an increase of breadth and a decrease of depth, without change of information. Induction is a certain increase of breadth without a change of depth, by an increase of believed information. Abstraction is a decrease of depth without any change of breadth, by a decrease of conceived information. Specification is commonly used (I should say unfortunately) for an increase of depth without any change of breadth, by an increase of asserted information. Supposition is used for the same process when there is only a conceived increase of information. Determination, for any increase of depth. Restriction, for any decrease of breadth; but more particularly without change of depth, by a supposed decrease of information. Descent, for a decrease of breadth and increase of depth, without change of information.
Let us next consider the effect of the different kinds of reasoning upon the breadth, depth, and area of the two terms of the conclusion.
In the case of deductive reasoning it would be easy to show, were it necessary, that there is only an increase of the extensive distinctness of the major, and of the comprehensive distinctness of the minor, without any change in information. Of course, when the conclusion is negative or particular, even this may not be effected.
Induction requires more attention. Let us take the following example:—
S′, S″, S‴, and Siv have been taken at random from among the M’s;
S′, S″, S‴, and Siv are P:
∴ any M is P.
We have here, usually, an increase of information. M receives an increase of depth, P of breadth. There is, however, a difference between these two increases. A new predicate is actually added to M; one which may, it is true, have been covertly predicated of it before, but which is now actually brought to light. On the other hand, P is not yet found to apply to anything but S′, S″, S‴, and Siv, but only to apply to whatever else may hereafter be found to be contained under M. The induction itself does not make known any such thing. Now take the following example of hypothesis:—
M is, for instance, P′, P″, P‴ and Piv;
S is P′, P″, P‴m and Piv:
∴ S is all that M is.
Here again there is an increase of information, if we suppose the premises to represent the state of information before the inferences. S receives an addition to its depth; but only a potential one, since there is nothing to show that the M’s have any common characters besides P′, P″, P‴, and Piv. M, on the other hand, receives an actual increase of breadth in S, although, perhaps, only a doubtful one. There is, therefore, this important difference between induction and hypothesis, that the former potentially increases the breadth of one term, and actually increases the