Charles S. Peirce
Chapter XI. On Logical Breadth and Depth
MS 233 (Robin 384): Writings 3, 98-102
As Logic is the study of the laws of signs so far as these denote thingsthose laws of signs which determine what things they denote and what they do notit is necessary in Logic to pay especial attention to those terms which denote signs. Such terms are genus species &c. No thing is a genus but as there are terms such as man and tree which denote some one thing leaving it more or less indeterminate what one so we may speak of whatever may be denoted by such a general term as a genus or class. Such terms are called "terms of second intention". The first intention is the mental act by which an object is conceived. The second intention is the mental act by which the first conception is made an object of conception in reference to its relation to its object. A term of second intention does not so much signify the sign itself as it signifies whatever is denoted by a sign of a certain description. As signs differ in their logical characters we may define an object by means of the logical characters of the sign which denotes it and in that case it is pointed out with a peculiar kind of generality which requires special attention. Two of the most important characters of general terms are their logical breadth and depth. The breadth of a term in general is that of which the term can be predicated. The depth of a term is that which can be predicated of it. The breadth therefore may be considered as a collection of objectsreal thingsthough it can also be considered as consisting of the terms which may be made subject of a true proposition of which the given term is the predicate. The depth of a term cannot be considered as a collection of things but can only be considered as a complex of terms or of attributes. The term attribute, character, mark, or quality is a term of second intention. Two things are alike in a certain respect; that is to say, the same predicate can be applied to either of them. Then the capacity of having that predicate applied to it with truth is called an attribute[,] that is[,] a thing to which it can be applied. The attribute is therefore an abstract term. Terms are divisible into concrete and abstract. The concrete are such as white virtuous &c. the abstract such as whiteness virtue, etc. Abstract terms do not denote any real thing but they denote fictitious things. An object's being white is conceived as being due to its being in some relation with a certain fictitious thing whiteness. In point of fact that the object is white may in a certain sense be said to be due to its connection with the sign or predicate white, that is to say it must be in such a relation to the name white that this name may be applied to it with truth or else it cannot be white. There is no falsity in this statement although it is more natural to state the matter in the inverse way and to say that its having that connection with that name is due to the fact that it is white. One statement is as true as the other. In the latter more natural mode of statement the existence of the thing is looked upon as the ultimate fact, but we have seen in the chapter upon reality that the final information is the ultimate fact, that final information consisting in applying a certain sign to certain objects in the predication and therefore it is perfectly correct to say that the thing's being white is due to and consists of the applicability of a certain predicate to a certain thing. A attribute or quality is not precisely the same as a predicate inasmuch as when we use the word predicate we have in mind the fact that the predicate is something extraneous to the thing which does not belong to it as it exists but belongs to it as it is thought whereas an attribute is considered as belonging to a thing whatever is thought. But upon our view of the nature of reality this is a distinction of very slight moment because existence is thus not independent of all thought and what is affirmed in the final judgment is the same as what really exists. Thus in considering the breadth and depth of terms it is desirable to make a number of distinctions. By the "informed breadth" of a term I shall mean all the real objects of which it is predicable with logical truth in the supposed state of information as our knowledge is never absolute but consists only of probabilities that all the information at hand must be taken into account and those things of which there is not on the whole reason to believe that the 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, S"s, and S'"s will constitute the informed breadth of T. If there be a second term T' which is predicable only of S' and S" and if it is not known that S'" is entirely included under S' and S" then T is considered to have a greater informed breadth than T'. If it is known that the S'"s are not all among the S's and S''s the excess of breadth is certain but if it is not known whether or not this is the case it is "doubtful". If certain S'"s are known to exist which are not known to be either S's or S"s , T is said to have a greater actual breadth than T' but if all the S'"s which are known to exist are also known to be S's and S''s though there are other S'"s which are not S' or S" then 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 then T may have a doubtful excess of breadth in one mind and no excess at all in the other mind. In that case the conception is said to be more extensively distinct to the latter mind.
By the informed depth of a term I mean all the real characters in contradistinction to mere synonymous names which can be predicated of it with logical truth in the supposed state of information no character being counted twice over knowingly. The depth like the breadth will be certainly doubtful 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 of knowledge in which no fact should be known but only the meanings of terms and, second, the state of information in which every fact should be known. This suggests two other sorts of breadth and depth corresponding to the two essential states of information which I shall term accordingly the essential and the substantial breadth and depth. The essential depth of a term which is sometimes called its essence consists of the really conceivable qualities predicated of it in its definition. This is one of the most important features of logic. Suppose the definition of the term T be this: "In T is at once P' P'' and P''' ". This sums up the whole meaning of T. It may not be known that there is no such thing as P' and therefore the meaning of T does not imply its existence. On the other hand we must know that P' P" and P'" are neither of them coextensive with the whole conception of being for we know the qualities of things only by comparison with their opposites hence we must know that there is something which is not P' and that this is not T, that there is something which is not P'' and that this is not T and that there is something which is not P'" and that this is not T. Accordingly if we define the essential breadth (if a term as "those real things of which according to its every meaning n term is predicable" then "not T" has an essential breadth that is to say its very meaning implies that there are things of which it is predicable. Thus T is a term which has essential depth but no essential breadth"not T" is a term which has essential breadth but not essential depth and all terms may be divided into two classes the "essential positive" and "essential negative" the former having essential depth but not essential breadth the latter having essential breadth but not essential depth. There are some terms which are affirmative in form but which according to this definition are essentially negative and vice versa. As examples of this we may allude particularly to the terms being and nothing both of which are terms of second intention. As every term has breadth and the breadth of one term is greater than that of another we may conceive of a term the breadth of which includes that of every other term so that it is predicable of anything whatever. This is the definition of the term "being". Its definition therefore gives it breadth but not depth and accordingly it is essentially negative. We may also conceive of a term whose depth includes the depth of all other terms so that anything whatever may be predicated of it without any falsity and this is the definition of the term "nothing". For you may say what you please of nothing and if it is clearly understood that what you speak of has no existence there is no falsity in what you assert because you have not made any assertion whatever. "Nothing" therefore is a term which has essential depth without any breadth and is according to our definition essentially affirmative. If two terms have the same essential breadth or the same essential depth logic recognizes no distinction between them. They are synonymous. They may differ rhetorically. One of these words may be associated in our minds with certain feelings with which the other is not associated but logic has nothing In do with such distinctions. But two terms may be indistinctly conceived so that it is not known whether they have the same essential bredth and depth or not and in this case the distinction must be admitted even in logic.
We come to the "substantial breadth and depth". The substantial breadth is the aggregate of real substance of which alone a term is predicable with absolute truth. Substantial depth is the real character as it exists in the object, which belongs to every thing of which a term is predicable with absolute truth.