Charles S. Peirce
Chap. IX: Of Relative Terms
MS 230 (Robin 387): Writings 3, 93-95
There are some reasonings in order to analyze which it is necessary to consider a proposition not in the simple form a is b, but in the form a is b to a c. For example, "Every man is indebted to a woman". This brings us to the subject of relative terms. A relative term is one which names nothing taken by itself but only in conjunction with another term, its correlate. Such are "father of", "lover of", "identical with", etc. We may express these by single letters and write their correlates directly after them so that lw may denote 'lover of a woman'. In studying relations, we shall do well to begin with those of individuals. It is true there are no individuals, strictly speaking, but nevertheless it is most useful in logic to consider what their properties would be if they existed. We may use the capital letters A, B, C, etc. for individual terms. The peculiarity of such terms is that if A<B then B<A.
Every individual will have a special relation to every other. Let us write (A:B) for the relative term which signifies the relation which A and A only has to B and B only. Then we shall have (A:B)B<A. But (A:B)C and (A:B)A will be absurd expressions and naming nothing.
We observe that such individual relatives will be of two kinds; those of the type (A: A) which signify the relation of some individual to itself, and those of the type (A:B) which signify the relation of some one individual to some other.
Since (B:C)C names the individual B and nothing else we may substitute this expression for B wherever the latter occurs. Then (A:B)B<A will become (A:B)(B:C)C-<A. But (A:C)C<A. Comparing these two expressions we are naturally led to consider (A:B)(B:C) which has received no signification as yet as the equivalent of (A:C). On the same principle, (A:B)(D:C), the letters in the middle not being the same, would be an absurdity and not equivalent to any relative.
Let us now pass to the consideration of general relative terms, first taking up those which are indeterminate among a finite number of individual cases. These are just as impossible as individual terms themselves. Let us suppose that l denotes either (A:B), (A:C), or (C:D). And let m denote either B, C, or D. Then lm will be one of these nine individuals
(A:B)B (A:C)B (C:D)B
(A:B)C (A:C)C (C:D)C
(A:B)D (A:C)D (C:D)D
Some of these expressions are absurd. The remainder are
(A:B)B, that is, A _____________ _____________
_____________ (A:C)C, that is, A _____________
_____________ _____________ (C:D)D, that is, C
Therefore lm denotes either A or C. And, in general, it is evident that xy will be indeterminate among all the cases which result from taking every case of x and every case of y. This holds even though the number of individual cases be innumerable. If therefore x1 <x and yi <y then x1y1<xy; and conversely if z<xy there must be some case x1 of x and some case y1 of y such that x1y1<z. For example, if to be a lover is to be a servant, and if to be a Negro is to be a man, then every lover of a Negro is a servant of a man. And if every jockey is a buyer of an animal then there must be some kind of a buyer, and some kind of an animal such that every such buyer of such an animal is a jockey.
Any expression of the form xyz may be considered as resulting from xy followed by z, or from yz preceded by x; because we have seen that this is true with individuals, and therefore it is true with every special case which xyz denotes.
If there are a finite number of individual relatives among which a general relative is indeterminate, they may be set out in an orderly manner in a table thus:
E | E:B E:C E:D
D | D:A D:E
C | C:B
B | B:C
A | A:B
A B C D E
If there is no finite number of individual cases the squares of the table must be made infinitesimally small and the table becomes a continuous surface and the blackened parts of it may show the nature of the relative. For example, let us represent in this way the relative "identical with". This is the relation which every individual bears to itself and nothing else