Charles S. Peirce

Chap. IX: Of Relative Terms

MS 230 (Robin 387):

Writings3, 93-95

Spring 1873

There are some reasonings in order to analyze which it is necessary to consider a proposition not in the simple form

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

(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

Any expression of the form

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