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
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 x
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