We may describe the result by saying that terms identical with the same term are identical with each other; and it is impossible to overlook the analogy to the first axiom of Euclid that “things equal to the same thing are equal to each other.” It has been very commonly supposed that this is a fundamental principle of thought, incapable of reduction to anything simpler. But I entertain no doubt that this form of reasoning is only one case of the general rule of inference. We have two propositions, A = B and B = C, and we may for a moment consider the second one as affirming a truth concerning B, while the former one informs us that B is identical with A; hence by substitution we may affirm the same truth of A. It happens in this particular case that the truth affirmed is identity to C, and we might, if we preferred it, have considered the substitution as made by means of the second identity in the first. Having two identities we have a choice of the mode in which we will make the substitution, though the result is exactly the same in either case.
Now compare the three following formulæ,
| (1) | A = B = C, hence A = C |
| (2) | A = B ~ C, hence A ~ C |
| (3) | A ~ B ~ C, no inference. |
In the second formula we have an identity and a difference, and we are able to infer a difference; in the third we have two differences and are unable to make any inference at all. Because A and C both differ from B, we cannot tell whether they will or will not differ from each other. The flowers and leaves of a plant may both differ in colour from the earth in which the plant grows, and yet they may differ from each other; in other cases the leaves and stem may both differ from the soil and yet agree with each other. Where we have difference only we can make no inference; where we have identity we can infer. This fact gives great countenance to my assertion that inference proceeds always through identity, but may be equally well effected in propositions asserting difference or identity.
Deferring a more complete discussion of this point, I will only mention now that arguments from double identity occur very frequently, and are usually taken for granted, owing to their extreme simplicity. In regard to the equivalence of words this form of inference must be constantly employed. If the ancient Greek χαλκός is our copper, then it must be the French cuivre, the German kupfer, the Latin cuprum, because these are words, in one sense at least, equivalent to copper. Whenever we can give two definitions or expressions for the same term, the formula applies; thus Senior defined wealth as “All those things, and those things only, which are transferable, are limited in supply, and are directly or indirectly productive of pleasure or preventive of pain.” Wealth is also equivalent to “things which have value in exchange;” hence obviously, “things which have value in exchange = all those things, and those things only, which are transferable, &c.” Two expressions for the same term are often given in the same sentence, and their equivalence implied. Thus Thomson and Tait say,[57] “The naturalist may be content to know matter as that which can be perceived by the senses, or as that which can be acted upon by or can exert force.” I take this to mean—
Matter = what can be perceived by the senses;
Matter = what can be acted upon by or can exert force.
For the term “matter” in either of these identities we may substitute its equivalent given in the other definition. Elsewhere they often employ sentences of the form exemplified in the following:[58] “The integral curvature, or whole change of direction of an arc of a plane curve, is the angle through which the tangent has turned as we pass from one extremity to the other.” This sentence is certainly of the form—
The integral curvature = the whole change of direction, &c. = the angle through which the tangent has turned, &c.
Disguised cases of the same kind of inference occur throughout all sciences, and a remarkable instance is found in algebraic geometry. Mathematicians readily show that every equation of the form y = mx + c corresponds to or represents a straight line; it is also easily proved that the same equation is equivalent to one of the general form Ax + By + C = 0, and vice versâ. Hence it follows that every equation of the form in question, that is to say, every equation of the first degree, corresponds to or represents a straight line.[59]