liability to error which I have shown1 to exist in “Immediate Inference by added Determinants,” as described by Dr. Thomson.2
Inference with Two Simple Identities.
One of the most common forms of inference, and one to which I shall especially direct attention, is practised with two simple identities. From the two statements that “London is the capital of England” and “London is the most populous city in the world,” we instantaneously draw the conclusion that “The capital of England is the most populous city in the world.” Similarly, from the identities
we infer
The general form of the argument is exhibited in the symbols
| B = A | (1) | |
|---|---|---|
| B = C | (2) | |
| hence | A = C. | (3) |
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