Now if we take the letter C to denote any third conceivable object, and use the sign ᔕ in its stated meaning of indefinite relation, then the general formula of all inference may be thus exhibited:—
or, in words—In whatever relation a thing stands to a second thing, in the same relation it stands to the like or equivalent of that second thing. The identity between A and B allows us indifferently to place A where B was, or B where A was; and there is no limit to the variety of special meanings which we can bestow upon the signs used in this formula consistently with its truth. Thus if we first specify only the meaning of the sign ᔕ, we may say that if C is the weight of B, then C is also the weight of A. Similarly
and so on ad infinitum.
We may also endow with special meanings the letter-terms A, B, and C, and the process of inference will never be false. Thus let the sign ᔕ mean “is height of,” and let
then it obviously follows since “3,590 feet is the height of Snowdon,” and “Snowdon = the highest mountain in England or Wales,” that, “3,590 feet is the height of the highest mountain in England or Wales.”
One result of this general process of inference is that we may in any aggregate or complex whole replace any part by its equivalent without altering the whole. To alter is to make a difference; but if in replacing a part I make no difference, there is no alteration of the whole. Many inferences which have been very imperfectly included in logical formulas at once follow. I remember the late Prof. De Morgan remarking that all Aristotle’s logic could not prove that “Because a horse is an animal, the head of a horse is the head of an animal.” I conceive that this amounts merely to replacing in the complete notion head of a horse, the term “horse,” by its equivalent some animal or an animal. Similarly, since
it follows that
and any event, circumstance or thing, which stands in a certain relation to the one will stand in like relation to the other. Milton reasons in this way when he says, in his Areopagitica, “Who kills a man, kills a reasonable creature, God’s image.” If we may suppose him to mean
it follows that “The killer of a man is the killer of some reasonable creature,” and also “The killer of God’s image.”
This replacement of equivalents may be repeated over and over again to any extent. Thus if person is identical in meaning with individual, it follows that
and if assemblage = meeting, we may make a new replacement and show that
We may in fact found upon this principle of substitution a most general axiom in the following terms13:—
If, for instance, exactly similar bricks and other materials be used to build two houses, and they be similarly placed in each house, the two houses must be similar. There are millions of cells in a human body, but if each cell of one person were represented by an exactly similar cell similarly placed in another body, the two persons would be undistinguishable, and would be only numerically different. It is upon this principle, as we