If it is preferred to put PQ for the indefinite some crystals, we have
PQ = PQB = PQBc.
The only difference is that the negative term c takes the place of C in the mood Darii.
Ellipsis of Terms in Partial Identities.
The reader will probably have noticed that the conclusion which we obtain from premises is often more full than that drawn by the old Aristotelian processes. Thus from “Sodium is a metal,” and “Metals conduct electricity,” we inferred (p. [55]) that “Sodium = sodium, metal, conducting electricity,” whereas the old logic simply concludes that “Sodium conducts electricity.” Symbolically, from A = AB, and B = BC, we get A = ABC, whereas the old logic gets at the most A = AC. It is therefore well to show that without employing any other principles of inference than those already described, we may infer A = AC from A = ABC, though we cannot infer the latter more full and accurate result from the former. We may show this most simply as follows:—
By the first Law of Thought it is evident that
AA = AA;
and if we have given the proposition A = ABC, we may substitute for both the A’s in the second side of the above, obtaining
AA = ABC . ABC.
But from the property of logical symbols expressed in the Law of Simplicity (p. [33]) some of the repeated letters may be made to coalesce, and we have