The mood Ferio is of exactly the same character as Darii or Barbara, except that it involves the use of a negative term. Take the example,
Assigning the letters as follows:—
Our argument is of the same form as before, and may be concisely stated in one line,
If it is preferred to put PQ for the indefinite some crystals, we have
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
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
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
Substituting again for ABC its equivalent A, we obtain
the desired result.
By a similar process of reasoning it may be shown that we can always drop out any term appearing in one member of a proposition, provided that we substitute for it the whole of the other member. This process was described in my first logical Essay,8 as Intrinsic Elimination, but it might perhaps be better entitled the Ellipsis of Terms. It enables us to get rid of needless terms by strict substitutive reasoning.
Inference of a Simple from Two Partial Identities.
Two terms may be connected together by two partial identities in yet another manner, and a case of inference then arises which is of the highest importance. In the two premises
the second member of each is the same; so that we can by obvious substitution obtain
Thus, in plain geometry we readily prove that “Every equilateral triangle is also an equiangular triangle,” and we can with equal ease prove that “Every equiangular triangle is an equilateral triangle.” Thence by substitution, as explained above, we pass to the simple identity,