acquainted with the fact that the German logician Beneke had employed the principle of substitution, and had used the word itself in forming a theory of the syllogism. My imperfect acquaintance with the German language had prevented me from acquiring a complete knowledge of Beneke’s views; but there is no doubt that Professor Lindsay is right in saying that he, and probably other logicians, were in some degree familiar with the principle.15 Even Aristotle’s dictum may be regarded as an imperfect statement of the principle of substitution; and, as I have pointed out, we have only to modify that dictum in accordance with the quantification of the predicate in order to arrive at the complete process of substitution.16 The Port-Royal logicians appear to have entertained nearly equivalent views, for they considered that all moods of the syllogism might be reduced under one general principle.17 Of two premises they regard one as the containing proposition (propositio continens), and the other as the applicative proposition. The latter proposition must always be affirmative, and represents that by which a substitution is made; the former may or may not be negative, and is that in which a substitution is effected. They also show that this method will embrace certain cases of complex reasoning which had no place in the Aristotelian syllogism. Their views probably constitute the greatest improvement in logical doctrine made up to that time since the days of Aristotle. But a true reform in logic must consist, not in explaining the syllogism in one way or another, but in doing away with all the narrow restrictions of the Aristotelian system, and in showing that there exists an infinite variety of logical arguments immediately deducible from the principle of substitution of which the ancient syllogism forms but a small and not even the most important part.
The Logic of Relatives.
There is a difficult and important branch of logic which may be called the Logic of Relatives. If I argue, for instance, that because Daniel Bernoulli was the son of John, and John the brother of James, therefore Daniel was the nephew of James, it is not possible to prove this conclusion by any simple logical process. We require at any rate to assume that the son of a brother is a nephew. A simple logical relation is that which exists between properties and circumstances of the same object or class. But objects and classes of objects may also be related according to all the properties of time and space. I believe it may be shown, indeed, that where an inference concerning such relations is drawn, a process of substitution is really employed and an identity must exist; but I will not undertake to prove the assertion in this work. The relations of time and space are logical relations of a complicated character demanding much abstract and difficult investigation. The subject has been treated with such great ability by Peirce,18 De Morgan,19 Ellis,20 and Harley, that I will not in the present work attempt any review of their writings, but merely refer the reader to the publications in which they are to be found.