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. [38] Ueberweg’s *System of Logic*, transl. by Lindsay, pp. 442–446, 571, 572. The anticipations of the principle of substitution to be found in the works of Leibnitz, Reusch, and perhaps other German logicians, will be noticed in the preface to this second edition. [39] *Substitution of Similars* (1869), p. 9. [40] *Port-Royal Logic*, transl. by Spencer Baynes, pp. 212–219. Part III. chap. x. and xi. *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,[41] De Morgan,[42] Ellis,[43] 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. [41] *Description of a Notation for the Logic of Relatives, resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic.* By C. S. Peirce. *Memoirs of the American Academy*, vol. ix. Cambridge, U.S., 1870. [42] *On the Syllogism No IV., and on the Logic of Relations.* By Augustus De Morgan. *Transactions of the Cambridge Philosophical Society*, vol. x. part ii., 1860. [43] *Observations on Boole’s Laws of Thought.* By the late R. Leslie Ellis; communicated by the Rev. Robert Harley, F.R.S. *Report of the British Association*, 1870. *Report of Sections*, p. 12. Also, *On Boole’s Laws of Thought*. By the Rev. Robert Harley, F.R.S., *ibid.* p. 14. CHAPTER II. TERMS.