The practical treatment of a science will seldom correspond closely to the ideal which its supporters propose to themselves, and still seldomer to that which its antagonists insist upon demanding from the supporters. If we were to take our account of the distinction between the two views of Logic expounded respectively by Hamilton and by Mill, from Mill and Hamilton respectively, we should certainly not find it easy to bring them under one common definition. By such a test, the material Logic would be regarded as nothing more than a somewhat arbitrary selection from the domain of Physical Science in general, and the conceptualist Logic nothing more than a somewhat arbitrary selection from the domain of Psychology. The former would omit all consideration of the laws of thought and the latter all consideration of the truth or falsehood of our conclusions. Of course, in practice, such extremes as these are soon seen to be avoidable, and in spite of all controversial exaggerations the expounders of the opposite views do contrive to retain a large area of speculation in common. I do not propose here to examine in detail the restrictions by which this accommodation is brought about, or the very real and important distinctions of method, aim, tests, and limits which in spite of all approach to agreement are still found to subsist. To attempt this would be to open up rather too wide an enquiry to be suitable in a treatise on one subdivision only of the general science of Inference. 2. One subdivision of this enquiry is however really forced upon our notice. It does become important to consider the restrictions to which the ultra-material account of the province of Logic has to be subjected, because we shall thus have our attention drawn to an aspect of the subject which, slight and fleeting as it is within the region of Induction becomes very prominent and comparatively permanent in that of Probability. According to this ultra-material view, Inductive Logic would generally be considered to have nothing to do with anything but objective facts: its duty is to start from facts and to confine itself to such methods as will yield nothing but facts. What is doubtful it either establishes or it lets alone for the present, what is unattainable it rejects, and in this way it proceeds to build up by slow accretion a vast fabric of certain knowledge.