implied. Both mediate and immediate inferences may be styled deductive as opposed to inductive. 3 This division may easily be shewn to be exhaustive. In any inference, we argue either to something already implied in the premisses or not; if the latter, the inference is inductive, if the former, deductive. If the deductive inference contain only a single premiss, it is immediate; if it contain two premisses and the conclusion be drawn from these jointly, it is mediate and is called 3 If we state explicitly all the assumptions made in the inductive process, the conclusion is contained in the premisses, and the form of the reasoning becomes deductive; but it is seldom that we do state our assumptions thus explicitly. The most essential distinction, however, between inductive and deductive reasoning consists not in the form of the inferences, but in the nature of the assumptions on which they rest. Deductive reasoning rests on certain assumptions with regard to language and co-existence (namely, the Law of Contradiction, the Law of Excluded Middle, and the Canons of Syllogism), while inductive reasoning assumes over and besides these laws the truth of the Laws of Universal Causation, of the Uniformity of Nature and, as implied in the latter, of the Conservation of Energy; or, if it be of the unscientific description which is known as Inductio per Enumerationem Simplicem, it merely assumes, instead of them, the vague and wide principle that the unknown resembles, or will resemble, the known. It hardly needs to be added that all reasoning alike assumes the trustworthiness of present consciousness and of memory. Amongst the assumptions or pre-suppositions of reasoning, I have not included the so-called Law of Identity; as to say that All A is A, or a thing is the same as itself, appears to me to be an utterly unmeaning proposition. Mr. Mill (Examination of Hamilton, ch. 21), in attempting to give a meaning to this maxim, really transforms it into a perfectly distinct proposition, namely, that Language may express the same idea in different forms of words. a syllogism. All deductive inferences which apparently contain more premisses than two admit of being analysed into a series of syllogisms. Note 1.-I am here departing from the ordinary scheme. of division adopted by logicians. Inferences are generally divided into mediate and immediate, and mediate inferences are subdivided into inductive and deductive. As however I regard inductions as more strongly contrasted with both syllogisms and immediate inferences than either of these classes is with the other, it seems preferable to make inductions one of the main members, rather than one of the subordinate members of the division. Nor is there any reason why an immediate inference should not be regarded as deductive. It should also be noticed that Sir W. Hamilton would deny the title of inferences to inductions (as they have been here explained), whereas Mr. Mill would deny that either a syllogism or an immediate inference can properly be called an inference. Mr. Mill maintains that all Inference is from the known to the unknown'; Sir W. Hamilton defines Inference as the 'carrying out into the last proposition what was virtually contained in the antecedent judgments.' Note 2.-The Aristotelian induction, in which the conclusion affirms or denies of a group what was in the premisses affirmed or denied of each member of the group severally, is, according to the above method of treatment, obviously regarded as a deductive inference. If I predicate some quality of each member of a group, and thence infer that all members of the group possess this quality, the conclusion is plainly contained in the premisses, and the inference is a syllogism. It may be represented in the form x, y, z are B, The individuals (or subordinate species) constituting group A are x, y, z; the ... The individuals (or subordinate species) constituting the group A are B. Such an inference is altogether different from what we now understand by an induction. On this subject the student may with advantage read Mr. Mill's chapter on 'Inductions improperly so called.' See Mill's Logic, Bk. III. ch. ii. An account of the Aristotelian induction will be found in Appendix G to Dr. Mansel's edition of Aldrich; in Sir W. Hamilton's Essay on Logic, and in his Lectures on Logic, Lect. xvii. and Appendix vii. These authors, as already noticed in the case of Sir W. Hamilton, regard inductions, in the modern sense of the word, as extra-logical. The advanced student may also consult with advantage Mr. De Morgan's chapter on 'Induction,' Formal Logic, ch. xi. 4 By Aristotle himself the inductive inference is analysed thus:x, y, z are B, x, y, z are (i. e. constitute) A; . A is B. The minor premiss, when stated in so peculiar a form, of course admits of simple conversion, and thus assumes the form given in the text. CHAPTER II On Immediate Inferences § 1. AN Immediate Inference may be formally defined as a combination of two propositions of which one is inferred from the other, the proposition inferred being virtually included in the proposition from which it is inferred. Of Immediate Inferences the most important forms are Oppositions, Conversions, Permutations'. § 2. On Oppositions. Two propositions are said to be opposed when they have the same subject and predicate, but differ in quantity or quality or both. An Opposition may be defined as an immediate inference in which from the truth or falsity of one proposition we infer either the truth or falsity of another, this proposition having the same subject and predicate as the former, but differing in quantity or quality or both. Thus from the proposition That all X is Y is true' we may infer the proposition That no X is Y is false,' or 'That some X is Y is true,' or 'That some X is not Y is false.' 1 It is the more common practice to speak of Opposition, Conversion, and Permutation, but I have adopted the plural number in order to draw attention to the fact that Logic is concerned with the results rather than with the processes by which they are arrived at. The opposition between A and E is called a Contrary Opposition. A and I, or E and O, a Subaltern Opposition. If A be true; E is false, I true, O false. If E be false; A is unknown, I true, O unknown. If O be true; A is false, E unknown, I unknown, It will be observed that it is only in a Contradictory Opposition (where the opposed terms differ both in quantity and quality) that from the truth or falsity of one proposition we can invariably infer the truth or falsity of another, the conclusion which we draw in this case being from the truth or falsity of the one propo |