449 CHAPTER XXXIV. INDUCTION-FORMAL AND MATERIAL-ANALOGY. § 574. According to the view of Categorical Reasoning which makes it dependent on the Law of Identity, or whole and part, it is obvious that we may reason not only from the whole or genus to the parts, but conversely from the parts to the whole. In the former case we have Deductive Categorical Reasoning, in the latter Inductive Categorical Reasoning. In the latter case we argue from "the notion of all the constituent parts discretively, to the notion of the constituted whole collectively. Its general laws are identical with those of the Deductive Categorical Syllogism, and it may be expressed, in like manner, either in the form of an Intensive or of an Extensive Syllogism." 1 § 575. Strictly formal induction has been named Perfect Induction or Perfect Enumeration, as compared with Imperfect Induction or Enumeration. In the former case, there is an enumeration of all the singulars under the species, or of all the species under the genus-i.e., under the universal in question. The latter founds merely on some of the singulars under the species, or some of the species under the genusi.e., under the universal in question. Aristotle recognised the distinction of reasoning either from singulars or from parts to the whole. He regards Induction as ἐπαγωγὴ ἡ ἀπὸ τῶν καθ ̓ ἕκαστον ἐπὶ τὰ καθόλου ἔφοδος, and as ἐκ τῶν κατὰ μέρος. Thus, to take singulars, we have Perfect Induction in the following: Mercury, Venus, the Earth, Mars, Jupiter, Saturn, Uranus, Neptune, are opaque bodies lit by the sun; 1 Hamilton, Logic, iii. p. 318. 2 An. Pos'., i. 18. These are all the primary planets; Therefore all the primary planets are opaque bodies lit by the sun. To take species : Gold, silver, copper, tin, lead, zinc, platinum, iron, are (all) the most malleable metals ; These are (all) the most useful; Therefore all the most malleable are the most useful metals. In Imperfect Induction we may reason thus:— Or This, that, and the other magnet attracts iron; This, that, and the other magnet represent all magnets; This, that, and the other criminal was about 25 years of age; This, that, and the other criminal represent the majority of criminals; Therefore criminals of about 25 years of age are the majority. $576. Aristotle recognised Formal Induction; and thus distinguished Syllogism and Induction. In propositions which have a middle term, syllogism takes place by this middle; in those which have not, it takes place by induction. We may thus say that induction is in some sort opposed to Syllogism; for this demonstrates the extreme of the third term through the middle; that demonstrates the extreme of the middle through the third term. Thus then the syllogism which is produced by a middle term is, in nature, prior and more known; but that which is formed by induction is for us more evident.1 § 577. To illustrate this by his own example: Or Every X Y Z is A; Therefore all B is A. This is a reason apparently in the Third Figure; but in it, according to the ordinary rule, it is illegitimate, because the conclusion is universal. But the conclusion is legitimated on the principle that when two terms are attributed wholly to a third, and when this third is reciprocal to the second of the two terms, the first of these terms is also attributable to the second. On this ground Aristotle may be supposed to rest the inductive syllogism as a valid independent form. No doubt he seems to suggest in (§ 4) the conversion of the minor premiss into All devoid of bile is man, horse, mule. We should thus have the inference in Barbara of the First Figure. Thus : Every man, horse, mule is long-lived; All devoid of bile is man, horse, mule; Therefore all devoid of bile is long-lived. But this is by no means conclusive, though through the emphasis given to the moods of the First Figure by subsequent logicians, the validity of the inductive form has been made unwarrantably to depend on its capability of reduction to this Figure. The validity of the inductive form obviously depends on the principle, which Aristotle himself elsewhere expressly disavows, of the universality of the predicate in an affirmative proposition-in fact, on the recently much-questioned form all is all. But this may be taken as an instance at once of its validity and utility. (a) Aristotle evidently recognises Material Induction when he tells us that "induction is a progress from singulars to the universal, as if the skilled pilot is the best, and the skilled charioteer, the skilled in every genus is the best; " and especially when he adds that "induction is more fitted for persuasion, and more certain as well as more evident to the sense and common to the many; but syllogism presses with a greater necessity and repels opponents with greater force."—(Top., i. 12.) Formal induction is, of course, as cogent as (Deductive) syllogism. We have also the recognition of Imperfect Induction as the basis of the reasoning from Example (see below, p. 484 et seq.) In the following passage, however, he refers obviously to that form of Induction in which the Universal is constituted through a complete enumeration of the parts. "There is, therefore, induction, and inference from induction, when we conclude one of the extremes of the middle by the other extreme. Thus, for example, if B is middle of A г, to demonstrate by г, that A is B; for this is how we make the induction. Let A be long-lived, B that which has not bile, and C all long-lived animals, as man, horse, mule, &c. Then A is in C all entire ; for all C is long-lived; but B also, that is, that which has no bile, is in all C; if, then, C is reciprocal to B, and does not exceed the middle, it is therefore necessary that A is in B; for it has been demonstrated that any two things being the attri butes of the same subject, if the extreme is reciprocal to one of them, it is necessary that the other attribute should also be in the reciprocal attribute. Further, it ought to be supposed that C is composed of all the particular cases; for induction comprehends all. Such is the syl. logism of the primitive and immediate proposition.”—(An. Pr., ii. 23.) There are other passages in which Aristotle referred to what we call material induction, as, for example, An. Post., i. 18; ii. 19. He tells us expressly that imperfect induction is only allowable, where there is no contrary instance (ěvotaσis).—(Top., vii. 8.) And he certainly prac tised it not without success in his History of Animals. In this use of the inductive method he but followed Hippocrates in medicine. But the truth is, there has been no time in the history of observational science in which Material Induction has not been followed more or less faithfully. Even Bacon, who signalised and emphasised the method— mistaking, at the same time, the place and scope of the Formal Induction and Deduction of Aristotle-had before him, as exemplifying the method, Copernicus, Kepler, and Galileo. Newton but took up the thread of the predecessors of Bacon, with the advantage of the illumination which Bacon had thrown on the method. Even Newton's deduction could be verified only by Bacon's observation and induction, as to coincidence with actual fact. § 578. Hamilton regards Induction as proceeding equally in Comprehension and Extension, and gives the following formulæ for Induction :— A. In Comprehension (1.) (The parts holding the place of the major term S.) X Y Z constitute M; M comprehends P; Therefore X Y Z comprehend P. (2.) (The parts holding the place of the middle term) S comprehends X Y Z; X Y Z constitute P; Therefore S comprehends P. B. In Extension (1.) (The parts holding the place of the major term P)X Y Z constitute M; S is contained under M; Therefore S is contained under X Y Z. (2.) (The parts holding the place of the middle term)— X Y Z are contained under P; X Y Z constitute S; Therefore S is contained under P. § 579. Perfect Induction may very properly be extended to cases in which there has been the observation or analysis of the individual constituent elements of a concrete, say physical whole. Thus we may reason:— Quartz, felspar, and mica are all the constituents of ordinary granite; Ordinary granite is an igneous rock; Therefore quartz, felspar, and mica are all the constituents of an (some) igneous rock. Or Cognition, feeling, desire, will, are all the phænomenal manifestations of mind in man; Mind in man is the only mind we directly know; Therefore cognition, feeling, desire, will, are all the phænomenal constituents of mind directly known to us. This principle applies very strictly to the constitution of geometrical figures, to all chemical analysis of bodies; and it serves to explain how, from a single analysis of a body or description of a figure, we are able to extend our analysis or description to all similars. Thus geometrical demonstration may be taken as a form of Perfect Induction, although in it we specify only a single figure. Exhibiting only a single diagram, we are able in a valid demonstration to draw a conclusion which is not only true, but necessarily true. As the latter it is universal, that is, applies to every figure of the same character. Thus, given a parallelogram, or a four-sided figure of which the opposite sides are parallel, it can be proved that the opposite sides |