It may be said that all the planets discovered since Kepler's time conform to his law, and thus confirm his discovery. This we grant: but they only confirm the discovery, they do not make it; they are not its groundwork. It was a discovery before these new cases were known; it was an inductive truth without them. Still, an objector might urge, if any one of these new planets had contradicted the law, it would have overturned the discovery. But this is too boldly said. A discovery which is so precise, so complex (in the phenomena which it explains), so supported by innumerable observations extending through space and time, is not so easily overturned. If we find that Uranus, or that Encke's comet, deviates from Kepler's and Newton's laws, we do not infer that these laws must be false; we say that there must be some disturbing cause in these cases. We seek, and we find these disturbing causes: in the case of Uranus, a new planet; in the case of Encke's comet, a resisting medium. Even in this case therefore, though the number of particulars is limited, the Induction was not made by a simple enumeration of all the particulars. It was made from a few cases, and when the law was discerned to be true in these, it was extended to all; the conversion and assumed universality of the proposition that "these are planets," giving us the proposition which we need for the syllogistic exhibition of Induction, "all the planets are as these."
I venture to say further, that it is plain, that Aristotle did not regard Induction as the result of simple enumeration. This is plain, in the first place, from his example. Any proposition with regard to a special class of animals, cannot be proved by simple enumeration: for the number of particular cases, that is, of animal species in the class, is indefinite at any period of zoological discovery, and must be regarded as infinite. In the next place, Aristotle says (§ 10 of the above extract), "We must conceive that C consists of a collection of all the particular cases; for induction is applied to all the cases." We must conceive (νοεῖν) that C in the major, consists of all the cases, in order that the conclusion may be true of all the cases; but we cannot observe all the cases. But the evident proof that Aristotle does not contemplate in this chapter an Induction by simple enumeration, is the contrast in which he places Induction and Syllogism. For Induction by simple enumeration stands in no contrast to Syllogism. The Syllogism of such Induction is quite logical and conclusive. But Induction from a comparatively small number of particular cases to a general law, does stand in opposition to Syllogism. It gives us a truth,—a truth which, as Aristotle says (§ 14), is more luminous than a truth proved syllogistically, though Syllogism may be more natural and usual. It gives us (§ 11) immediate propositions, obtained directly from observation, and not by a chain of reasoning: "first truths," the principles from which syllogistic reasonings may be deduced. The Syllogism proves by means of a middle term (§ 13) that the extreme is true of a third thing: thus, (acholous being the middle term):
Acholous animals are long-lived:
All elephants are acholous animals:
Therefore all elephants are long-lived.
But Induction proves by means of a third thing (namely, particular cases) that the extreme is true of the mean; thus (acholous, still being the middle term)
Elephants are long-lived:
Elephants are acholous animals:
Therefore acholous animals are long-lived.
It may be objected, such reasoning as this is quite inconclusive: and the answer is, that this is precisely what we, and as I believe, Aristotle, are here pointing out. Induction is inconclusive as reasoning. It is not reasoning: it is another way of getting at truth. As we have seen, no reasoning can prove such an inductive truth as this, that all planets describe ellipses. It is known from observation, but it is not demonstrated. Nevertheless, no one doubts its universal truth, (except, as aforesaid, when disturbing causes intervene). And thence, Induction is, as Aristotle says, opposed to syllogistic reasoning, and yet is a means of discovering truth: not only so, but a means of discovering primary truths, immediately derived from observation.
I have elsewhere taught that all Induction involves a Conception of the mind applied to facts. It may be asked whether this applies in such a case as that given by Aristotle. And I reply, that Aristotle's instance is a very instructive example of what I mean. The Conception which is applied to the facts in order to make the induction possible is the want of the gall-bladder;—and Aristotle supplies us with a special term for this conception; acholous[349]. But, it may be said, that the animals observed, the elephant, horse, mule, &c., are acholous, is a mere fact of observation, not a Conception. I reply that it is a Selected Fact, a fact selected and compared in several cases, which is what we mean by a Conception. That there is needed for such selection and comparison a certain activity of the mind, is evident; but this also may become more clear by dwelling a little further on the subject. Suppose that Aristotle, having a desire to know what class of animals are long-lived, had dissected for that purpose many animals; elephants, horses, cows, sheep, goats, deer and the like. How many resemblances, how many differences, must he have observed in their anatomy! He was very likely long in fixing upon any one resemblance which was common to all the long-lived. Probably he tried several other characters, before he tried the presence and absence of the gall-bladder:—perhaps, trying such characters, he found them succeed for a few cases, and then fail in others, so that he had to reject them as useless for his purpose. All the while, the absence of the gall-bladder in the long-lived animals was a fact: but it was of no use to him, because he had not selected it and drawn it forth from the mass of other facts. He was looking for a mean term to connect his first extreme, long-lived, with his second, the special cases. He sought this middle term in the entrails of the many animals which he used as extremes: it was there, but he could not find it. The fact existed, but it was of no use for the purpose of Induction, because it did not become a special Conception in his mind. He considered the animals in various points of view, it may be, as ruminant, as horned, as hoofed, and the contrary; but not as acholous and the contrary. When he looked at animals in that point of view,—when he took up that character as the ground of distinction, he forthwith imagined that he found a separation of long-lived and short-lived animals. When that Fact became a Conception, he obtained an inductive truth, or, at any rate, an inductive proposition.
He obtained an inductive proposition by applying the Conception acholous to his observation of animals. This Conception divided them into two classes; and these classes were, he fancied, long-lived and short-lived respectively. That it was the Conception, and not the Fact which enabled him to obtain his inductive proposition, is further plain from this, that the supposed Fact is not a fact. Acholous animals are not longer-lived than others. The presence or absence of the gall-bladder is no character of longevity. It is true, that in one familiar class of animals, the herbivorous kind, there is a sort of first seeming of the truth of Aristotle's asserted rule: for the horse and mule which have not the gall-bladder are longer-lived than the cow, sheep, and goat, which have it. But if we pursue the investigation further, the rule soon fails. The deer-tribe that want the gall-bladder are not longer-lived than the other ruminating animals which have it. And as a conspicuous evidence of the falsity of the rule, man and the elephant are perhaps, for their size, the longest-lived animals, and of these, man has, and the elephant has not, the organ in question. The inductive proposition, then, is false; but what we have mainly to consider is, where the fallacy enters, according to Aristotle's analysis of Induction into Syllogism. For the two premisses are still true; that elephants, &c., are long-lived; and that elephants, &c., are acholous. And it is plain that the fallacy comes in with that conversion and generalization of the latter proposition, which we have noted as necessary to Aristotle's illustration of Induction. When we say "All acholous animals are as elephants, &c.," that is, as those in their biological conditions, we say what is not true. Aristotle's condition (§ 8) is not complied with, that the middle term shall not extend beyond the extreme. For the character acholous does extend beyond the elephant and the animals biologically resembling it; it extends to deer, &c., which are not like elephants and horses, in the point in question. And thus, we see that the assumed conversion and generalization of the minor proposition, is the seat of the fallacy of false Inductions, as it is the seat of the peculiar logical character of true Inductions.
As true Inductive Propositions cannot be logically demonstrated by syllogistic rules, so they cannot be discovered by any rule. There is no formula for the discovery of inductive truth. It is caught by a peculiar sagacity, or power of divination, for which no precepts can be given. But from what has been said, we see that this sagacity shows itself in the discovery of propositions which are both true, and convertible in the sense above explained. Both these steps may be difficult. The former is often very laborious: and when the labour has been expended, and a true proposition obtained, it may turn out useless, because the proposition is not convertible. It was a matter of great labour to Kepler to prove (from calculation of observations) that Mars moves elliptically. Before he proved this, he had tried to prove many similar propositions:—that Mars moved according to the "bisection of the eccentricity,"—according to the "vicarious hypothesis,"—according to the "physical hypothesis,"—and the like; but none of these was found to be exactly true. The proposition that Mars moves elliptically was proved to be true. But still, there was the question, Is it convertible? Do all the planets move as Mars moves? This was proved, (suppose,) to be true, for the Earth and Venus. But still the question remains, Do all the planets move as Mars, Earth, Venus, do? The inductive generalizing impulse boldly answers, Yes, to this question; though the rules of Syllogism do not authorize the answer, and though there remain untried cases. The inductive Philosopher tries the cases as fast as they occur, in order to confirm his previous conviction; but if he had to wait for belief and conviction till he had tried every case, he never could have belief or conviction of such a proposition at all. He is prepared to modify or add to his inductive truth according as new cases and new observations instruct him; but he does not fear that new cases or new observations will overturn an inductive proposition established by exact comparison of many complex and various phenomena.
Aristotle's example offers somewhat similar reflections. He had to establish a proposition concerning long-lived animals, which should be true, and should be susceptible of generalized conversion. To prove that the elephant, horse and mule are destitute of gall-bladder required, at least, the labour of anatomizing those animals in the seat of that organ. But this labour was not enough; for he would find those animals to agree in many other things besides in being acholous. He must have selected that character somewhat at a venture. And the guess was wrong, as a little more labour would have shown him; if for instance he had dissected deer: for they are acholous, and yet short-lived. A trial of this kind would have shown him that the extreme term, acholous, did extend beyond the mean, namely, animals such as elephant, horse, mule; and therefore, that the conversion was not allowable, and that the Induction was untenable. In truth, there is no relation between bile and longevity[350], and this example given by Aristotle of generalization from induction is an unfortunate one.