CHAPTER VI.

MIXED INDUCTIONS.

We know by intuition that if certain things are true, certain other things are also true. When, therefore, one of these facts of the first class has been established by observation, one of the facts of the second class can be established by making a syllogism, of which one premise is known to be true by intuition, and the other by observation; the conclusion will be a Mixed Induction.

We know, mathematically, that if the masts of ships appear before their hulls, the surface of the sea must be not flat but curved. We observe that the masts do actually appear first. The conclusion, that the surface of the sea is curved, is a mixed induction.

The nature of mixed inductions is well illustrated in the famous discoveries of Sir Isaac Newton. We quote from Mr. Mill:

"Newton began by an assumption, that the force which at each instant deflects a planet from its rectilineal course, and makes it describe a curve round the sun, is a force tending directly towards the sun. He then proved that, if it be so, the planet will describe, as we know by Kepler's first law it does describe, equal areas in equal times; and, lastly, he proved that if the force acted in any other direction whatever, the planet would not describe equal areas in equal times. It being thus shown that no other hypothesis could accord with the facts, the assumption was proved; the hypothesis became a law, established by the method of difference. Not only did Newton ascertain by this hypothetical process the

direction of the deflecting force; he proceeded in exactly the same manner to ascertain the law of variation of the quantity of that force. He assumed that the force varied inversely as the square of the distance; showed that from this assumption the remaining two of Kepler's laws might be deduced; and, finally, that any other law of variation would give results inconsistent with those laws, and inconsistent, therefore, with the real motions of the planets, of which Kepler's laws were known to be a correct expression.”1

That is, Newton showed mathematically that if the planets move in a given manner, they must be affected by a force acting toward the sun and varying inversely as the square of the distance; Kepler had shown that the planets do move in the given manner; the mixed induction was therefore established that there is such a force.

It will be seen that Mr. Mill introduces this as an example of hypothesis, but it will also be seen that it was wholly unnecessary for Newton to make any conjecture or assumption. All he had to do was to ask, The motions being as they are observed to be, what, mathematically, must be the direction and law of the force? It is not necessary to form an hypothesis that the surface of the sea is curved and then test that hypothesis by looking at an incoming ship. All that is necessary is to state the mathematical possibilities and then observe the facts; the conclusion follows of

course.

We take another fine illustration from Sir John Herschel :

"It had been objected to the doctrine of Copernicus, that, were it true, Venus (and, it might have been added, Mercury, as the other inferior planet) should appear sometimes horned like the

1 Logic, p. 351.

moon. To this he answered by admitting the conclusion, and averring that, should we ever be able to see its actual shape, it would appear so. It is easy to imagine with what force the application would strike every mind when the telescope confirmed this prediction, and showed the planet just as both the philosopher and his objectors had agreed it ought to appear." 1

Having considered the three kinds of induction, we are now ready to answer several questions proposed by Mr. Mill:

"In order to a better understanding of the problem which the logician must solve if he would establish a scientific theory of induction, let us compare a few cases of incorrect inductions with others which are acknowledged to be legitimate. Some, we know, which were believed for centuries to be correct, were, nevertheless, incorrect. That all swans are white, cannot have been a good induction, since the conclusion has turned out to be erroneous. The experience, however, on which the conclusion rested was genuine. From the earliest records, the testimony of all the inhabitants of the known world was unanimous on the point. The uniform experience of the inhabitants of the known world, agreeing in a common result, is not always sufficient to establish a general conclusion.... When a chemist announces the existence and properties of a newly discovered substance, if we confide in his accuracy, we feel assured that the conclusions he has arrived at will hold universally, although the induction be founded but on a single instance. We do not withhold our assent, waiting for a repetition of the experiment; or if we do, it is from a doubt whether the one experiment was properly made, not whether, if properly made, it would be conclusive. Here, then, is a general law of nature, inferred without hesitation from a single instance; an universal proposition from a singular one. Now, mark another case and contrast it with this. Not all the instances which have been observed since the beginning of the world, in support of the general proposition that all crows are black, would be deemed a sufficient presumption of the truth of the proposition, to outweigh

1 Discourse on the Study of Natural Philosophy, § 299.

the testimony of one unexceptionable witness who should affirm that in some region of the earth not fully explored, he had caught and examined a crow, and had found it to be gray.

"Why is a single instance, in some cases, sufficient for a complete induction, while in others, myriads of concurring instances, without a single exception known or presumed, go such a very little way towards establishing an universal proposition? Whoever can answer this question knows more of the philosophy of logic than the wisest of the ancients, and has solved the great problem of induction.” 1

Our discussion up to this point has prepared the student to answer Mr. Mill's question, and to claim the proud distinction of "knowing more of the philosophy of logic than the wisest of the ancients." It is plain that when a chemist determines for the first time the specific gravity of a new substance, rubidium, for example, he combines this one observation deductively with the acknowledged primary induction that chemical and physical properties of the several natural kinds are constant, and thus reaches at once the secondary induction, that the specific gravity of rubidium will be always found 1.5, or whatever the determination may be. Whenever a single instance leads to an induction, it is a secondary induction or a mixed induction. Bacon called such instances "crucial instances," from the Latin crux, a finger-post; since they point out the line of uniformity. No single instance can give a primary induction. In investigating the color of swans and crows we start with the well-established primary induction that color is, in animals, an uncertain quality. Combining this with the observation that these crows are black, we, of course, reach no conclusion. We have,

1 Logic, p. 227.

however, made a primary induction that all English crows are black; and this is correct. This leads us to remark that, in making an induction, it is necessary to define correctly the field under investigation. Having seen a thousand Chinamen in California, we conclude by induction that all Chinamen are, on the average, shorter than Americans. But when we learn that these men all came from one province, that of which HongKong is the port, we change, not the induction, but the area of it; it concerns not Chinamen but one sort of Chinamen. So the induction “ All crows are black" was correct for England, but not certainly for the whole world.