men? May we not argue that because some men die therefore he must? Is it requisite to ascend by induction to the general proposition "all men must die," and then descend by deduction from that general proposition to the case of Mr. Gladstone? My answer undoubtedly is that we must ascend to general propositions. The fundamental principle of the substitution of similars gives us no warrant in affirming of Mr. Gladstone what we know of other men, because we cannot be sure that Mr. Gladstone is exactly similar to other men. Until his death we cannot be perfectly sure that he possesses all the attributes of other men; it is a question of probability, and I have endeavoured to explain the mode in which the theory of probability is applied to calculate the probability that from a series of similar events we may infer the recurrence of like events under identical circumstances. There is then no such process as that of inferring from particulars to particulars. A careful analysis of the conditions under which such an inference appears to be made, shows that the process is really a general one, and that what is inferred of a particular case might be inferred of all similar cases. All reasoning is essentially general, and all science implies generalisation. In the very birth-time of philosophy this was held to be so: "Nulla scientia est de individuis, sed de solis universalibus," was the doctrine of Plato, delivered by Porphyry. And Aristotle1 held a like opinionΟὐδεμία δὲ τέχνη σκοπεῖ τὸ καθ ̓ ἕκαστον ... τὸ δὲ καθ' ἕκαστον ἄπειρον καὶ οὐκ ἐπιστητόν. “No art treats of particular cases; for particulars are infinite and cannot be known." No one who holds the doctrine that reasoning may be from particulars to particulars, can be supposed to have the most rudimentary notion of what constitutes reasoning and science. At the same time there can be no doubt that practically what we find to be true of many similar objects will probably be true of the next similar object. This is the result to which an analysis of the Inverse Method of Probabilities leads us, and, in the absence of precise data from which we may calculate probabilities, we are usually obliged to make a rough assumption that similars in some 1 Aristotle's Rhetoric, Liber I. 2. 11. respects are similars in other respects. Thus it comes to pass that a large part of the reasoning processes in which scientific men are engaged, consists in detecting similarities between objects, and then rudely assuming that the like similarities will be detected in other cases. Distinction of Generalisation and Analogy. There is no distinction but that of degree between what is known as reasoning by generalisation and reasoning by analogy. In both cases from certain observed resemblances we infer, with more or less probability, the existence of other resemblances. In generalisation the resemblances have great extension and usually little intension, whereas in analogy we rely upon the great intension, the extension being of small amount (p. 26). If we find that the qualities A and B are associated together in a great many instances, and have never been found separate, it is highly probable that on the next occasion when we meet with A, B will also be present, and vice versa. Thus wherever we meet with an object possessing gravity, it is found to possess inertia also, nor have we met with any material objects possessing inertia without discovering that they also possess gravity. The probability has therefore become very great, as indicated by the rules founded on the Inverse Method of Probabilities (p. 257), that whenever in the future we meet an object possessing either of the properties of gravity and inertia, it will be found on examination to possess the other of these properties. This is a clear instance of the employment of generalisation. In analogy, on the other hand, we reason from likeness in many points to likeness in other points. The qualities or points of resemblance are now numerous, not the objects. At the poles of Mars are two white spots which resemble in many respects the white regions of ice and snow at the poles of the earth. There probably exist no other similar objects with which to compare these, yet the exactness of the resemblance enables us to infer, with high probability, that the spots on Mars consist of ice and snow. In short, many points of resemblance imply many more. From the appearance and behaviour of those white spots we infer that they have all the chemical and physical properties of frozen water. The inference is of course only probable, and based upon the improbability that aggregates of many qualities should be formed in a like manner in two or more cases, without being due to some uniform condition or cause. In reasoning by analogy, then, we observe that two . . . . have objects ABCDE... and A'B'C'D'E' . . . . . many like qualities, as indicated by the identity of the letters, and we infer that, since the first has another quality, X, we shall discover this quality in the second case by sufficiently close examination. As Laplace says, 1 "Analogy is founded on the probability that similar things have causes of the same kind, and produce the same effects. The more perfect this similarity, the greater is this probability." The nature of analogical inference is aptly described in the work on Logic attributed to Kant, where the rule of ordinary induction is stated in the words, "Eines in vielen, also in allen," one quality in many things, therefore in all; and the rule of analogy is " Vieles in einem, also auch das übrige in demselben "2 many (qualities) in one, therefore also the remainder in the same. It is evident that there may be intermediate cases in which, from the identity of a moderate number of objects in several properties, we may infer to other objects. Probability must rest either upon the number of instances or the depth of resemblance, or upon the occurrence of both in sufficient degrees. What there is wanting in extension must be made up by intension, and vice versa. Two Meanings of Generalisation. The term generalisation, as commonly used, includes two processes which are of different character, but are often closely associated together. In the first place, we generalise when we recognise even in two objects a common nature. We cannot detect the slightest similarity without opening the way to inference from one case to the other. If we compare a cubical crystal with a regular octahedron, there is little apparent similarity; but, as soon as we perceive 1 Essai Philosophique sur les Probabilités, p. 86. that either can be produced by the symmetrical modification of the other, we discover a groundwork of similarity in the crystals, which enables us to infer many things of one, because they are true of the other. Our knowledge of ozone took its rise from the time when the similarity of smell, attending electric sparks, strokes of lightning, and the slow combustion of phosphorus, was noticed by Schönbein. There was a time when the rainbow was an inexplicable phenomenon-a portent, like a comet, and a cause of superstitious hopes and fears. But we find the true spirit of science in Roger Bacon, who desires us to consider the objects which present the same colours as the rainbow; he mentions hexagonal crystals from Ireland and India, but he bids us not suppose that the hexagonal form is essential, for similar colours may be detected in many transparent stones. Drops of water scattered by the oar in the sun, the spray from a water-wheel, the dewdrops lying on the grass in the summer morning, all display a similar phenomenon. No sooner have we grouped together these apparently diverse instances, than we have begun to generalise, and have acquired a power of applying to one instance what we can detect of others. Even when we do not apply the knowledge gained to new objects, our comprehension of those already observed is greatly strengthened and deepened by learning to view them as particular cases of a more general property. A second process, to which the name of generalisation is often given, consists in passing from a fact or partial law to a multitude of unexamined cases, which we believe to be subject to the same conditions. Instead of merely recognising similarity as it is brought before us, we predict its existence before our senses can detect it, so that generalisation of this kind endows us with a prophetic power of more or less probability. Having observed that many substances assume, like water and mercury, the three states of solid, liquid, and gas, and having assured ourselves by frequent trial that the greater the means we possess of heating and cooling, the more substances we can vaporise and freeze, we pass confidently in advance of fact, and assume that all substances are capable of these three forms. Such a generalisation was accepted by Lavoisier and Laplace before many of the corroborative facts now in our possession were known. The reduction of a single comet beneath the sway of gravity was considered sufficient indication that all comets obey the same power. Few persons doubted that the law of gravity extended over the whole heavens; certainly the fact that a few stars out of many millions manifest the action of gravity, is now held to be sufficient evidence of its general extension over the visible universe. Value of Generalisation. It might seem that if we know particular facts, there can be little use in connecting them together by a general law. The particulars must be more full of useful information. than an abstract general statement. If we know, for instance, the properties of an ellipse, a circle, a parabola, and hyperbola, what is the use of learning all these properties over again in the general theory of curves of the second degree? If we understand the phenomena of sound and light and water-waves separately, what is the need of erecting a general theory of waves, which, after all, is inapplicable to practice until resolved again into particular cases? But, in reality, we never do obtain an adequate knowledge of particulars until we regard them as cases of the general. Not only is there a singular delight in discovering the many in the one, and the one in the many, but there is a constant interchange of light and knowledge. Properties which are unapparent in the hyperbola may be readily observed in the ellipse. Most of the complex relations which old geometers discovered in the circle will be reproduced mutatis mutandis in the other conic sections. The undulatory theory of light might have been unknown at the present day, had not the theory of sound supplied hints by analogy. The study of light has made known many phenomena of interference and polarisation, the existence of which had hardly been suspected in the case of sound, but which may now be sought out, and perhaps found to possess unexpected interest. The careful study of water-waves shows how waves alter in form and velocity with varying depth of water. Analogous changes may some time be detected in sound waves. Thus there is mutual interchange of aid. |