CHAPTER V.

On the relation of Induction to Deduction, and on Verification.

THE results of our inductions are summed up in general propositions, which are not unfrequently stated in the shape of mathematical formula. These general propositions, the results of inductive reasoning, become, in turn, the data from which deductive reasoning proceeds. Though the major premiss of any particular deductive argument may itself be the result of deduction, it will invariably be found, as pointed out long ago by Aristotle', that the ultimate major premiss of a chain of deductive reasoning is a result of induction. There must be some limit to the generality of the propositions under which our deductive inferences can be subsumed, and, when we have reached this limit, the only evidence on which the ultimate major premiss can repose, if it depend on evidence at all, must be inductive. Thus, most of the deductions in the science of Astronomy, and many

1 Ἡ μὲν δὴ ἐπαγωγὴ ἀρχή ἐστι καὶ τοῦ καθόλου, ὁ δὲ συλλογισμὸς ἐκ τῶν καθόλου. Εἰσὶν ἄρα ἀρχαὶ ἐξ ὧν ὁ συλλογισμὸς, ὧν οὐκ ἔστι συλλογισμός· ἐπαγωγὴ ἄρα.—Εth. Nic. vi. 3 (3). Cf. Etb. Nic. vi. 6. 8 (9); Metaphysics, i. I; Posterior Analytics, ii. 19.

of those in the science of Mechanics, depend ultimately on the Law of Universal Gravitation; but this Law itself is the result of an induction based upon a variety of facts, including both the fall of bodies to the earth and the motion of the planets in their orbits. Again, a large number of geometrical deductions may be traced up to the ultimate major premiss: Things that are equal to the same thing are equal to one another.' But this proposition, if not referred directly to induction, is classed under the head of intuitive conceptions, the most probable, though perhaps not the most commonly received, explanation of which is that which derives them. from the accumulated experience of generations, transmitted hereditarily from father to son.

A Deductive Inference combines the results of previous inductions or deductions, and evolves new propositions as the consequence, or, to put the matter in a slightly different point of view, as expressing the total result, of these combinations. We append a few easy examples of the manner in which the results of induction are employed in a deductive argument.

To begin with a very simple instance, but one which will serve as a good illustration of the stage at which our investigations cease to be inductive and become deductive; suppose we have ascertained, by previous inductions, that A produces a, B produces b, C produces, D produces, and E produces, we know, by calculation, that is, by deductive reasoning, that the total effect of A, B, C, D, E is b + 4. In this case the

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simple rules of Algebra, governing the addition and subtraction of quantities, combined with the special data here furnished, are the premisses from which our deductive reasoning proceeds.

The proposition proved in Euclid, Book i. Prop. 38, that 'Triangles upon equal bases, and between the same parallels, are equal to one another,' is derived from, or is the total result of, the previous deductions (1) that 'Parallelograms upon equal bases, and between the same parallels, are equal to one another,' (2) that 'Triangles formed by the diagonal of a parallelogram are each of them equal to half the parallelogram' (i. 34), and (3) the previous induction that 'the halves of equal things are equal.'

What is called the Hydrostatic Paradox, namely, that a man standing on the upper of two boards, which form the ends of an air-tight leather bag, and blowing through a small tube opening into the space between the board, can easily raise his own weight, is a combination of two propositions, both gained from experience by means of induction, these propositions being (1) that fluids transmit pressure equally in all directions, (2) that the greater the pressure brought to bear on any surface from below, the greater the weight which it will sustain (otherwise expressed by the Mechanical Law that action and reaction are equal).

To take another very simple instance of a similar kind. One of the earliest and easiest problems in the Science of Optics is the following: 'A conical pencil of rays is

incident upon a plane reflecting surface; to determine the form of the reflected pencil.' The solution, that the reflected pencil will be a cone having for its vertex a certain imaginary point, which can be geometrically determined, on the other side of the surface, is derived from a combination of the experimental truth, gained by induction, that the angle of reflexion is equal to the angle of incidence' with the geometrical propositions stated in Euclid i. 8 and i. 29.

In the Science of Political Economy, Ricardo's Theory of Rent, namely that 'the rent of land represents the pecuniary value of the advantages which such land possesses over the worst land in cultivation,' is an easy deduction from two principles which are supplied by every one's experience, namely, (1) that land varies in value, and (2) that there is some land either so bad or so disadvantageously situated as to be not worth the cultivating2.

Professor Cairnes' work on the Slave Power furnishes a remarkable example of the successful application of the deductive method to the determination of economical questions. The economical effects of slavery are thus traced. We learn from observation and induction that slave labour is subject to certain characteristic defects: it is given reluctantly; it is unskilful; and, lastly, it is wanting in versatility. As a consequence of these characteristics, it can only be employed with profit when

2 The student will find an easy exposition of this Theory in Fawcett's Manual of Political Economy, Bk. II. ch. iii. ad init.

it is possible to organise it on a large scale. It requires constant supervision, and this for small numbers or for dispersed workmen would be too costly to be remunerative. The slaves must, consequently, be worked in large gangs. Now there are only four products which repay this mode of cultivation, namely, cotton, sugar, tobacco, and rice. Hence a country in which slave labour prevails is practically restricted to these four products, for it is another characteristic of slave labour, under its modern form, that free labour cannot exist side by side with it. But, besides restricting cultivation to these four products, some or all of which have a peculiar tendency to exhaust the soil, slave labour, from its want of versatility, imposes a still further restriction. The difficulty of teaching the slave anything is so great -the result of the compulsory ignorance in which he is kept, combined with want of intelligent interest in his work that the only chance of rendering his labour profitable is, when he has once learned a lesson, to keep him to that lesson for life. Accordingly where agricultural operations are carried on by slaves, the business of each gang is always restricted to the raising of a single product. Whatever crop be best suited to the character of the soil and the nature of slave industry, whether cotton, tobacco, sugar, or rice, that crop is cultivated, and that crop only. Rotation of crops is thus precluded by the conditions of the case. The soil is tasked again and again to yield the same product, and the inevitable.

result follows. After a short series of years its fertility