merely recollecting the feelings and conduct of the same person in some previous instance, or from considering how we should feel or act ourselves. It is not only the village matron who, when called to a consultation upon the case of a neighbour's child, pronounces on the evil and its remedy simply on the recollection and authority of what she accounts the similar case of her Lucy." 56. Mr. Mill expresses as clearly as it is well possible that we argue in common life, as he thinks, not by the syllogism, but directly from instance. to instance by the similarity observed between the instances. But this argument from similars to similars is the identical process which I have called the substitution of similars, and which I have shown to be capable of explaining the syllogism itself, and much more. In fact we find Mr. Mill enunciating this principle himself in another chapter, where he is treating of argument from analogy or resemblance. After noticing the stricter meaning of analogy as a resemblance of relations, he continues: "It is on the whole more usual, however, to extend the name of analogical evidence to arguments from any sort of resemblance, provided they do not amount to a complete induction: without 1 "System of Logic," vol. ii. p. 86. peculiarly distinguishing resemblance of relations. Analogical reasoning, in this sense, may be reduced to the following formula;-Two things resemble each other in one or more respects; a certain proposition is true of the one; therefore it is true of the other. But we have nothing here by which to discriminate analogy from induction, since this type will serve for all reasoning from experience. In the most rigid induction, equally with the faintest analogy, we conclude, because A resembles B in one or more properties, that it does so in a certain other property." 57. If this be, as Mr. Mill so clearly states, the type of all reasoning from experience, it follows that the principle of inductive reasoning is actually identical with that which I have shown to be sufficient to explain the forms of deductive reasoning. The only difference I apprehend is, that in deductive reasoning we know or assume a similarity or identity to be certainly known, and the conclusion from it is therefore equally certain; but in inductive arguments from one instance to another we never can be sure that the similarity of the instance is so deep and perfect as to warrant our substitution of one for the other. Hence the conclusion is never certain, and possesses only a degree of probability, greater or less according to the circumstances of the case; and the theory of probabilities is our only resource for ascertaining this degree of probability, if ascertainable at all. 58. It is instructive to contrast mathematical induction with the induction as employed in the experimental sciences. The process by which we arrive at a general proof of a problem in Euclid's "Elements of Geometry" is really a process of generalization presenting a striking illustration of our principle. To prove that the square on the hypothenuse of a right-angled triangle is equal to the sum of the squares on the sides containing the right angle, Euclid takes only a single example of such a triangle, and proves this to be true. He then trusts to the reader perceiving of his own accord that all other right-angled triangles resemble the one accidentally adopted in the points material to the proof, so that any one right-angled triangle may be indifferently substituted for any other. Here the process from one case to another is certain, because we know that one case exactly resembles another. In physical science it is not so, and the distinction has been expressed, as it seems to me, with admirable insight by Professor Bowen in his well-known "Treatise on Logic, or the Laws of Pure Thought." He says of mathe1 Cambridge, United States, 1866, p. 354. F matical figures:-"The same measure of certainty which the student of nature obtains by intuition respecting a single real object, the mathematician. acquires respecting a whole class of imaginary objects, because the latter has the assurance, which the former can never attain, that the single object which he is contemplating in thought is a perfect representative of its whole class: he has this assurance, because the whole class exists only in thought, and are therefore all actually before him, or present to consciousness. For example: this bit of iron, I find by direct observation, melts at a certain temperature; but it may well happen that another piece of iron, quite similar to it in external appearance, may be fusible only at a much higher temperature, owing to the unsuspected presence with it of a little more or a little less carbon in composition. But if the angles at the base of this triangle are equal to each other, I know that a corresponding equality must exist in the case of every other figure which conforms to the definition of an isosceles triangle; for that definition excludes every disturbing element. The conclusion in this latter case, then, is universal, while in the former it can be only singular or particular." This passage perfectly supports my view that all reasoning consists in taking one thing as a representative, that is to say, as a substitute, for another, and the only difficulty is to estimate rightly the degree of certainty, or of mere probability, with which we make the substitution, The forms and methods of induction and the calculus of probabilities are necessary to guide us rightly in this; but to show that the principle of substitution is really present and active throughout inductive logic is more than I can undertake to show in this tract, although I believe it to be so. 59. Though I have pointed out how consistent are many of Mr. Mill's expressions with the view of logic here put forward, and how clearly in one place he describes the principle of substitution itself, I cannot but feel that his system is full of anomalies and breaches of consistency. These arise, I believe, from the profound error into which he has fallen, of undervaluing the logical discovery of the quantification of the predicate. Of Sir W. Hamilton's views he says: "If I do not consider the doctrine of the quantification of the predicate a valuable accession to the art of logic, it is only because I consider the ordinary rules of the syllogism to be an adequate test, and perfectly sufficient to exclude all inferences which do not follow from the premises. Considered, however, 1 "System of Logic," fifth ed. vol. i. p. 196 note. |