us to explain some expressions which might otherwise be misunderstood. Socrates, in the concluding part of this Sixth Book of the Republic, says, that this kind of knowledge is "that of which the Reason (Aóyos) takes hold, in virtue of its power of reasoning." Here we are plainly not to understand that we arrive at First Principles by reasoning: for the very opposite is true, and is here taught; -namely, that First Principles are not what we reason to, but what we reason from. The meaning of this passage plainly is, that First Principles are those of which the Reason takes hold in virtue of its power of reasoning ;—they are the conditions which must exist in order to make any reasoning possible :-they are the propositions which the Reason must involve implicitly, in order that we may reason explicitly;-they are the intuitive roots of the dialectical power. In accordance with the views now explained, Plato's Diagram may be thus further expanded. The term loéa is not used in this part of the Republic; but, as is well known, occurs in its peculiar Platonic sense in the Tenth Book. APPENDIX D. CRITICISM OF ARISTOTLE'S ACCOUNT OF INDUCTION. (Cam. Phil. Soc. FEB. 11, 1850.) THE Cambridge Philosophical Society has willingly admitted among its proceedings not only contributions to science, but also to the philosophy of science; and it is to be presumed that this willingness will not be less if the speculations concerning the philosophy of science which are offered to the Society involve a reference to ancient authors. Induction, the process by which general truths are collected from particular examples, is one main point in such philosophy and the comparison of the views of Induction entertained by ancient and modern writers has already attracted much notice. I do not intend now to go into this subject at any length; but there is a cardinal passage on the subject in Aristotle's Analytics, (Analyt. Prior. 11. 25) which I wish to explain and discuss. I will first translate it, making such emendations as are requisite to render it intelligible and consistent, of which I shall afterwards give an account. I will number the sentences of this chapter of Aristotle in order that I may afterwards be able to refer to them readily. § 1. "We must now proceed to observe that we have to examine not only syllogisms according to the aforesaid figures,-syllogisms logical and demonstrative, but also rhetorical syllogisms,-and, speaking generally, any kind of proof by which belief is influenced, following any method. § 2. "All belief arises either from Syllogism or from Induction: [we must now therefore treat of Induction.] §3. "Induction, and the Inductive Syllogism, is when by means of one extreme term we infer the other extreme term to be true of the middle term. §4. Thus if A, C, be the extremes, and B the mean, we have to show, by means of C, that A is true of B. GG § 5. "Thus let A be long-lived; B, that which has no gallbladder; and C, particular long-lived animals, as elephant, horse, mule. "Then every C is A, for all the animals above named are § 6. long-lived. § 7. "Also every C is B, for all those animals are destitute of gall-bladder. § 8. "If then B and C are convertible, and the mean (B) does not extend further than extreme (C), it necessarily follows that every B is A. § 9. "For it was shown before, that, if any two things be true of the same, and if either of them be convertible with the extreme, the other of the things predicated is true of the convertible (extreme). § 10. " "But we must conceive that C consists of a collection of all the particular cases; for Induction is applied to all the cases. § 11. "But such a syllogism is an inference of a first truth and immediate proposition. § 12. "For when there is a mean term, there is a demonstrative syllogism through the mean; but when there is not a mean, there is proof by Induction. § 13. "And in a certain way, Induction is contrary to Syllogism; for Syllogism proves, by the middle term, that the extreme is true of the third thing: but Induction proves, by means of the third thing, that the extreme is true of the mean. §14. "And Syllogism concluding by means of a middle term is prior by nature and more usual to us; but the proof by Induction, is more luminous." I think that the chapter, thus interpreted, is quite coherent and intelligible; although at first there seems to be some confusion, from the author sometimes saying that Induction is a kind of Syllogism, and at other times that it is not. The amount of the doctrine is this. When we collect a general proposition by Induction from particular cases, as for instance, that all animals destitute of gallbladder (acholous), are long-lived, (if this proposition were true, of which hereafter,) we may express the process in the form of a Syllogism, if we will agree to make a collection of particular cases our middle term, and assume that the proposition in which the second extreme term occurs is convertible. Thus the known propositions are Elephant, horse, mule, &c., are long-lived. But if we suppose that the latter proposition is convertible, we shall have these propositions: Elephant, horse, mule, &c., are long-lived. All acholous animals are elephant, horse, mule, &c., from whence we infer, quite rigorously as to form, All acholous animals are long-lived. This mode of putting the Inductive inference shows both the strong and the weak point of the illustration of Induction by means of Syllogism. The strong point is this, that we make the inference perfect as to form, by including an indefinite collection of particular cases, elephant, horse, mule, &c., in a single term, C. The Syllogism then is All Care long-lived. All acholous animals are C. Therefore all acholous animals are long-lived. The weak point of this illustration is, that, at least in some instances, when the number of actual cases is necessarily indefinite, the representation of them as a single thing involves an unauthorized step. In order to give the reasoning which really passes in the mind, we must say Elephant, horse, &c., are long-lived. All acholous animals are as elephant, horse, &c., This "as" must be introduced in order that the "all C" of the first proposition may be justified by the "C" of the second. This step is, I say, necessarily unauthorized, where the number of particular cases is indefinite; as in the instance before us, the species of acholous animals. We do not know how many such species there are, yet we wish to be able to assert that all acholous animals are long-lived. In the proof of such a proposition, put in a syllogistic form, there must necessarily be a logical defect; and the above discussion shows that this defect is the substitution of the proposition, "All acholous animals are as elephant, &c.," for the converse of the experimentally proved proposition, "elephant, &c., are acholous." In instances in which the number of particular cases is limited, the necessary existence of a logical flaw in the syllogistic translation of the process is not so evident. But in truth, such a flaw exists in all cases of Induction proper: (for Induction by mere enumeration can hardly be called Induction). I will, however, consider for a moment the instance of a celebrated proposition which has often been taken as an example of Induction, and in which the number of particular cases is, or at least is at present supposed to be, limited. Kepler's laws, for instance the law that the planets describe ellipses, may be regarded as examples of Induction. The law was inferred, we will suppose, from an examination of the orbits of Mars, Earth, Venus. And the syllogistic illustration which Aristotle gives, will, with the necessary addition to it, stand thus, Mars, Earth, Venus describe ellipses. Mars, Earth, Venus are planets. Assuming the convertibility of this last proposition, and its universality, (which is the necessary addition in order to make Aristotle's syllogism valid) we say All the planets are as Mars, Earth, Venus. Whence it follows that all the planets describe ellipses. If, instead of this assumed universality, the astronomer had made a real enumeration, and had established the fact of each particular, he would be able to say Saturn, Jupiter, Mars, Earth, Venus, Mercury, describe ellipses. Saturn, Jupiter, Mars, Earth, Venus, Mercury are all the planets. And he would obviously be entitled to convert the second proposition, and then to conclude that All the planets describe ellipses. But then, if this were given as an illustration of Induction by means of syllogism, we should have to remark, in the first place, that the conclusion that "all the planets describe ellipses," adds nothing to the major proposition, that " S., J., M., E., V., m., do so." It is merely the same proposition expressed in other words, so long as S., J., M., E., V., m., are supposed to be all the planets. And in the next place we have to make a remark which is more important; that the minor, in such an example, must generally be either a very precarious truth, or, as appears in this case, a transitory error. For that the planets known at any time are all the planets, must always be a doubtful assertion, liable to be overthrown to-night by an astronomical observation. And the assertion, as received in Kepler's time, has been overthrown. For Saturn, Jupiter, Mars, Earth, Venus, Mercury, are not all the planets. Not only have several new ones been discovered at intervals, as Uranus, Ceres, Juno, Pallas, Vesta, but we have new ones discovered every day; and any conclusion depending upon this premiss that A, B, C, D, E, F, G, H, to Z are all the planets, is likely to be falsified in a few years by the discovery of A', B', C', &c. If, therefore, this were the syllogistic analysis of Induction, Kepler's discovery rested upon a false proposition; and even if the analysis were now made conformable to our present knowledge, that induction, analysed as above, would still |