unnaturally led by the place which is assigned to axioms at the beginning of the elements of geometry, and by the manner in which they are afterwards referred to in demonstrating the propositions. "Since A (it is said) is equal to B, and B to C, A is equal to C; for, things which are equal to one and the same thing, are equal to one another." This place, I have little doubt, has been occupied by mathematical axioms, as far back, at least, as the foundation of the Pythagorean school; and Aristotle's fundamental axiom will be found to be precisely of the same description. Instead, therefore, of saying, with Dr. Gillies, that "on the basis of one single truth Aristotle has reared a lofty and various structure of abstract science,❞—it would be more correct to say, that the whole of this science is comprised or implied in the terms of one single axiom. Nor must it be forgotten (if we are to retain Dr. Gillies's metaphor) that the structure may, with much more propriety, be considered as the basis of the axiom, than the axiom of the structure.

When it is recollected that the greater part of our best philosophers (and among the rest Dr. Reid) still persevere, after all that Locke has urged on the opposite side of the question, in considering axioms as the ground work of mathematical science, it will not appear surprising that Aristotle's demonstrations should have so long continued to maintain their ground in books of logic. That this idea is altogether erroneous, in so far as mathematics is concerned, has been already sufficiently shown; the whole of that science resting ultimately, not on axioms, but on definitions or hypotheses. By those who have examined my reasonings on this last point, and who take the pains to combine them with the foregoing remarks, I trust it will be readily allowed, that the syllogistic theory furnishes no exception to the general doctrine concerning demonstrative evidence, which I formerly endeavored to establish; its pretended demonstrations being altogether nugatory, and terminating at last (as must be the case with every process of thought involving no data but what are purely axiomatical) in the very proposition from which they originally set out.

The idea that all demonstrative science must rest ultimately on axioms, has been borrowed, with many other erroneous maxims, from the logic of Aristotle; but is now, in general, stated in a manner much more consistent (although perhaps not nearer to the truth) than in the works of that philosopher. According to Dr. Reid, the degree of evidence which accompanies our conclusions, is necessarily determined by the degree of evidence which accompanies our first principles; so that, if the latter be only probable, it is perfectly impossible that the former should be certain. Agreeing, therefore, with Aristotle, in considering axioms as the basis of all demonstrative science, he was led, at the same time, in conformity with the doctrine just mentioned, to consider them as eternal and immutable truths, which are perceived, to be such by an intuitive judgment of the understanding. This, however, is not

the language of Aristotle; for, while he tells us, that there is no demonstration but of eternal truths,* he asserts, that the first principles which are the foundation of all demonstration, are got by induction from the informations of sense. In what manner this apparent contradiction is to be reconciled, I leave to the consideration of his future commentators.

For my own part, I cannot help being of opinion with Lord Monboddo (who certainly was not wanting in a due respect for the authority of Aristotle) that the syllogistic theory would have accorded much better with the doctrine of Plato concerning general ideas, than with that held on the same subject by the founder of the Peripatetic school. (Ancient Metaphysics, vol. v. pp. 184, 195.) To maintain that, in all demonstration, we argue from generals to particulars, and at the same time, to assert, that the necessary progress of our knowledge is from particulars to generals, by a gradual induction from the informations of sense, do not appear, to an ordinary understanding, to be very congruous parts of the same system; and yet the last of these tenets has been eagerly claimed as a discovery of Aristotle by some of the most zealous admirers of his logical demonstrations. (See Dr. Gillies's Analysis of Aristotle's works, passim.||)

Φανερον δε και, εαν ωσιν &ι προτάσεις καθόλον εξ ὦν ὁ συλλογισμος ότι αναγκη και το συμπερασμα αίδιον είναι της τοιαυτης αποδείξεως, και της (άπλως ειπείν) αποδείξεως. ουκ εστιν αρα αποδειξις των φθαρτων, ουδ' επιστημη άπλως, αλλ' ούτως, worey zata ovußeßzos. Analyt. Post. lib. i. cap. viii.

† Εκ μεν ουν αισθήσεως γιγνεται μνήμη. εκ δε μνήμης πολλακις του αυτού γινομένης, εμπειρία. in γαρ πολλαι μνημαι το αριθμο, εμπειρία μια εστιν εκ δ' εμπειρίας η εκ παντος ηρεμησαντος του καθολου εν τη ψυχή, του ένος παρά τα πολλα, ό αν εν άπασιν εν ενη εκείνοις το αυτό, τεχνης αρχη και επιστήμης. εαν μεν περί γενεσιν, τεχνης, Ear de regi to Oy, εлoτηus. (Analyt. Post. lib. ii. cap. xix.) The whole chapter may be read with advantage by those who wish for a fuller explanation of Aristotle's opinion on this question. His illustration of the intellectual process by which general principles are obtained from the perceptions of sense, and from reiterated acts of memory resolving into one experience, is more particularly deserving of attention.

It may perhaps be asked, Is not this the very mode of philosophizing recommended by Bacon, first, to proceed analytically from particulars to generals, and then to reason synthetically from generals to particulars? My reply to this question (a question which will not puzzle any person at all acquainted with the subject) I must delay, till I shall have an opportunity, in the progress of my work, of pointing out the essential difference between the meanings annexed to the word induction, in the Aristotelian, and in the Baconian logic-Upon the present occasion, it is sufficient to observe, that Bacon's plan of investigation was never supposed to be applicable to the discovery of principles which are necessary and eternal.

|| In this learned, and on the whole very instructive performance, I find several doctrines ascribed to Aristotle, which appear not a little at variance with each other. The following passages (which I am led to select from their connexion with the present argument) strike me as not only widely different, but completely contradictory, in their import.

"According to Aristotle, definitions are the foundations of all science; but those fountains are pure only when they originate in an accurate examination, and patient comparison of the perceptible qualities of individual objects." Vol. i. p. 77. "Demonstrative truth can apply only to those things which necessarily exist after a certain manner, and whose state is unalterable and we know those things when we know their causes: thus we know a mathematical proposition when we know the causes that make it true; that is, when we know all the intermediate

In this point of view, Lord Monboddo has certainly conducted, with greater skill, his defence of the syllogistic theory; inasmuch as he has entirely abandoned the important conclusions of Aristotle concerning the natural progress of human knowledge; and has attempted to entrench himself in (what was long considered as one of the most inaccessible fastnesses of the Platonic philosophy) the very ancient theory, which ascribes to general ideas an existence necessary and eternal. Had he, upon this occasion, after the example of Aristotle, confined himself solely to abstract principles, it might, not have been an easy task to refute to the satisfaction of common readers, his metaphysical arguments. Fortunately however, he has favored us with some examples and illustrations, which render this undertaking quite unnecessary; and which, in my opinion, have given to the cause which he was anxious to support, one of the most deadly blows which it has ever received. The following panegyric, in particular, on the utility of logic, while it serves to show that, in admiration of the Aristotelian demonstrations, he did not yield to Dr. Gillies, forms precisely such a comment as I myself could have wished for, on the leading propositions which I have now been attempting to establish.

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"In proof of the utility of logic," says Lord Monboddo, will give an example of an argument to prove that man is a substance; which argument, put into the syllogistic form, is this:

"Every animal is a substance;

Every man is an animal;

Therefore every man is a substance.

"There is no man, I believe, who is not convinced of the truth of the conclusion of this syllogism: but, how he is convinced of this, and for what reason he believes it to be true, no man can tell, who has not learned, from the logic of Aristotle, to know what a proposition, and what a syllogism is. There he will learn, that every proposition affirms or denies something of some other thing.

propositions, up to the first principles or axioms, on which it is ultimately built." Ibid. pp 95, 96.

It is almost superfluous to observe, that while the former of these quotations founds all demonstrative evidence on definitions, the latter founds it upon axioms. Nor is this all. The former, as is manifest from the second clause of the sentence, can refer only to contingent truths; inasmuch as the most accurate examination of the perceptible qualities of individual objects can never lead to the knowledge of things which necessarily exist after a certain manner. The latter as obviously refers, and exclusively refers, to truths which resemble mathematical theorems.

As to Aristotle's assertion, that definitions are the first principles of all demonstrations (ai agyar Tor anoderžtor of ógrouor,) it undoubtedly seems, at first view, to coincide exactly with the doctrine which I was at so much pains to inculcate, in treating of that peculiar evidence which belongs to mathematics. I hope, however, I shall not, on this account, be accused of plagiarism, when it is considered, that the commentary upon these words, quoted above from Dr. Gillies, absolutely excludes mathematics from the number of those sciences to which they are to be applied. On this point, too, Aristotle's own language is decisive. E avayzulov αρα συλλογισμός εστιν ή αποδειξις. Analyt. Poster. lib. i. cap. iv.

What is affirmed or denied is called the predicate; and that of which it is affirmed or denied, is called the subject. The predicate being a more general idea than the subject of which it is predicated, must contain or include it, if it be an affirmative proposition; or if it be a negative proposition, it must exclude it. This is the nature of propositions: and as to syllogism, the use of it is to prove any proposition that is not self-evident. And this is done by finding out what is called a middle term; that is, a term connected with both the predicate and the subject of the proposition to be proved. Now the proposition to be proved here is, that man is a substance; or, in other words, that substance can be predicated of man: and the middle term, by which this connexion is discovered, is animal, of which substance is predicated; and this is the major proposition of the syllogism, by which the major term of the proposition to be proved, is predicated of the middle term. Then animal is predicated of man; and this is the minor proposition of the syllogism, by which the middle term is predicated of the lesser term, or subject of the proposition to be proved. The conclusion, therefore, is, that as substance contains animal, and man is contained in animal, or is part of animal, therefore substance contains man. And the conclusion is necessarily deduced from the axiom I have mentioned, as the foundation of the truth of the syllogism, that the whole is greater than any of its parts, and contains them all.' So that the truth of the syllogism is as evident as when we say, that if A contain B, and B contain C, then A contains C.

"In this manner Aristotle has demonstrated the truth of the syllogism. But a man, who has not studied his logic, can no more tell why he believes the truth of the syllogism above mentioned, concerning man being a substance, than a joiner, or any common mechanic, who applies a foot or a yard to the length of two bodies, and finds that both agree exactly to that measure, and are neither longer or shorter, can give a reason why he believes the bodies to be equal, not knowing the axiom of Euclid, that two things, which are equal to a third thing, are equal to one another.'

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"By this discovery Aristotle has answered the question, which Pontius Pilate, the Roman Governor, asked of our Savior, what truth is? The answer to which appears now to be so obvious, that I am persuaded Pilate would not have asked it as a question, which he no doubt thought very difficult to be answered, if he had studied the logic of Aristotle."* (Ancient Metaphysics, vol. v. pp. 152-154.)

*I have quoted this passage at length, because I consider it as an instructive example of the effects likely to be produced on the understanding by scholastic studies, where they become a favorite and habitual object of pursuit. The author (whom I knew well, and for whose men ory I entertain a sincere respect) was a man of no common mental powers. Besides possessing a rich fund of what is commonly called learning, he was distinguished by natural acuteness; by a more than ordinary share of wit; and, in the discharge of his judicial func

After perusing the above exposition of Aristotle's demonstration, the reader, if the subject be altogether new to him, will be apt to imagine, that the study of logic is an undertaking of much less difficulty than he had been accustomed formerly to apprehend; the whole resolving ultimately into this axiom, "that if A contains B, and B contains C, then A contains C." In interpreting this axiom, he will probably figure to himself A, B, and C, as bearing some resemblance to three boxes, the sizes of which are so adapted to each other, that B may be literally put into the inside of A, and C into the inside of B. Perhaps it may be reasonably doubted, if there is one logician in a hundred, who ever dreamed of understanding it in any other sense. When considered in this light, it is not surprising that it should instantly command the assent of the merest novice: nor would he hesitate one moment longer about its truth, if, instead of being limited (in conformity to the three terms of a syllogism) to the three letters, A, B, C, it were to be extended from A to Z; the series of boxes corresponding to the series of letters, being all conceived to be nestled, one with another, like those which we sometimes see exhibited in the hands of a juggler.

If the curiosity of the student, however, should lead him to inquire a little more accurately into Aristotle's meaning, he will soon have the mortification to learn, that when one thing is said by the logician, to be in another, or to be contained in another, these words are not to be understood in their ordinary and most obvious sense, but in a particular and technical sense, known only to adepts; and about which, we may remark by the way, adepts are not, to this day, unanimously agreed. "To those," says Lord Monboddo, "who know no more of logic nor of ancient philosophy than Mr. Locke did, it will be necessary to explain in what sense one idea can be said to contain another, or the idea less general can be said to be a part of the more general. And, in the first place, it is not in the sense that one body is said to be a part of another, or the greater body to contain the lesser; nor is it as one number is said to contain another; but it is virtually or potentially that the more general idea contains the less general. In this way the genus contains the species; for the genus may be predicated of every species under it, whether existing or not existing; so that virtually it contains all the specieses under it, which exist or may exist. And not

tions, by the singular correctness, gravity, and dignity of his unpremeditated elocution; and yet, so completely had his faculties been subdued by the vain abstractions and verbal distinctions of the schools, that he had brought himself seriously to regard such discussions as that which I have here transcribed from his works, not only as containing much excellent sense, but as the quintessence of sound philosophy. As for the mathematical and physical discoveries of the Newtonians, he held them in comparative contempt, and was probably prevented, by this circumstance, from ever proceeding farther than the first elements of these sciences. Indeed, his ignorance of both was wonderful, considering the very liberal education which he has received, not only in his own country, but at a foreign university.