attempts to find the relative attribute of Major or Minor, especially in the Second and Third figures, by a minute, complex, and excursive inquiry into the material relations of the collated notions. His opinions are elaborately recorded by the Aphrodisian, and refuted ;-but not on the simple and sufficient ground, that, as material, they are extra-logical. (See Alexander on the first book of the Prior Analytics, especially on chapters v. and vi.) Waitz sometimes appears to approximate in doctrine to Herminus. See his commentary on the Organon, i. 379.

iii.) The next doctrine (I do not say, chronologically) was held by "The Commentators" by pre-eminence,—by Alexander and Averroes. It maintains that the Major term is the Predicate in the Problem, while the subject in the Problem constitutes the Minor.

iv.) A kindred opinion had, however, perhaps previously, been entertained, for it is explicitly redargued by the Aphrodisian. It holds, that the Major term is what is predicated, the Minor term what is subjected in the Conclusion. This theory is held by Ammonius Hermia, Philoponus, and others; indeed the Problem having been long thrown out of view, this has become the prevalent, if not the exclusive doctrine.

v.) Some, however, and with good reason, combine the last two opinions. This is done by the anonymous Greek author of the treatise "On Syllogisms.”

What is common to these three opinions (iii. iv. v.), and at the same time of principal importance, is this,—that Aristotle's distinction of the Major and Minor terms by relative position they interpret by their relative dignity, and what he states of these terms lying closer to, or farther from, the Middle term, in the Second or Third figures, they explain by their nearer or more remote propinquity to it by nature. And thus; Aristotle speaking of the Second figure says, that in this form the Major extreme is that which by position lies nearer to the Middle term, the Minor library, logical treatises of lower Byzantines; and if Ehinger did not scruple to father upon Psellus, without the slightest authority, a Greek version of the Summulæ of Hispanus, (see p. 127), we need not be slack in believing, that his friend Wegelin should lightly append, like so many editors before him, both the diagrams of Ammonius, and the western commutations into Greek of Barbara, Celarent, &c. -But the MSS. should be compared.

extreme that which by position lies farther from it; this, on their doctrine, means, that the Middle term being predicate in both premises, is more closely allied to that extreme which is once at least predicated, than to the other which is not predicated even once. In like manner, when speaking of the Third figure, the Philosopher says, that by position the Major extreme is that which lies farther from, the Minor that which lies nearer to, the Middle term; this they expound, that the Middle term being in both premises the subject, is more akin to that extreme which is once subjected, than to that which is not subjected at all.-This doctrine is best explicated by Ammonius. Stated, long after, by Pachymeres, it is, I see, misapprehended by Waitz, (t. i. p. 387.)

vi.) The definitions by Aristotle (i.) are, if superficially considered, sufficiently arbitrary. But a far more arbitrary doctrine was to be introduced by Boëthius; for he, in opposition even to the Philosopher, who held that either premise might be indifferently enounced first or second, actually defines the Major and Minor term, the Major and Minor proposition, of a syllogism, from the accident of its priority or posteriority in expression. (De Syll. Categ. L. ii. Opera, pp. 592, 593, ed. 1570.) Nor is he even consistent herein. But though arbitrary in itself, and historically contradicted by the practice of the earlier Latins and of all the Greeks and Orientals, (see p. 692,) this opinion obtained, at least a vulgar popularity in the western world, subsequently to the period of Boëthius.

vii.) According to my own view:-1°, The Majority and Minority of the syllogistic terms are determined by the counter quantities of Breadth and Depth; the term which is Major in the one quantity being Minor in the other. According therefore as we regard the syllogism from the point of view of Breadth or of Depth, must we denominate its terms and propositions.—2°, There is, formally or logically, no Major or Minor, be it term or premise, in the Unfigured syllogism or in the Second or Third figures of the Figured; for in these forms, the extremes are either in no quantity or in the same. This distinction, accordingly, is limited to the First figure; and here, either extreme may be Major or Minor, according as we make the one quantity or the other decisive. In subordination to this, the distinction in the counter quantities coincides, mutatis mutandis, with the three. kindred views previously enumerated (iii. iv. v.), and more especially with the last.

APPENDIX II. LOGICAL

(B.) ON AFFIRMATION AND NEGATION,-ON PROPOSITIONAL FORMS, ON BREADTH AND DEPTH,-ON SYLLOGISTIC, AND SYLLOGISTIC NOTATION, &c.

THE present article consists of observations made in reference to a memoir by Professor De Morgan, entitled, " On the Symbols of Logic, the Theory of the Syllogism," &c., read, in February 1850, to the Cambridge Philosophical Society, and published in their Transactions, (vol. ix.) The author, (with whom I had previously been involved in a logical discussion, more, however, of personal than of scientific concernment,) politely transmitted to me a copy of this paper, during the following summer; and the character of its contents induced me, forthwith, to address the following letter to the Editor of the Athenæum. This letter, I was compelled to limit to a single point, in consequence of the others leading me into a field of discussion too extensive: but, as I now find that my observations upon these were more fully written out than I had recollected,-as the unexclusive controversy involves some questions of scientific novelty,—and tends withal to shew of what value are the mathematical improvements of Logic, now proposed; on second thoughts, I here append the whole discussion, with a few verbal amplifications, and two supplementary notes. I regret, indeed, that the necessity of vindicating what, to me, is the cause of truth, should have given to these comments a character so controversial; constraining me to combat, from first to last, the logical speculations of one who ranks deservedly among the highest of our British Mathematicians. In fact, if I be not radically wrong, with the exception of two doctrines,-which are themselves, indeed, only borrowed, -there is not, in the whole compass of Mr De Morgan's "Logical Systems," a single logical novelty which is not a logical blunder. Of other errors, I say nothing. This, Mr De Morgan himself has not only warranted, but called on, me to shew. For,

*

though casting no blame on the aggressive purport of his paper, it will, at least, be allowed, that the attack is from too respectable a quarter not, on my part, to justify,—even, perhaps, to necessitate, a defence: and blame, assuredly, I cast neither on Professor De Morgan nor on the Philosophical Society of Cambridge; for the love of truth is always, of itself, polemical, (Πόλεμος ἅπαντων, καὶ τῆς ̓Αληθείας, πατήρ); whilst reason and experience concur in shewing, that Mathematics and Logic, like Love and Majesty,

"Haud bene conveniunt, nec in una sede morantur."

But it comes to this:-If, as has been said, Mr De Morgan's Memoir may represent the Transactions, the Transactions the Society, and the Society the University of Cambridge, then, either is the knowledge of Logic,-even of "Logic not its own," —in that seminary now absolutely null, or I am publicly found ignorant of the very alphabet of the science I profess. The alternative I am unable to disown; the decision I care not to avoid; and the discussion, I hope, may have its uses.

Edinburgh, 7th August 1850. SIR,-May I request the favour of being permitted, through your journal, to say a few words on a somewhat abstract subject, and in answer to Professor De Morgan's paper " On the Symbols of Logic," &c., in the volume of the "Transactions of the Philosophical Society of Cambridge," which has just appeared. [Wrong; the volume was not then published.] With that gentleman's logical theories, in general, I should not have thought of interfering; and even his errors concerning my own doctrines I would have willingly left to refute themselves. Not that I entertain a low opinion of Mr De Morgan's talent. In so far as I am qualified to judge, he well deserves the high reputation as a mathematician which he enjoys. But as a writer on the theory of reasoning, I cannot think that he has done his talent justice.

* The Philosophical Society of Cambridge ought not, however, to be so entitled, if we take the word Philosophy in the meaning attached to it everywhere out of Britain. (See above, p. 276.) I may add, as another example, that the recent edition, by the learned Erdmann, of the "Opera Philosophica" of Leibnitz, precisely omits, as non-philosophical, the matters which in Cambridge are styled Philosophy;-to wit, Physics and Mathematics. Philosophy is not, however, formally excluded from the "Philosophical Society of Cambridge," as it is from the "Philosophical Society of London." Mr De Morgan's paper is an example.

I am persuaded, indeed, that had he studied Mathematics as he has studied Logic, and were the members of the "Cambridge Philosophical Society" as competent judges in the one science as in the other, his character as a mathematician would rank very differently from what it does, nor would their "Transactions" have introduced his logical speculations to the world. It is because Mr De Morgan has not merely erred himself, but put into my mouth his own rudimentary mistakes; and because, so far from these mistakes being detected when his paper was read and discussed, that paper has been deemed by the Philosophical Society a contribution worthy of publication as a part of its proceedings-these special causes now principally constrain me to a brief exposition of the unintentional misrepresentation.

The present comments relate exclusively to Mr De Morgan's strictures on my abstract Notation of Syllogistic Forms, a specimen of which has been published by Mr Thomson in his " Outline of the Laws of Thought." But though that fragment contain only affirmations, and of these only the naked symbols, Mr De Morgan excogitating the negative forms, translates them into concrete language, according to his conception of what they ought to express; and then, without a word of explanation, makes me their author.-Farther: Finding that these expressions, as those which he attributes to logicians in general, are repugnant to "common thought," to "common language," he might have fairly added, and to common sense, he has swelled a memoir of more than fifty quarto pages with objections to Aristotle's doctrine and to mine; but radically misapprehending both, the illustration of his errors at once dispels the objections themselves, and therewith the two novel "Systems" reared on the same imaginary foundation.

Mr De Morgan says:

"The following phrase of Sir William Hamilton's system, 'All A is not some B,' [!] is very forced, both in order and phraseology; one who sees it for the first time finds it hard to make English or sense of it. The meaning is,' Each A is not any one among certain of the B's,' [!] and in its place in the system alluded to, the uncouth expression helps to produce system, and the perception of uniform laws of inference."—(P. 5.) And again: "The logician, who must have forms, has to make a choice, and he has invented cumular expressions which do not suit the genius of common thought or common language. All man is not fish,' [!] is the form in which a logician denies that any man is a fish. Sir William Hamilton says, 'All man is not all fish. [!] Common language would deny the first by saying, 'No, nor