can be invalid which does not violate one or more of the six Rules first enounced.

After the usual manner of logicians, the foregoing Rules have been summed up in these mnemonic hexameters :

Distribuas medium, nec quartus terminus adsit;

Utraque nec præmissa negans, nec particularis;
Sectetur partem conclusio deteriorem,

Et non distribuat nisi cum præmissa, negetve.

But the application of these rules may become a matter of considerable complexity, when it is considered that, from the same naked (unquantified) Terms, a great variety of different Syllogisms may be formed. Each of the three Terms may be either Particular or Universal; each of the three Judgments, either Affirmative or Negative; the Judgments may be placed in any order with respect to each

negative; then its Predicate is distributed; and Rule 6, taken in conjunction with what has just been stated respecting the number of distributed Terms in the Premises, requires one of these Premises to be Universal.

Again, if either Premise is Particular, the Conclusion must be Particular. For the Subject of a Universal Affirmative Conclusion must be Universal; therefore, in the Premise wherein this Subject appears, it must, by Rule 6, be Universal, and the Middle Term, which is therein joined with it, must consequently be Particular, since it must be the Predicate of an Affirmative Judgment. Then the Middle Term, in order to be once distributed, must be the Universal Subject of the other Premise. Hence, if the Conclusion is Universal Affirmative, both Premises must be Universal.

And if the Conclusion is Universal Negative, both Premises must also be Universal. For both Terms of the Conclusion are then distributed; and as the Middle Term must also be distributed, there must be at least three Terms distributed in the Premises. But this cannot be, unless both Premises are Universal, since both of them, by Rule 4, cannot be Negative. Hence, whether the Conclusion is Affirmative or Negative, if it be Universal, both Premises must be Universal. Then, if either Premise is Particular, the Conclusion must be Particular.

But according to Rule 5, if either Premise is Negative, the Conclusion is Negative. Then, the Conclusion must follow the weaker part; — that is, it must be Particular, if either Premise is Particular, and Negative, if either Premise is Negative.-Q. E. D.

other, and for three Judgments, six different orders of position are possible; and each of the three Terms may be either Subject or Predicate in either or both of the Premises, the two principal Terms also assuming either place in the Conclusion. The larger portion of the numerous Syllogisms thus formed, it is true, are invalid, as offending against one or more of the preceding Rules. We need some more succinct mode than that of severally applying to each Syllogism all these Rules, before we can be satisfied that it is impeccable. Many of these Syllogistic forms, moreover, are equivalents of each other; that is, the Reasoning may be changed from one form to another, without impairing its validity, or even changing its signification in any essential respect. But of these equivalent forms some are more natural and obvious than the others; the mind seeks for these by preference; and when the process of reasoning appears in one of these natural and preferred forms, its validity is determined with ease and in a moment. The application of the Rules to such cases is made with the quickness of instinct, and may be reduced almost to a mechanical process.

A highly ingenious, though artificial, system has been contrived of classifying these numerous Syllogistic forms under a few heads, throwing out at once all that are illegitimate, immediately recognizing the remainder, and then transmuting those which are valid in substance, but unnatural and obscure in form, into the easy and familiar types in which the mind quickly perceives their legitimacy. The study of this system, a ready use of which may be said to constitute the art of Syllogizing, is facilitated by a series of mnemonic contrivances, many of them of marvellous ingenuity and completeness. The notation and most of the operations are of an algebraic character; and attempts have not been wanting of late years to enlarge and perfect the system by a further introduction of mathe

matical signs and processes. The failure of such an undertaking is not to be wondered at, for it proceeds, as it seems to me, upon a mistaken opinion as to the relative position of the two sciences. Logic is not a department of mathematics. Rather the reverse is true. Mathematics is the

science of pure quantity, — of reasoning about dimensions and numbers in the abstract, or as unmodified by any of the differences of quality by which all the objects of thought are actually distinguished; and it is, therefore, only a department, or a special application, of the far more comprehensive science which has for its object Reasoning itself and all its subsidiary processes, and thus covers the whole domain of Pure Thought. All computation is reasoning; but all reasoning is not computation, and therefore cannot be carried on by the processes, or be made subject to the special laws, of pure mathematics.

Syllogistic forms are classified with respect to Mood and Figure, the former having regard to the value of the three component Judgments, and the latter to the relative position of the three Terms in these Judgments. It will be convenient, then, to have a uniform mode of designating these three Terms. In future, S will stand for the Subject, and P for the Predicate, of the Conclusion, and M for the Middle Term. The Consequence, or what we usually express by the words therefore, consequently, &c., will be indicated by three dots placed thus .. For example:

Mis P;
S is M;

.. S is P.

To facilitate reference, the Logicians have given special names to these several Terms and Judgments. The Predicate of the Conclusion is called the Major Term, and its Subject the Minor Term. The Premise in which the Major is compared with the Middle Term is called the Major Premise, and that in which the Minor is compared

with the Middle, is the Minor Premise. These names have reference to the Quantity of Extension only, and are founded upon the received doctrine, that the natural order of predication is that in which the Genus is predicated of the Species, the Species of the Individual, and, generally, the Extensive whole of its part. Then the more Extensive Term, the Major, usually occupies, at least in Affirmative Judgments, the Predicate's place. "This," says Dr. Thomson, "is the natural, though not invariable, order; and it is worthy of remark, that, even in Negative Judgments, where, from the negation, the two Terms cannot be set together to determine their respective Extension, if, apart from the Judgment, we know that the one is a small and the other a large class, — the one a clearly determined and the other a vague notion, we naturally take the small and clearly determined Concept for the Subject. Thus, it is more natural to say that the Apostles are not deceivers, than that No deceivers are Apostles. So that, if our minds are not influenced by some previous thought to give greater prominence to the wider notion, and so make it the Subject," thus reversing the primary and natural order, the Term of major Extension will always be the Predicate, and that of minor Extension, the Subject.

As these names Major, Middle, and Minor-thus correctly indicate the comparative Extension of the three Terms, an Affirmative Syllogism in which these Terms occupy their natural place is conveniently symbolized by three concentric circles, of which the outermost and largest indicates the Predicate of the Conclusion, or the Major Term; the innermost and smallest, the Subject of the Conclusion, or the Minor; and the intermediate one, the Middle Term. Thus:

All mammals are viviparous;
All whales are mammals;

.. All whales are viviparous.

P

Here the reasoning is, that S, which is a part of M, must also be a part of P, since M is a part of P. We are thus led to another mode of enunciating the governing principle of all Syllogisms, that a part of a part is a part of the whole; or, as Leibnitz expresses it, contentum contenti est contentum continentis. This principle agrees in every essential respect with the famous Dictum of Aristotle, usually called the Dictum de omni et nullo, that whatever is predicated (affirmed or denied) universally of any Class (i. e. of any whole), may be also predicated of any part of that Class. Both principles have been already recognized and applied in the doctrine of Subalternation. The name of this Dictum is derived from the two forms which it assumes as applied either to affirmative or negative Conclusions ; the Dictum de omni being thus expressed, Quicquid de omni valet, valet etiam de quibusdam et singulis; and the Dictum de nullo being, Quicquid de nullo valet, nec de quibusdam nec de singulis valet. Both of these principles are evidently of a secondary or derivative character, their affirmative and negative forms being grounded respectively upon the two Axioms of Identity and Non-Contradiction; for as a whole is identical with the sum of all its parts, whatever is affirmed or denied (distributively) of the whole is thereby affirmed or denied of each of its parts. Burgersdyck remarks, that, for the purpose of applying the Dictum to Syllogisms, it may more conveniently be thus expressed: Whatever Predicate is universally affirmed or denied of any Middle Term or Part is also affirmed or denied of any Subject which is contained under that intermediate Term or Part.

The mode of symbolizing the mutual relations of the three Terms of a Syllogism, which is applied above to a Universal Affirmative, may be extended to Negatives and Particulars. The total disagreement of two Terms with each other, which is expressed by a Negative Judgment, is