This process seems to be sufficiently simple and obvious. But the mood Bocardo was famous in the schools, and, even more than Baroco, was the opprobrium of the scholastic system of reduction. Says Hamilton, "So intricate, in fact, was this mood considered that it was looked upon as a trap, into which if you once got, it was no easy matter to find an exit. Bocardo was, during the Middle Ages, the name given in Oxford to the Academical Jail, or Carcer, for refractory students,- -a name which still remains as a relique of the ancient logical glory of that venerable seminary." Perhaps the perplexity arose somewhat from the process being considered and named as a kind of reduction. Many logicians of the present day continue to speak of it as "indirect reduction." But obviously it is not a reduction at all, and to call it so is mere confusion. It is, as already indicated, only an indirect test of the validity of the reasoning when it occurs in these moods. And it may be well to add that all the other moods can be tested by the same process. This is elaborately but uselessly exhibited by Schuyler."

But the test, even in the case of Baroco and Bocardo, is of no practical value, and is superfluous. We have inherited it from the old logicians, who, as has been said, supposed that these two moods could not be ostensively reduced. In this they were mistaken. Mark Duncan, as early as 1612, and after him Noldius, in 1666, showed that by the use of contraposition Baroco could be reduced to Ferio, and Bocardo to Darii. Noldius proposed to call the former Facrono, and the latter Docamroc. Whately called attention to this method, but did not observe a defect in the name Facrono, and rendered the other defective by omitting the terminal letter. Hamilton recognizes that the reduction may be made, but blunders sadly in the attempt to reduce Baroco, which his editor admits. We have proposed the names Fakofo and Dokamok as alternates, or as substitutes, and have already exhibited in an example the reduction of the former to Ferio. The admission of these substitutes would not affect the metre of the mnemonic lines, and we could then dismiss from technical Logic this operose, indirect, and practically useless test per impossibile.

The Latin mnemonic hexameters, it must be confessed, are a marvel of ingenuity. De Morgan calls them "magic lines, more full of meaning than any others that ever were written." Hamilton calls them "cabalistical verses," and says that "taking them on their own

ground, there are few human inventions which display a higher ingenuity." They were, so far as relates to the first three figures, the invention of Petrus Hispanus, already referred to as the author of the propositional symbols. He was a native of Lisbon, became Pope John XXII in 1277, and died the same year. The corresponding Greek lines are much less ingenious, as the names of the moods in them mark only the order, the quantity, and the quality of the propositions, not indicating any method of reduction, and, indeed, not even the equivalent moods. They were the invention of Nicephorus Blemmidas, who was nominated Patriarch of Constantinople. Which had priority in this invention is uncertain. It is curious, however, to note that these two logicians attained the two highest places, the one in the Roman, the other in the Greek hierarchy; but as the one had hardly begun to reign when he was killed by the fall of his palace, so the other, declining the nomination, did not enter on the office at all. The several works of the Pope and the Patriarch were for many centuries the text-books on Logic, the one in the Latin, the other in the Greek schools.'

But it may very properly be asked, why should we have reduction ? Reasoning certainly does not become more cogent by being reduced to the first figure; but, says Bowen, it becomes more elegant and perspicuous. That depends on whether a given case naturally belongs in the first figure. If so, then this is true. But if the case naturally belongs in some other figure, then its reduction to the first renders it more or less awkward and obscure. The answer more usually given assumes that the system of reduction is a method for testing the validity of reasonings. As the dicta of Aristotle are directly applicable only to the first figure, instead of inventing other dicta for the other figures, we reduce them to the first, and then apply the dicta de omni et nullo. Thus we become assured of the validity of our reasoning, and any fallacies in it, which might otherwise escape notice, become at once apparent. This answer is clear, but unsatisfactory. It views Logic as an art. If such be the object of reduction, it is not worth an hour's study; for in actual argumentation this test is never used by the initiated, and the uninitiated never err for lack of it. The mind practically grasps with more ease an argument in its familiar condensed modes of presentation, and sees in them more clearly and certainly its validity or invalidity, than when expressed in these pro

Hamilton's Logic, p. 308.

lix scholastic forms. The answer we would prefer to give is as follows: Logic is a science. The system of reduction serves the purpose of showing that all reasoning is governed by the same principle, that these processes of thought, whatever shapes they may take naturally and spontaneously, are in all cases fundamentally one and the We are thus enabled to comprehend in a single grasp movements of intellect which otherwise would seem multifarious and perplexed. We attain that clear unity which is the end of all science. Our practice is improved by such investigation, but its direct object is not skill, it is knowledge.

same.

§ 6. In the mnemonic hexameters, moods of the same figure occur together. We present on the opposite page a scheme in which the equivalent moods are arranged together. Equivalent moods are those reducible to each other. Their names have the same initial letter. The three methods of notation are also exhibited. This scheme brings to light several important facts, among others the following:

Equivalent moods are not merely reducible to the same mood of Fig. 1, but are reducible to each other. That is, a syllogism in any mood may by reduction be expressed in any of its equivalents. This is evinced by the symbolic notation being similar for all equivalents. But a syllogism cannot be reduced to another mood not equivalent. Thus it appears that the variation by figure, as well as the order of premises, is in a sense unessential, accidental, external; whereas the variation by mood, which depends on the quantity and quality of the premises, is essential and internal. Hence it would seem to be logically more accurate to consider the syllogism as containing under it moods, the equivalents being the species, under which we find varieties in figure, and then we reach the individual moods which have received proper names. A subspecies also might be formed of those equivalent moods requiring only simple conversion in order to reduction. Such are absolutely equivalent, and appear in each of the four figures. They constitute groups in the scheme.

Moods which have the same initial letter, that is, equivalent moods, conclude the same formal judgment. Moods in B conclude A; those in C conclude E; those in D conclude I; those in F conclude O. The exceptions are Bramantip and Dokamok.

The linear and circular notation are symbolic. A different circular diagram may be made for each individual mood, the relative positions of the circles being varied so as in most cases to express the individual

differences. This, however, presents no advantages. The linear notation, which is not thus variable, is on this account rather to be preferred. The graphic notation is not symbolic, but consists of arbitrary signs. It expresses all the accidental variations in external form, whereas the linear expresses only the internal, essential feature, i. e., the mood. The graphic, used in the scheme to express extension, may express also intension. In extension the copula points to the predicate, in intension to the subject; in general, the copula of the conclusion always points to the major term. In comparing the several notations, we must not forget, especially in case of moods containing m, that C and I are indifferent, and therefore interchangeable.

Arnauld, after detailing what Hamilton calls "the disgusting rules for reduction," pronounces them superfluous, and proposes to supersede them by one General Rule for Reduction, as follows: If the terms of the syllogism do not appear in the order required by the first figure, make them assume this order by any legitimate conversion, also transposing, if need be, the premises.

8

§ 7. We are now prepared to examine the Fourth Figure. Its legitimacy has been disputed by many logicians. Feeling it to be awkward, they reject it as an encumbrance, assigning various reasons. Hamilton hotly denounces it as "a monster undeserving of toleration, far less of countenance and favor." He argues that it is unnatural and useless, because the premises are in intension while the conclusion is in extension, and that passing from one of these quantities to the other in the same syllogism is violative of the order of thought, and to no purpose. To this we object, first, that his assumption that the premises are in intension is grounded solely upon their order, which, we repeat, is arbitrary, and hence indicates nothing inherent in the reasoning. We object, secondly, that such alternations of quantity occur very frequently in the other figures, are often to good purpose, and in some cases seem essential (i, § 3). If so, we may grant they occur in Fig. 4, without furnishing a ground for rejecting it. Indeed, as has been said, these quantities cannot stand apart. Every logical judgment, every reasoning is in both at once, and their alternate predominance is not, in any important sense, a change of thought.

Other logicians have thought so well of Fig. 4 that it has withstood these attacks and taken deep root in the literature of Logic,

8 Logic, p. 303.