It will be seen that of the sixty-four moods, when referred to the four figures, there are only six in each which have not been rejected. It now remains further to test these moods in the second, third, and fourth figures by reducing them to moods in the first. Reduction. As we have adopted no canon for the second, third, and fourth figures, we have as yet no positive proof that the six moods remaining in each of those figures are valid; we merely know that they do not offend against any of the syllogistic rules. But, if we can reduce them, i. e. bring them back to the first figure, by shewing that they are only different statements of its moods, their validity will be proved beyond question. There are two methods of performing this operation: 1st. that called Ostensive Reduction, which consists in employing one or more of the processes of conversion, permutation, and transposition of premisses; 2nd. that called Reductio per impossibile, which consists in shewing, by means of the first figure and the laws of opposition, that the contradictory of the conclusion is false, and therefore the conclusion itself true. Either of these methods is applicable to all the eighteen moods, and the result is that all are proved to be valid. We shall give instances of the application of each method. By ostensive reduction we shall test E A O in the fourth, IA I in the third, A EE and A OO in the second figures. Fig. 2. A All A is B. Some C is not B. O.. Some C is not A. The mark Fig. 1. ... No B is A. (Simple Conversion.) ... Some C is B. (Conversion per acc.) Some C is not A. Fig. 1. All B is C. ... Some A is B. (Simple conversion.) Some A is C. ... Some C is A. (Simple Conversion.) Fig. 1. ... No B is C. (Simple Conversion.) All A is B. No A is C. ... No C is A. (Simple Conversion.) ... No A is not-B. (Permutation.) shews that the premisses are transposed; the sign.. on the right-hand side of the page is here appropriated to express the employment of conversion or permutation. The last example is interesting, because AOO in fig. 2, and OAO in fig. 3, inasmuch as they contain O premisses, cannot be reduced by the ordinary methods of transposition of premisses and conversion. Hence the older logicians (who, with few exceptions, did not recognise permutation) applied to them the tedious method of reductio per impossibile (or, if we write it in full, H reductio per deductionem ad impossibile). This method is equally applicable to all the imperfect moods, as the moods of the three last figures are often called. We now proceed to give an example of it, and shall select AAI in the third figure. This conclusion must be true; for, if not, suppose it to be false, Then its contradictory must be true, i. e. No C is A. But (from the premisses) All B is C. .. (By figure 1) No B is A. But (from the premisses) All B is A. Syll. II. Now these two (being contrary propositions) cannot both be true. But the proposition All B is A is assumed to be true. ... The proposition No B is A must be false. Hence, either the reasoning of Syll. II. is faulty, or one of the premisses is untrue. But the reasoning (being in the first figure) must be valid. ... One of the premisses is false. Now the premiss 'All B is C,' being one of the premisses of the original syllogism, is assumed to be true. ... The other premiss (No C is A) must be false. ... Its contradictory (Some C is A) is true. Q. E. D. As the positive test of reduction confirms in every case the negative test of the syllogistic rules, it follows that six moods (though not the same six moods) are valid in each figure. These moods may be remembered by means of the mnemonic lines: Barbara, Celarent, Darii, Ferioque, prioris: Nomen habent nullum, nec, si bene colligis, usum. In the above lines, the initial consonants, B, C, D, F, shew that the mood in the second, third, or fourth figure to which they are prefixed is to be reduced to the mood correspondingly marked in the first. Thus Disamis, when reduced, will become Darii. The vowels shew the moods; thus Disamis represents IAI in the third figure. The letters, when it occurs after a vowel, shews that the proposition for which that vowel stands is to be converted simply, the letter p that it is to be converted per accidens. The letter m shews that the premisses are to be transposed, k that the mood is to be reduced per impossibile. It will be noticed that k occurs only in two moods, Baroko and Bokardo, but we have shewn that the per impossibile method is equally applicable to all imperfect moods, and that these two moods can be reduced ostensively by means of permutation, so that any imperfect mood may be reduced either ostensively or per impossibile. The initial B in Baroko and Bokardo shews that the per impossibile method, in their case, assumes the validity of Barbara, but in other cases the operation may assume the validity of some one of the other moods in the first figure; thus, in the particular instance we have taken above, it is performed by means of Celarent. It is perhaps needless to add that all letters, not already explained, in the mnemonic lines, are non-significant. The nature of the subaltern moods has already been explained. They are, AAI, EAO in fig. 1, EAO, AEO in fig. 2, and AEO in fig. 4, included respectively in AAA, EAE, EAE, AEE, AEE. They cannot properly be regarded as illegitimate, inasmuch as the conclusions are valid, but they are superfluous, inasmuch as they infer less than is justified by the premisses. The Special Rules. Besides the general syllogistic rules, already enunciated and proved, certain Special Rules have been enunciated for each figure. We give them below as generally stated. Those for the first figure have been proved in establishing its canon; those for the other figures the student may verify for himself by applying the rules, already laid down, on the distribution of terms. |