Fig. 1. (a) The major premiss must be universal. Fig. 2. (a) The major premiss must be universal. Fig. 3. (a) The minor premiss must be affirmative. Fig. 4. (a) When the major premiss is affirmative, the minor must be universal. (8) When the minor premiss is affirmative, the conclusion must be particular. (y) In negative moods the major premiss must be universal. Note.-If, leaving out of consideration the conclusion, we regard the number of possible figures as three and that of possible moods as sixteen, we may proceed as follows. Having enunciated the canon of the first figure, we may constitute the moods Barbara, Celarent, Darii, and Ferio. The subaltern moods, AAI and EAO, are not admissible, as the question here before us is not 'What conclusions are legitimated by such and such premisses,' but To what conclusions do such and such premisses lead?' Now, from this point of view, the conclusions. of the subaltern moods are not directly inferred from the premisses, but are inferred by subalternation from the universal conclusions to which the premisses directly lead. The same observation will of course apply to the subaltern moods in the other figures. If we take no account of the conclusion, we have no means of determining which is the major and which is the minor term. Consequently, the premisses may lead to two kinds of conclusions: 1st, those in which the predicate of the first premiss is predicated of the subject of the second; 2nd, those in which the subject of the second premiss is predicated of the predicate of the first. Now the canon of the first figure applies only to the first case; consequently we are bound to ask if any conclusions, falling under the second head, may be inferred from the premisses. These cannot be determined directly, but must be determined in the same manner as conclusions in the second and third figures. Here we proceed by a method similar to that employed in the text. The syllogistic rules exclude seven of the sixteen possible moods, viz. EE, EO, OE, OO, II, IO, OI. When the moods are referred to their several figures, we find that, where the extreme employed in the first premiss becomes the predicate, and the extreme employed in the second premiss the subject of the conclusion, the results are, in the second figure, Cesare, Camestres, Festino, Baroko; in the third, Darapti, Disamis, Datisi, Felapton, Bokardo, Ferison, the subaltern moods of the second figure EAO, AEO, being inadmissible. Where the extreme employed in the first premiss becomes the subject, and the extreme employed in the second premiss the predicate of the conclusion, the results are, in the first figure, AAI, AEO, AII, EAE, IEO, which, when we transpose the premisses, become repectively Bramantip, Fesapo, Dimaris, Camenes, and Fresison in the fourth figure. Hence, according to this mode of treatment, the moods of the fourth figure are regarded as indirect moods of the first. Similarly, in the second figure we may constitute the indirect moods AEE, EAE, IEO, OAO. These, if we transpose the premisses, are merely a repetition of the ordinary moods of the second figure. This is also the case with the indirect moods of the third figure, viz. AAI, AEO, AII, AOO, IAI, IEO. It will therefore be seen that, with the exception of rejecting the subaltern moods, which even there we regarded as superfluous, we arrive practically at the same results as in the text. The moods of the fourth figure are recognised, but, instead of being regarded as moods of a distinct figure, they are treated as indirect moods of the first. By the expression 'indirect moods,' it will be seen, we mean moods in which the extreme employed in the first premiss becomes the subject, and the extreme employed in the second premiss the predicate of the conclusion. CHAPTER IV. On Trains of Reasoning. (The Sorites.) SYLLOGISMS may be combined in what is called a Train of Reasoning. Thus the major and minor premisses, or either, of our ultimate syllogisms may themselves be proved by syllogisms; the major and minor premisses of these, or either, by other syllogisms, and so on, till at last we come to premisses not admitting of syllogistic proof. Such premisses are either assumed without any proof at all, or they are the result either of direct observation or of the testimony of others or of Induction. In a train of reasoning, any syllogism proving a premiss of a subsequent syllogism is called with reference to the subsequent syllogism a Pro-Syllogism, and the subsequent syllogism with reference to it an Epi-Syllogism. It is obvious that the very same syllogism in different relations may be called a Pro-Syllogism or an Epi-Syllogism. The Sorites is a common instance of a train of reasoning in a compressed form. It consists of a series of propositions, the predicate of each becoming the subject of the next. The conclusion predicates the last predicate of the first subject. Thus, All A is B, All B is C, All C is D, All D is E; ... All A is E. When expanded, the Sorites contains as many syllogisms as there are propositions intermediate between the first proposition and the conclusion. These syllogisms are in the first figure, and the conclusion of each becomes the minor premiss of the next. Thus, the above Sorites contains three syllogisms, viz. (1) All B is C, All A is B; ... All A is C. (2) All C is D, All A is C; ... All A is D. (3) All D is E, All A is D; .. All A is E. In a Sorites, only one premiss can be particular, viz. the first; and only one negative, viz. the last1. 1 The first premiss, if particular, may be stated in the form 'Some B is A,' instead of in the usual form 'Some A is B ;' the first syllogism of the expanded Sorites will then be in the third figure instead of the first. Similarly, the last premiss, if negative, may be stated in the form 'No E is D,' instead of in the form 'No D is E,' which will make the last |