The conclusion being affirmative, the premisses must be so too (by Rule 7). Therefore the minor term is undistributed in the minor premiss, where it is predicate. Hence it cannot be distributed in the conclusion (by Rule 4). Therefore the affirmative conclusion must be particular. § 620. Proof of Rule 6.--_Neither of the premisses can lie a, PARTICULAR NEGATIVE_. If the major premiss were a particular negative, the conclusion would be negative. Therefore the major term would be distributed in the conclusion. But the major premiss being particular, the major term could not be distributed there. Therefore we should have an illicit process of the major term. If the minor premiss were a particular negative, then, since the major must be affirmative (by Rule 5), we should have undistributed middle. CHAPTER XIV _Of the Determination of the Moods that are valid in the Four Figures._ § 621. By applying the special rules just given we shall be able to determine how many of the eleven legitimate moods are valid in the four figures. $622. These eleven legitimate moods were found to be AAA. AAI. AEE. AEO. AII. AOO. EAE. EAO. EIO. IAI. OAO. FIGURE 1. § 623. The rule that the major premiss must be universal excludes the last two moods, IAI, OAO. The rule that the minor premiss must be affirmative excludes three more, namely, AEE, AEO, AOO. Thus we are left with six moods which are valid in the first figure, namely, AAA. EAE. AII. EIO. AAI. EAO. FIGURE II. § 624. The rule that one premiss must be negative excludes four moods, namely, AAA, AAI, AII, IAI. The rule that the major must be universal excludes OAO. Thus we are left with six moods which are valid in the second figure, namely, EAE. AEE. EIO. AOO. EAO. AEO. FIGURE III. § 625. The rule that the conclusion must be particular confines us to eight moods, two of which, namely AEE and AOO, are excluded by the rule that the minor premiss must be affirmative. Thus we are left with six moods which are valid in the third figure, namely, AAI. IAI. AII. EAO. OAO. EIO. FIGURE IV. § 626. The first of the eleven moods, AAA, is excluded by the rule that the conclusion cannot be a universal affirmative. Two more moods, namely AOO and OAO, are excluded by the rule that neither of the premisses can be a particular negative. AII violates the rule that when the major premiss is affirmative, the minor must be universal. EAE violates the rule that, when the minor premiss is affirmative, the conclusion must be particular. Thus we are left with six moods which are valid in the fourth figure, namely, AAI. AEE. IAI. EAO. EIO. AEO. § 627. Thus the 256 possible forms of syllogism have been reduced to two dozen legitimate combinations of mood and figure, six moods being valid in each of the four figures. FIGURE I. AAA. EAE. AII. EIO. (AAI. EAO.) FIGURE II. EAE. AEE. EIO. AGO. (EAO. AEO.)