§ 604. We will now arrive at the same result by a shorter and more scientific method. This method consists in first determining what pairs of premisses are valid in accordance with Rules 6 and g, and then examining what conclusions may be legitimately inferred from them in accordance with the other rules of syllogism. § 605. The major premiss may be either A, E, I or O. If it is A, the minor also may be either A, E, I or O. If it is E, the minor can only be A or I. If it is I, the minor can only be A or E. If it is O, the minor can only be A. Hence there result 9 valid pairs of premisses. AA. AE. AI. AO. EA. EI. IA. IE. OA. Three of these pairs, namely AA, AE, EA, yield two conclusions apiece, one universal and one particular, which do not violate any of the rules of syllogism; one of them, IE, yields no conclusion at all; the remaining five have their conclusion limited to a single proposition, on the principle that the conclusion must follow the weaker part. Hence we arrive at the same result as before, of II legitimate moods-- AAA. AAI. AEE. AEO. EAE. EAO. AII. AOO. EIO. IAI. OAO. CHAPTER XIII. _Of the Special Rules of the Four Figures_. § 606. Our next task must be to determine how far the 11 moods which we arrived at in the last chapter are valid in the four figures. But before this can be done, we must lay down the _Special Rules of the Four Figures_. FIGURE 1. Rule 1, The minor premiss must be affirmative. Rule 2. The major premiss must be universal. FIGURE II. Rule 1. One or other premiss must be negative. Rule 2. The conclusion must be negative. Rule 3. The major premiss must be universal. FIGURE III. Rule 1. The minor premiss must be affirmative. Rule 2. The conclusion must be particular. FIGURE IV. Rule 1. When the major premiss is affirmative, the minor must be universal. Rule 2. When the minor premiss is particular, the major must be negative. Rule 3, When the minor premiss is affirmative, the conclusion must be particular. Rule 4. When the conclusion is negative, the major premiss must be universal. Rule 5. The conclusion cannot be a universal affirmative. Rule 6. Neither of the premisses can be a particular negative. § 607. The special rules of the first figure are merely a reassertion in another form of the Dictum de Omni et Nullo. For if the major premiss were particular, we should not have anything affirmed or denied of a whole class; and if the minor premiss were negative, we should not have anything declared to be contained in that class. Nevertheless these rules, like the rest, admit of being proved from the position of the terms in the figure, combined with the rules for the distribution of terms (§ 293). _Proof of the Special Rules of the Four Figures._ FIGURE 1. § 608. Proof of Rule 1.--_The minor premiss must be affirmative_. B--A C--B C--A