§ 616. Proof of Rule 2.--_When the minor premiss is particular, the major must be negative._
This rule is the converse of the preceding, and depends upon the same principle.
§ 617. Proof of Rule 3.--_When the minor premiss is affirmative, the conclusion must be particular._
If the conclusion were universal, there would be illicit process of the minor.
§ 618. Proof of Rule 4.--_When the conclusion is negative, the major premiss must_ be universal.
If the major premiss were particular, there would be illicit process of the major.
§ 619. Proof of Rule 5.--_The conclusion CANNOT be A UNIVERSAL affirmative_.
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.
_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.
§ 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,
§ 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,