Fig.  I.—­A.A.A., E.A.E., A.I.I., E.I.O. (A.A.I., E.A.O., Subaltern);

Fig.  II.—­E.A.E., A.E.E., E.I.O., A.O.O. (E.A.O., A.E.O., Subaltern);

Fig.  III.—­A.A.I., I.A.I., A.I.I., E.A.O., O.A.O., E.I.O.;

Fig.  IV.—­A.A.I., A.E.E., I.A.I., E.A.O., E.I.O. (A.E.O., Subaltern).

Thus, including subaltern Moods, there are six valid in each Figure.  In Fig.  III. alone there is no subaltern Mood, because in that Figure there can be no universal conclusion.

Sec. 5.  Special Canons of the several Figures, deduced from the Common Canons, enable us to arrive at the same result by a somewhat different course.  They are not, perhaps, necessary to the Science, but afford a very useful means of enabling one to thoroughly appreciate the character of formal syllogistic reasoning.  Accordingly, the proof of each rule will be indicated, and its elaboration left to the reader.  There is no difficulty, if one bears in mind that Figure is determined by the position of the middle term.

Fig.  I., Rule (a):  The minor premise must be affirmative.

For, if not, in negative Moods there will be illicit process of the major term.  Applying this rule to the eleven possible Moods given in Sec. 4, as remaining after application of the Common Canons, it eliminates A.E.E., A.E.O., A.O.O.

(b) The major premise must be universal.

For, if not, the minor premise being affirmative, the middle term will be undistributed.  This rule eliminates I.A.I., O.A.O.; leaving six Moods, including two subalterns.

Fig.  II. (a) One premise must be negative.

For else neither premise will distribute the middle term.  This rule eliminates A.A.A., A.A.I., A.I.I., I.A.I.

(b) The major premise must be universal.

For else, the conclusion being negative, there will be illicit process of the major term.  This eliminates I.A.I., O.A.O.; leaving six Moods, including two subalterns.

Fig.  III. (a) The minor premise must be affirmative.

For else, in negative moods there will be illicit process of the major term.  This rule eliminates A.E.E., A.E.O., A.O.O.

(b) The conclusion must be particular.

For, if not, the minor premise being affirmative, there will be illicit process of the minor term.  This eliminates A.A.A., A.E.E., E.A.E.; leaving six Moods.

Fig.  IV. (a) When the major premise is affirmative, the minor must be universal.

For else the middle term is undistributed.  This eliminates A.I.I., A.O.O.

(b) When the minor premise is affirmative the conclusion must be particular.

Otherwise there will be illicit process of the minor term.  This eliminates A.A.A., E.A.E.

(c) When either premise is negative, the major must be universal.

For else, the conclusion being negative, there will be illicit process of the major term.  This eliminates O.A.O.; leaving six Moods, including one subaltern.