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.
§ 6. Reduction is either—(1) Ostensive or (2) Indirect. Ostensive Reduction consists in showing that an argument given in one Mood can also be stated in another; the process is especially used to show that the Moods of the second, third, and fourth Figures are equivalent to one or another Mood of the first Figure. It thus proves the validity of the former Moods by showing that they also essentially conform to the Dictum, and that all Categorical Syllogisms are only superficial varieties of one type of proof.
To facilitate Reduction, the recognised Moods have all had names given them; which names, again, have been strung together into mnemonic verses of great force and pregnancy:
Barbara, Celarent, Darii, Ferioque prioris:
Cesare, Camestres, Festino, Baroco, secundæ:
Tertia, Darapti, Disamis, Datisi, Felapton,
Bocardo, Ferison, habet: Quarta insuper addit
Bramantip, Camenes, Dimaris, Fesapo, Fresison.
In the above verses the names of the Moods of Fig. I. begin with the first four consonants B, C, D, F, in alphabetical order; and the names of all other Moods likewise begin with these letters, thus signifying (except in Baroco and Bocardo) the mood of Fig. I., to which each is equivalent, and to which it is to be reduced: as Bramantip to Barbara, Camestres to Celarent, and so forth.
The vowels A, E, I, O, occurring in the several names, give the quantity and quality of major premise, minor premise, and conclusion in the usual order.