Sec. 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, secundae: 
    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.

The consonants s and p, occurring after a vowel, show that the proposition which the vowel stands for is to be converted either (s) simply or (p) per accidens; except where s or p occurs after the third vowel of a name, the conclusion:  then it refers not to the conclusion of the given Mood (say Disamis), but to the conclusion of that Mood of the first Figure to which the given Mood is reduced (Darii).

M (mutare, metathesis) means ‘transpose the premises’ (as of Ca_m_estres).

C means ’substitute the contradictory of the conclusion for the foregoing premise,’ a process of the Indirect Reduction to be presently explained (see Baroco, Sec. 8).

The other consonants, r, n, t (with b and d, when not initial), occurring here and there, have no mnemonic significance.

What now is the problem of Reduction?  The difference of Figures depends upon the position of the Middle Term.  To reduce a Mood of any other Figure to the form of the First, then, we must so manipulate its premises that the Middle Term shall be subject of the major premise and predicate of the minor premise.