Sec. 8.  Indirect reduction is the process of proving a Mood to be valid by showing that the supposition of its invalidity involves a contradiction.  Take Baroco, and (since the doubt as to its validity is concerned not with the truth of the premises, but with their relation to the conclusion) assume the premises to be true.  Then, if the conclusion be false, its contradictory is true.  The conclusion being in O., its contradictory will be in A. Substituting this A. for the minor premise of Baroco, we have the premises of a syllogism in Barbara, which will be found to give a conclusion in A., contradictory of the original minor premise; thus: 

Baroco.  Barbara.

All P is M;      -----------------> All P is M;

Some S is not M:  <-----\    /-----> All S is P: 
\  /
contradictory \/
/\ contradictory
/  \
.’.  Some S is not P  ------/    \------ .’.  All S is M.

But the original minor premise, Some S is not M, is true by hypothesis; and therefore the conclusion of Barbara, All S is M, is false.  This falsity cannot, however, be due to the form of Barbara, which we know to be valid; nor to the major premise, which, being taken from Baroco, is true by hypothesis:  it must, therefore, lie in the minor premise of Barbara, All S is P; and since this is contradictory of the conclusion of Baroco Some S is not P, that conclusion was true.

Similarly, with Bocardo, the Indirect Reduction proceeds by substituting for the major premise the contradictory of the conclusion; thus again obtaining the premises of a syllogism in Barbara, whose conclusion is contradictory of the original major premise.  Hence the initial B in Baroco and Bocardo:  it points to a syllogism in Barbara as the means of Indirect Reduction (Reductio ad impossibile).

Any other Mood may be reduced indirectly:  as, for example, Dimaris.  If this is supposed to be invalid and the conclusion false, substitute the contradictory of the conclusion for the major premise, thus obtaining the premises of Celarent: 

Dimaris.  Celarent.

contradictory
Some P is M;       <---------    -------->      No S is P;
\  /
\/
All M is S:         -----------/\--------->      All M is S: 
/  \
contradictory/    \
.’.  Some S is P. -----------      --------  .’.  No M is P}
} simply converted
.’.  No P is M}

The conclusion of Celarent, simply converted, contradicts the original major premise of Dimaris, and is therefore false.  Therefore the major premise of Celarent is false, and the conclusion of Dimaris is true.  We might, of course, construct mnemonic names for the Indirect Reduction of all the Moods:  the name of Dimaris would then be Cicari.