"; and in defence of the above alteration it
may be said that they now deserve that praise still more.

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: