on All S is in M. If then the original conclusion
is denied, it follows that All S is in M. But this contradicts the
Minor Premiss, which has been admitted to be true. It is thus shown
that an opponent cannot admit the premisses and deny the conclusion
without contradicting himself.

The same process may be applied to Bokardo.

Some M is not in P.
All M is in S.
Some S is not in P.

Deny the conclusion, and you must admit that All S is in P. Syllogised
in Barbara with All M is in S, this yields the conclusion that All M
is in P, the contradictory of the Major Premiss.

The beginner may be reminded that the argument _ad absurdum_ is not
necessarily confined to Baroko and Bokardo. It is applied to them
simply because they are not reducible by the ordinary processes to the
First Figure. It might be applied with equal effect to other Moods,
DI_m_A_r_I_s_, _e.g._, of the Third.

Some M is in P.
All M is in S.
Some S is in P.

Let Some S is in P be denied, and No S is in P must be admitted. But
if No S is in P and All M is in S, it follows (in Celarent) that No M
is in P, which an opponent cannot hold consistently with his admission
that Some M is in P.

The beginner sometimes asks: What is the use of reducing the Minor
Figures to the First? The reason is that it is only when the relations
between the terms are stated in the First Figure that it is at once
apparent whether or not the argument is valid under the Axiom or
_Dictum de Omni_. It is then undeniably evident that if the Dictum
holds the argument holds. And if the Moods of the First Figure hold,
their equivalents in the other Figures must hold too.

Aristotle recognised only two of the Minor Figures, the Second and
Third, and thus had in all only fourteen valid moods.

The recognition of the Fourth Figure is attributed by Averroes to
Galen. Averroes himself rejects it on the ground that no arguments
expressed naturally, that is, in accordance with common usage, fall
into that form. This is a sufficient reason for not spending time upon
it, if Logic is conceived as a science that has a bearing upon the
actual practice of discussion or discursive thought. And this was
probably the reason why Aristotle passed it over.

If however the Syllogism of Terms is to be completed as an abstract
doctrine, the Fourth Figure must be noticed as one of the forms of
premisses that contain the required relation between the extremes.
There is a valid syllogism