a more systematic way of demonstrating the valid moods is desired,
the simplest method is to deduce from the Canons special rules for
each Figure. Aristotle arrived at these special rules by simple
inspection, but it is easier to deduce them.

I. In the First Figure, the Major Premiss must be Universal, and the
Minor Premiss affirmative.

To make this evident by the Canons, we bear in mind the Scheme or
Figure--

M in P
S in M--

and try the alternatives of Affirmative Moods and Negative Moods.
Obviously in an affirmative mood the Middle is undistributed unless
the Major Premiss is Universal. In a negative mood, (1) If the Major
Premiss is O, the Minor must be affirmative, and M is undistributed;
(2) if the Major Premiss is I, M may be distributed by a negative
Minor Premiss, but in that case there would be an illicit process of
the Major--P being distributed in the conclusion (Canon V.) and not in
the Premisses. Thus the Major Premiss can neither be O nor I, and must
therefore be either A or E, _i.e._, must be Universal.

That the Minor must be affirmative is evident, for if it were
negative, the conclusion must be negative (Canon V.) and the Major
Premiss must be affirmative (Canon IV.), and this would involve
illicit process of the Major, P being distributed in the conclusion
and not in the Premisses.

These two special rules leave only four possible valid forms in the
First Figure. There are sixteen possible combinations of premisses,
each of the four types of proposition being combinable with itself and
with each of the others.

AA EA IA OA
AE EE IE OE
AI EI II OI
AO EO IO OO

Special Rule I. wipes out the columns on the right with the particular
major premisses; and AE, EE, AO, and EO are rejected by Special Rule
II., leaving BA_rb_A_r_A, CE_l_A_r_E_nt_, DA_r_II and FE_r_IO.

II. In the Second Figure, only Negative Moods are possible, and the
Major Premiss must be universal.

Only Negative moods are possible, for unless one premiss is negative,
M being the predicate term in both--

P in M
S in M--

is undistributed.

Only negative moods being possible, there will be illicit process of
the Major unless the Major Premiss is universal, P being its subject
term.

These special rules reject AA and AI, and the two columns on the
right.

To get rid of EE and EO, we must call in the general Canon IV.;
which leaves us with EA, AE, EI, and AO-