It may facilitate the recollection of this most important point if we schematise the figures thus: 

  I. II.  III.  IV.

M—–­P P—–­M M—–­P P—–­M \ | | /
  \ | | /
   \ | | /
S—–­M S—–­M M—–­S M—–­S

The horizontal lines represent the premises, and at the angles formed with them by the slanting or by the perpendicular lines the middle term occurs.  The schema of Figure IV. resembles Z, the last letter of the alphabet:  this helps one to remember it in contrast with Figure I., which is thereby also remembered.  Figures II. and III. seem to stand back to back.

Sec. 3.  The Moods of each Figure are the modifications of it which arise from different combinations of propositions according to quantity and quality.  In Figure I., for example, four Moods are recognised:  A.A.A., E.A.E., A.I.I., E.I.O.

    A. All M is P;
    A. All S is M: 
    A. .’.  All S is P.

    E. No M is P;
    A. All S is M: 
    E. .’.  No S is P.

    A. All M is P;
    I. Some S is M: 
    I. .’.  Some S is P.

    E. No M is P;
    I. Some S is M: 
    O. .’.  Some S is not P.

Now, remembering that there are four Figures, and four kinds of propositions (A.  I. E. O.), each of which propositions may be major premise, minor premise, or conclusion of a syllogism, it appears that in each Figure there may be 64 Moods, and therefore 256 in all.  On examining these 256 Moods, however, we find that only 24 of them are valid (i.e., of such a character that the conclusion strictly follows from the premises), whilst 5 of these 24 are needless, because their conclusions are ‘weaker’ or less extensive than the premises warrant; that is to say, they are particular when they might be universal.  Thus, in Figure I., besides the above 4 Moods, A.A.I. and E.A.O. are valid in the sense of being conclusive; but they are superfluous, because included in A.A.A. and E.A.E.  Omitting, then, these 5 needless Moods, which are called ‘Subalterns’ because their conclusions are subaltern (chap. vii.  Sec. 2) to those of other Moods, there remain 19 Moods that are valid and generally recognised.

Sec. 4.  How these 19 Moods are determined must be our next inquiry.  There are several ways more or less ingenious and interesting; but all depend on the application, directly or indirectly, of the Six Canons, which were shown in the last chapter to be the conditions of Mediate Inference.