It will be found that in each figure there are twelve valid moods, which are neither strengthened nor weakened. This result may be established by either of the two alternative methods which follow. 382
I. We may enquire what various combinations of premisses will yield conclusions of the forms A, Y, E, I, O, η, respectively.
It will suffice, as we have already seen, to consider some one figure. We may, therefore, take figure 1, so that the position of the terms will be—
| M | P |
| S | M |
| ⎯⎯⎯⎯ | |
| S | P |
(i) To prove SaP, both premisses must be affirmative; and, in order to avoid illicit minor, the minor premiss must be SaM. It follows that the major must be MaP or there would be undistributed middle. Hence AAA is the only valid mood yielding an A conclusion.
(ii) To prove SyP, both premisses must be affirmative; and, in order to avoid illicit major, the major premiss must be MyP. It follows that the minor must be SyM, in order to avoid undistributed middle. Hence YYY is the only valid mood yielding a Y conclusion.
(iii) To prove SeP, the major must be (1) MeP or (2) MyP or (3) MoP in order to avoid illicit major. If (1), the minor must be SaM or there would be either two negative premisses or illicit minor; if (2), it must be SeM or there would be undistributed middle or illicit minor; if (3), it must be affirmative and distribute both S and M, which is impossible. Hence EAE and YEE are the only valid moods yielding an E conclusion.
(iv) To prove SiP, both premisses must be affirmative, and since SaM would necessarily be a strengthened premiss, the minor must be (1) SiM or (2) SyM. If (1), the major must be MaP or there would be undistributed middle; and if (2), it must be MiP or there would be a strengthened premiss. Hence AII and IYI are the only valid (unstrengthened and unweakened) moods yielding an I conclusion.
(v) To prove SoP, the major must be (1) MeP or (2) MyP or (3) MoP or there would be illicit major. If (1), the minor must be SiM or there would be a strengthened premiss; if (2), it must be SoM or there would be either two affirmative premisses with a negative conclusion or undistributed middle or a 383 strengthened premiss; and if (3), it must be SyM or there would be two negative premisses or undistributed middle. Hence EIO, YOO, OYO are the only valid (unstrengthened and unweakened) moods yielding an O conclusion.
(vi) To prove SηP, the minor must be (1) SeM or (2) SaM or (3) SηM or there would be illicit minor. If (1), the major must be MiP or there would be a strengthened premiss; if (2), the major must be MηP or there would be undistributed middle or two affirmative premisses with a negative conclusion or a strengthened premiss; and if (3), the major must be MaP or there would be undistributed middle or two negative premisses. Hence IEη, ηAη, Aηη are the only valid (unstrengthened and unweakened) moods yielding an η conclusion.