400 The following is another method of stating and solving all three problems: To determine in what cases it is possible to obtain two incompatible trios of propositions, each trio containing three and only three terms and each including a proposition which is identical with a proposition in the other and also a proposition which is the contradictory of a proposition in the other.

Let the propositions be P, Q, Rʹ and P, Qʹ, T ; and let P contain the terms X and Y ; Q and Qʹ the terms Y and Z ; R and T, the terms Z and X. Suppose Q to be affirmative, and Qʹ negative.
Then since one of each trio of propositions must be negative, and not more than one can be so (as shewn in section [214]), P and T must be affirmative, and Rʹ negative.
Again, since each of the terms X, Y, Z must be distributed once at least in each trio of propositions (as shewn in section 214), and since Y must be undistributed either in Q or in Qʹ, Y must be distributed in P.
Hence P = YaX.
X, being undistributed in P, must be distributed in Rʹ and T.
Hence T = XaZ.
Z, being undistributed in T, must be distributed in Qʹ, and therefore undistributed in Q, and distributed in Rʹ.
Hence Q = YaZ or YiZ or ZiY ;
Qʹ = YoZ or YeZ or ZeY ;
Rʹ = XeZ or ZeX.
We have then the following solution of our problem:—

YaZ, YaZ or YiZ or ZiY, XeZ or ZeX ;
YaZ, YoZ or YeZ or ZeY, XaZ.

345. Numerical Moods of the Syllogism.[437]—The following are examples of numerical moods in the different figures of the syllogism:—401

  Figure 1. (i) All M’s are P’s,
At least n S’s are M’s,
therefore,At least n S’s are P’s ;
(ii) Less than n M’s are P’s,
All S’s are M’s,
therefore,Less than n S’s are P’s ;
(iii) Less than n M’s are P’s,
At least n S’s are M’s,
therefore,Some S’s are not P’s ;
  Figure 2. (iv) All P’s are M’s,
Less than n S’s are M’s,
therefore,Less than n S’s are P’s ;
(v) Less than n P’s are M’s,
All S’s are M’s,
therefore,Less than n S’s are P’s ;
(vi) Less than n P’s are M’s,
At least n S’s are M’s,
therefore,Some S’s are not P’s ;
  Figure 3. (vii) Less than n M’s are P’s,
At least n M’s are S’s,
therefore,Some S’s are not P’s ;
(viii) All M’s are P’s,
At least n M’s are S’s,
therefore,At least n S’s are P’s ;
(ix) At least n M’s are P’s,
All M’s are S’s,
therefore,At least n S’s are P’s ;
  Figure 4. (x) At least n P’s are M’s,
All M’s are S’s,
therefore,At least n S’s are P’s ;
(xi) All P’s are M’s,
Less than n M’s are S’s,
therefore,Less than n S’s are P’s ; 402
(xii) Less than n P’s are M’s,
At least n M’s are S’s,
therefore,Some S’s are not P’s.

[437] This section was suggested by the following question of Mr Johnson’s:—“Shew the validity of the following syllogisms: (i) All M’s are P’s, At least n S’s are M’s, therefore, At least n S’s are P’s; (ii) All P’s are M’s, Less than n S’s are M’s, therefore, Less than n S’s are P’s; (iii) Less than n M’s are P’s, At least n M’s are S’s, therefore, Some S’s are not P’s. Deduce from the above the ordinary non-numerical moods of the first three figures.”

The above moods may be established as follows:—
(i) From All M’s are P’s, it follows that Every S which is M is also P, and since At least n S’s are M’s, it follows further that At least n S’s are P’s.
Denoting the major premiss of (i) by A, the minor by B, and the conclusion by C, we obtain immediately the following syllogisms:—

A, Cʹ,
Cʹ, B,
⎯ ⎯
∴ Bʹ ; ∴ Aʹ ;

and these are respectively equivalent to (iv) and (vii).
(v) is obtainable from (iv) by transposing the premisses and converting the conclusion;
(ii) from (v) by converting the major premiss;
(iii) from (vii) by converting the minor premiss;
(vi) from (iii) by converting the major premiss;
(viii) from (i) by converting the minor premiss;
(ix) from (viii) by transposing the premisses and converting the conclusion;
(x) from (i) by transposing the premisses and converting the conclusion;
(xi) from (iv) by converting the minor premiss;
(xii) from (vii) by converting the major premiss.

The ordinary non-numerical moods of the different figures may be deduced from the above results as follows:—
Figure 1. (i) Putting n = total number of S’s, we have MaP, SaM, ∴ SaP, that is, Barbara ; and putting n = 1, we have MaP, SiM, ∴ SiP, that is, Darii.
(ii)  Putting n = 1, MeP, SaM, ∴ SeP (Celarent).
(iii) Putting n = 1, MeP, SiM, ∴ SoP (Ferio).
AAI and EAO follow à fortiori.