269. Scheme of the Valid Moods of Figure l.—So far as the nature of the reasoning involved is concerned, there is practically no distinction between Barbara and Darii, or between Celarent and Ferio. For in each case, if S is the minor term, the S’s referred to in the conclusion are precisely the same S’s as those referred to in the minor premiss.
Again, the only difference between Barbara and Celarent, or between Darii and Ferio, is that the universal rule which the minor premiss enables us to apply to a particular case is in Barbara and Darii a universal affirmation, while in Celarent and Ferio it is a universal denial.
We may, therefore, sum up all four moods in the following scheme:[366]
| All B is C (or is not C), | (Rule) | |
| All (or some) A is B, | (Case) | |
| therefore, | All (or some) A is C (or is not C). | (Result) |
[366] Compare C. S. Peirce in the Johns Hopkins Studies in Logic, p. 148, and Sigwart, Logic, i. p. 354. Sigwart gives the following formula:
| If anything is M it is P (or is not P), | |
| Certain subjects S are M, | |
| therefore, | They are P (or are not P). |
This way of setting out the valid moods of figure 1 shews clearly how they are all included under the dictum de omni et nullo.
270. Scheme of the Valid Moods of Figure 2.—Applying the principle of indirect reduction, we may immediately obtain from the scheme given in the last preceding section the following scheme, summing up the valid moods of figure 2:[367] 337
| All B is C (or is not C), | (Rule) | |
| Some (or all) A is not C (or is C), | (Denial of Result) | |
| therefore, | Some (or all) A is not B. | (Denial of Case) |
[367] Sigwart’s way of putting it (Logic, i. p. 354) is that in figure 2, instead of inferring from ground to consequence, we infer from invalidity of consequence to invalidity of ground; and he gives the following scheme: