The subaltern moods are,—
In Figure 1, AAI, EAO ;
In Figure 2, EAO, AEO ;
In Figure 4, AEO.
It is obvious that there can be no weakened conclusion in Figure 3, since in no case is it possible to infer more than a particular conclusion in this figure.
AAI in Figure 4 is sometimes spoken of as a subaltern mood. But this is a mistake. With the premisses All P is M, All M is S, the conclusion Some S is P is certainly in one sense weaker than the premisses would warrant since the universal conclusion All P is S might have been inferred. But All P is S is not the universal corresponding to Some S is P. The subjects of these two propositions are different; and we infer all that we possibly can about S when we say that some S is P. In other words, regarded as a mood of figure 4, this mood is not a subaltern. AAI in figure 4 is thus differentiated from AAI in figure 1, and its inclusion in the mnemonic verses justified.
246. Strengthened Syllogisms.—If in a syllogism the same conclusion can still be obtained although for one of the premisses we substitute its subaltern, the syllogism is said to be a strengthened syllogism. A strengthened syllogism is thus a syllogism with an unnecessarily strengthened premiss.[336]
[336] Compare De Morgan, Formal Logic, pp. 91, 130. De Morgan calls a syllogism fundamental, when neither of its premisses is stronger than is necessary to produce the conclusion (Formal Logic, p. 77).
For example, the conclusion of the syllogism—
| All M is P, | |
| All M is S, | |
| therefore, | Some S is P, |
could equally be obtained from the premisses All M is P, Some M is S ; or from the premisses Some M is P, All M is S.
By trial we may find that every syllogism in which there 315 are two universal premisses with a particular conclusion is a strengthened syllogism, with the single exception of AEO in the fourth figure.[337]
[337] A general proof of this proposition will be given in section [351].