In a full enumeration there are two strengthened syllogisms in each figure:—
In Figure 1, AAI, EAO ;
In Figure 2, EAO, AEO ;
In Figure 3, AAI, EAO ;
In Figure 4, AAI, EAO.
It will be observed that in figures 1 and 2, a syllogism having a strengthened premiss may also be regarded as a syllogism having a weakened conclusion, and vice versâ ; but that in figures 3 and 4, the contrary holds in both cases. The only syllogism with a weakened conclusion in either of these figures is AEO in figure 4; and in this mood no conclusion is obtainable if either of the premisses is replaced by its subaltern.
If syllogisms containing either a strengthened premiss or a weakened conclusion are omitted, we are left with 15 valid moods, namely, 4 in each of the first three figures and 3 in figure 4.
247. The peculiarities and uses of each of the four figures of the syllogism.[338]—Figure 1. In this figure it is possible to prove conclusions of all the forms A, E, I, O; and it is the only figure in which a universal affirmative conclusion can be proved. This alone makes it by far the most useful and important of the syllogistic figures. All deductive science, the object of which is to establish universal affirmatives, tends to work in AAA in this figure.
[338] On the distinctive characteristics of the different figures, see also sections [269] to 271.
Another point to notice is that only in this figure is it the case that both the subject of the conclusion is subject in the premisses, and the predicate of the conclusion predicate in the premisses; in figure 2 the predicate of the conclusion is subject in the major premiss; in figure 3 the subject of the conclusion is predicate in the minor premiss; and in figure 4 there is a double inversion.[339] This no doubt partly 316 accounts for the fact that a reasoning expressed in figure 1 so often seems more natural than the same reasoning expressed in any other figure.[340]
[339] The double inversion in figure 4 is one of the reasons given by Thomson for rejecting that figure altogether. Compare section [262].
[340] Compare Solly, Syllabus of Logic, pp. 130 to 132.
Figure 2. In this figure, only negatives can be proved; and therefore it is chiefly used for purposes of disproof. For example, Every real natural poem is naïve ; those poems of Ossian which Macpherson pretended to discover are not naïve (but sentimental); hence they are not real natural poems (Ueberweg, System of Logic, § 113). It has been called the exclusive figure; because by means of it we may go on excluding various suppositions as to the nature of something under investigation, whose real character we wish to ascertain (a process called abscissio infiniti). For example, Such and such an order has such and such properties, This plant has not those properties ; therefore, It does not belong to that order. A syllogism of this kind may be repeated with a number of different orders till the enquiry is so narrowed down that the place of the plant is easily determined. Whately (Elements of Logic, p. 92) gives an example from the diagnosis of a disease.