The contradictory of (iii) is SeP, from which we obtain PaSʹ. Combining with (i), we have—
| PaSʹ, | ||
| MaP, | ||
| ⎯⎯ | ||
| ∴ | MaSʹ | in Barbara. |
But from this conclusion we may obtain SeM, which is the contradictory of (ii).
[349] It has been maintained, that this reduction is unnecessary, and that, to all intents and purposes, Darii is Barbara, since the “some S” in the minor is, and is known to be, the same some as in the conclusion. Compare section [269].
261. Is Reduction an essential part of the Doctrine of the Syllogism?—According to the original theory of reduction, the object of the process is to be sure that the conclusion is a valid inference from the premisses. The validity of a syllogism in figure 1 may be directly tested by reference to the dictum de omni et nullo: but this dictum has no direct application to syllogisms in the remaining three figures. Thus, Whately says, “As it is on the dictum de omni et nullo that all reasoning ultimately depends, so all arguments may be in one way or other brought into some one of the four moods in the first figure: and a syllogism is, in that case, said to be reduced” (Elements of Logic, p. 93). Professor Fowler puts the same position somewhat more guardedly, “As we have adopted no canon for the 2nd, 3rd, and 4th figures, we have as yet 326 no positive proof that the six moods remaining in each of those figures are valid: we merely know that they do not offend against any of the syllogistic rules. But if we can reduce them, i.e., bring them back to the first figure, by shewing that they are only different statements of its moods, or in other words, that precisely the same conclusions can be obtained from equivalent premisses in the first figure, their validity will be proved beyond question” (Deductive Logic, p. 97).
Reduction is, on the other hand, regarded by some logicians as both unnecessary and unnatural. It is, in the first place, said to be unnecessary, on the ground that the dictum de omni et nullo has no claim to be regarded as the paramount law for all valid inference.[350] In sections [270] to 272 it will be shewn that dicta can be formulated for the other figures, which may be regarded as making them independent of the first, and putting them on a level with it. It may also be maintained that in any mood the validity of a particular syllogism is as self-evident as that of the dictum de omni et nullo itself; and that, therefore, although axioms of syllogism are useful as generalisations of the syllogistic process, they are needless in order to establish the validity of any given syllogism. This view is indicated by Ueberweg.
[350] Compare Thomson, Laws of Thought, p. 172.
Reduction is, in the second place, said to be unnatural, inasmuch as it often involves the substitution of an unnatural and indirect for a natural and direct predication. Figures 2 and 3 at any rate have their special uses, and certain reasonings fall naturally into these figures rather than into the first figure.[351]
[351] Compare a quotation from Lambert (Neues Organon, §§ 230, 231) given by Sir W. Hamilton (Logic, II. p. 438).
The following example is given by Thomson (Laws of Thought, p. 174): “Thus, when it was desirable to shew by an example that zeal and activity did not always proceed from selfish motives, the natural course would be some such syllogism as the following. The Apostles sought no earthly reward, the Apostles were zealous in their work; therefore, 327 some zealous persons seek not earthly reward.” In reducing this syllogism to figure 1, we have to convert our minor into “Some zealous persons were Apostles,” which is awkward and unnatural.