349. Given that the middle term is distributed twice in the premisses of a syllogism, determine directly (i.e., without any reference to the mnemonic verses or the special rules of the figures) in what different moods it might possibly be. [K.]
The premisses must be either both affirmative, or one affirmative and one negative.
In the first case, both premisses being affirmative can distribute their subjects only. The middle term must, therefore, be the subject in each, and both must be universal. This limits us to the one syllogism,— 405
| All M is P, | |
| All M is S, | |
| therefore, | Some S is P. |
In the second case, one premiss being negative, the conclusion must be negative and will, therefore, distribute the major term. Hence, the major premiss must distribute the major term, and also (by hypothesis) the middle term. This condition can be fulfilled only by its being one or other of the following,—No M is P or No P is M. The major being negative, the minor must be affirmative, and in order to distribute the middle term must be All M is S.
In this case we get two syllogisms, namely,—
| No M is P, | |
| All M is S, | |
| therefore, | Some S is not P ; |
| No P is M, | |
| All M is S, | |
| therefore, | Some S is not P. |
The given condition limits us, therefore, to three syllogisms (one affirmative and two negative); and by reference to the mnemonic verses we may identify these with Darapti and Felapton in figure 3, and Fesapo in figure 4.
350. If the major premiss and the conclusion of a valid syllogism agree in quantity, but differ in quality, find the mood and figure. [T.]
Since we cannot have a negative premiss with an affirmative conclusion, the major premiss must be affirmative and the conclusion negative. It follows immediately that, in order to avoid illicit major, the major premiss must be All P is M (where M is the middle term and P the major term). The conclusion, therefore, must be No S is P (S being the minor term); and this requires that, in order to avoid undistributed middle and illicit minor, the minor premiss should be No S is M or No M is S. Hence the syllogism is in Camestres or in Camenes.
351. Given a valid syllogism with two universal premisses and a particular conclusion, such that the same conclusion cannot be inferred, if for either of the premisses is substituted its subaltern, determine the mood and figure of the syllogism. [K.]
Let S, M, P be respectively the minor, middle, and major terms of the given syllogism. Then, since the conclusion is particular, it must be either Some S is P or Some S is not P. 406
First, if possible, let it be Some S is P.
The only term which need be distributed in the premisses is M. But since we have two universal premisses, two terms must be distributed in them as subjects.[438] One of these distributions must be superfluous; and it follows that for one of the premisses we may substitute its subaltern, and still get the same conclusion.
The conclusion cannot then be Some S is P.
Secondly, if possible, let the conclusion be Some S is not P.
If the subject of the minor premiss is S, we may clearly substitute its subaltern without affecting the conclusion. The subject of the minor premiss must therefore be M, which will thus be distributed in this premiss. M cannot also be distributed in the major, or else it is clear that its subaltern might be substituted for the minor and nevertheless the same conclusion inferred. The major premiss must, therefore, be affirmative with M for its predicate. This limits us to the syllogism—