and in order to establish the desired conclusion we must be able to infer at least one of the following,—Some Q is not S, Some S is not Q.
But neither of these propositions can be inferred; for they distribute respectively S and Q, and neither of these terms is distributed in the given premisses. The question is, therefore, to be answered in the negative.
303 211. If it be known concerning a syllogism in the Aristotelian system that the middle term is distributed in both premisses, what can we infer as to the conclusion? [C.]
If both premisses are affirmative, they can between them distribute only two terms, and by hypothesis the middle term is distributed twice in the premisses; hence the minor term cannot be distributed in the premisses, and it follows that the conclusion must be particular.
If one of the premisses is negative, there may be three distributed terms in the premisses; these must, however, be the middle term twice (by hypothesis) and the major term (since the conclusion must now be negative and will therefore distribute the major term); hence the minor term cannot be distributed in the premisses, and it again follows that the conclusion must be particular.
But either both premisses will be affirmative, or one affirmative and the other negative; in any case, therefore, we can infer that the conclusion will be particular.
212. Shew directly in how many ways it is possible to prove the conclusions SaP, SeP ; point out those that conform immediately to the Dictum de omni et nullo ; and exhibit the equivalence between these and the remainder. [W.]
(1) To prove All S is P.
Both premisses must be affirmative, and both must be universal.
S being distributed in the conclusion must be distributed in the minor premiss, which must therefore be All S is M.
M not being distributed in the minor must be distributed in the major, which must therefore be All M is P.
SaP can therefore be proved in only one way, namely,
| All M is P, | |
| All S is M, | |
| therefore, | All S is P ; |
and this syllogism conforms immediately to the Dictum.
(2) To prove No S is P.
Both premisses must be universal, and one must be negative while the other is affirmative; i.e., one premiss must be E and the other A.
First, let the major be E, i.e., either No M is P or No P is M. In each case the minor must be affirmative and must distribute S ; therefore, it will be All S is M.
304 Secondly, let the minor be E, i.e., either No S is M or No M is S. In each case the major must be affirmative and must distribute P ; therefore, it will be All P is M.
We can then prove SeP in four ways, thus,—
| (i) | MeP, | (ii) | PeM, | (iii) | PaM, | (iv) | PaM, |
| SaM, | SaM, | SeM, | MeS, | ||||
| ⎯⎯ | ⎯⎯ | ⎯⎯ | ⎯⎯ | ||||
| SeP. | SeP. | SeP. | SeP. |
Of these, (i) only conforms immediately to the dictum, and we have to shew the equivalence between it and the others.
The only difference between (i) and (ii) is that the major premiss of the one is the simple converse of the major premiss of the other; they are, therefore, equivalent. Similarly the only difference between (iii) and (iv) is that the minor premiss of the one is the simple converse of the minor premiss of the other; they are, therefore, equivalent.
Finally, we may shew that (iv) is equivalent to (i) by transposing the premisses and converting the conclusion.
213. Given that the major term is distributed in the premisses and undistributed in the conclusion of a valid syllogism, determine the syllogism. [C.]