| All P is M, | |
| No M is S, | |
| therefore, | Some S is not P ; |
and this syllogism, which is AEO in figure 4, does fulfil the given conditions, for it becomes invalid if either of the premisses is made particular.
The above amounts to a general proof of the proposition laid down in section [246]:—Every syllogism in which there are two universal premisses with a particular conclusion is a strengthened syllogism with the single exception of AEO in figure 4.
[438] We here include the case in which the middle term is itself twice distributed.
352. Given two valid syllogisms in the same figure in which the major, middle, and minor terms are respectively the same, shew, without reference to the mnemonic verses, that if the minor premisses are subcontraries, the conclusions will be identical. [K.]
The minor premiss of one of the syllogisms must be O, and the major premiss of this syllogism must, therefore, be A and the conclusion O. The middle and the major terms having then to be distributed in the premisses, this syllogism is determined, namely,—
| All P is M, | |
| Some S is not M, | |
| therefore, | Some S is not P. |
407 Since the other syllogism is to be in the same figure, its minor premiss must be Some S is M ; the major must therefore be universal, and in order to distribute the middle term it must be negative. This syllogism therefore is also determined, namely,—
| No P is M, | |
| Some S is M, | |
| therefore, | Some S is not P. |
The conclusions of the two syllogisms are thus shewn to be identical.
353. Find out in which of the valid syllogisms the combination of one premiss with the subcontrary of the conclusion would establish the subcontrary of the other premiss. [J.]