(3) AYI in figure 1, some being used in its ordinary logical sense, is equivalent to AAI in figure 3 in the ordinary syllogistic scheme, and is valid. But it is invalid if some is used in the sense of some only, for the conclusion now implies that S and P are partially excluded from each other as well as partially coincident, whereas this is not implied by the premisses. With 380 this use of some, the correct conclusion can be expressed only by stating an alternative between SuP, SaP, SyP, and SiP. This case may serve to illustrate the complexities in which we should be involved if we were to attempt to use some consistently in the sense of some only.[412]
[412] Compare Monck, Logic, p. 154.
(4) ηUO in figure 2 is valid:—
| No P is some M, | |
| All S is all M, | |
| therefore, | Some S is not any P. |
Without the use of quantified predicates, we can obtain the same conclusion in Bocardo, thus,—
| Some M is not P, | |
| All M is S, | |
| therefore, | Some S is not P. |
It will be observed that both (3) and (4) are strengthened syllogisms.
(5) AUA in figure 2 runs as follows,—
| All P is some M, | |
| All S is all M, | |
| therefore, | All S is some P. |
Here we have neither undistributed middle nor illicit process of major or minor, nor is any rule of quality broken, and yet the syllogism is invalid.[413] Applying the rule given above that “whenever one of the premisses is U, the conclusion may be obtained by substituting S or P (as the case may be) for M in the other premiss,” we find that the valid conclusion is Some S is all P. More generally, it follows from this rule of substitution that if one premiss is U while in the other premiss the middle term is undistributed, then the term combined with the middle term in the U premiss must be undistributed in the conclusion. This appears to be the one additional syllogistic rule required if we recognise U propositions in syllogistic reasonings.