(1) Let every proposition be interpreted as implying the existence both of its subject and of its predicate.[421] In this case, the existence of the major, middle, and minor terms is in every case guaranteed by the premisses, and therefore no further assumption with regard to existence is required in order that the conclusion may be legitimately obtained.[422] We may regard the above supposition as that which is tacitly made in the ordinary doctrine of the syllogism.
[421] It will be observed that this is not quite the same as supposition (1) in section [156].
[422] If, however, we are to be allowed to proceed as in section [206] (where from all P is M, all S is M, we inferred some not-S is not-P) we must posit the existence not merely of the terms directly involved, but also of their contradictories.
(2) Let every proposition be interpreted as implying the existence of its subject. Under this supposition an affirmative proposition ensures the existence of its predicate also; but not so a negative proposition. It follows that any mood will be valid unless the minor term is in its premiss the predicate of a negative proposition. This cannot happen either in figure 1 or in figure 2, since in these figures the minor is always subject in its premiss; nor in figure 3, since in this figure the minor 391 premiss is always affirmative. In figure 4, the only moods with a negative minor are Camenes and its weakened form AEO. Our conclusion then is that on the given supposition every ordinarily recognised mood is valid except these two.[423]
[423] Reduction to figure 1 appears to be affected by this supposition, since it makes the contraposition of A and the conversion of E in general invalid. The contraposition of A is involved in the direct reduction of Baroco (Faksoko). The process is, however, in this particular case valid, as the existence of not-M is given by the minor premiss. The conversion of E is involved in the reduction of Cesare, Camestres, and Festino from figure 2; and of Camenes, Fesapo, and Fresison from figure 4. Since, however, one premiss must be affirmative the existence of the middle term is thereby guaranteed, and hence the simple conversion of E in the second figure, and in the major of the fourth becomes valid. Also the conversion of the conclusion resulting from the reduction of Camestres is legitimate, since the original minor term is subject in its premiss. Hence Camenes (and its weakened form) are the only moods whose reduction is rendered illegitimate by the supposition under consideration. This result agrees with that reached in the text.
(3) Let no proposition be interpreted as implying the existence either of its subject or of its predicate. Taking S, M, P, as the minor, middle, and major terms respectively, the conclusion will imply that if there is any S there is some P or not-P (according as it is affirmative or negative). Will the premisses also imply this? If so, then the syllogism is valid; but not otherwise.
It has been shewn in section [212] that a universal affirmative conclusion, All S is P, can be proved only by means of the premisses, All M is P, All S is M ; and it is clear that these premisses themselves imply that if there is any S there is some P. On our present supposition, then, a syllogism is valid if its conclusion is universal affirmative.
Again, as shewn in section [212], a universal negative conclusion, No S is P, can be proved only in the following ways:—
| (i) | No M is P (or No P is M), |
| All S is M, | |
| ⎯⎯⎯⎯ | |
| therefore, | No S is P ; |
| (ii) | All P is M, |
| No S is M (or No M is S), | |
| ⎯⎯⎯⎯ | |
| therefore, | No S is P. |
392 In (i) the minor premiss implies that if S exists then M exists, and the major premiss that if M exists then not-P exists. In (ii) the minor premiss implies that if S exists then not-M exists, and the major premiss that if not-M exists then not-P exists (as shewn in section [158]). Hence a syllogism is valid if its conclusion is universal negative.