It must be remembered that SPʹ = 0 carries with it the implications Either S = 0 or P > 0, Either Pʹ = 0 or Sʹ > 0.
Let particulars be interpreted as implying, while universals are not interpreted as implying, the existence of their subjects. 223 We then have:—
| A,—SPʹ = 0; |
| E,—SP = 0; |
| I,—SP > 0; |
| O,—SPʹ > 0. |
158. Immediate Inferences and the Existential Import of Propositions.—It has been already suggested that before coming to any decision in regard to the existential import of propositions, it will be well to enquire how certain logical doctrines are affected by the different existential assumptions upon which we may proceed. This discussion will as far as possible be kept distinct from the enquiry as to which of the assumptions ought normally to be adopted. The latter question is of a highly controversial nature, but the logical consequences of the various suppositions ought to be capable of demonstration, so as to leave no room for differences of opinion.
We shall in the present section enquire how far different hypotheses regarding the existential import of propositions affect the validity of obversion and conversion and the other immediate inferences based upon these. In the next section we shall consider inferences connected with the square of opposition.
We may take in order the suppositions formulated in section 156.
(1) Let every proposition he understood to imply the existence of both its subject and its predicate and also of their contradictories.
It is clear that on this hypothesis the validity of conversion, obversion, contraposition, and inversion will not be affected by existential considerations. The terms of the original proposition together with their contradictories being in each case identical with the terms of the inferred proposition together with their contradictories, the latter cannot possibly contain any existential implication that is not already contained in the original proposition.[229]
[229] The reader may be reminded that in our first working out of these immediate inferences we provisionally assumed, apart from any implication contained in the propositions themselves, that the terms involved and also their contradictories represented existing classes.
224 (2) Let every proposition he understood to imply simply the existence of its subject.
(a) The validity of obversion is not affected.
(b) The conversion of A is valid, and also that of I. If All S is P and Some S is P imply directly the existence of S, then they clearly imply indirectly the existence of P ; and this is all that is required in order that their conversion may be legitimate. The conversion of E is not valid; for No S is P implies neither directly nor indirectly the existence of P, whilst its converse does imply this.
(c) The contraposition of E is valid, and also that of O. No S is P and Some S is not P both imply on our present supposition the existence of S, and since by the law of excluded middle every S is either P or not-P, it follows that they imply indirectly the existence of not-P. The contraposition of A is not valid; for it involves the conversion of E, which we have already seen not to be valid.[230]
(d) The process of inversion is not valid; for it involves in the case of both A and E the conversion of an E proposition.[231]
If along with an E proposition we are specially given the information that P exists, or if this is implied in some other proposition given us at the same time, then the E proposition may of course be converted. In corresponding circumstances the contraposition and inversion of A and the inversion of E may be valid.[232] Or again, given simply No S is P, we may infer Either P is non-existent or no P is S ; and similarly in other cases.
[230] Or we might argue directly that the contraposition of A is not valid, since All S is P does not imply the existence of not-P, whilst its contrapositive does imply this.