This is an interesting case to notice in connexion with the discussion raised in sections [158] and [159].
We have
SaP = SePʹ = PʹeS ;
SʹaP = SʹePʹ = PʹeSʹ = PʹaS.
The given propositions come out, therefore, as contraries.
On the view that we ought not to enter into any discussion concerning existence in connexion with immediate inference, we must, I suppose, rest content with this statement of the case. It seems, however, sufficiently curious to demand further investigation and explanation. We may as before take different suppositions with regard to the existential import of propositions.
(1) If every proposition implies the existence of both subject and predicate and their contradictories, then it is at once clear that the two propositions cannot both be true together; for between them they deny the existence of not-P.
(2) On the view that propositions imply simply the existence of their subjects, it has been shewn in section [158] that we are not justified in passing from All not-S is P to All not-P is S unless we are given independently the existence of not-P. But it will be observed that in the case before us the given propositions make this impossible. Since all S is P and all not-S is P, and everything is either S or not-S by the law of excluded middle, it follows that 247 nothing is not-P. In order, therefore, to reduce the given propositions to such a form that they appear as contraries (and consequently[267] as inconsistent with each other) we have to assume the very thing that taken together they really deny.
(3) and (4). On the view that at any rate universal propositions do not imply the existence of their subjects, we have found in section [159] that the propositions No not-P is S, All not-P is S, are not necessarily inconsistent, for they may express the fact that P constitutes the entire universe of discourse. But this fact is just what is given us by the propositions in their original form.
Under each hypothesis, then, the result obtained is satisfactorily accounted for and explained.
[267] It will be remembered that under suppositions (1) and (2) the ordinary doctrine of contrariety holds good.
166. “The boy is in the garden.”
“The centaur is a creation of the poets.”
“A square circle is a contradiction.”
Discuss the above propositions as illustrating different functions of the verb “to be”; or as bearing upon the logical conception of different universes of discourse or of different kinds of existence. [C.]
167. Discuss the existential import of singular propositions.
“The King of Utopia did not die on Tuesday last.” Examine carefully the meaning to be attached to the denial of this proposition. [K.]
168. Some logicians hold that from All S is P we may infer Some not-S is not-P. Take as an illustration, All human actions are foreseen by the Deity. [C.]
169. Discuss the validity of the following inference:—All trespassers will be prosecuted, No trespassers have been prosecuted, therefore, There have been no trespassers. [C.]