161 What then do we know concerning P? The three cases give us respectively,—(i) All P is S ; (ii) Some P is S and Some P is not S ; (iii) No P is S. But (i) and (iii) are inconsistent with one another. Hence nothing can be affirmed of P that is true in all three cases indifferently.
(4) To illustrate the more complicated forms of immediate inference. Taking, for example, the proposition All S is P, we may ask, What does this enable us to assert about not-P and not-S respectively? We have one or other of these cases,—
As regards not-P, these yield respectively (i) No not-P is S ; (ii) No not-P is S. And thus we obtain the contrapositive of the given proposition.
As regards not-S, we have (i) No not-S is P, (ii) Some not-S is P and some not-S is not P.[168] Hence in either case we may infer Some not-S is not P.
[168] It is assumed in the use of Euler’s diagrams that S and P both exist in the universe of discourse, while neither of them exhausts that universe. This assumption is the same as that upon which our treatment of immediate inferences in the preceding chapter has been based.
E, I, O may be dealt with similarly.
(5) To illustrate the joint force of a pair of complementary or contra-complementary or sub-complementary propositions (compare section [100]). Thus, the pair of complementary propositions, SaP and PaS, taken together, limit us to
