But, again, if it be false that All men are wise, it is always true that Some are not wise; for though in denying that ‘wise’ is a predicate of ‘All men’ we do not deny it of each and every man, yet we deny it of ‘Some men.’  Of ‘Some men,’ therefore, by the principle of Excluded Middle, ‘not-wise’ is to be affirmed; and Some men are not-wise, is by obversion equivalent to Some men are not wise.  Similarly, if it be false that No men are wise, which by obversion is equivalent to All men are not-wise, then it is true at least that Some men are wise.

By extending and enforcing the doctrine of relative terms, certain other inferences are implied in the contrary and contradictory relations of propositions.  We have seen in chap. iv. that the contradictory of a given term includes all its contraries:  ‘not-blue,’ for example, includes red and yellow.  Hence, since The sky is blue becomes by obversion, The sky is not not-blue, we may also infer The sky is not red, etc.  From the truth, then, of any proposition predicating a given term, we may infer the falsity of all propositions predicating the contrary terms in the same relation.  But, on the other hand, from the falsity of a proposition predicating a given term, we cannot infer the truth of the predication of any particular contrary term.  If it be false that The sky is red, we cannot formally infer, that The sky is blue (cf. chap. iv.  Sec. 8).

Sec. 8.  Sub-contrariety is the relation of two propositions, concerning the same matter that may both be true but are never both false.  This is the case with I. and O. If it be true that Some men are wise, it may also be true that Some (other) men are not wise.  This follows from the maxim in chap. vi.  Sec. 6, not to go beyond the evidence.

For if it be true that Some men are wise, it may indeed be true that All are (this being the subalternans):  and if All are, it is (by contradiction) false that Some are not; but as we are only told that Some men are, it is illicit to infer the falsity of Some are not, which could only be justified by evidence concerning All men.

But if it be false that Some men are wise, it is true that Some men are not wise; for, by contradiction, if Some men are wise is false, No men are wise is true; and, therefore, by subalternation, Some men are not wise is true.

Sec. 9.  The Square of Opposition.—­By their relations of Subalternation, Contrariety, Contradiction, and Sub-contrariety, the forms A. I. E. O. (having the same matter) are said to stand in Opposition:  and Logicians represent these relations by a square having A. I. E. O. at its corners: 

A. Contraries E.