Such propositions cannot both be true; but they may both be false, for some men may be wise and some not.

They cannot both be true; for, by the principle of Contradiction, if _wise_ may be affirmed of _All men, not-wise_ must be denied; but _All men are not-wise_ is the obverse of _No men are wise_, which therefore may also be denied.
At the same time we cannot apply to A.and E.the principle of Excluded Middle, so as to show that one of them must be true of the same matter.
For if we deny that _All men are wise_, we do not necessarily deny the attribute 'wise' of each and every man: to say that _Not all are wise_ may mean no more than that _Some are not_.

This gives a proposition in the form of O.; which, as we have seen, does not imply its subalternans, E.
If, however, two Singular Propositions, having the same matter, but differing in quality, are to be treated as universals, and therefore as A.and E., they are, nevertheless, contradictory and not merely contrary; for one of them must be false and the other true.
Sec.7.Contradiction is a relation between two propositions analogous to that between contradictory terms (one of which being affirmed of a subject the other is denied)--such, namely, that one of them is false and the other true.

This is the case with the forms A.and O., and E.
and I., in the same matter.

If it be true that _All men are wise_, it is false that _Some men are not wise_ (equivalent by obversion to _Some men are not-wise_); or else, since the 'Some men' are included in the 'All men,' we should be predicating of the same men that they are both 'wise' and 'not-wise'; which would violate the principle of Contradiction.