sense of
affirmatives.

Again, having obtained the obverse of a given proposition, it may be
desirable to recover the obvertend; or it may at any time be requisite
to change a given infinite proposition into the corresponding direct
affirmative or negative; and in such cases the process is still
obversion. Thus, if _No S is not-P_ be given us to recover the obvertend
or to find the corresponding affirmative; the proposition being formally
negative, we apply the rule for obverting negatives: 'Remove the
negative sign, and for the predicate substitute its contradictory.' This
yields the affirmative _All S is P_. Similarly, to obtain the obvertend
of _All S is not-P_, apply the rule for obverting Affirmatives; and this
yields _No S is P_.

Sec. 6. Contrariety.--We have seen in chap. iv. Sec. 8, that contrary terms
are such that no two of them are predicable in the same way of the same
subject, whilst perhaps neither may be predicable of it. Similarly,
Contrary Propositions may be defined as those of which no two are ever
both true together, whilst perhaps neither may be true; or, in other
words, both may be false. This is the relation between A. and E. when
concerned with the same matter: as A.--_All men are wise_; E.--_No men
are wise_. 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 de