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 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.  Similarly, No men are wise, being by obversion equivalent to All men are not-wise, is incompatible with Some men are wise, by the same principle of Contradiction.