| No S is P, | |
| therefore, | No P is S ; |
for, if not, then by the law of contradiction, Some P is S ; and we have this syllogism,—
| No S is P, | |
| Some P is S, | |
| therefore, | Some P is not P, |
a reductio ad absurdum.[161]
[161] Compare Mansel’s Aldrich, p. 62. The conversion of A and the conversion of I may be established similarly.
(2) It may be plausibly maintained that in Aristotle’s proof of the conversion of E (given in section [99]), there is an implicit syllogism: namely,—Q is P, Q is S, therefore, Some S is P.
(3) The contraposition of A may be established by means of a syllogism in Camestres as follows:—
| Given | All S is P, | |
| we have also | No not-P is P, | by the law of contradiction, |
| therefore, | No not-P is S.[162] |
[162] Similarly, granting the validity of obversion, the contraposition of O may be established by a syllogism in Datisi as follows:—
Given Some S is not P, then we have