O cannot be converted either simply or per accidens. Some S is not P does not enable you to make any converse assertion about P. All P may be S, or No P may be S, or Some P may be not S. All the three following diagrams are compatible with Some S being excluded from P.
(c) Another mode of Conversion, known by mediæval logicians following Boethius as Conversio per contra positionem terminorum, is useful in some syllogistic manipulations. This Converse is obtained by substituting for the predicate term its Contrapositive or Contradictory, not-P, making the consequent change of Quality, and simply converting. Thus All S is P is converted into the equivalent No not-P is S.[6]
Some have called it "Conversion by Negation," but "negation" is manifestly too wide and common a word to be thus arbitrarily restricted to the process of substituting for one term its opposite.
Others (and this has some mediæval usage in its favour, though not the most intelligent) would call the form All not-P is not-S (the Obverse or Permutation of No not-P is S), the Converse by Contraposition. This is to conform to an imaginary rule that in Conversion the Converse must be of the same Quality with the Convertend. But the essence of Conversion is the interchange of Subject and Predicate: the Quality is not in the definition except by a bungle: it is an accident. No not-P is S, and Some not-P is S are the forms used in Syllogism, and therefore specially named. Unless a form had a use, it was left unnamed, like the Subalternate forms of Syllogism: Nomen habent nullum: nec, si bene colligis, usum.
Table of Contrapositive Converses.
All S is P No S is P Some S is not P Some S is P | Con. Con. No not-P is S Some not-P is S Some not-P is S None. |
When not-P is substituted for P, Some S is P becomes Some S is not not-P, and this form is inconvertible.