No A is B. .'. All A is not-B (by permutation). .'. Some not-B is A (by conversion per accidens). .'. Some not-B is not not-A (by permutation).
§ 527. In its unanalysed form conversion by contraposition may be defined generally as--A form of conversion in which both subject and predicate are replaced by their contradictories.
§ 528. Conversion by contraposition differs in several respects from conversion by negation.
(1) In conversion by negation the converse differs in quality from the convertend: whereas in conversion by contraposition the quality of the two is the same.
(2) In conversion by negation we employ the contradictory either of the subject or predicate, but in conversion by contraposition we employ the contradictory of both.
(3) Conversion by negation involves only two steps of immediate inference: conversion by contraposition three.
§ 529. Conversion by contraposition cannot be applied to the ordinary E proposition except by limitation (§ 526).
From 'No A is B' we cannot infer 'No not-B is not-A.' For, if we could, the contradictory of the latter, namely, 'Some not-B is not-A' would be false. But it is manifest that this is not necessarily false. For when one term is excluded from another, there must be numerous individuals which fall under neither of them, unless it should so happen that one of the terms is the direct contradictory of the other, which is clearly not conveyed by the form of the expression 'No A is B. 'No A is not-A' stands alone among E propositions in admitting of full conversion by contraposition, and the form of that is the same after it as before.
§ 530. Nor can conversion by contraposition be applied at all to I.
From 'Some A is B' we cannot infer that 'Some not-B is not-A.' For though the proposition holds true as a matter of fact, when A and B are in part mutually exclusive, yet this is not conveyed by the form of the expression. It may so happen that B is wholly contained under A, while A itself contains everything. In this case it will be true that 'No not-B is not-A,' which contradicts the attempted inference. Thus from the proposition 'Some things are substances' it cannot be inferred that 'Some not-substances are not-things,' for in this case the contradictory is true that 'No not-substances are not-things'; and unless an inference is valid in every case, it is not formally valid at all.