|No S are P.
Therefore, no non-P are non-S.
|Some S are P.
Therefore, some non-P are non-S.
|Similar Validating Forms|
|All P are Q.
Therefore, all non-Q are non-P.
|Some P are not Q.
Therefore, some non-Q are not non-P.
|No conservatives are liberals.
Therefore, no non-liberals are non-conservatives.
|Some dogs are pets.
Therefore, some non-pets are non-dogs.
|No dogs are cats.
Therefore, no non-cats are non-dogs.
|Some physical things are invisible.
Therefore, some visible things are non-physical.
"Illicit contraposition" is not a scandal from the Reagan administration. Rather, "contraposition" refers to the process of switching the subject and predicate terms of a categorical proposition, and negating each. For instance, the contraposition of "all bats are mammals" is "all non-mammals are non-bats".
Contraposition is a validating form of immediate inference for A- and O-type categorical propositions. However, contraposition is a non-validating form of inference for E- or I-type propositions. Hence, contraposing an E- or I-type proposition is Illicit Contraposition.
The Venn Diagrams above show that the two forms of Illicit Contraposition are not validating. The diagram for the E-type proposition shows that no S is P―which is indicated by the blue shading of the overlap of the S and P circles―but it fails to show that there is nothing in the area outside both circles, which is the overlap of the non-S and non-P areas. A question mark in an area means that it might or might not be empty. Similarly, the diagram for the I-type proposition shows that some S is P, but fails to show that there is anything in the area outside both circles.
Moreover, the Counter-Examples above show that the forms are not validating, since each has an instance with a true premiss and a false conclusion. It's harder to find a Counter-Example to the contraposition of I propositions than E ones since, in order for the conclusion of such an I inference to be false, the two classes must jointly exhaust the universe but not be mutually exclusive. In the Counter-Example, the premiss is true because there are invisible physical things, such as atoms; but the conclusion is false because there are no visible things that are not physical.
Patrick J. Hurley, A Concise Introduction to Logic (Fifth Edition) (Wadsworth, 1994), p. 214.
Acknowledgment: Thanks to John Congdon for the "contra" joke in the Exposition.