Illicit

Forms  

E  I 
No S are P.
Therefore, no nonP are nonS. 
Some S are P.
Therefore, some nonP are nonS. 
Venn Diagrams  
Similar Validating Forms  
A  O 
All P are Q.
Therefore, all nonQ are nonP. 
Some P are not Q.
Therefore, some nonQ are not nonP. 
Examples  
No conservatives are liberals.
Therefore, no nonliberals are nonconservatives. 
Some dogs are pets.
Therefore, some nonpets are nondogs. 
CounterExamples  
No dogs are cats.
Therefore, no noncats are nondogs. 
Some physical things are invisible.
Therefore, some visible things are nonphysical. 
"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 nonmammals are nonbats".
Contraposition is a validating form of immediate inference for A and Otype categorical propositions. However, contraposition is a nonvalidating form of inference for E or Itype propositions. Hence, contraposing an E or Itype proposition is Illicit Contraposition.
The Venn Diagrams above show that the two forms of Illicit Contraposition are not validating. The diagram for the Etype 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 nonS and nonP areas. A question mark in an area means that it might or might not be empty. Similarly, the diagram for the Itype proposition shows that some S is P, but fails to show that there is anything in the area outside both circles.
Moreover, the CounterExamples 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 CounterExample 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 CounterExample, 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.