§ 758. The fallacy of denying the antecedent assumes the following form—
If A is B, C is D,
.'. If A is not B, C is not D,
equivalent to—
All cases of A being B are cases of C being D.
.'. Whatever is not a case of A being B is not a case of C being D.
This is the same as to argue—
All A is B,
.'. All not-A is not-B,
an erroneous form of immediate inference for which there is no special name, but which involves the vice of simple conversion of A, since 'All not-A is not-B' is the contrapositive, not of 'All A is B,' but of its simple converse 'All B is A.'
§ 759. The above-mentioned form of immediate inference, however (namely, the employment of contraposition without conversion), is valid in the case of the U proposition; and so also is simple conversion. Accordingly we are able, as we have seen, in dealing with a proposition of that form, both to deny the antecedent and to assert the consequent with impunity—
If A is B, then only C is D,
.'. A not being B, C is not D;
and again, C being D, A must be B.