Denying the antecedent
If A lives in London, [then] A lives in England. A does not live in London. ∴ A does not live in England.
Even if the premises hold true, one cannot conclude that from a negative of the condition follows a negative consequence. In this example: There are also other places in England where A could live.
For comparison, the following table contrasts common valid forms of inference with the fallacy:
Origin of the term
The name indicates that, in contrast to the (valid) modus tollens, it is not the consequent that is negated but the antecedent, which leads to an invalid conclusion.
When are such inferences valid?
Thus, if it can be proved that in addition to
A → B, also
A ≡ B is valid, it follows that
B → A and we have a valid modus tollens.