Invalid argument:
If a number is divisible by 4, it is even.
But it is not divisible by 4.
Therefore it is not even.
The number 6 refutes this: it is even although it is not divisible by 4.
Therefore denying the antecedent does not imply denying the consequent. The original conditional only licenses the direction from A to B; a converse requires separate grounds.
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Exceptive Syllogism: Valid Inferences from Connected and Disjunctive Conditionals
Invalid argument:
If a number is divisible by 4, it is even.
But it is not divisible by 4.
Therefore it is not even.
The number 6 refutes this: it is even although it is not divisible by 4.
Therefore denying the antecedent does not imply denying the consequent. The original conditional only licenses the direction from A to B; a converse requires separate grounds.