Denying the antecedent

Denying the antecedent, sometimes also called ‘fallacy of the inverse’, is a formal fallacy of inferring the inverse from the original statement. It is committed by reasoning in the form:

If P, then Q.
Not P.
Therefore, not Q.

Arguments of this form are invalid. The name denying the antecedent derives from the premise “not P“, which denies the “if” clause of the conditional premise.

One way to demonstrate the invalidity of this argument form is with a counterexample with true premises but an obviously false conclusion. For example:

If it is raining, then the grass is wet.
It is not raining.
Therefore, the grass is not wet.

The conclusion is invalid because there are other reasons why the grass could be wet at the time (someone could have watered it).

That argument is obviously bad, but arguments of the same form can sometimes seem superficially convincing, as in the following example offered, with apologies for its lack of logical rigour, by Alan Turing in the article ‘Computing Machinery and Intelligence’:

If each man had a definite set of rules of conduct by which he regulated his life he would be no better than a machine. But there are no such rules, so men cannot be machines.

However, men could still be machines that do not follow a definite set of rules. Thus this argument (as Turing intends) is invalid.


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Filed under Logical fallacies

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