# Denying the antecedent

Denying the antecedent, sometimes also called inverse error or 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.
Therefore, if not P, then not Q.

which may also be phrased as

${\displaystyle P\rightarrow Q}$ (P implies Q)
${\displaystyle \therefore \neg P\rightarrow \neg Q}$ (therefore, not-P implies not-Q)

Arguments of this form are invalid. Informally, this means that arguments of this form do not give good reason to establish their conclusions, even if their premises are true.

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 argument is invalid because there are other reasons for which the grass could be wet (being sprayed with water by a hose, for example). Another example:

If Queen Elizabeth is a French citizen, then she is a human being.
Queen Elizabeth is not a French citizen.
Therefore, Queen Elizabeth is not a human being.

That argument is intentionally 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.[1]

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

It is possible that an argument that denies the antecedent could be valid, if the argument instantiates some other valid form. For example, if the claims P and Q express the same proposition, then the argument would be trivially valid, as it would beg the question. In everyday discourse, however, such cases are rare, typically only occurring when the "if-then" premise is actually an "if and only if" claim (i.e., a biconditional/equality). For example:

If I am President of the United States, then I can veto Congress.
I am not President.
Therefore, I cannot veto Congress.

The above argument is not valid, but would be if the first premise ended thus: "...and if I can veto Congress, then I am the U.S. President" (as is in fact true). More to the point, the validity of the new argument stems not from denying the antecedent, but modus tollens (denying the consequent).