Affirming the consequent

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Affirming the consequent, sometimes called converse error, is a formal fallacy, committed by reasoning in the form:

If P, then Q.

Q.

Therefore, P.

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

The name affirming the consequent derives from the premise Q, which affirms the "then" 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 Bill Gates owns Fort Knox, then he is rich.

Bill Gates is rich.

Therefore, Bill Gates owns Fort Knox.

Arguments of the same form can sometimes seem superficially convincing, as in the following example:

If I have the flu, then I have a sore throat.

I have a sore throat.

Therefore, I have the flu.

But many illnesses cause sore throat, such as the common cold or strep throat.

However, it is possible that an argument that affirms the consequent could be valid, if the argument instantiates some other valid form. For example, if claims P and Q express the same proposition, then the argument would be trivially valid, as it would beg the question.

If P, then P.

P.

Therefore, P.

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). For example:

If he's not inside, then he's outside.

He's outside.

Therefore, he's not inside.

The above argument may be valid, but only if the claim "if he's outside, then he's not inside" follows from the first premise. More to the point, the validity of the argument stems not from affirming the consequent, but affirming the antecedent.

Although affirming the consequent is an invalid inference, it is defended by some as a type of inductive reasoning, sometimes under the name "inference to the best explanation". That is, in some cases, reasoners argue that the antecedent is the best explanation, given the truth of the consequent. For example, someone considering the results of a scientific experiment may reason in the following way:

Theory P predicts that we will observe Q.

Experimental observation shows Q.

Therefore theory P is true.

However, such reasoning is still affirming the consequent and still logically weak. (Let, e.g., P = geocentrism and Q = sunrise and sunset.) The strength of such reasoning as an inductive inference depends on the likelihood of alternative hypotheses, which shows that such reasoning is based on additional premises, not merely on affirming the consequent.