Affirming the consequent

Affirming the consequent is a formal fallacy, committed by reasoning in the form:


 * If P, then Q.
 * Q.
 * Therefore, P.

Arguments of this form are invalid (except in the rare cases where such an argument also instantiates some other, valid, form). 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 affirming the consequent derives from the term for the "then" clause of a conditional claim.

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. Thus this argument is weak at best.

As noted above, 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. In everyday discourse, however, such valid cases of affirming the consequent 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. But even in such a case, the validity stems not from affirming the consequent, but from the form modus ponens.

Although affirming the consequent is an invalid inference, it is defended by some as an acceptable type of inductive reasoning, sometimes under the name "inference to the best explanation".