Affirming the consequent

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

1. If P, then Q.
2. Q.
3. Therefore, P.

Arguments of this form are invalid, in that the conclusion (3) does not have to follow even when statements 1 and 2 are true. The simple reason for this is that P was never asserted as the only sufficient condition for Q, so, in general, any number of other factors could account for Q (while P was false).

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.

Owning Fort Knox is not the only way to be rich. There are any number of other ways to be rich.

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.

Having the flu is not the only cause of a sore throat since many illnesses cause sore throat, such as the common cold or strep throat.

The following is a more subtle version of the fallacy embedded into conversation.

A: All Republicans are pro-life.

B: That's not true. My uncle's pro-life and he's not a Republican.

B attempts to falsify A's conditional statement ("if Republican then pro-life") by providing evidence he believes would contradict its implication. However, B's example of his uncle does not contradict A's statement, which says nothing about non-Republicans. What would be needed to disprove A's assertion are examples of Republicans who are not pro-life.

Cases where affirming the consequent is valid

Tautologies

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.

This is also the case for definitions. For example.

If a man is a bachelor, then he's an unmarried male

A man is an unmarried male.

Therefore, the man is a bachelor.

In everyday discourse, however, such cases are rare. The validity of such definitions is due to the fact that definitions can be expressed as an if and only if (see below). Clearly if the definition of "bachelor" is "an unmarried male", then the propositional statement: "A is a bachelor" if and only if "A is an unmarried male", must be true. In normal speech it is awkward to use the phrase "if and only if", so we substitute the valid but less complete "if", giving the conventional form which is similar to the form of the formal fallacy.

If and only if

The reason the conclusion of an argument that affirms the consequent does not follow is the lack of a unique cause for Q. However, if it is explicitly stated that the consequent could only have one cause (known as an "if and only if" statement or biconditional), the argument becomes valid. 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.

Such if and only if statements often make their way into detective mysteries.

The only way he could have gotten into the bedroom without leaving marks in the hall is the window.

No marks were found in the hall.

The cigar ends show he was in the room.

Therefore, he used the window.

Use of the fallacy in science

Although affirming the consequent is an invalid inference, it is defended in some contexts as a type of abductive 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. (e.g., Let 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.

References

• Affirming the Consequent in The Fallacy Files

See also

• Modus ponens
• Modus tollens
• Post hoc ergo propter hoc
• Denying the antecedent
• Fallacy of the undistributed middle
• Inference to the best explanation
• ELIZA effect

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