Denying the antecedent

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Denying the antecedent (also known as vacuous implication) is a type of logical fallacy.

Suppose in an argument one were to deny the "if" part of a conditional (the antecedent) first, and conclude with the denial of "then" part (the consequent).

If P, then Q.

P is false.

Therefore, Q is false.

This argument form has the name denying the antecedent, because in arguing this way one does indeed deny the antecedent in the second premise. This is a non sequitur (irrelevant conclusion). If we argue this way, we make a mistake. One can see this with an example:

If I'm asleep, my eyes are closed.

I'm not asleep.

Therefore, my eyes are not closed.

Clearly, the fact that my eyes are closed when I sleep does not exclude the possibility that I might close my eyes while being fully awake.

When embedded in a more complicated argument, an invalid inference of this kind may be convincing because of confusion between the meanings of if and if and only if. Denying the antecedent is valid if the first premise asserts "if and only if" rather than "if", such that I not only always close my eyes when I'm asleep, but I also only close my eyes when I'm asleep (not when I'm awake).

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