Denying the antecedent fallacy { Philosophy Index }

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Denying the antecedent

Denying the antecedent is a logical fallacy, committed by an invalid argument form "If P then Q. Not P. Therefore, not Q.".

The denying the antecedent fallacy may be expressed formally as follows:

α → β, ¬α ∴ ¬β

The argument is invalid because β for some reason other than α.

Denying the antecedent example

If the car breaks down, we'll be late for the movie.
The car won't break down.
So, we won't be late for the movie.

This argument can be demonstrated as invalid by the case that we're late for the movie for some reason other than the car breaking down, such as a traffic jam.

Biconditional version

It should be mentioned that, if the conditional is replaced by a biconditional (if and only if) operator, then the argument becomes valid:

α ↔ β, ¬α ∴ ¬β

In this case, β cannot be true for any reason other than α, so we can infer β's falsity from ¬α. For example:

We'll be late for the movie only if the car breaks down.
The car won't break down.
So, we won't be late for the movie.

This differs from the affirming the consequent fallacy because it eliminates the possibility of being late for the movie for any other reason in the first premise, by stating that it will happen if and only if the car breaks down.

A biconditional proof of this nature is essentially a case of denying the consequent (or modus tollens) rather than denying the antecedent, since α ↔ β is equivalent to (α → β) ∧ (β → α) [see biconditional introduction rule].