The Denying the antecedent reference article from the English Wikipedia on 24-Jul-2004
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Denying the antecedent

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Denying the antecedent 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. If we argue this way, we make a mistake. One can see this with an example:
If there is fire here, then there is oxygen here. (Since oxygen is required for fire.)
There is no fire here.
Therefore, there is no oxygen here.

This form of argument 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".

See also: modus ponens, modus tollens, affirming the consequent.