Denying the antecedent

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 antecedent).

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 logical fallacy. 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.

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