Denying the Antecedent

If A, then B.
Not A.
Therefore, not B.

This is an invalid argument. Wrong. Bad. It is also a commonly asserted deduction in human debate.

Consider:

If I created a new writeup, I will gain experience.
I did not create a new writeup.
Therefore, I did not gain experience.

This one is obvious: There are other ways to gain experience. What if someone just cooled one of your old writeups?

This one's a bit tougher:

If I have faith in an organized religion, I can gain greater peace of mind.
I do not have faith in organized religion.
Therefore, I cannot gain greater peace of mind.

Same logic. And it is wrong for exactly the same reason -- no more, no less.

(Note: This type of argument can be valid if and only if you use a biconditional if)

A logical fallacy in which, as artfuldodger clearly states, the implication logical operator is abused. The problem is that implication only states that if A is true, then B must be true. If A is false, we know nothing about the value of B (as opposed to B being false, so we know A is false; see Denying the consequent/Modus Tollens).

To prove that the antecedent is being denied, show that the relation is an implication and that the premise corresponds to NOT A. You may do this by showing that something else besides A may cause B.