It is important to know why contradiction is such a bad thing in logic. If you allow one contradiction, then anything, and everything, is true. From a contradiction, one can "prove" any statement in logic.
Let's start with our contradiction. Assume P and not-P are both true.
For any statement P, is is true that "P or x" is true. An "or" statement is called a disjunction. Only one of the terms of a disjunction has to be true for the statement as a whole to be true. So, if it is true that I like Meat Beat Manifesto, it is also true that I like Meat Beat Manifesto or getting punched in the face. I can just pull the second term out of nowhere: this is known as addition. Both terms of a disjunction can be true also, of course. So it is true that I like Meat Beat Manifesto or Johnny Cash when it so happens I like them both. But for the first statement, if I tell you that I do not, in fact, enjoy being punched in the face, you can deduce that I must like Meat Beat Manifesto. This is a disjunctive syllogism. A or B. Not B. Therefore A.
Anyone by now must see how contradictions allow you to prove anything. Assume she loves me and she loves me not. If she loves me, then she loves me, and monkeys fly out of my ass. But she loves me not! Therefore monkeys fly out of my ass.
Which is to say:
If P and not P,
then P or X,
This is all known as Ex Falso Quodlibet: What follows from the false. The reductio ad absurdum argument ends in a contradiction- showing how an argument must be false, because it leads to the absurdity of a contradiction. This picks up where it leaves off.
and given not A.
If A, it is true that A or B by Addition
If A or B and not A, then B, by Disjunctive Syllogism
In other words: Banana
and not banana: therefore car