Why does a proof by contradiction work? I know what a proof by contradiction is and how to use it, but I don't understand why a proof by contradiction proves the assertion. For example, to prove $A\implies B$, I suppose that $B$ is false and arrive at a contradiction. Why does that prove that $A\implies B$ is true ? 
 A: To prove $A\implies B$ by contradiction, you suppose $A$ AND not $B$ (i.e. $A\wedge \neg B$). Therefore, if you arrive to a contradiction, you proved that $A$ AND not $B$ is false, i.e. that not $A$ OR $B$ is true (i.e. that $\neg A\vee B$ is true), which is in fact $A\implies B$.
A: We may understand "$A$ contradicts $A'$" as "$A$ and $A'$ cannot hold simultaneously", which implies that $A$ implies the negation of $A'$; take $A' :=$ the negation of $B$.
A: Assume one wants to show $A \implies B$ and one is allowed to assume $A$ is true. In a proof by contradiction rather than deriving $B$ one tries to show that from $\lnot B$ one can derive $\lnot A$. 
Note that $A$ and $\lnot A$ contradict each other. 
One wants to reject contradictions because in classical logic from a contradiction anything follows. When anything follows then everything is true and one has what is called trivialism. To avoid trivialism, reject the assumption, $\lnot B$, when one can derive a contradiction from it. 
Classical logic assumes that any statement has only one truth value, that is, $B$ is either true or false from the principle of bivalence. So one can derive $B$ from $\lnot B$ being false. It is the only other truth value available. From there one can derive using conditional introduction or the deduction theorem that $A \implies B$.
This is what happens as defined by inference rules in classical logic when one derives a contradiction. In paraconsistent logics some contradictions are tolerated without leading to trivialism. But that would be a separate topic.
