Can you apply proof by contrapositive on proof by contradiction In a proof by contradiction, a false statement implies a contradiction. What happens if you take the contrapositive of this? A not-contradiction implies a not-false statement. I.e. a tautology implies a true statement. Is this right?
 A: What you are saying is:
Suppose you want to prove $A \rightarrow B$. Then a proof by contradiction is of the form: $A \wedge \neg B \Rightarrow$ "contradiction". Then "non-contradiction"$\Rightarrow \neg(A \wedge \neg B)$. But realise that $\neg(A \wedge \neg B)$ is really just $(\neg A \vee B) \equiv A\rightarrow B$. Hence if the proof by contradiction holds, $A\rightarrow B$ follows from any tautology.
But what does this even mean?


*

*It's a "well known fact" that you can prove anything from a contradiction. On the other hand, if you start off with a tautology, you can only prove other true statements. This is because if $p \Rightarrow q$ and $p$ is true, then $q$ must be true. This makes sense... but what does it mean in practice?

*Suppose you want to prove $A\rightarrow B$, but you've decided a contradiction isn't your style. You decide you want to use the contrapositive of a contradiction. This means you must prove $T \Rightarrow (A\rightarrow B)$ where $T$ is any tautology. But as explained above, this is true only if $A\rightarrow B$ is true. Hence you now set off to prove $A\rightarrow B$... hey wait a minute! This is just a direct proof! Unfortunately there is nothing "useful" in taking the contrapositive of a proof by contradiction.
It's definitely an interesting thing to think about however!
A: Both are forms of indirect proof. As you would expect, a proof by contradiction makes use of a contradiction. A proof by contrapositive does not.
You can use proof by contradiction to infer that a statement $X$ is true by first assuming $X$ is false and then deriving a contradiction of the form $Y \land \neg Y$ or $Y\iff \neg Y$. 
You can use proof by contrapositive to infer that an implication of the form $X\implies Y$ is true by first assuming $Y$ is false and then proving that $X$ is false. No contradiction is required.
Of course, you can use both methods several times in the same proof. 
