Are proofs by contradiction really logical? Let's say that I prove statement $A$ by showing that the negation of $A$ leads to a contradiction. 
My question is this: How does one go from "so there's a contradiction if we don't have $A$" to concluding that "we have $A$"?
That, to me, seems the exact opposite of logical. It sounds like we say "so, I'll have a really big problem if this thing isn't true, so out of convenience, I am just going to act like it's true". 

Another way to see it is that a proof by contradiction is a simplification where you "forget", for a while, that $A$ can be true.
An usual scheme of proof by contradiction is the following:


*

*Assume $\neg A$

*Prove $B$

*Prove $\neg B$

*This is a contradiction, then $A$ is true.


If you want to follow the same pattern but without assuming anything false, you can do:


*

*Start with $A\vee \neg A$ (always true)

*With the same arguments as before, prove $A\vee (\neg A \wedge B)$

*With the same arguments as before, prove $A\vee (\neg A \wedge B \wedge \neg B)$

*Since $B \wedge \neg B$ is false, then you have just proven the following statements: $A\vee (\neg A \wedge$ false $)$, then $A\vee ($ false $)$ and finally $A$.


This approach manipulates only true statements, which is arguably comfortable in a proof, but the cost is that we always take into account that "$A$ can also be true". Then, the bottom line is basically "Either $A$ is true, or nothing".
