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". 

It's just case analysis with the assumption that one of the cases must be valid. Proof that all cases but one are invalid. The one left over therefore has to be valid. The only thing you can challenge is the assumption doesn't hold.
