Is proof by contradiction always a sufficient proof technique ?
A proof by contradiction has the form:
Let $P$ and $Q$ be statements. If $ P \rightarrow Q \land \lnot Q $ then you can conclude $\lnot P$.
However just because $P$ lead to a contradiction, how can one be sure that $\lnot P$ is true ? What if the axiomatic system allows both $P$ and $\lnot P$ to be false in different context? I mean you could prove $P$ is true by showing $\lnot P$ leads to a contradiction, but what if a different context you can show $\lnot P$ is true by showing $P$ leads to a contradiction?
Is the theory of mathematics always constructed so $P$ and $\lnot P$ cannot both be false in different context?