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- Are proofs by contradiction really logical? 12 answers
First of all I'd like to say that I have looked for the answers to my specific question and have not found it in the existing topics.
The question is fairly simple. Say, we need to prove statement P by the method of contradiction. Assuming that $\lnot P$ holds, using the list of statements proven earlier to hold or derived by us during the proof, we arrive to P being $true$.
$$\lnot P \to A_1 \to\ ... \ \to A_n \to P$$ $$\lnot P \to P \iff \lnot(\lnot P) \lor P \iff P $$
We can therefore add P to the list of our proven statements, because it was derived. Most of the proofs contain something in the lines of "the obtained contradiction proves that our initial assumption ($\lnot P$) was wrong and so $P$ holds".
What I don't understand is, if the initial assumption ($\lnot P$) is thus proven to be false, then why can we be sure that anything derived from it holds (in particular, that P holds)? On the other hand, if it cannot be derived then the assumption ($\lnot P$) can in fact be true.
Can someone explain why this type of argument cannot be used?