This could just be a lack of understanding on my part as I have never really properly learnt first order logic. If that is the case, then please let me know and I will delete the question. I will lead this question with an example, but that is not the focus of the question. This question will probably sound really convoluted, but I hope you read through it.
My question is, how do we know all the things that we use in writing proofs to be true. For example, do we have to prove that a proof by contradiction is a proof? How do we know for sure that if we arrive at a contradiction then our hypothesis must be false and the negation of that hypothesis must be true then? The part that the hypothesis makes sense but the part that the negation must be true is bit hard to accept.
What I am trying to get at is: Do we just use these techniques because we think that they should hold or do we have to prove that these techniques work and are true all the time?
A more familiar phrasing is how one might believe that the Collatz conjecture holds for every natural number because they tried it for a lot of numbers, but proving it holds for every natural number is a different thing. Is this is what is going on with the things that we students are taught? Another example is: Proving a statement by proving its contrapositive. It makes sense morally that this should be true, but can we prove that always proves the statement?
I am assuming this might have something to do with proof theory(although I know nothing about this) as I keep saying the word proof but I don't have a definition of what is a proof.
So my question is: Do we have to prove all the things we use to write proofs or are those things just taken for granted?