It seems to me, these assumptions are necessary to demonstrate proof by contradiction:
i) Every proposition must belong to $T$ or $F$.
ii) No proposition belongs to both $T$ and $F$
iii) If having $Y$ belong to $T$ implies that some proposition $Z$ may violate i) or ii), then $Y$ can not belong to $T$
Now I seem to think that all three assumptions are necessary for proof by contradiction is be legitimate:
let $b$ belong to $T$.
Suppose $¬a$ is in $T$.
It is the case that we can show $¬a → ¬b$
Now it follows $¬b$ is in $T$. So $b$ is in $F$. By ii), $b$ can not both be in $T$ and $F$.
We may apply iii) on $¬a$, and we attain that $¬a$ can not be in $T$.
By i), if $¬a$ is not in $T$, it must be in $F$. So $a$ is in $T$.
Now to the questions.
Firstly, is the construction correct?
Secondly, are all the assumptions necessary, i.e. could proof by contradiction be achieved with less? i.e. how can these assumptions/axioms be broken down to simpler ones.
Thirdly, how could such assumptions, if all necessary, be possibly justified?
It seems:
"i)" assumes that every proposition one can formulate must be true or false. There can not be a proposition that has no undefined truthfulness or somehow belongs to neither. Surely there are propositions unprovable by the method of contradiction, as there are propositions unprovable in general, e.g. axioms.
"ii)" it seems a bit like we are resolving our issues in a system by starting on the assumption "this system will be issue-free". Surely not can not toss out the possibility that there may be propositions one may both confirm or deny depending on the approach. Why would one assume, a priori, such will not be the case.
"iii)" assumes that our system will have full internal consistency. Similar to before, we seem to assume a conclusion we would like to demonstrate! If propositions are assumed to have inter-consistency, doesn't that beat the point of trying to show that mathematics is harmonious?
So in conclusion, I have questions in 3 areas. 1) was the working correct. 2) how can it / the assumptions be simplified. 3) How are such assumptions, if necessary, typically justified.
The smallest assumption that you need to allow proof by contradiction is Peirce's law. $((P→Q)→P)→P$. From peirce’s law you can derive the law of the excluded middle as well as doing RAA proofs.