The principle of explosion states that "from a contradiction, anything follows."
The proof for the principle of explosion thus needs to show that if we have a contradiction, we can then suppose anything.
The most common method of proof is as follows:
- P and not P (assumption)
- P &E 1
- P v Z vI 2
- ~P &E1
- Z DS 3,4
each step is a valid inference. Walking through them,
- P and not P by assumption -- we can start our proof with any contradiction, so we arbitrarily pick P and not P
- P by &E1 (we can always reduce a conjunction to one of its terms without changing the truth value since P & Q is true iff P is true and Q is true.)
- P v Z by vI2 (we can always introduce a disjunction when we have a true element because the truth value of P v Q is true is if either P or Q is true).
- ~P by &E1 (see 2)
- Z by DS 3,4 (given a disjunction, if we know one half is false, then the other half must be true for the whole to be true).
You suggest in a comment on Mauro's great answer that
1. We start [our proof] by assuming [P and ~P] are not [mutually exclusive].
2. Our use of DS assumes that they are exclusive.
I don't think we have to commit ourselves to the assumption that P and ~P are "mutually exclusive" or not per se. Here's a few reasons:
First, there's no rule in proving that you have to look at every variable simultaneously. You can work with whichever variables and inference rules are available. Thus, there's nothing illicit about the fact that our proof builds on the P branch in 2 and 3 and then ~P branch in 4.
Second, there's no need to resolve with they are "mutually exclusive" or not in the proof for the principle of explosion. Instead, we are merely trying to show that their conjunction is explosive -- and for that reasons.
Thus, there's no reason to believe the conclusion is illicit.
You do make an interesting suggestion -- reject the possibility of DS, but this seems problematic since DS is logically equivalent to MT.
RAA can be seen as premised on the principle of explosion. Or to put it another way, explosion is the proof for RAA. Proof for the principle of explosion proves that contradictions explode. RAA accepts that it explodes and thus backs out of the assumption.
In RAA, we have a subproof with a contradiction, which means that within that subproof, we arrived at a point where we can draw any conclusion we want. In other words, if the initial assumption of the subproof is true, then we can reach the conclusion of the proof or its opposite or something completely unrelated.