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The proof for the Principle of Explosion starts by assuming a contradiction.
When we use reductio ad absurdum, we establish a proof by reaching a contradictory conclusion in sub-argument and then refusing to accept a contradiction.
In classical logic, we refuse to accept a contradiction while at the same time accepting a statement derived from a contradiction. But how is this justified?
In classical and intuitionistic logic, the Principle of Explosion is often a basic law of inference.
Wiki's entry deduces it from Disjunctive syllogism:
Assume P as true; then (by Disjunction introduction) we have: P ∨ Q, with Q whatever.
But we have also ¬P. Thus, we may conclude with Q.
This is what happens in classical and intuitionsitic logic when we assume as true a contradiction.
Conclusion: contradictions are never true.
We have to be clear about the difference regarding: assuming something as true, and proving something.
But it is allowed to assume something (not known to be true or false) as true and see "what happens".
If a contradiction follows by way of the correct "inference procedure" (i.e. having applied correctly the inference rules) then, because contradictions can never be true, this mean that our starting assumption is wrong e we have to reject it.
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:
each step is a valid inference. Walking through them,
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.
