# The Principle of Explosion v. Reductio ad Absurdum

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 intuitionsitic logic contradictions are never true. But it is allowed to assume something (not known to be true or false) as true and see "what happens". If a contradiction follows then, because contradicitions can never be true and we have folowed the correct "inference procedure" (i.e. aaplied correctly the rules) this mean that our starting assumption is wrong. Apr 2, 2017 at 8:49
• @MauroALLEGRANZA That last comment is worth expanding into an answer: I think it addresses OP's core misapprehension. And the current answer is dense and confusing. Apr 2, 2017 at 11:09
• I'm not sure how to better word your third paragraph, but that largely depends on us getting really clear on what you're finding problematic in this. Apr 3, 2017 at 5:16

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.

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.

• In the proof I link to, we want to "see what happens". In RaA, we're trying to figure out if the assumption is true or false. OK. Apr 2, 2017 at 16:00
• I've been having trouble with the use of the disjunctive syllogism there. It assumes P and ~P are mutually exclusive, but in that proof we start by assuming they aren't. It seems to invalidate the disjunctive syllogism as an inference rule. Apr 3, 2017 at 0:06

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:

1. P and not P (assumption)
2. P &E 1
3. P v Z vI 2
4. ~P &E1
5. Z DS 3,4

each step is a valid inference. Walking through them,

1. P and not P by assumption -- we can start our proof with any contradiction, so we arbitrarily pick P and not P
2. 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.)
3. 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).
4. ~P by &E1 (see 2)
5. 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.

• I'm not talking about using the Principle of Explosion, I'm talking about demonstrating it, which the article I link to does. (I really could have been more clear about that.) It seems like a good idea to demonstrate when introducing the concept because it isn't intuitive. People commonly ask why we accept it. I think that's why the proof is in there. Apr 2, 2017 at 15:55
• I'm not following your comment. "the article" (i.e. the wikipedia article) doesn't say anything contrary to what I'm saying here. The argument wrt explosion is simply that we can derive anything from a contradiction (in part because we can use RAA with any assumption to get the negation of that assumption). Apr 2, 2017 at 23:02
• I should have used the word "prove" instead of "demonstrate". It feels weird calling it a proof though because the PoE is an inference rule rather than a theorem. Am I making more sense now or less? I appreciate your patience, anyway. Apr 3, 2017 at 0:35
• I think I may finally be grasping your question. You have a question about how we prove (or demonstrate) the principle of explosion in classical logic? Particularly you think it involves a dubious inference somewhere. Apr 3, 2017 at 1:22
• But what's dubious in any of its inferences? and how does RAA not stand in line with it? Apr 3, 2017 at 1:36