Rather than give you the correct technical answer (edit: okay I ended up going into some technical details, oops), which has been provided, I think I'll try and diagnose where you're intuitions have led you astray.

It seems to me that you think that if a formal system can derive (P & ¬P), then we have to accept it and thus accept the resulting explosion. If I'm right, what you're doing is assuming that whenever we write a logical statement, we're endorsing it. But we're not, we're reasoning meta-logically. That is, to write "S ⊢ Q" is to say " suppose S ⊢ Q is a true statement in the logical system" not asserting it as true, and then trying to make it false later having already said it's true. If we were doing the latter, then yes, we'd have to accept the consequence of every logical statement we write.

But we don't have to, once we get the contradiction, accept the contradiction's truth. Instead, we can (actually, we have to) accept that whatever we used to prove the contradiction can't be true (nor derivable). What we've done is shown that some part of the set of statements IS NOT derivable because we assumed it was derivable and got contradiction. That is, what we've got is the following inference from our exercise:

Let S be a finite set of statements and Q be what we've derived from S. The question is, is S itself and instance of "⊢ S" or not? Suppose S ⊢ Q. But then suppose Q ⊢ (P & ¬P). You think that now we've got (P & ¬P), and given (P & ¬P) ⊢ A ( where A = any statement and its negation), we've now exploded the logic. But on the contrary, *because* assuming that "⊢ S" leads to S ⊢ (P & ¬P), we know that the assumption "⊢ S" must have been false. And because it's false, we know that the explosion "⊢ A" is also false. To accept (P & ¬P) as true would lead to explosion. But instead we've reasoned that, axiomatically, ⊢ ¬(P & ¬P), therefore anything deriving (P & ¬P) can't be derivable. Then we've shown that, were S derivable, (P & ¬P) would also be derivable. But this contradicts our axiom that ⊢ ¬(P & ¬P). From this it follows that S was never derivable in the first place. But if S is not derivable, then nor is anything of the form S ⊢ Q (Q standing for anything derived from S). But S ⊢ A is an instance of S ⊢ Q. So A is not derivable from S.

For the sake of completeness (in the non technical sense), lets also note that for every possible set of statements, if that set of statements derives A (still symbolising every possible statement, ie: explosion), the same proof above applies. Therefore, there is no possible set of statements that derive A. So not only do we know that a given set of statements S doesn't derive A, but that no set of statements derive A. Therefore "⊢ A" is ALWAYS false, and as such, this proves that explosion is impossible for our logical system. Isn't that a satisfying result we've now got ourselves to put our minds at rest?

When we write "⊢ S" in this (ludicrously informal) proof, we're not committed to it, nor any of its consequences. We just write it down as if it were the case, and as soon as we derive something impossible within the axiom schema and rules of inference our system is using from pretending its true, then we have a direct way of knowing it's false.

I hope I haven't left you in even more confusion.

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– Geoffrey Thomas – 2019-09-13T08:44:49.117No, logic will not work without disjunction elimination. It is part of 'introduction/removal harmony' which most folks would not discard. If I must turn left, then I must turn either right or left, and that is not negotiable to normal humans. Intuitionist reject his #3, because it leads to proofs where you get a result without a path. A combing of a hairy ball either does or does not exist. A ball with no such combing can't exist. Therefore every ball has such a combing -- but

we can't construct itbecause it fell out of a rule, and was not the result of a reasoning process. – None – 2019-09-13T19:49:17.7973Please stop editing the question further. If you have another question, completely changing one with some good answers already given is not the way we do things here. – Philip Klöcking – 2020-06-22T22:31:17.580

@PhilipKlöcking I think that I have it just about perfect. Improving my questions is my only chance to get my question asking restored. I – polcott – 2020-06-22T22:33:52.980

2That maybe the case, but you should also consider that there is a difference between perfectionalism and steadily shifting the goalposts of a question so that relevant, existing answers do not fully match the question anymore. This has also to do with respect towards thos who answered and this question here is certainly one you should not be worrying about as much as others. – Philip Klöcking – 2020-06-22T22:36:41.537

@PhilipKlöcking Try studying it. I think that I came up with something new. – polcott – 2020-06-22T22:36:51.423

@PhilipKlöcking I tried to keep that in mind when I made a slight shifts of focus. – polcott – 2020-06-22T22:37:53.770

5Coming up with something is not what questions are for. It just highlights that there is no genuine question, only presumption. – Philip Klöcking – 2020-06-22T22:37:57.900

@PhilipKlöcking At this point I could back out of most of the changes and provide my answer as an answer. I could also reedit my question so that it much more closely corresponds to a question and the existing answers. Does that sound good? – polcott – 2020-06-22T22:48:43.347

@PhilipKlöcking I rolled it back to a much earlier version that is consistent with the current answers. – polcott – 2020-06-23T05:19:10.783