You asked (I am making your question a bit crisper): "What formal guarantee is there that it cannot happen that both $\lnot p$ and $p$ lead to a contradiction?" You seem to worry that if logic is inconsistent, then proof by contradiction is problematic. But this is not the case at all.
If logic ...
I will be honest, I am not 100% sure what I feel of realizing that inconsistency actually just makes formal system trivial. Everything is just true because we can conclude anything from a false premise, is that so? I am trying to understand, are you saying that if there is a inconsistency, I should instead embrace it because it makes everything true automagically?
Also, I think you forgot to address my most important point. Once a contradiction has been shown from $\neg A$, what exactly guarantees that if we applied the same computational process to search for a proof (say with a super Turning Machine) that this process will not reach a contradiction. Or do we just assume it won't? Or is your answer just, we don't care because if it does find a contradiction the it proves $\neg A$, $A$ and $A \wedge \neg A$ are all true, which is gives us a lot more true values than what we were after...in fact it shoes True=False too, no?
@hobbs if I understand correctly, the issue with your assertion "ex falso sequitur quodlibet" is that it uses contradiction, the method I am question, to conclude weird things won't happen. In in this context, you are saying since $\neg A$ has been shown false via a contradiction then $A$ must be true. But if $A$ were indeed false then you would be able to conclude everything, but what I don't understand is if your assertion is true then why with "incorrect" assumptions $\neg A$ is true do we not conclude the pope is batman? Is it because we can't do that until we reach a contradiction?
@hobbs what I'm still confused is once $\neg A$ has been shown false, what guarantees that we won't also reach a contradiction if we were to apply the inference rules on $\neg \neg A = A$? Right now the only thing that makes sense is that since $A$ and all those statements are dealt with in maths as propositional logic (even if their content is say, about real analysis) then we won't reach a contradiction because proposition logic is complete, sound and computable (but can't capture arithmetic). I don't see any other reason that makes sense to me why accept proof by contradiction right now.
@mpasko256 whats the point you are trying to make? Sorry I just don't understand the relation of your comment and my comments/questions...I am aware anything follows from a contradiction by the principle of explosion (which Andrej was kind enough to remind me). My question is essentially, we know $\neg A$ can't be true because of a contradiction, but does that truly mean $A$ is True (and not False)? My worry is that if we applied the rules of inference similarly on $A$ that we would also reach a contradiction, but for some reason that I am struggling with is that we aren't worried about this.
@CharlieParker I tried to address rather answer, not question nor comments. I just noticed that there could be only a typho, as 1+1=2 is already true sentence. Btw, I just found some source: ceadserv1.nku.edu/longa//classes/mat385_resources/docs/…
@mpasko256 I know your trying to help and I appreciate it, I just don't want your efforts to go to waste. I can't identify what your addressing so the comments just seem random too me right now. Thanks for your patience.
@CharlieParker: (regarding @mpasko256's comments) The common example given in philosophy classes is "Given that 1=2, I'll prove that I'm the king of France" -- to illustrate that given a false logical statement, any statement can be proven (it's not 1+1=2, that must have been a typo).
@RoyTinker I forget if I argued it here, but I am saying is that is not true. A falsehood does not imply that. The reason according to my understanding is because modus ponens won't let you conclude B from a false premise A. Even if "if 1=2 => I am the pope" is true, it does not imply "I am the pope". For that to be true we need contradiction so that we can use the principle of explosion. If "1=2 AND 1!=2" then we can say "1=2 OR I am the pope" is true because "1=2". Then consider separately "1=2 or I am the pope" and recall "1!=2" is True. But since we know "1=2 or I am the pope" is True
and we also know that "1!=2" we must be able to infer that "I am the pope". This is just me regurgitating the principle of explosion. What I am saying is that as far as I know, you can't conclude that "I am the pope" from a falsehood, you can only do that if you have a contradiction (or for some reason define False to be a contradiction). Hope that makes sense. Or maybe you are saying that any implication is true if the premise is false. I agree but that doesn't mean the consequent is true. We need modus ponens for that. As far as I understand.
@CharlieParker I think you're misunderstanding the proof's method. If I recall correctly, it works by adding 'person' units and assigning them to real people. A somewhat informal summary: 1=2; 1=1+1; 1person = 1person + 1person; 1person = (me) + (the KoF); I and the KoF are 1 person; I am the KoF
I wrote $1 + 1 = 2$ on purpose, as "if $1 = 2$ then you are the next pope" is true even without an inconsistency. In any case, I think the rather large amount of misunderstanding present in this question suggests that everyone would be better served by moving over to math.stackexchange.com. It's getting counter-productive. The OP is still confounding many different concepts.
@AndrejBauer While I think that this question should be answered by experts in Logic, I was wondering whether they are more 'common' on CS.SE or on Math.SE. Do you think there are more of them on Math.SE?
@AndrejBauer Actually, given that Math.SE is generally larger, that question isn't hard. But I take it you believe the signal to noise ratio would be better?
The question has been answered by an expert in logic. The problem here is that the OP is not listening, or does not understand what is being said. He has not realized yet that his question contains implicit misunderstanding of what is what. He is using words in an imprecise and wrong way. But it's difficult to get through to him. In any case, I am done answering and participating. This has gone way beyond the point of diminishing returns.
Discussion on answer by Andrej Bauer:…
Discussion on answer by Andrej Bauer:…