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I know a few proofs of consistency of propositional logic, and all of them are based on very similar things.

We are showing our axioms are tautologies and our inference rules are preserving truth, so we can only prove the tautologies. Since $\left(A\wedge\lnot A\right)$ is not a tautology, we can't prove $\left(A\wedge\lnot A\right)$, and since inconsistent systems can prove all statements, propositional logic is consistent.

Here is my question; what axiomatic system did we use to prove the consistency of propositional logic, and how do we know that that axiomatic system is consistent? How can we be sure that propositional logic is actually consistent?

I know inconsistent systems can prove their consistency if we write the formula that states "this axiomatic system is consistent" with its language, since they can prove every statement. Thus, proving that an axiomatic system is consistent with its own axioms is not enough to actually prove the consistency...

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    $\begingroup$ Your title seems to ask a different question than the body of your post. Do you need clarification on what consistency is (if so, what exactly are you unsure about?), or are you interested in how the proof of consistency proceeds? $\endgroup$ – lemontree Aug 8 '20 at 22:47
  • $\begingroup$ What do we learn when we prove the consistency in that way? $\endgroup$ – Ali Dursun Aug 8 '20 at 23:09
  • $\begingroup$ In what way? Aren't you asking how consistency is proved in the first place? $\endgroup$ – lemontree Aug 8 '20 at 23:20
  • $\begingroup$ We proved it, but we used another axiomatic system to prove it, I think. But if that axiomatic system is inconsistent, this means propositional calculus could be inconsistent as well. $\endgroup$ – Ali Dursun Aug 8 '20 at 23:23
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    $\begingroup$ @AliDursun Yes, it's certainly correct to say that you proved it in a 'another system' (i.e. not in the system of propositional logic you're proving things about) though I doubt you proved it in any kind of formalized system, but rather just using "ordinary mathematical reasoning". And yes, if we are skeptical of the consistency (or more specifically, the correctness) of the portion of this "system" that was used in the proof, it might lead us to doubt the result that propositional logic is consistent. $\endgroup$ – spaceisdarkgreen Aug 9 '20 at 23:22
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"Here is my question; what axiomatic system did we use to prove the consistency of propositional logic, and how do we know that that axiomatic system is consistent?"

A formal axiomatic system probably didn't get used. Probably informal reasoning got used. I don't know of any guarantee that such informal reasoning is consistent.

"How can we be sure that propositional logic is actually consistent?"

I don't think that we can be absolutely sure of this. However, no mistakes come as known in any of the consistency meta-proofs of propositional logic. Also, were it not the case that propositional logic were consistent, then it would be inconsistent. So, it would have both $\vdash$A and $\vdash$$\lnot$A for some well-formed formula A. But, propositional logic is sound, so A is true. But, $\lnot$A is true also by inconsistency, so A is false. Thus, A is both true and false, and propositional logic would be unsound. But, propositional logic is sound, so propositional logic is not inconsistent, and thus propositional logic is consistent.

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    $\begingroup$ But can we actually use the soundness? Since we proved the soundness in a different axiomatic system which we don't know if it is consistent... If it is, it is very clear that propositional logic is consistent. $\endgroup$ – Ali Dursun Aug 27 '20 at 18:31
  • $\begingroup$ @AliDursun Soundness can get used, since it got proved for the system we are talking about. $\endgroup$ – Doug Spoonwood Aug 27 '20 at 18:37
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    $\begingroup$ no, because we used another system to prove, and we don't know if that system is sound. $\endgroup$ – Ali Dursun Aug 27 '20 at 22:29
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    $\begingroup$ @AliDursun We weren't evaluating the other system though. We were evaluating the system of propositional calculus under inspection, which we've supposed as demonstrated as sound. $\endgroup$ – Doug Spoonwood Aug 27 '20 at 22:55
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    $\begingroup$ @AliDursun I agree with you that the soundness proof is on much the same ground as the consistency proof (in fact in this case they're pretty much the same thing). However, I'd add that the proof is so transparent and constructive that I don't think you could find a person who both fully understands it and would seriously doubt that it is meaningful and correct. (Or in other words the 'system' in which the proof could be carried out should we choose to formalize it is very weak and transparently correct and trustworthy.) So yes, if you doubt the metatheory, doubt the result, but nobody does. $\endgroup$ – spaceisdarkgreen Aug 27 '20 at 23:13
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The question is: "What does consistency of propositional logic means?"

My reply . There are two classical notions of consistency: consistency in the traditional sense and consistency in Post's sense (or consistency in the absolute sense) .

According to the definitions of these notions:

consistency of propositional logic in the traditional sense means that there does not exist such well-built formula that this formula and its negation belong simultaneously to the set of consequence of propositional logic;

consistency of propositional logic in Post's sense (or in the absolute sense) means that the set of consequence of propositional logic, is not equal to the set of all well-built formulas (the set of propositional formulas) of propositional logic.

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