What does consistency of propositional logic means? 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...
 A: "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.
A: 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.
