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...