Take for example the Hilbert-style axiomatizations of the propositional and first-order calculus. Since a crucial point when operating with a proof system is that no contradictions must be found in it, it seems to me reasonable to look for a consistency proof of those calculus. But I googled it and I could not find anything about it.
So, my question is:
Is there a consistency proof for propositional and first-order logic?
and if there is,
How is this proof carried out?
the reason I ask this is because, since the tools for any proof is an logical apparatus itself, a consistency proof of a logical system X seems to assume that X is already consistent! Or this proof is carried out using second-order logic such like Gentzen's consistency proof of PA?
I appreciate if you shed some light into these topics or indicate some reference on the subject.