In my course on propositional logic, we are given the axioms:
And, of course, modus ponens.
Why are these three chosen? Yes, you can use them to prove the completeness theorem, so in that respect they are useful. But I've never seen a proof that you can't deduce one of the three from the other two? (Without using the completeness theorem, obviously). Are these three independent? Are these three the maximum independent set of tautologies?