How does one create axiom schemata that will be able to generate all tautologies in the system?

Question

To be more specific, how do I know that the axiom schemata(together with modus ponens) in the Hilbert calculus is able to generate all valid formulas in propositional calculus?

Answer 1

The easiest way I know to consistently create complete axiom sets is to have some rule of detachment, have a conditional connective, and use functorial variables (except authors often use propositonal calculi too weak to have functorial variables!). Then you basically embed the truth tables in the axioms, have a constant tautology which you can call '1' as an axiom, and have some axiom which basically says that if something holds for each truth value, then it holds for an arbitrary variable. I made some notes on this a while back building on Prior's and other's work.

This doesn't cover completeness for an equivalential calculus, a Sheffer Stroke (NAND) system, or one using Peirce's arrow (NOR).