I don't know if this is the right place for this question, so if it isn't let me know.
So I initially learned symbolic logic from Introducing Symbolic Logic by Robert Martain where we learned to do proofs using a certain set of rules. After the course, I started to look into modal logic in A New Introduction to Modal Logic by Hughes and Cresswell, but there they used axioms rather than rules for proofs. Are these two distinct methods, or are they reducible to each other? Like, can you take any set of axioms and turn them into rules, and vice versa? And if they are distinct, I am confused on how two distinct methods of proof can be invented and guaranteed to be valid.