Propositional systems have axioms and inference rules. They bifurcate into two types, Hilbert-style where the number of inference rules are minimised; and Natural Deduction where the number of axioms are minimsed.
In fact it is possible to have Natural Deduction systems that have no axioms, I'm not sure if this is in fact the usual ones looked at.
In general Hilbert-Style systems usually have just one inference rule: modus ponens.
Obviously it is possible to have a Hilbert-style formal system with no inference rules, the question is whether they are useful in anyway?
(Perhaps if we consider the category of all Hilbert-style formal systems they need to be included to get good formal properties for the category?)