I'm sorry if this question is not clearly formulated: An axiomatization, or an axiomatic system, usually means a set of axioms (i.e. a theory). A formal theory is such a set of formulas in some formal language (e.g. Peano axioms, in the language of first order arithmetic).
My question is, what does it mean to axiomatize a logic (e.g predicate logic)? It does not seem to mean the same thing because, for example, the Hilbert style of axiomatization of predicate logic includes rules of inference and those are not formulas in a formal language. And what is supposed to be the formal language here in the axiomatization of predicate logic? Is there such a language as "predicate logic"? What is its signature, does it include an infinite number of arbitrary constants, functions and relations?
I understand the difference between a schema and a formula. But I don't see how that difference answers my question.