TITLE: Is the modus ponens is an axiom in formal logic?
QUESTION [5 upvotes]: Modus Ponens is a rule of inference, which is not exactly an axiom. As far as I understand an axiom is a starting point for a proof, while a rule of inference is tools used to make conclusions out the axioms given. So doesnt that make rules of inference axioms themselves? Correct me if im wrong.

REPLY [4 votes]: $(Q)$ : [Are] rules of inference axioms themselves? 

Rules of inference formalize the "basic" inference steps, i.e. the "legal" moves from one or more sentences (asserted or assumed) : the premises, to a new one : the conclusion.
Axioms are sentences assumed to be true.
We can see :

Stephen Cole Kleene, Introduction to Metamathematics (1952), page 80, for the discussion about deductive rules or transformation rules, and page 83 :


Carnap [The logical syntax of language, 1934] brings [axioms and rules] under the common term "transformatiion rules", by considering the axioms as the result of transformation from zero premises.

Thus, we can say instead that axioms are "a sort of" rules ...