&gt; Things are not that simple, because you can often convert axioms to rules of inference or vice versa, without changing the set of derivable consequences.<p>Yes, but the derivations themselves will change.<p>&gt; As an example, consider pure first-order logic (FOL). As one extreme, you can present FOL with just one rule of inference (Modus Ponens), see for example [1]. The other extreme is Gentzen&#x27;s sequent calculus [2] which has only one axiom (A |- A), everything else being a rule of inference. Most presentations of FOL are between these extremes.<p>Yep. I&#x27;m aware of the phenomenon that a single mathematical object of type T (say, infinity-categories) may admit multiple presentations by objects of type T&#x27; (say, model categories). But, just because two objects of type T&#x27; present the same object of type T (e.g., two Quillen-equivalent categories), it doesn&#x27;t mean that they are equal in all respects (e.g., the category of simplicial sets is much nicer than the category of topological spaces).