<p><pre><code>    Axioms and rules of inference are fundamentally different:

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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>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>[1] A mathematical introduction to logic, by H. Enderton<p>[2] Untersuchungen über das logische Schliessen I, by G. Gentzen