In hmakholm's answer to this question, the following is written:
If it has no axioms then there is no way to begin a proof in the empty theory, and without rules of inference all that could be proven would be the axioms themselves.
I don't see how either of these assertions can be true. Tackling the first; let's say I have a deductive system which has the following rule, among other:
$$\Gamma \vdash \theta \ \text{and} \ \Gamma \vdash \phi, \ \text{then} \ \Gamma \vdash \theta \land \phi $$
Well, whenever I wind up with the two premises, I am able to prove $\theta \land \phi$, right?
Now, to tackle the second; let's say my logic has the following axiom:
$$P \lor \neg P$$
Let's say I find out, or assume, that $n$ is even. Well, now I have to premises; an axiom and a fact/assumption.
$$P \lor \neg P \\ 2 \mid n \\ \therefore \neg(2 \not \mid n)$$
Surely, $\neg(2 \not \mid n)$ is not an axiom? It follows from premises, unlike axioms, so it cannot be an axiom, right?
(1) Where did $\Gamma$ come from? Ultimately it must have come from an axiom (if it is not an axiom itself).
(2) You are using a rule of inference: from $\Phi$ deduce $\neg(\neg\Phi)$.
And BTW, "$n$ is even" is expressed by $2\mid n$, not $n\mid2$.