Proving equivalence of natural deduction and Hilbert systems for classical logic

Question

I was reading this question which asks how to prove equivalence of natural deduction and Hilbert systems in the sense:

For all formulas, $M$, $\hspace{0.3cm} \Gamma \vdash_{H} M$ iff $\Gamma \vdash_N M$ is derivable, where $\Gamma$ is finite and $M$ contains at most one formula.

That question deals with intuitionistic logic. I had a similar question about classical logic. The answer there describes a proof strategy involving deriving each system's axioms and rules in the other. Wouldn't it be easier to just use soundness and completeness:

If $\Gamma \vdash_H \varphi$, then $\Gamma \models \varphi$ by soundness of the Hilbert system $H$. By completeness of the natural deduction system $N$, we have $\Gamma \vdash_N \varphi$. The reverse direction can be proven likewise.

This should be possible since the meaning of $\models$ does not rely on the deduction system. Moreover, it would work for any logic where both soundness and completeness are available, like classical and intuitionistic logic. Am I missing something? Why is the long winded approach in the other answer better?

Answer 1

Sure, if you already have soundness and completeness theorems, you're happy with using them, and you primarily care just about the brute fact that the derivability relations are equivalent, then the approach you suggest works fine. Now let's see why each of these assumptions might not hold.

Obviously, the first problem is that you may not have proven soundness or completeness yet, and, in general, you may not even have a completeness result. Regardless, proving completeness is likely to be more involved than directly proving the equivalence of the derivability relations.

One philosophical issue with using soundness and completeness is that it means going through a semantics. Typically semantics is formulated in a much more powerful language than the languages you're talking about, usually set theory. A direct proof, on the other hand, is basically just a matter of replacing parts of a proof with fragments of proofs in the other language. This can be done in very weak frameworks (e.g. primitive recursive arithmetic (PRA)). Using semantics is then looks like pulling out a sledgehammer to crack a nut. More precise delineation of the requirements of theorems is also useful for reasons unrelated to philosophy. At a more pragmatic level a direct proof can be done without needing to even introduce any of the machinery of semantics. It's certainly not worth introducing semantics just for this.

Finally, a direct proof will almost certainly be constructive and informative as it will e.g. involve proving the deduction theorem explicitly. That is, not only will you know that a proof can be translated, you'll know what the translation looks like. Now, it's not hard to find proofs of the completeness theorems that are constructive, i.e. that will (if you unfold enough) produce an actual proof. In that case, the route through soundness and completeness will produce an actual explicit translation. It will, however, be terrible. All structure of the original proof that is being "translated" will be obliterated. The completeness theorem has no way of knowing about the original proof; it will output the same proof no matter what you started with. And those proofs are likely to be ugly to boot. For example, the proof of completeness (of just propositional logic) might do a case analysis over all variable assignments as the first proof here does. This means all produced proofs are exponentially large relative to the number of propositional variables.