Which FOL consequence relation is better (to teach)?

Question

Backstory: I had problems of accepting Mendelson's rule of Generalization (it's $(A(x) \vdash \forall x, A(x)$).

Surfing around the net, I found this post which explained the situation for me: http://www.cs.nyu.edu/pipermail/fom/2004-September/008483.html (there are two different consequence relations; the other one gives GEN rule that I like much more: if $\vdash A(x)$ then $\vdash \forall x, A(x)$).

My reasons to believe that "truth relation consequence" is better:

It's used when writing standard proofs (at school, highschool and also in all proof assistants for formal proofs that I know - Coq, Lean and probably much more). There's no reason to teach something that wouldn't be used practically later. (At least for intro courses of Mathematical Logic, as I believe.)
It has beautiful correspondence between consequence relation and usual implication connective.
No more weirdness of Deduction theorem for proofs that has used GEN inference rule (this is probably subjective and related to 2nd point).
Truth consequence relation is much more related to usual type theory, I believe. I don't know what's the status of Mendelson's consequence relation in this aspect, but there must be a reason why it seems not used in normal mathematical work. (I may be wrong.)

I believe that truth consequence relation should be teached as primary. The other kind can be interpreted by adding all missing universal quantifiers to formulas and treating it as "theorem inference rule" (?) that gives one more theorem from given theorems.

So there's the question:

What's the consensus in academia on the pedagogical and practical usefulness of Mendelson's (et al.) variation of FOL? Is it true that "truth consequence rule" is more natural?

I agree with you; there are proof systems (Hilbert-style) without Gen rule; see Herbert Enderton, A Mathematical Introduction to Logic (2nd ed - 2001) or Martin Goldstern and Haim Judah, The Incompleteness Phenomenon (1995). — Mauro ALLEGRANZA, Mar 13 '16 at 11:39
See this related post: elliott-mendelson-introduction-to-mathematical-logic and also: what-is-the-common-definition-of-model-in-first-order-logic. — Mauro ALLEGRANZA, Mar 13 '16 at 11:41
Nice to see that I'm not the only one with such opinion.
If I need I can work with Mendelson's variation - it has different inference rules, that's all. Using them formally won't be difficult (just add those implicit quantifiers in your mind) and I understand them, but they're unnatural.

I will have to do this during next year studies and I dislike this fact a bit. :( — zaa, Mar 13 '16 at 11:56
But things are not "so bad" ... The "preferred" generalization (meta-)ule is "if $\Gamma \vdash A(x)$, then $\Gamma \vdash \forall x A(x)$, provided that $x$ is not free in $\Gamma$". In "usual" math life, $\Gamma$ must be the set of axioms of a theory; in this case, we have no need of free variables: we can start from axioms that are (implicitly) universally quantified. — Mauro ALLEGRANZA, Mar 13 '16 at 13:40
Probably you're right. On the other side, I will still have to study about Mendelson's variation next year - with the teacher who, if I can believe him, didn't know about the other possibility (other kind of GEN rule, at least). — zaa, Mar 15 '16 at 11:51

Which FOL consequence relation is better (to teach)?

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