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zpavlinovic
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In model theory, does compactness easily imply completeness?
@JoelDavidHamkins My understanding of tatulogical validities was wrong. Sorry for that.
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In model theory, does compactness easily imply completeness?
A bit late, but I have a comment on the $MM$ system. In Enderton's book, he basically makes the proof (for FOL) $\Sigma \vdash \alpha$ iff $\Sigma \cup \Delta$ tautologically implies $\alpha$. $\Delta$ are his axioms. The proof he gives is just as the one here for system $MM$. But, tautological validity is computable and provable, so why does his proof do not constitute a proof of completeness for FOL also?
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Deduction theorem
Deduction theorem also does not hold in epistemic logic.
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Deduction theorem
I am probably late, but consider the first system you described. Say you assume $A$ and you want to prove $B$. You could easily do it by your rule $A \vdash B$. However, this proof is not ok since $B$ does not have to be true. Hence, this system is unsound. Am I maybe misinterpreting your text?