In general, when introducing logic, we present students with a given formal system (e.g. Hilbert-style axioms, natural deduction, sequence calculi, etc.) and then, after presenting the standard semantics for first-order logic, we proceed to show that the system is adequate with respect to this semantics, i.e. it is both correct and complete. In some cases, we may even present students with several such systems, comparing their pros and cons (one system may be more cumbersome to work with, but makes it easier to prove correctness and completeness, etc.). There is something a bit unsatisfying about this since it may give the impression that it is just luck, or a historical contingency, that the systems we do have are indeed adequate to the standard semantics. Hence my question:
Can we isolate a set of necessary and sufficient conditions such that any system which satisfies them will be adequate with respect to the usual semantics?
I think I may have stumbled upon an answer to my question as I was perusing Smullyan's First-Order Logic (especially Chapter VI "A Unifying Principle"). Keisler discusses a similar idea in his Model Theory for Infinitary Logic (with reference to Smullyan) and so does Hinman in his Fundamentals of Mathematical Logic (pp. 63ff for the propositional case and pp. 216ff for the first-order case). I will discuss here the propositional case since it is easier.
Consider a formal system $F$, which we may suppose is sound (with respect to the standard semantics). Since it is sound, it will generate a notion of consistency. In particular, we may say that a given set of sentences $\Sigma$ is $F$-consistent iff there is at least one formula $\phi$ such that $\Sigma \not \vdash \phi$. Now, we say that this $F$-consistency notion is a consistency property iff for each $F$-consistent set $\Sigma$, the following holds:
(1) If $\phi \in \Sigma$, then $\neg \phi \not \in \Sigma$;
(2) If $\neg \neg \phi \in \Sigma$, then $\Sigma \cup \{\phi\}$ is also $F$-consistent;
(3) If $\phi \vee \psi \in \Sigma$, then either $\Sigma \cup \{\phi\}$ is $F$-consistent or $\Sigma \cup \{\psi\}$ is $F$-consistent;
(4) If $\phi \wedge \psi \in \Sigma$, then $\Sigma \cup \{\phi\}$ and $\Sigma \cup \{\psi\}$ are also $F$-consistent.
We then have the following theorem:
Theorem: If $F$-consitency is a consistency property, then any $F$-consistent set can be extended to a maximal complete set. In particular, every $F$-consistent set has a model. (See any of the references given in the first paragraph.)
If I'm not mistaken, this means that the above gives a set of necessary and sufficient conditions for a (classical) system to be adequate.