Modern treatments of (classical) propositional calculus first set up a language $\mathsf{L}$, which consists of formulas built up recursively from a set of propositional letters $\{p,q,r,\dots \}$, and a certain number of connectives $\{ \neg, \lor, \land \}$.
The semantics of $\mathsf{L}$ is a Boolean valuation of the propositional letters, whereby each letter is mapped to either $T$ or $F$, and the valuation is extended to all formulas of $\mathsf{L}$, such that the connectives are interpreted in the usual way, as truth-functional connectives. We say that the formula $\phi$ semantically entails formula $\Phi$, or $\phi \vDash \Phi$, if every valuation of $L$ which maps $\phi$ to $T$ must map $\Phi$ to $T$ also.
If we set up a proof system (say Gentzen's natural deduction) for $\mathsf{L}$, we say that $\phi$ syntactically entails $\Phi$ if there is a proof of $\Phi$ from $\phi$ ("proof" would of course depend on the proof system used).
Now it can be proved that, for $\mathsf{L}$ equipped with a proof system, $\phi \vDash \Phi$ if and only if $\phi \vdash \Phi$. The delineation between semantic and syntactic entailment is emphasised very much in most texts; in certain logic books I've consulted, the authors even make it a point that truth tables are a purely semantical concept. I do not have any problems with the distinction.
However, Chang and Keisler's book on Model Theory has cast a serious doubt on the syntax-semantics dichotomy. Their definition of the semantics of propositional calculus is as before, but look at how they define syntactic consequence (my paraphrase):
Let $\phi$ be a formula, and $p_0, p_2, \dots, p_n$ be all the propositional letters occuring in $\phi$. We say that $ \vdash \phi$ if $\phi$ has the value $T$ for every valuation of $p_0, p_1, \dots, p_n$.
And now to my question proper:
With this definition, I no longer see much difference between semantic entailment and syntactic entailment. They claim that the method of truth tables is purely syntactic, which some books vehemently disagree with. This definition renders the completeness theorem of propositional logic pretty much useless in my opinion, because the original motivation of the theorem was to show that we can mechanically derive all propositional tautologies from a small number of axioms with some inference rules. I would appreciate answers which address this doubt.
Besides the completeness theorem, is there a reason why logicians came up with the theory of semantics of formal languages? The semantics of propositional logic only involves assigning each propositional letter to an element of $\{0,1\}$, and does not seem to be address the deeper issues of "truth".
