TITLE: Why should we care about syntactic proofs if we can show semantically that statements are true?
QUESTION [31 upvotes]: I am talking about classical logic here.
I admit this might be a naive question, but as far as I understand it:
Syntactic entailment means there is a proof using the syntax of the language, while on the other hand semantic entailment does not care about the syntax, it simply means that a statement must be true if a set of other statements are also true.
That being said, isn't semantic entailment sufficient to know whether or not a statement is true? Why do we need syntactic proofs?
Granted I know that in the case of boolean logic, proving statements by truth tables gets intractable very fast, but in essence, isn't semantic entailment "superior"? As it does not rely on how we construct the grammar?
Thank you
Edit: Suppose it wasn't the case that finding a satisfying assignment to an arbitrary boolean statement is an exponentially increasing problem, then would we even need syntactic entailment?

REPLY [2 votes]: Here are two more reasons why syntactic proofs are interesting. (I figured I'd make this a separate post because it makes a completely different point than my other answer).

1. Syntactic proofs can help us capture better why, rather than just that, an argument is valid
While truth tables can tell us that a certain validity or entailment holds, they typically don't really tell us why. Sure, the entailment holds because the conclusion is true under all interpretations under which all the premises are true, but why is that? Truth tables (or arguments about valuations) don't usually give much insight into the patterns that underly logical reasoning.
The calculus of natural deduction has its name because it is intended to formalize how mathematicians would reason naturally in informal proofs. If you already know all about natural deduction, you can skip the next paragraph.

For instance, consider the following entailment: $A \lor B, B \to C \vDash A \lor C$.
Naturally, we would carry out a proof by cases on the disjunction in the first premise. There are two cases to consider:

$A$ holds. Then surely, either $A$ or $C$ holds, i.e. $A \lor C$. We can express this step of reasoning with a bar from the assumption to the new observation:
$\dfrac{A}{A \lor C}$
$B$ holds. By the second premise, we know that if $B$, then $C$. The combination of these two pieces of knowledge allows to conclude us that $C$:
$\dfrac{B \quad B \to C}{C}$
Analogous to above, we may conclude that either $A$ or $C$:
$\dfrac{\dfrac{B \quad B \to C}{C}}{A \lor C}$

Now we proved that in both cases, $A \lor C$ follows. We don't know which of the disjuncts is true, but we know that at least one is, so we can conclude $A \lor C$ for sure. At this point, the actual truth values of each of the disjuncts no longer matter, so we can discharge the temporary assumptions $A$ and $B$, and the final conclusion only depends on the premises $A \lor B$ and $B \to C$. Puzzling the first premise and the two subproofs together, we get the following structure of the entire argument:
$\dfrac{A \lor B \quad 
\dfrac{A\hspace{-6pt}\times}{A \lor C} \quad
\dfrac{\dfrac{B\hspace{-7.5pt}\times \quad B \to C}{C}}{A \lor C}
}
{A \lor C}$

And that's precisely what a tree-style natural deduction looks like. This presentation reveals several insights about the way reason logically:

Arguments can be composed stepwise from elementary rules of reasoning.
The structure of an argument can be thought of as a tree shape, where we dissect and reassemble assumptions to new formulas to eventually reach a single conclusion.
We can make local assumptions, for example in a proof by cases, and, under some conditions, later resolve the dependency on the assumptions and discharge them.

All of this a simple truth table can not provide -- but this is precisely the kind of issues that are interesting: The task of logic is not primarily to exhaustively list the valid theorems, but rather to investigate the elementary building blocks and structural properties of logical arguments.

While natural deduction is particularly suited to mimic the structure of intuitive reasoning, other syntactic calculi have their merits as well.
The tableau calculus may look like not more than a fancy truth table at first sight, but what makes it interesting is the ability to directly construct a (counter) model from an open branch. That way, we are provided with a formal means to analyze what structures make a particular formula satisfiable or invalid. This works even for predicate logic, where truth tables don't exist.
Sequent calculus provides a new perspective because it essentially makes inferences about inferences: The sequent rules express from which derivability claim we may infer another one.
And last but not least, syntactic proof systems are interesting on their own -- this is what proof theory studies. There are entire books out there just about natural deduction because there is a lot to say about the properties of proofs as mathematical objects, normalization being one important keyword.

2. Syntactic proof systems can be taught to computers
When talking about tableaus, it shouldn't be left unmentioned that they are a completely algorithmic, deterministic procedure and thus very straightforward to implement, thereby providing a computational mechanism for both theorem proving and model generation, at least in principle (though this procedure is among the quicker ones to blow up on larger formulas).
Among the tools employed in real-life computing, resolution and unificatoin are powerful procedures that can be expressed as a syntactic calculus.
Axiomatic systems in a Hilber-style manner due to their simplicity are particularly useful in carrying proofs about decidability and computability issues in theoretical computer science.
Truth tables may be straightforward to implement, but end at propositional logic, which is not very useful for many real-life problems, and that's where syntactic calculi come in handy.
The possibility to treat predicate logic with computers of course has its own perks. You can do theorem proving to verify mathematical arguments, software verification, SAT solving for configuration checkers, ..., and of course, again, the ifs and hows of doing mathematics with computers is interesting in itself for mathematics and theoretical computer science.