TITLE: Why truth table is not used in logic?
QUESTION [12 upvotes]: One day, I bought Principia Mathematica and saw a lot of proofs of logical equations, such as $\vdash p \implies p$ or $\vdash \lnot (p \wedge \lnot p)$. (Of course there's bunch of proofs about rel&set in later)
After reading these proofs, I suddenly thought that "why they don't use the truth table?". I know this question is quite silly, but I don't know why it's silly either (just my feeling says that).
My (discrete math) teacher says that "It's hard question, and you may not understand until you'll become university student," which I didn't expected (I thought the reason would be something easy).
Why people don't use truth table to prove logical equations? (Except for study logic (ex: question like "prove this logic equation using truth table"), of course.)
PS. My teacher is a kind of people who thinks something makes sense iff something makes sense mathematically.

REPLY [10 votes]: Your teacher may be referring to the difference between semantics and syntax. A syntactic proof is a finite formal derivation of a sentence from the axioms of a theory using the logical axioms and the rules of inference of a logic. A proof by a truth table is a semantic proof; in allowing truth tables you are tacitly assuming the completeness theorem of propositional logic. Essentially, a priori, we don't know that everything we can prove by studying the models of a theory (i.e. truth tables, in the case of propositional logic) can be proven syntactically, or even for that matter vice versa. It's a non-trivial result in logic, which I will state below:

Definition. Let $T$ be a theory in some logic. We say the logic is sound if every sentence $P$ which can be formally derived from the axioms of $T$ is true in every model of $T$, i.e. $T \vdash P$ implies $T \vDash P$. We say the logic is complete if when a sentence $P$ is true for every model of $T$, $P$ can also be derived formally from the axioms of $T$, i.e. $T \vDash P$ implies $T \vdash P$.
Theorem. Classical propositional logic is sound and complete.
Theorem (Gödel). Classical first-order logic is sound and complete.

And yes, it is that Gödel. There are also incomplete logics; you don't have to go very far to find them, since even second-order logic (under the usual semantics) is incomplete.