Why truth table is not used in logic? 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.

In Principia, the authors wanted to produce an explicit list of purely logical ideas, including an explicit finite list of axioms and rules of inference, from which all of mathematics could be derived.  The method of truth tables is not such a finite list, and in any case would only deal with propositional logic. 
The early derivations of Principia are quite tedious, and could have been eliminated by adopting a more generous list of initial axioms.  But for reasons of tradition, the authors wanted their list to be as small as possible. 
Remark: Principia is nowadays only of historical interest, since the subject has developed in quite different directions from those initiated by Russell and Whitehead. The idea of basing mathematics (including the development of the usual integers, reals, function spaces) purely on "logic" has largely been abandoned in favour of set-theory based formulations.  And Principia does not have a clear separation between syntax and semantics.  Such a separation is essential to the development of Model Theory in the past $80$ years.
