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.
Best Answer
First, of course, mathematicians and logicians often do use truth tables, so it is incorrect to suggest that they are not used. (I won't comment on your reading of Principia Mathematica, beyond saying that that text is not meant to be taken as a piece of pedagogy.)
But second, the truth table method is not a feasible method for large propositions. This is because for a formula with $n$ free variables, the truth table is an object with $2^n$ rows, exponentially large in size. But many logical expressions nevertheless have comparatively short derivations showing them to be tautological.
For example, $$(p_0\vee \neg p_0)\wedge(p_1\vee \neg p_1)\wedge\cdots\wedge(p_{100}\vee\neg p_{100})$$ has a truth table with $2^{101}$ rows, but there is a simple, obvious derivation that allows us to see it as taugological without calculating this table.
Thus, although when there are just two propositional variables, I don't mind computing a truth table, or even for three, it is often easier with more to use more focused reasoning.