Logic - What does ∴ mean in a truth table?

Solution 1:

The symbol means "therefore". In common usage, "therefore" is a stronger statement than "if/then". When we say "$p$ therefore $q$", we mean both "if $p$ then $q$" and "$p$ is true". Thus we assure that $q$ is true, which the "if/then" statement alone does not. Mathematically:

$p \therefore q \equiv ((p\rightarrow q) \land p)$

Solution 2:

It is a semantic statement rather than a syntactic one. Syntax is the level of propositional calculus in which $A,B, A\wedge B$ live. Semantics is at a higher level, where we assign truth values to propositions based on interpreting them in a larger universe.

Your (1), $(A\wedge B) \to C$, is a proposition. It may be true or false. However $(A\wedge B) \therefore C$ cannot be false. It can be a valid proof, or invalid, which is again at a a semantic level rather than a syntactic one.

For example, if $P$ holds then $\neg(\neg P)$ must hold. We have $P \therefore \neg(\neg P)$. This is a different meaning than $P\to \neg(\neg P)$, which is a proposition that happens to be always true (a tautology). Asserting that $P\to \neg(\neg P)$ is logically equivalent to a tautology, is equivalent to $P \therefore \neg(\neg P)$.