Logic - What does ∴ mean in a truth table? I see the symbol used, and I've never seen it logically defined. In words, It's defined as a symbol meaning 'therefore'. Because of a lack of definition, I have no idea why this is false:
A
B
∴C

For me, this statemnet could mean one of two things.


*

*'If A and B thus C':
(A^B) -> C
i.e same as conditional

*That ∴ is the same as the conditional, only without a truth value for when the premises is false (conclusions cannot draw upon a false premise) 
'A thus B', perhaps? I've read that a valid argument is only so if it is (logically/symbolically/algebraically) impossible that the conclusion be false, given that premise is true.
I am not certain on that, and any correction would be appreciated. However, regardless, I still can't find a strict, logical definition or equivalent of ∴, and it's not that I haven't looked, the ∴ only ever seems to be talked about, never strictly defined.
 A: 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:
$p \therefore q \equiv ((p\rightarrow q) \land p)$
A: 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 $P \therefore \neg(\neg P)$.
