Why aren't vacuous truths just undefined? I am struggling to understand this. According to truth tables, if $P$ is false, it doesn't matter whether $Q$ is true or not: Either way, $P \implies Q$ is true.
Usually when I see examples of this people make up some crazy premise for $P$ as a way of showing that $Q$ can be true or false when $P$ is something outrageous and obviously untrue, such as "If the moon is made of bacon-wrapped apple-monkey carburetors, then I am a better wakeborder than Gauss."
$P$ is clearly false, but $P \implies Q$ is true no matter what the state of $Q$ is, and I don't understand why.
Are we saying "If $P$ is false, then all bets are off and $Q$ can be anything, either true or false, and not contradict our earlier claim, and if it isn't false, it must be true"?
Otherwise why can't we say that if $P$ is false, then we can't make any claims one way or the other on whether or not it implies anything at all? 

Consider the statement:

All multiples of 4 are even.

You would say that statement is true, right?
So let's formulate that in formal logic language:

$\forall x: 4|x \implies 2|x$

(Here "$a|b$" means "$a$ divides $b$", that is, $b$ is a multiple of $a$.)
Now a $\forall$ statement is true if it is true whatever you insert for the quantified variable (after all,that's "for all" means). So let's try to insert $3$:

$4|3 \implies 2|3$

But wait, $4|3$ is false! Moreover, $2|3$ is also false. So the only way for the original statement to be true is that the implication $\text{false}\implies\text{false}$ gives true.
A similar argument can be done for $\text{false}\implies\text{true}$.
