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 any proof by contradiction, for example Euclid's proof of prime infinity. 

If $x$ is the largest prime number ("P"), then $y=\prod_{p\,\,prime} p+ 1$ is a prime ("Q").

This is a valid implication that can be shown with perfectly valid simple arguments. However, the premise is never satisfied. But within the proof, the reader may not yet see that it is ''nonsense'' at this step. You just follow the logic step by step. The fact that the premise is never satisfied does not make the logic (used to derive the claim above) less valid. So it is reasonable to assume that the claim above is True.
