TITLE: Difficulty understanding why $ P \implies Q$ is equivalent to P only if Q.
QUESTION [6 upvotes]: I have difficulties understanding why $ P \implies Q$ is equivalent to P only if Q. I do understand that in the statement "P only if Q", it means if $ \lnot Q \implies \lnot P$". 
Regarding this example, Equivalence of $a \rightarrow b$ and $\lnot a \vee b$
If I win the lottery, then I will give you \$1 billion. 
This statement has the form $P \implies Q$.
But saying P only if Q, means "I win the lottery only if I give you $1 billion" doesn't sound so right. Is there anything I'm missing here?

REPLY [2 votes]: It really helps me to think about propositions as light bulbs. If a proposition is true, its  light bulb is on. If a proposition is false, its light bulb is off. So $p \implies q$ means that whenever $p$ is on, $q$ is on too. If $p$ is off, $q$ could be either on or off. So now let's think about "$p$ only if $q$". If $q$ is off, then $p$ is DEFINITELY off, because if $p$ was on, the $q$ would've been on too. That means that $p$ can only be on when $q$ is on.