Difficulty understanding why $ P \implies Q$ is equivalent to P only if Q. 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?
 A: $P$ only if $Q$ means that when $Q$ is false, then $P$ is false, i.e., ~$Q \implies$ ~$P$.
But the statement of ~$Q \implies$ ~$P$ is equivalent to the statement $P \implies Q$.  Specifically, the former is the contrapositive of the latter.
Applying this to the example you gave:
"I win the lottery only if I give you \$1 billion" is equivalent to "if I don't give you \$1 billion, then I won't win the lottery", and the latter statement is the contrapositive to "if I win the lottery, then I will give you \$1 billion."
If you have doubts about why the contrapositive of a statement is an equivalent statement, have you tried making a truth table?
A: "$P$ only if $Q$" means that the only way $P$ can true be true is if $Q$ is true. 
Now assume "$P$ only if $Q$." Assume also that $P$ is true. Then the only way this can be the case is if $Q$ is true. So $Q$ is true. Hence $P \rightarrow Q$.
Conversely, assume $P \rightarrow Q$. Then from the truth of $P$, we may infer the truth of $Q$. So the only way $P$ can be true is if $Q$ is true. So "$P$ only if $Q$."
A: 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.
A: I find it easier to think of the signs $$\implies,\impliedby,\iff$$ as inequalities between the truth values of statements. For instance, $$A\implies B$$ is the same as saying $$[A]\leq[B],$$
where $[S]$ denotes the truth value of the statement $S$.
If $[B]=0$, then $[A]\leq[B]\implies[A]=0$, so $A$ is true only (but not necessarily) if $B$ is true.
