interpreting words as if-then statements

In my book it is stated the $P \rightarrow Q$ is used to interpret $P$ only if $Q$.

So, in the statement "$x$ divides 4 only if $x$ divides 8" should the symbolic form not be

$P: x \text{ divides }4$

$Q: x \text{ divides }8$

$P \rightarrow Q$ then, where $P$ is the antecedent and $Q$ is the consequent following the interpretation given.

However, the answer is stated as the opposite ($Q$ is antecedent...), which makes sense and I agree with. I just don't see how this follows from the interpretation of english to symbols given.

• Which book are you using? – Conor O'Brien Sep 2 '15 at 1:29
• Stated in comments of answer below, with a link. – Pythonista Sep 2 '15 at 1:31

I think you are mixing up the definition here. So $$P \rightarrow Q$$ translates to $P$ implies $Q$ or $Q$ only if $P$. So if $P$ is false, then the statement is true, but $Q$ is true only if $P$ is true beforehand. That's why the answer in the book is reversed, because the statement "$x$ divides 4 only if $x$ divides 8" translates, with your predicates, to $$Q\rightarrow P$$.
• @Slayer Yeah, it should be $Q \rightarrow P$ ($Q$ only if $P$) because $Q$ is true only if $P$ is true as explained in the answer. Did this help? – SalmonKiller Sep 1 '15 at 2:19