# 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 $$Q\rightarrow P$$.
• Right, and I agree that makes logical sense. Maybe I'm missing something simple or understanding just hasn't sunken in yet... I'm just having trouble why the interpretation is stated the way it is in the book. As a direct excerpt Use P => Q to interpret: P only if Q. Should this not be Q only if P for the literal translation of English to symbols? – Pythonista Sep 1 '15 at 2:18
• @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