TITLE: same variable bound by different quantifiers
QUESTION [5 upvotes]: In studying first-order logic, I have come across this sentence:
$\exists x\, P(x)\land\exists x\, R(x)$
If there is some $x$ such that $P(x)$, and there is some $y$ such that $R(y)$, is this sentence true? Are the two $x$'s different? I apologize for asking something so simple, but my book is not clear on this and a search did not turn up any questions sufficiently similar for me to avoid asking.

REPLY [4 votes]: Yes to your first question, and yes to the second. We are talking about the existence of something such that $P(\text{that something}),$ and the existence of something else such that $R(\text{something else}).$ The formula is equivalent to $\exists x P(x)\land \exists yR(y)$
If, on the other hand, we have $\exists x \Big(P(x) \land R(x)\Big)$, then we would be talking about some $x$ such that both $P(x)$ and $R(x)$ are true for that $x$.