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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.

flakmonkey
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1 Answers1

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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 $x$.

amWhy
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    Thank you! So, to clarify, we treat any occurrence of a variable within the scope of a quantifier as distinct from the occurrences of that variable outside the scope of the quantifier? – flakmonkey Jun 24 '14 at 23:31
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    Exactly, flakmonkey! – amWhy Jun 25 '14 at 00:01