Show that one cannot prove the following formula by natural deduction

Question

Show that one cannot prove the following formula by natural deduction:

$$\exists x \; \forall y\;R_0(x,y)\to \forall x \; \exists y\; R_0(x,y)$$

So I have to find a case where I get truth values $1 \rightarrow 0$, right?

Answer 1

Hint: If this were true, what would happen if the truth or falsehood of $R0(x,y)$ did not depend on $y$ at all?

Your (deleted) example was a good start, but in basic logic, $x$ and $y$ have to be the same "type" - you can't have $x$ range amongst people and $y$ range amongst money.

Instead, what if $x,y$ ranged amongst people, and $R0(x,y)$ means

$x=y$ or $x$ gave $y$ money