First order logic question about whether variables in the same sentence are bound

Question

Is my intuition right that $$((\exists x)Px \land (\exists x)Gx)$$ is equivalent to $$((\exists x)Px \land (\exists y)Gy)$$ or is it actually equivalent to $$(\exists x)(Px \land Gx)$$

Any help would be great.

Answer 1

The first two formulas are equivalent, but they are not equivalent to the third one.

Consider the following counter-example :

"(There is an Even number) and (there is an Odd number)" vs "There is a number (that is Even and Odd)".

A bound variable, like $x$ in $\exists x Px$ and "no life" outside the context of the formula where it occurs. This is the reason why we can reuse the same bound variable in different contexts, like in $\exists x Px \land \exists x Qx$ (also if can be more clear to avoid it : we have infinite many variables in our formal language).

In a nuthsell, this means that the object that $\exists xPx$ asserts to exist is not necessarily the same object whose existence is asserted by $\exists x Qx$. This is obviously not the case with $\exists x (Px \land Qx)$, where the object whose existence is asserted is one and the same for both $P$ and $Q$.