How should $\exists x P(x)\land\exists y Q(y)$ be interpreted?

Question

If I encounter an statement like the following: $$\exists x P(x)\land\exists y Q(y)$$

Should this be interpreted as if x and y refer necessarily to different objects or it is to be interpreted as they may or may not be the same object?

Is it therefore equivalent to: $$\exists x P(x)\land\exists x Q(x)$$ ?

Answer 1

just to be clear, in this kind of statements, the scope of the first existential quantifier finishes at the $∧$ symbol?

Yes, the convention is that quantifiers apply to as little as construeable; so, parentheses are required when the quantification is meant to apply beyond this. For example, \begin{align}\big(\forall x\, A(x)\big)\to B(x)\quad\equiv\quad\forall x \, A(x)\to B(x)\quad\not\equiv\quad\forall x\;\big(A(x)\to B(x)\big).\end{align}

(For unambiguity and good practice, the first, second, fourth, and fifth occurrences of $x$ in the above ought to be replaced with $y.$)

Is $$\exists x P(x)\land\exists y Q(y)\tag1$$ equivalent to $$\exists x P(x)\land\exists x Q(x)\;?\tag2$$

Yes, sentences $(1)$ and $(2)$ are logically equivalent to each other.

Should this be interpreted as if $x$ and $y$ refer necessarily to different objects, or it is to be interpreted as they may or may not be the same object?

The latter. For example, if predicates $P$ and $Q$ symbolise “is Taiwanese” and “likes to read”, respectively, and Brigitte is a Taiwanese bookworm, then both variables $x$ and $y$ in sentence $(1)$ can refer to her.

Answer 2

They are equivalent statements in theory . because $ x$ and $y$ concern only the statement they are in (they are called bound variables as mentioned in the comments).

So in theory you can name both of them $ x$

However in practice , you should note them differently, it is necessary to do so actually .

Otherwise during the reasoning for example you can't differentiate the $ x$ that satisfies $ P$ from the one that satisfies $ Q$ which will lead to an unnecessary ambiguity .