I have this sentence: "All parrots are black and there exist a parrot which is in the cage." It is probably written in a predicate logic like this:
$(\forall x) b(x)\land(\exists y) c(y)$
My question is: Why there has to be two variables (x and y) istead of one?
Variables that are in the scope of a quantifier have no "life" outside their context, i.e. outside the sub-formula that is the scope of the quantifier.
Thus, we can re-use a variable in different contexts [but this does not mean that it refers to the same object; consider e.g. $\exists x \ \text {Odd}(x) \land \exists x \ \text {Even}(x)$ ].
This means that the above formula is equivalent to: $(∀x)b(x) ∧ (∃x)c(x)$ as well as to: $(∀y)b(y)∧(∃y)c(y)$.
In turn, the above formulas are also equivalent to: $(∀x)(∃y)[b(x)∧c(y)]$ (see Prenex normal form).
The case of: $(∀x)(∃x)(b(x) ∧ c(x))$ is different.
This formula is equivalent to $(∃x)(b(x) ∧ c(x))$; adding a new quantifier $∀x$ to a formula that has no variable $x$ free does not change the meaning of the formula.