I couldn't find a proper explanation to several questions i have about the scope of quantifiers, free and bound variables.
1) $\exists X(p(X, Y) \mathop{\&} \exists Yq(X, Y))$ - I am wondering about the second X. Is it still in the scope of the $\exists X$ quantifier,because it's actually nested under ∃Y? If it is not in the scope of $\exists X$ then is it a free variable? I am thinking it's should be bound to $\exists X$. 2) $\exists X(p(X, Y) \mathop{\&} \exists Xq(Y, Z))$ - is the second ∃X needed at all? Does it change anything for the free variables Y and Z?
I have more questions of this type but those two bother me the most currently. And answering them might make me get the idea in general and answer myself the next questions.
Also if you have a resource where stuff of this type are explained in detail I will be thankful!
Thank you in advance!
1) The second $X$ in $\exists X(p(X, Y) \mathop{\&} \exists Y\,q(X, Y))$ is bound by the $\exists X$ at the beginning of the formula. $X$ is free in $\exists Y\,q(X, Y)$ and in both of its occurrences in $p(X, Y) \mathop{\&} \exists Y\,q(X, Y)$. These two occurrences the become bound when you form $\exists X(p(X, Y) \mathop{\&} \exists Y\,q(X, Y))$.
2) If $X$ does not appear free in $\phi$, then $\exists X\phi$ is logically equivalent to $\phi$. $\exists X(p(X, Y) \mathop{\&} \exists X\,q(Y, Z))$ logically equivalent to $\exists X(p(X, Y) \mathop{\&} q(Y, Z))$.