I was having a discussion with some classmates about what the following means:
$$\exists x \forall x P(x)$$
As far as I understand, a variable can only be bound by one quantifier. I claimed that the innermost one should bind it, because that simply made most sense to me, but some argued that the existential should always bind it because it was the "weaker" of the two.
The way I see it, the $x$ is being quantified by the universal, so the existential is meaningless in this situation, so
$$\exists x \forall x P(x) \Leftrightarrow \exists y \forall x P(x) \Leftrightarrow \forall x P(x)$$
It is similar to saying "there exists a number $x$ such that all numbers $x$ are equal to themselves", what this means is simply that all numbers are equal to themselves. The "exists a number $x$ such that" is unnecessary, because you're not doing anything with that number $x$.
Is this correct?
The standard convention is that the inner-most quantifier binds the variable. That is, you are correct in the sense that most logicians would read $\exists x \forall x P(x)$ as equivalent to $\exists y \forall x P(x)$.
However, this is really just convention. This choice happens to be relatively convenient, but it is ultimately not important. Another convention you could take is that $\exists x \forall x P(x)$ is not even a well-formed formula; you could postulate that if $\phi$ is a formula and $x$ is a variable, then $\exists x \phi$ and $\forall x \phi$ are only formulas if $x$ does not occur as a bound variable in $\phi$.