I have a question about mathematical English in these statements we get in basic quantification problems, it sounds to me like $x$ is some particular constant named $x$ so under this interpretation $x$ -> some number for which $P(x)$ is true, implying to me (at least) that it's unique. I assume this an incorrect reading of the statement?
Is the correct reading that there exists (at least one value of the variable in our domain of discourse) $x$ such that $P(x)$, so an existentially quantified statement? Allowing for the existence of multiple values.
If you just have a proposition like () without any quantifiers then is a free variable. It is not a constant.
The value of is contingent upon the value of . That is different than the existential quantifier which asserts that there is an that makes the proposition true.
For example, let $P(x)$ be $x+5=7$. Then, assuming we are talking about integer addition, the value $x=2$ makes the proposition true and other values of $x$ make it false.
That is different than $\exists xP(x)$ which asserts that there is an element that makes the proposition true.