Why does the order of quantifiers matter in a logical statement?

In logic, the order of quantifiers matters because it determines the logical priority and dependency between variables, which fundamentally changes the meaning of a statement.1

In symbolic logic, the scope of a quantifier indicates the range of the variable it binds, but it also establishes "priority scope," which dictates how quantifiers interact. When quantifiers are nested, the one appearing earlier in the sequence typically has logical priority over those that follow, meaning the subsequent variables may depend on the values chosen for the preceding ones.1

For example, consider the difference between these two expressions:
* $(\forall x)(\exists y)(x\text{ loves }y)$: This asserts that for every person $x$, there exists some person $y$ whom $x$ loves. Here, the choice of $y$ can depend on $x$.1
* $(\exists y)(\forall x)(x\text{ loves }y)$: This asserts that there exists a specific person $y$ who is loved by everyone. In this case, the existential quantifier does not depend on the universal one, resulting in a much stronger claim.1

While sequences of identical quantifiers (all universal or all existential) can be rearranged without changing the truth-value of the statement, mixing them creates distinct logical dependencies that are not equivalent.21