which bind strong or which come first in definition of precedent ¬, ∀, and ∃

Question

in first-order language Why doesn't the definition of precedent cover which of ¬, ∀, and ∃ bind the most strongly or which come first in priority?

Answer 1

Because precedence only becomes an issue at all when at least one $\leq 2$-place operator (such as $\land$) is involved. Between only unary (1-place) operators like $\neg, \forall, \exists$, there is no ambiguity -- there are no two ways to read a statement like $\neg \forall x \exists y \phi$. Because all the operators only apply to one argument, namely the one that comes immediately next, the linear order of the operators fully determines the structure of the formula.

That being said, one does sometimes define that quantifiers have precedence over all connectives, which entails that $\forall, \exists$ have precedence over $\neg$.