Meaning of duplicated predicate quantifiers What is the meaning of duplicated predicate quantifiers? Examples: $$ ∃x\ ∃x\ ∀x\ ∀x\ P(xy) \\\ ∀y\ ∀x\ ∃x\ ∀x\ ∃x\ ∃y\ ∃x\ P(xy) \\\ ∃y\ ∀y\ ∀x\ ∃x\ ∃y\ ∀y\ ∀x\ ∃x\ ∃x\ P(xy) $$

The quantifier closest to the quantified variable dominates any previous quantifiers.

The first is equivalent to $\forall x P(x, y)$, with $y$ unbound.

The second, then, is equivalent to $\exists y \exists x P(x, y)$.