Scope Dominance with Upward Monotone Quantifiers

We give a complete characterization of the class of upward monotone generalized quantifiers Q₁ and Q₂ over countable domains that satisfy the scheme Q₁ x Q₂ y φ → Q₂ y Q₁ x φ. This generalizes the characterization of such quantifiers over finite domains, according to which the scheme holds iff Q₁ is ∃ or Q₂ is ∀ (excluding trivial cases). Our result shows that in infinite domains, there are more general types of quantifiers that support these entailments.