I was reading through Type Theory and Formal Proofs (recommended by @parsonsmatt), and there was some info about how to represent "exists" (Ǝ).
In the type-theory version of classical logic, "exists" can represented as a higher-order function using "forall" like the following:
Ǝ x : S. P x is defined as ∀ A : *. (∀ x : S. P x -> A) -> A