What are type quantifiers?

Question

Many statically typed languages have parametric polymorphism. For example in C# one can define:

T Foo(T x){ return x; }

In a call site

Answer 1

if forall is lambda ..., then what is exists

Why, tuple of course!

In Martin-Löf type theory you have Π types, corresponding to functions/universal quantification and Σ-types, corresponding to tuples/existential quantification.

Their types are very similar to what you have proposed (I am using Agda notation here):

Π : (A : Set) -> (A -> Set) -> Set
Σ : (A : Set) -> (A -> Set) -> Set

Indeed, Π is an infinite product and Σ is infinite sum. Note that they are not "intersection" and "union" though, as you proposed because you can't do that without additionally defining where the types intersect. (which values of one type correspond to which values of the other type)

From these two type constructors you can have all of normal, polymorphic and dependent functions, normal and dependent tuples, as well as existentially and universally-quantified statements:

-- Normal function, corresponding to "Integer -> Integer" in Haskell
factorial : Π ℕ (λ _ → ℕ)

-- Polymorphic function corresponding to "forall a . a -> a"
id : Π Set (λ A -> Π A (λ _ → A))

-- A universally-quantified logical statement: all natural numbers n are equal to themselves
refl : Π ℕ (λ n → n ≡ n)


-- (Integer, Integer)
twoNats : Σ ℕ (λ _ → ℕ)

-- exists a. Show a => a
someShowable : Σ Set (λ A → Σ A (λ _ → Showable A))

-- There are prime numbers
aPrime : Σ ℕ IsPrime

However, this does not address parametricity at all and AFAIK parametricity and Martin-Löf type theory are independent.

For parametricity, people usually refer to the Philip Wadler's work.