+In fact, unlike in the simply-typed lambda calculus,
+it is even possible to give a type for ω in System F.
+
+<code>ω = λx:(∀α.α->α). x [∀α.α->α] x</code>
+
+In order to see how this works, we'll apply ω to the identity
+function.
+
+<code>ω id ==</code>
+
+ (λx:(∀α.α->α). x [∀α.α->α] x) (Λα.λx:α.x)
+
+Since the type of the identity function is `∀α.α->α`, it's the
+right type to serve as the argument to ω. The definition of
+ω instantiates the identity function by binding the type
+variable `α` to the universal type `∀α.α->α`. Instantiating the
+identity function in this way results in an identity function whose
+type is (in some sense, only accidentally) the same as the original
+fully polymorphic identity function.
+
+So in System F, unlike in the simply-typed lambda calculus, it *is*
+possible for a function to apply to itself!
+
+Does this mean that we can implement recursion in System F? Not at
+all. In fact, despite its differences with the simply-typed lambda
+calculus, one important property that System F shares with the
+simply-typed lambda calculus is that they are both strongly
+normalizing: *every* expression in either system reduces to a normal
+form in a finite number of steps.
+
+Not only does a fixed-point combinator remain out of reach, we can't
+even construct an infinite loop. This means that although we found a
+type for ω, there is no general type for Ω ≡ ω
+ω. Furthermore, it turns out that no Turing complete system can
+be strongly normalizing, from which it follows that System F is not
+Turing complete.
+
+
+## Polymorphism in natural language
+
+Is the simply-typed lambda calclus enough for analyzing natural
+language, or do we need polymorphic types? Or something even more expressive?
+
+The classic case study motivating polymorphism in natural language
+comes from coordination. (The locus classicus is Partee and Rooth
+1983.)
+
+ Ann left and Bill left.
+ Ann left and slept.
+ Ann and Bill left.
+ Ann read and reviewed the book.
+
+In English (likewise, many other languages), *and* can coordinate
+clauses, verb phrases, determiner phrases, transitive verbs, and many
+other phrase types. In a garden-variety simply-typed grammar, each
+kind of conjunct has a different semantic type, and so we would need
+an independent rule for each one. Yet there is a strong intuition
+that the contribution of *and* remains constant across all of these
+uses. Can we capture this using polymorphic types?
+
+ Ann, Bill e
+ left, slept e -> t
+ read, reviewed e -> e -> t
+
+With these basic types, we want to say something like this:
+
+ and:t->t->t = λl:t. λr:t. l r false
+ and = Λα.Λβ.λl:α->β.λr:α->β.λx:α. and [β] (l x) (r x)
+
+The idea is that the basic *and* conjoins expressions of type `t`, and
+when *and* conjoins functional types, it builds a function that
+distributes its argument across the two conjuncts and conjoins the two
+results. So `Ann left and slept` will evaluate to `(\x.and(left
+x)(slept x)) ann`. Following the terminology of Partee and Rooth, the
+strategy of defining the coordination of expressions with complex
+types in terms of the coordination of expressions with less complex
+types is known as Generalized Coordination.
+
+But the definitions just given are not well-formed expressions in
+System F. There are three problems. The first is that we have two
+definitions of the same word. The intention is for one of the
+definitions to be operative when the type of its arguments is type
+`t`, but we have no way of conditioning evaluation on the *type* of an
+argument. The second is that for the polymorphic definition, the term
+*and* occurs inside of the definition. System F does not have
+recursion.
+
+The third problem is more subtle. The defintion as given takes two
+types as parameters: the type of the first argument expected by each
+conjunct, and the type of the result of applying each conjunct to an
+argument of that type. We would like to instantiate the recursive use
+of *and* in the definition by using the result type. But fully
+instantiating the definition as given requires type application to a
+pair of types, not to just a single type. We want to somehow
+guarantee that β will always itself be a complex type.
+
+So conjunction and disjunction provide a compelling motivation for
+polymorphism in natural language, but we don't yet have the ability to
+build the polymorphism into a formal system.
+
+And in fact, discussions of generalized coordination in the
+linguistics literature are almost always left as a meta-level
+generalizations over a basic simply-typed grammar. For instance, in
+Hendriks' 1992:74 dissertation, generalized coordination is
+implemented as a method for generating a suitable set of translation
+rules, which are in turn expressed in a simply-typed grammar.
+
+Not incidentally, we're not aware of any programming language that
+makes generalized coordination available, despite is naturalness and
+ubiquity in natural language. That is, coordination in programming
+languages is always at the sentential level. You might be able to
+evaluate `(delete file1) and (delete file2)`, but never `delete (file1
+and file2)`.
+
+We'll return to thinking about generalized coordination as we get
+deeper into types. There will be an analysis in term of continuations
+that will be particularly satisfying.
+
+
+#Types in OCaml
+
+