+## 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* (the one defined in the first line)
+conjoins expressions of type `t`. But when *and* conjoins functional
+types (the definition in the second line), it builds a function that
+distributes its argument across the two conjuncts and conjoins the two
+results. The intention is that `Ann left and slept` will evaluate to
+`(\x.and(left x)(slept x)) ann`. Following the terminology of Partee
+and Rooth, this 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. We know how to handle some
+cases of using a function name inside of its own definition in the
+untyped lambda calculus, but System F does not have
+recursion. [Exercise: convince yourself that the fixed-point
+combinator `Y` can't be typed in System F.]
+
+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, so that
+"<code>and [β]</code>" evaluates to the kind of *and* that
+coordinates expressions of type β. But fully instantiating the
+definition as given requires type application to a *pair* of types,
+not to just to a single type. We want to somehow guarantee that β
+will always itself be a complex type. This goes beyond the expressive
+power of System F.
+
+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.
+
+There is some work using System F to express generalizations about
+natural language: Ponvert, Elias. 2005. Polymorphism in English Logical
+Grammar. In *Lambda Calculus Type Theory and Natural Language*: 47--60.
+
+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
+