System F:
---------
- types τ ::= c | 'a | τ1 -> τ2 | ∀'a. τ
- expressions e ::= x | λx:τ. e | e1 e2 | Λ'a. e | e [τ]
+ types τ ::= c | α | τ1 -> τ2 | ∀'a. τ
+ expressions e ::= x | λx:τ. e | e1 e2 | Λα. e | e [τ]
In the definition of the types, "`c`" is a type constant. Type
constants play the role in System F that base types play in the
simply-typed lambda calculus. So in a lingusitics context, type
-constants might include `e` and `t`. "`'a`" is a type variable. The
+constants might include `e` and `t`. "α" is a type variable. The
tick mark just indicates that the variable ranges over types rather
than over values; in various discussion below and later, type variable
can be distinguished by using letters from the greek alphabet
(α, β, etc.), or by using capital roman letters (X, Y,
etc.). "`τ1 -> τ2`" is the type of a function from expressions of
-type `τ1` to expressions of type `τ2`. And "`∀'a. τ`" is called a
+type `τ1` to expressions of type `τ2`. And "`∀α. τ`" is called a
universal type, since it universally quantifies over the type variable
-`'a`. You can expect that in `∀'a. τ`, the type `τ` will usually
-have at least one free occurrence of `'a` somewhere inside of it.
+`'a`. You can expect that in `∀α. τ`, the type `τ` will usually
+have at least one free occurrence of `α` somewhere inside of it.
In the definition of the expressions, we have variables "`x`" as usual.
Abstracts "`λx:τ. e`" are similar to abstracts in the simply-typed lambda
type. Applications "`e1 e2`" are just like in the simply-typed lambda calculus.
In addition to variables, abstracts, and applications, we have two
-additional ways of forming expressions: "`Λ'a. e`" is called a *type
+additional ways of forming expressions: "`Λα. e`" is called a *type
abstraction*, and "`e [τ]`" is called a *type application*. The idea
is that <code>Λ</code> is a capital <code>λ</code>: just
like the lower-case <code>λ</code>, <code>Λ</code> binds
<code>Λ α (λ x:α . x)</code>
-the <code>Λ</code> binds the type variable `'a` that occurs in
+the <code>Λ</code> binds the type variable `α` that occurs in
the <code>λ</code> abstract. Of course, as long as type
variables are carefully distinguished from expression variables (by
tick marks, Grecification, or capitalization), there is no need to
ready to apply this identity function to, say, a variable of type
boolean, just do this:
-<code>(Λ 'a (λ x:'a . x)) [t]</code>
+<code>(Λ α (λ x:α . x)) [t]</code>
This type application (where `t` is a type constant for Boolean truth
-values) specifies the value of the type variable `'a`. Not
+values) specifies the value of the type variable `α`. Not
surprisingly, the type of this type application is a function from
Booleans to Booleans:
-<code>((Λ 'a (λ x:'a . x)) [t]): (b -> b)</code>
+<code>((Λ α (λ x:α . x)) [t]): (b -> b)</code>
Likewise, if we had instantiated the type variable as an entity (base
type `e`), the resulting identity function would have been a function
of type `e -> e`:
-<code>((Λ 'a (λ x:'a . x)) [e]): (e -> e)</code>
+<code>((Λ α (λ x:α . x)) [e]): (e -> e)</code>
-Clearly, for any choice of a type `'a`, the identity function can be
-instantiated as a function from expresions of type `'a` to expressions
-of type `'a`. In general, then, the type of the uninstantiated
+Clearly, for any choice of a type `α`, the identity function can be
+instantiated as a function from expresions of type `α` to expressions
+of type `α`. In general, then, the type of the uninstantiated
(polymorphic) identity function is
-<code>(Λ 'a (λ x:'a . x)): (∀ 'a . 'a -> 'a)</code>
+<code>(Λ α (λ x:α . x)): (∀ α . α -> α)</code>
Pred in System F
----------------