`Λ_T`

,
which is the smallest set such that
* each type `t` has an infinite set of distinct variables, {x^t}_1,
{x^t}_2, {x^t}_3, ...
* If a term `M` has type σ --> τ, and a term `N` has type
σ, then the application `(M N)` has type τ.
* If a variable `a` has type σ, and term `M` has type τ,
then the abstract `λ a M`

has type σ --> τ.
The definitions of types and of typed terms should be highly familiar
to semanticists, except that instead of writing σ --> τ,
linguists (following Montague, who followed Church) write <σ,
τ>. We will use the arrow notation, since it is more iconic.
Some examples (assume that `x` has type `o`):
x o
\x.x o --> o
((\x.x) x) o
Excercise: write down terms that have the following types:
o --> o --> o
(o --> o) --> o --> o
(o --> o --> o) --> o
#Associativity of types versus terms#
As we have seen many times, in the lambda calculus, function
application is left associative, so that `f x y z == (((f x) y) z)`.
Types, *THEREFORE*, are right associative: if `f`, `x`, `y`, and `z`
have types `a`, `b`, `c`, and `d`, respectively, then `f` has type `a
--> b --> c --> d == (a --> (b --> (c --> d)))`.
It is a serious faux pas to associate to the left for types. You may
as well use your salad fork to stir your tea.
#The simply-typed lambda calculus is strongly normalizing#
If `M` is a term with type τ in Λ_T, then `M` has a
normal form. The proof is not particularly complex, but we will not
present it here; see Berendregt or Hankin.
Since Ω does not have a normal form, it follows that Ω
cannot have a type in Λ_T. We can easily see why:
Ω = (\x.xx)(\x.xx)
Assume Ω has type τ, and `\x.xx` has type σ. Then
because `\x.xx` takes an argument of type σ and returns
something of type τ, `\x.xx` must also have type σ -->
τ. By repeating this reasoning, `\x.xx` must also have type
(σ --> τ) --> τ; and so on. Since variables have
finite types, there is no way to choose a type for the variable `x`
that can satisfy all of the requirements imposed on it.
In general, there is no way for a function to have a type that can
take itself for an argument. It follows that there is no way to
define the identity function in such a way that it can take itself as
an argument. Instead, there must be many different identity
functions, one for each type.
#Typing numerals#
Version 1 type numerals are not a good choice for the simply-typed
lambda calculus. The reason is that each different numberal has a
different type! For instance, if zero has type σ, then `false`
has type τ --> τ --> τ, for some τ. Since one is
represented by the function `\x.x false 0`, one must have type (τ
--> τ --> τ) --> σ --> σ. But this is a different
type than zero! Because each number has a different type, it becomes
impossible to write arithmetic operations that can combine zero with
one. We would need as many different addition operations as we had
pairs of numbers that we wanted to add.
Fortunately, the Church numberals are well behaved with respect to
types. They can all be given the type (σ --> σ) -->
σ --> σ.