Rather, the problem is that the definition of the function requires
subterms that can't be simply-typed. We'll illustrate with our
-implementation of the predecessor, sightly modified in inessential
-ways to suit present purposes:
+implementation of the predecessor function, based on the discussion in
+Pierce 2002:547:
let zero = \s z. z in
let snd = \a b. b in
let pair = \a b. \v. v a b in
let succ = \n s z. s (n s z) in
- let collect = \p. p (\a b. pair (succ a) a)
- let pred = \n. n collect (pair zero zero) snd in
+ let shift = \p. p (\a b. pair (succ a) a)
+ let pred = \n. n shift (pair zero zero) snd in
+
+Note that `shift` applies its argument p ("p" for "pair") to a
+function that ignores its second argument---why does it do that? In
+order to understand what this code is doing, it is helpful to go
+through a sample computation, the predecessor of 3:
+
+ pred (\s z.s(s(s z)))
+ (\s z.s(s(s z))) (\n.n shift (\f.f 0 0) snd)
+ shift (shift (shift (\f.f 0 0))) snd
+ shift (shift ((\f.f 0 0) (\a b.pair(succ a) a))) snd
+ shift (shift (\f.f 1 0)) snd
+ shift (\f. f 2 1) snd
+ (\f. f 3 2) snd
+ 2
+
+At each stage, `shift` sees an ordered pair that contains two numbers
+related by the successor function. It can safely discard the second
+element without losing any information. The reason we carry around
+the second element at all is that when it comes time to complete the
+computation---that is, when we finally apply the top-level ordered
+pair to `snd`---it's the second element of the pair that will serve as
+the final result.
Let's see how far we can get typing these terms. `zero` is the Church
encoding of zero. Using `N` as the type for Church numbers (i.e.,
-<code>N == (σ -> σ) -> σ -> σ</code> for some
-σ, `zero` has type `N`. `snd` takes two numbers, and returns
-the second, so `snd` has type `N -> N -> N`. Then the type of `pair`
-is `N -> N -> (type(snd)) -> N`, that is, `N -> N -> (N -> N -> N) ->
-N`. Likewise, `succ` has type `N -> N`, and `collect` has type `pair
--> pair`, where `pair` is the type of an ordered pair of numbers,
-namely, <code>pair ≡ (N -> N -> N) -> N</code>. So far so good.
+<code>N ≡ (σ -> σ) -> σ -> σ</code> for
+some σ, `zero` has type `N`. `snd` takes two numbers, and
+returns the second, so `snd` has type `N -> N -> N`. Then the type of
+`pair` is `N -> N -> (type(snd)) -> N`, that is, `N -> N -> (N -> N ->
+N) -> N`. Likewise, `succ` has type `N -> N`, and `shift` has type
+`pair -> pair`, where `pair` is the type of an ordered pair of
+numbers, namely, <code>pair ≡ (N -> N -> N) -> N</code>. So far
+so good.
The problem is the way in which `pred` puts these parts together. In
particular, `pred` applies its argument, the number `n`, to the
-`collect` function. Since `n` is a number, its type is `(σ ->
-σ) -> σ -> σ`. This means that the type of
-`collect` has to match `σ -> σ`. But we concluded above
-that the type of `collect` also had to be `pair -> pair`. Putting
-these constraints together, it appears that `σ` must be the type
-of a pair of numbers. But we already decided that the type of a pair
-of numbers is `(N -> N -> N) -> N`. Here's the difficulty: `N` is
-shorthand for a type involving `σ`. If `σ` turns out to
-depend on `N`, and `N` depends in turn on `σ`, then `σ` is a proper
-subtype of itself, which is not allowed in the simply-typed lambda
-calculus.
-
-The way we got here is that the pred function relies on the right-fold
-structure of the Church numbers to recursively walk down the spine of
-its argument. In order to do that, the argument number had to take
-the operation in question as its first argument. And the operation
-required in order to build up the predecessor must be the sort of
-operation that manipulates numbers, and the infinite regress is
+`shift` function. Since `n` is a number, its type is <code>(σ
+-> σ) -> σ -> σ</code>. This means that the type of
+`shift` has to match <code>σ -> σ</code>. But we
+concluded above that the type of `shift` also had to be `pair ->
+pair`. Putting these constraints together, it appears that
+<code>σ</code> must be the type of a pair of numbers. But we
+already decided that the type of a pair of numbers is `(N -> N -> N)
+-> N`. Here's the difficulty: `N` is shorthand for a type involving
+<code>σ</code>. If <code>σ</code> turns out to depend on
+`N`, and `N` depends in turn on <code>σ</code>, then
+<code>σ</code> is a proper subtype of itself, which is not
+allowed in the simply-typed lambda calculus.
+
+The way we got here is that the `pred` function relies on the built-in
+right-fold structure of the Church numbers to recursively walk down
+the spine of its argument. In order to do that, the argument had to
+apply to the `shift` operation. And since `shift` had to be the
+sort of operation that manipulates numbers, the infinite regress is
established.
Now, of course, this is only one of myriad possible implementations of
possibly be simply-typeable? It turns out that this can't be done.
See the works cited by Oleg for details.
+Because lists are (in effect) a generalization of the Church numbers,
+computing the tail of a list is likewise beyond the reach of the
+simply-typed lambda calculus.
+
+This result is not obvious, to say the least. It illustrates how
+recursion is built into the structure of the Church numbers (and
+lists). Most importantly for the discussion of the simply-typed
+lambda calculus, it demonstrates that even fairly basic recursive
+computations are beyond the reach of a simply-typed system.
+
+
+## Montague grammar is based on a simply-typed lambda calculus
+
+Systems based on the simply-typed lambda calculus are the bread and
+butter of current linguistic semantic analysis. One of the most
+influential modern semantic formalisms---Montague's PTQ
+fragment---included a simply-typed version of the Predicate Calculus
+with lambda abstraction.
+
+Montague called the semantic part of his PTQ fragment *Intensional
+Logic*. Without getting too fussy about details, we'll present the
+popular Ty2 version of the PTQ types, roughly as proposed by Gallin
+(1975). [See Zimmermann, Ede. 1989. Intensional logic and two-sorted
+type theory. *Journal of Symbolic Logic* ***54.1***: 65--77 for a
+precise characterization of the correspondence between IL and
+two-sorted Ty2.]
+
+We'll need three base types: `e`, for individuals, `t`, for truth
+values, and `s` for evaluation indicies (world-time pairs). The set
+of types is defined recursively:
+
+ the base types e, t, and s are types
+ if a and b are types, <a,b> is a type
+
+So `<e,<e,t>>` and `<s,<<s,e>,t>>` are types. As we have mentioned,
+this paper is the source for the convention in linguistics that a type
+of the form `<a, b>` corresponds to a functional type that we will
+write here as `a -> b`. So the type `<a,b>` is the type of a function
+that maps objects of type `a` onto objects of type `b`.
+
+Montague gave rules for the types of various logical formulas. Of
+particular interest here, he gave the following typing rules for
+functional application and for lambda abstracts:
+
+* If *α* is an expression of type *<a, b>*, and *β* is an
+expression of type b, then *α(β)* has type *b*.
+
+* If *α* is an expression of type *a*, and *u* is a variable of type *b*, then *λuα* has type <code><b, a></code>.
+
+When we talk about monads, we will consider Montague's treatment of
+intensionality in some detail. In the meantime, Montague's PTQ is
+responsible for making the simply-typed lambda calculus the baseline
+semantic analysis for linguistics.