determine which expressions can combine with which other expressions.
If a word is a member of the category of prepositions, it had better
not try to combine (merge) with an expression in the category of, say,
-an auxilliary verb, since *under has* is not a well-formed constituent
+an auxilliary verb, since \**under has* is not a well-formed constituent
in English. Likewise, types in formal languages will determine which
expressions can be sensibly combined.
not correspond to any salient syntactic distinctions (as in any
analysis that involves silent type-shifters, such as Herman Hendriks'
theory of quantifier scope, in which expressions change their semantic
-type without any effect on the syntactic expressions they can combine
+type without any effect on the expressions they can combine
with syntactically). We will consider again the relationship between
syntactic types and semantic types later in the course.
convention. This is ok, however, because he also reverses the order,
so that `te` is a function from objects of type `e` to objects of type
`t`. Cool paper! If you ever want to see Church numerals in their
-native setting--but I'm getting ahead of my story. Pedantic off.]
+native setting--but we're getting ahead of our story. Pedantic off.]
There's good news and bad news: the good news is that the simply-typed
lambda calculus is strongly normalizing: every term has a normal form.
(o -> o) -> o -> o
(o -> o -> o) -> o
+#A first glipse of the connection between types and logic
+
+In the simply-typed lambda calculus, we write types like <code>σ
+-> τ</code>. This looks like logical implication. We'll take
+that resemblance seriously when we discuss the Curry-Howard
+correspondence. In the meantime, note that types respect modus
+ponens:
+
+<pre>
+Expression Type Implication
+-----------------------------------
+fn α -> β α ⊃ β
+arg α α
+------ ------ --------
+(fn arg) β β
+</pre>
+
+The implication in the right-hand column is modus ponens, of course.
+
+
#Associativity of types versus terms#
As we have seen many times, in the lambda calculus, function
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. Some of those types can be functions,
-and some of those functions can be (type-restricted) identity
-functions; but a simply-types identity function can never apply to itself.
+In fact, we can't even type the parts of Ω, that is, `ω
+\equiv \x.xx`. 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. Some of
+those types can be functions, and some of those functions can be
+(type-restricted) identity functions; but a simply-types identity
+function can never apply to itself.
#Typing numerals#
## Predecessor and lists are not representable in simply typed lambda-calculus ##
-As Oleg Kiselyov points out, [[predecessor and lists can't be
-represented in the simply-typed lambda
-calculus|http://okmij.org/ftp/Computation/lambda-calc.html#predecessor]].
+
This is not because there is any difficulty typing what the functions
involved do "from the outside": for instance, the predecessor function
is a function from numbers to numbers, or τ -> τ, where τ
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 fst = \x y. x in
+ let snd = \x y. y in
+ let pair = \x y . \f . f x y 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. pair (succ (p fst)) (p fst) in
+ let pred = \n. n shift (pair zero zero) snd in
+
+Note that `shift` takes a pair `p` as argument, but makes use of only
+the first element of the pair. 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 3
+ 3 shift (pair zero zero) snd
+ (\s z.s(s(s z))) shift (pair zero zero) snd
+ shift (shift (shift (\f.f 0 0))) snd
+ shift (shift (pair (succ ((\f.f 0 0) fst)) ((\f.f 0 0) fst))) snd
+ shift (shift (\f.f 1 0)) snd
+ shift (\f. f 2 1) snd
+ (\f. f 3 2) snd
+ snd 3 2
+ 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
the predecessor function in the lambda calculus. Could one of them
possibly be simply-typeable? It turns out that this can't be done.
-See the works cited by Oleg for details.
+See Oleg Kiselyov's discussion and works cited there for details:
+[[predecessor and lists can't be represented in the simply-typed
+lambda
+calculus|http://okmij.org/ftp/Computation/lambda-calc.html#predecessor]].
+
+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,
+Montague's 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, which match the rules
+for the simply-typed lambda calculus exactly:
+
+* 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.