-<!-- λ Λ ∀ ≡ α β γ ρ ω Ω ○ μ η δ ζ ξ ⋆ ★ • ∙ ● 𝟎 𝟏 𝟐 𝟘 𝟙 𝟚 𝟬 𝟭 𝟮 ¢ ⇧ -->
+<!-- λ Λ ∀ ≡ α β γ ρ ω Ω ○ μ η δ ζ ξ ⋆ ★ • ∙ ● ⚫ 𝟎 𝟏 𝟐 𝟘 𝟙 𝟚 𝟬 𝟭 𝟮 ⇧ (U+2e17) ¢ -->
-[[!toc levels=2]]
-The Reader Monad
-================
+[[!toc levels=2]]
-The goal for this part is to introduce the Reader Monad, and present
-two linguistics applications: binding and intensionality. Along the
+The goal for these notes is to introduce the Reader Monad, and present
+two linguistic applications: binding and intensionality. Along the
way, we'll continue to think through issues related to order, and a
related notion of flow of information.
At this point, we've seen monads in general, and three examples of
-monads: the identity monad (invisible boxes), the Maybe monad (option
-types), and the List monad.
+monads: the Identity monad (invisible boxes), the Option/Maybe monad (option
+types), and the List monad.
-We've also seen an application of the Maybe monad to safe division.
-The starting point was to allow the division function to return an int
-option instead of an int. If we divide 6 by 2, we get the answer Just
-3. But if we divide 6 by 0, we get the answer Nothing.
+We've also seen an application of the Option/Maybe monad to safe division.
+The starting point was to allow the division function to return an `int
+option` instead of an `int`. If we divide `6` by `2`, we get the answer `Some
+3`. But if we divide `6` by `0`, we get the answer `None`.
The next step was to adjust the other arithmetic functions to teach
-them what to do if they received Nothing instead of a boxed integer.
-This meant changing the type of their input from ints to int options.
+them what to do if they received `None` instead of a boxed integer.
+This meant changing the type of their input from `int`s to `int option`s.
But we didn't need to do this piecemeal; rather, we "lift"ed the
ordinary arithmetic operations into the monad using the various tools
-provided by the monad.
+provided by the monad.
We'll go over this lifting operation in detail in the next section.
-## Tracing the effect of safe-div on a larger computation
+## Tracing the effect of `safe_div` on a larger computation
So let's see how this works in terms of a specific computation.
/ 6
</pre>
-No matter which arithmetic operation we begin with, this computation
-should eventually reduce to 13. Given a specific reduction strategy,
-we can watch the order in which the computation proceeds. Following
-on the lambda evaluator developed during the previous homework, let's
+No matter what order we evaluate it in, this computation
+should eventually reduce to `13`. Given a specific reduction strategy,
+we can watch the order in which the computation proceeds. Following
+on the evaluator developed during the previous homework, let's
adopt the following reduction strategy:
- In order to reduce an expression of the form (head arg), do the following in order:
- 1. Reduce head to h'
- 2. Reduce arg to a'.
- 3. If (h' a') is a redex, reduce it.
+> In order to reduce an expression of the form (head arg), do the following in order:
+
+> 1. Reduce `head` to `head'`
+> 2. Reduce `arg` to `arg'`.
+> 3. If `(head' arg')` is a redex, reduce it.
There are many details left unspecified here, but this will be enough
-for today. The order in which the computation unfolds will be
-
- 1. Reduce head (+ 1) to itself
- 2. Reduce arg ((* ((/ 6) 2)) 3)
- 1. Reduce head (* ((/ 6) 2))
- 1. Reduce head *
- 2. Reduce arg ((/ 6) 2)
- 1. Reduce head (/ 6) to itself
- 2. Reduce arg 2 to itself
- 3. Reduce ((/ 6) 2) to 3
- 3. Reduce (* 3) to itself
- 2. Reduce arg 4 to itself
- 3. Reduce ((* 3) 4) to 12
- 3. Reduce ((+ 1) 12) to 13
+for today. The order in which the computation unfolds will be
+
+> 1. Reduce head `(+ 1)` to itself
+> 2. Reduce arg `((* ((/ 6) 2)) 4)`
+> 1. Reduce head `(* ((/ 6) 2))`
+> 1. Reduce head `*` to itself
+> 2. Reduce arg `((/ 6) 2)`
+> 1. Reduce head `(/ 6)` to itself
+> 2. Reduce arg `2` to itself
+> 3. Reduce `((/ 6) 2)` to `3`
+> 3. Reduce `(* 3)` to itself
+> 2. Reduce arg `4` to itself
+> 3. Reduce `((* 3) 4)` to `12`
+> 3. Reduce `((+ 1) 12)` to `13`
This reduction pattern follows the structure of the original
expression exactly, at each node moving first to the left branch,
processing the left branch, then moving to the right branch, and
-finally processing the results of the two subcomputation. (This is
-called depth-first postorder traversal of the tree.)
+finally processing the results of the two subcomputation. (This is
+called a depth-first postorder traversal of the tree.)
[diagram with arrows traversing the tree]
type tree = Leaf of contents | Branch of tree * tree
Never mind that these types will allow us to construct silly
-arithmetric trees such as `+ *` or `2 3`. Note that during the
+arithmetric trees such as `+ *` or `2 3`. Note that during the
reduction sequence, the result of reduction was in every case a
-well-formed subtree. So the process of reduction could be animated by
-replacing subtrees with the result of reduction on that subtree, till
-the entire tree is replaced by a single integer (namely, 13).
+well-formed subtree. So the process of reduction could be animated by
+replacing subtrees with the result of reduction on that subtree, until
+the entire tree is replaced by a single integer (namely, `13`).
-Now we replace the number 2 with 0:
+Now we replace the number `2` with `0`:
<!--
\tree ((((+) (1)) (((*) (((/) (6)) (0))) (4))))
When we reduce, we get quite a ways into the computation before things
break down:
- 1. Reduce head (+ 1) to itself
- 2. Reduce arg ((* ((/ 6) 0)) 3)
- 1. Reduce head (* ((/ 6) 0))
- 1. Reduce head *
- 2. Reduce arg ((/ 6) 0)
- 1. Reduce head (/ 6) to itself
- 2. Reduce arg 0 to itself
- 3. Reduce ((/ 6) 0) to ACKKKK
+> 1. Reduce head `(+ 1)` to itself
+> 2. Reduce arg `((* ((/ 6) 0)) 4)`
+> 1. Reduce head `(* ((/ 6) 0))`
+> 1. Reduce head `*` to itself
+> 2. Reduce arg `((/ 6) 0)`
+> 1. Reduce head `(/ 6)` to itself
+> 2. Reduce arg `0` to itself
+> 3. Reduce `((/ 6) 0)` to ACKKKK
-This is where we replace `/` with `safe-div`.
-Safe-div returns not an int, but an int option. If the division goes
-through, the result is Just n, where n is the integer result.
-But if the division attempts to divide by zero, the result is Nothing.
+This is where we replace `/` with `safe_div`. `safe_div` returns not an `int`, but an `int option`. If the division goes
+through, the result is `Some n`, where `n` is the integer result.
+But if the division attempts to divide by zero, the result is `None`.
We could try changing the type of the arithmetic operators from `int
-> int -> int` to `int -> int -> int option`; but since we now have to
-anticipate the possibility that any argument might involve division by
+anticipate the possibility that *any* argument might involve division by
zero inside of it, it would be better to prepare for the possibility
-that any subcomputation might return here is the net result for our
-types. The easy way to do that is to change (only) the type of num
-from int to int option, leaving everying else the same:
+that any subcomputation might return `None` here. So our operators should have
+the type `int option -> int option -> int option`. Let's bring that about
+by just changing the type `num` from `int` to `int option`, leaving everying else the same:
type num = int option
type contents = Num of num | Op2 of (num -> num -> num)
type tree = Leaf of contents | Branch of tree * tree
The only difference is that instead of defining our numbers to be
-simple integers, they are now int options; and so Op is an operator
-over int options.
+simple integers, they are now `int option`s; and so `Op` is an operator
+over `int option`s.
-At this point, we bring in the monadic machinery. In particular, here
-is the ⇧ and the map2 function from the notes on safe division:
+At this point, we bring in the monadic machinery. In particular, here
+is the `⇧` and the `map2` function from the notes on safe division:
- ⇧ (a: 'a) = Just a;;
+ ⇧ (x : 'a) = Some x
- map2 (g : 'a -> 'b -> 'c) (u : 'a option) (v : 'b option) =
- match u with
+ map2 (f : 'a -> 'b -> 'c) (xx : 'a option) (yy : 'b option) =
+ match xx with
| None -> None
| Some x ->
- (match v with
+ (match yy with
| None -> None
- | Some y -> Some (g x y));;
+ | Some y -> Some (f x y))
+
+Then we lift the entire computation into the monad by applying `⇧` to
+the integers, and by applying `map2` to the operators. Only, we will replace `/` with `safe_div`, defined as follows:
-Then we lift the entire computation into the monad by applying ⇧ to
-the integers, and by applying `map2` to the operators:
+ safe_div (xx : 'a option) (yy : 'b option) =
+ match xx with
+ | None -> None
+ | Some x ->
+ (match yy with
+ | None -> None
+ | Some 0 -> None
+ | Some y -> Some ((/) x y))
<!--
\tree ((((map2 +) (⇧1)) (((map2 *) (((map2 /) (⇧6)) (⇧0))) (⇧4))))
| |
___|____ ⇧0
| |
- map2 / ⇧6
+ safe_div ⇧6
</pre>
With these adjustments, the faulty computation now completes smoothly:
- 1. Reduce head ((map2 +) ⇧1)
- 2. Reduce arg (((map2 *) (((map2 /) ⇧6) ⇧2)) ⇧3)
- 1. Reduce head ((map2 *) (((map2 /) ⇧6) ⇧2))
- 1. Reduce head *
- 2. Reduce arg (((map2 /) ⇧6) ⇧0)
- 1. Reduce head ((map2 /) ⇧6)
- 2. Reduce arg ⇧0
- 3. Reduce (((map2 /) ⇧6) ⇧0) to Nothing
- 3. Reduce ((map2 *) Nothing) to Nothing
- 2. Reduce arg ⇧4
- 3. Reduce (((map2 *) Nothing) ⇧4) to Nothing
- 3. Reduce (((map2 +) ⇧1) Nothing) to Nothing
-
-As soon as we try to divide by 0, safe-div returns Nothing.
-Thanks to the details of map2, the fact that Nothing has been returned
-by one of the arguments of a map2-ed operator guarantees that the
-map2-ed operator will pass on the Nothing as its result. So the
-result of each enclosing computation will be Nothing, up to the root
+> 1. Reduce head `((map2 +) ⇧1)`
+> 2. Reduce arg `(((map2 *) ((safe_div ⇧6) ⇧0)) ⇧4)`
+> 1. Reduce head `((map2 *) ((safe_div ⇧6) ⇧0))`
+> 1. Reduce head `(map2 *)`
+> 2. Reduce arg `((safe_div ⇧6) ⇧0)`
+> 1. Reduce head `(safe_div ⇧6)`
+> 2. Reduce arg `⇧0`
+> 3. Reduce `((safe_div ⇧6) ⇧0)` to `None`
+> 3. Reduce `((map2 *) None)`
+> 2. Reduce arg `⇧4`
+> 3. Reduce `(((map2 *) None) ⇧4)` to `None`
+> 3. Reduce `(((map2 +) ⇧1) None)` to `None`
+
+As soon as we try to divide by `0`, `safe_div` returns `None`.
+Thanks to the details of `map2`, the fact that `None` has been returned
+by one of the arguments of a `map2`-ed operator guarantees that the
+`map2`-ed operator will pass on the `None` as its result. So the
+result of each enclosing computation will be `None`, up to the root
of the tree.
It is unfortunate that we need to continue the computation after
-encountering our first Nothing. We know immediately at the result of
-the entire computation will be Nothing, yet we continue to compute
-subresults and combinations. It would be more efficient to simply
-jump to the top as soon as Nothing is encoutered. Let's call that
-strategy Abort. We'll arrive at an Abort operator later in the semester.
+encountering our first `None`. We know immediately at the result of
+the entire computation will be `None`, yet we continue to compute
+subresults and combinations. It would be more efficient to simply
+jump to the top as soon as `None` is encoutered. Let's call that
+strategy Abort. We'll arrive at an `Abort` operator later in the semester.
-So at this point, we can see how the Maybe/option monad provides
+So at this point, we can see how the Option/Maybe monad provides
plumbing that allows subcomputations to send information from one part
-of the computation to another. In this case, the safe-div function
+of the computation to another. In this case, the `safe_div` function
can send the information that division by zero has been attempted
-throughout the rest of the computation. If you think of the plumbing
+upwards to the rest of the computation. If you think of the plumbing
as threaded through the tree in depth-first, postorder traversal, then
-safe-div drops Nothing into the plumbing half way through the
-computation, and that Nothing travels through the rest of the plumbing
+`safe_div` drops `None` into the plumbing half way through the
+computation, and that `None` travels through the rest of the plumbing
till it comes out of the result faucet at the top of the tree.
+
## Information flowing in the other direction: top to bottom
We can think of this application as facilitating information flow.
-In the save-div example, a subcomputation created a message that
+In the `safe_div` example, a subcomputation created a message that
propagated upwards to the larger computation:
<pre>
/ 6
</pre>
+(The message was implemented by `None`.)
+
We might want to reverse the direction of information flow, making
information available at the top of the computation available to the
subcomputations:
<pre>
- [λx]
+ [λn]
___________ |
| | |
_|__ ___|___ |
| | |
* __|___ |
| | |
- _|__ x <----|
+ _|__ n <----|
| |
/ 6
</pre>
We've seen exactly this sort of configuration before: it's exactly
what we have when a lambda binds a variable that occurs in a deeply
-embedded position. Whatever the value of the argument that the lambda
+embedded position. Whatever the value of the argument that the lambda
form combines with, that is what will be substituted in for free
occurrences of that variable within the body of the lambda.
+
## Binding
So our next step is to add a (primitive) version of binding to our
-computation. We'll allow for just one binding dependency for now, and
+computation. We'll allow for just one binding dependency for now, and
then generalize later.
Binding is independent of the safe division, so we'll return to a
-situation in which the Maybe monad hasn't been added. One of the nice
+situation in which the Option/Maybe monad hasn't been added. One of the nice
properties of programming with monads is that it is possible to add or
-subtract layers according to the needs of the moment. Since we need
-simplicity, we'll set the Maybe monad aside for now.
+subtract layers according to the needs of the moment. Since we need
+simplicity, we'll set the Option/Maybe monad aside for now.
-So we'll go back to the point where the num type is simple int, not
-int options.
+So we'll go back to the point where the num type is simple `int`, not
+`int option`s.
type num = int
-And we'll start with the computation the map2 or the ⇧ from the option
-monad.
+And we won't be using the `map2` or `⇧` from the Option/Maybe monad anymore.
As you might guess, the technique we'll use to arrive at binding will
-be to use the Reader monad, defined here in terms of m-identity and bind:
-
- α ==> int -> α (* The ==> is a Kleisli arrow *)
- ⇧a = \x.a
- u >>= f = \x.f(ux)x
- map f u = \x.f(ux)
- map2 f u v = \x.f(ux)(vx)
+be to use the Reader monad, defined here in terms of `mid`/`⇧` and `mbind`/`>>=`:
+<pre>
+type <u>α</u> = int -> α
+⇧x = \n. x
+xx >>= k = \n. k (xx n) n
+map f xx = \n. f (xx n)
+ff ¢ xx = \n. (ff n) (xx n)
+map2 f xx yy = map f xx ¢ yy = \n. f (xx n) (yy n)
+</pre>
A boxed type in this monad will be a function from an integer to an
-object in the original type. The unit function ⇧ lifts a value `a` to
+value in the original type. The `mid`/`⇧` function lifts a value `x` to
a function that expects to receive an integer, but throws away the
-integer and returns `a` instead (most values in the computation don't
-depend on the input integer).
+integer and returns `x` instead (most values in the computation don't
+depend on the input integer).
-The bind function in this monad takes a monadic object `u`, a function
-`f` lifting non-monadic objects into the monad, and returns a function
-that expects an integer `x`. It feeds `x` to `u`, which delivers a
-result in the orginal type, which is fed in turn to `f`. `f` returns
-a monadic object, which upon being fed an integer, returns an object
-of the orginal type.
+The `mbind`/`>>=` function in this monad takes a monadic value `xx`, a function
+`k` taking non-monadic values into the monad, and returns a function
+that expects an integer `n`. It feeds `n` to `xx`, which delivers a
+result in the orginal type, which is fed in turn to `k`. `k` returns
+a monadic value, which upon being fed an integer, also delivers a result
+in the orginal type.
-The map2 function corresponding to this bind operation is given
-above. It should look familiar---we'll be commenting on this
-familiarity in a moment.
+The `map`, `¢`, and `map2` functions corresponding to this `mbind` are given
+above. They may look familiar --- we'll comment on this in a moment.
Lifing the computation into the monad, we have the following adjusted
types:
-type num = int -> int
+ type num = int -> int
-That is, `num` is once again replaced with the type of a boxed int.
-When we were dealing with the Maybe monad, a boxed int had type `int
-option`. In this monad, a boxed int has type `int -> int`.
+That is, `num` is once again replaced with the type of a boxed `int`.
+When we were dealing with the Option/Maybe monad, a boxed `int` had type `int
+option`. In this monad, a boxed int has type `int -> int`.
<!--
\tree ((((map2 +) (⇧1)) (((map2 *) (((map2 /) (⇧6)) (x))) (⇧4))))
map2 / ⇧6
</pre>
-It remains only to decide how the variable `x` will access the value input
-at the top of the tree. Since the input value is supposed to be the
-value put in place of the variable `x`. Like every leaf in the tree
+It remains only to decide how the variable `n` will access the input value supplied
+at the top of the tree. The input value is supposed to be the
+value that gets used for the variable `n`. Like every leaf in the tree
in argument position, the code we want in order to represent the
-variable will have the type of a boxed int, namely, `int -> int`. So
+variable will have the type of a boxed `int`, namely, `int -> int`. So
we have the following:
- x = (\fun (i:int) = i)
+ var_n = fun (n : int) -> n
-That is, variables in this system denote the indentity function!
+That is, variables in this system denote the identity function!
The result of evaluating this tree will be a boxed integer: a function
-from any integer `x` to `(+ 1 (* (/ 6 x)) 4)`.
+from any integer `n` to `(+ 1 (* (/ 6 n)) 4)`.
-Take a look at the definition of the reader monad again. The
-midentity takes some object `a` and returns `\x.a`. In other words,
-`⇧a = Ka`, so `⇧ = K`. Likewise, the reason the `map2` function
-looked familiar is that it is essentially the `S` combinator.
+Take a look at the definition of the Reader monad again. The
+`mid` takes some object `x` and returns `\n. x`. In other words,
+`⇧x = Kx`, for our familiar combinator **K**, so `⇧ = K`. Likewise, the `map` operation
+is just function composition, and the `mapply`/`¢` operation is our friend the **S** combinator. `map2 f xx yy` for the Reader monad is `(f ○ xx) ¢ yy` or `S (f ○ xx) yy`.
We've seen this before as a strategy for translating a binding
-construct into a set of combinators. To remind you, here is a part of
+construct into a set of combinators. To remind you, here is a part of
the general scheme for translating a lambda abstract into Combinatory
-Logic. The translation function `[.]` translates a lambda term into a
+Logic. The translation function `[.]` translates a lambda term into a
term in Combinatory Logic:
[(MN)] = ([M] [N])
The reason we can make do with this subset of the full translation
scheme is that we're making the simplifying assumption that there is
-at most a single lambda involved. So once again we have the identity
-function I as the translation of the bound variable, K as the function
+at most a single lambda involved. So once again we have the identity
+function **I** as the translation of the bound variable, **K** as the function
governing expressions that don't contain an instance of the bound
-variable, S as the operation that manages the combination of complex
-expressions.
+variable, **S** as an operation that manages `¢`, that is, the applicative combination of complex expressions.
+
## Jacobson's Variable Free Semantics as a Reader Monad
to show that Jacobson's Variable Free Semantics (e.g., Jacobson 1999,
[Towards a Variable-Free
Semantics](http://www.springerlink.com/content/j706674r4w217jj5/))
-implements a reader monad.
+implements a Reader monad.
More specifically, it will turn out that Jacobson's geach combinator
-*g* is exactly our `lift` operator, and her binding combinator *z* is
-exactly our `bind` (though with the arguments reversed).
+*g* is exactly our `map` operator, and her binding combinator *z* is
+exactly our `mbind` (though with the arguments reversed).
-Jacobson's system contains two main combinators, *g* and *z*. She
-calls *g* the Geach rule, and *z* performs binding. Here is a typical
-computation. This implementation is based closely on email from Simon
+Jacobson's system contains two main combinators, *g* and *z*. She
+calls *g* the Geach rule, and *z* performs binding. Here is a typical
+computation. This implementation is based closely on email from Simon
Charlow, with beta reduction as performed by the on-line evaluator:
<pre>
; Jacobson's analysis of "Everyone_i thinks he_i left"
-let g = \f u. \x. f (u x) in
-let z = \f u. \x. f (u x) x in
-let he = \x. x in
-let everyone = \P. FORALL x (P x) in
+let g = \f xx. \n. f (xx n) in ; or `f ○ xx`
+let z = \k xx. \n. k (xx n) n in ; or `S (flip k) xx`, or `Reader.(>>=) xx k`
+let he = \n. n in
+let everyone = \P. FORALL i (P i) in
everyone (z thinks (g left he))
-~~> FORALL x (thinks (left x) x)
+~~> FORALL i (thinks (left i) i)
</pre>
Two things to notice: First, pronouns once again denote identity
The basic recipe in Jacobson's system is that you transmit the
dependence of a pronoun upwards through the tree using *g* until just
before you are about to combine with the binder, when you finish off
-with *z*. Here is an example with a longer chain of *g*'s:
+with *z*. Here is an example with a longer chain of *g*'s:
<pre>
-everyone (z thinks (g (t bill) (g said (g left he))))
+; "Everyone_i thinks that Bill said he_i left"
+
+everyone (z thinks (g (T bill) (g said (g left he))))
+; or `everyone (thinks =<< T bill ○ said ○ left ○ I)`
-~~> FORALL x (thinks (said (left x) bill) x)
+~~> FORALL i (thinks (said (left i) bill) i)
</pre>
If you compare Jacobson's values for *g* and *z* to the functions in
the reader monad given above, you'll see that Jacobson's *g*
-combinator is exactly our `map` operator. Furthermore, Jacobson's *z*
-combinator is identital to `>>=`, except with the order of the
-arguments reversed (i.e., `(z f u) == (u >>= f)`).
-
-(The `t` combinator in the derivations above is given by `t x =
-\xy.yx`; it handles situations in which English word order reverses
-the usual function/argument order.)
+combinator is exactly our `map` operator. Furthermore, Jacobson's *z*
+combinator is identical to our Reader `>>=`, except with the order of the
+arguments reversed (i.e., `(z k xx) == (xx >>= k)`).
+
+(The `T` combinator in the derivations above is given by `T x <~~> \f. f x`;
+it handles situations in which English word order reverses
+the usual function/argument order. `T` is what Curry and Steedman call this
+combinator. Jacobson calls it "lift", but it shouldn't be confused with the
+`mid` and `map` operations that lift values into the Reader monad we're focusing
+on here. It does lift values into a *different* monad, that we'll consider in
+a few weeks.)
In other words,
Jacobson's variable-free semantics is essentially a Reader monad.
## The Reader monad for intensionality
-[This section has not been revised since 2010, so there may be a few
-places where it doesn't follow the convetions we've adopted this time;
-nevertheless, it should be followable.]
-
Now we'll look at using the Reader monad to do intensional function application.
In Shan (2001) [Monads for natural
First, the familiar linguistic problem:
- Bill left.
- Cam left.
- Ann believes [Bill left].
- Ann believes [Cam left].
+ Bill left.
+ Cam left.
+ Ann believes that Bill left.
+ Ann believes that Cam left.
We want an analysis on which the first three sentences can be true at
-the same time that the last sentence is false. If sentences denoted
+the same time that the last sentence is false. If sentences denoted
simple truth values or booleans, we have a problem: if the sentences
*Bill left* and *Cam left* are both true, they denote the same object,
and Ann's beliefs can't distinguish between them.
The traditional solution to the problem sketched above is to allow
sentences to denote a function from worlds to truth values, what
-Montague called an intension. So if `s` is the type of possible
+Montague called an intension. So if `s` is the type of possible
worlds, we have the following situation:
The main difference between the intensional types and the extensional
types is that in the intensional types, the arguments are functions
-from worlds to extensions: intransitive verb phrases like "left" now
-take so-called "individual concepts" as arguments (type s->e) rather than plain
-individuals (type e), and attitude verbs like "think" now take
-propositions (type s->t) rather than truth values (type t).
+from worlds to extensions: intransitive verb phrases like *left* now
+take so-called "individual concepts" as arguments (type `s->e`) rather than plain
+individuals (type `e`), and attitude verbs like *think* now take
+propositions (type `s->t`) rather than truth values (type `t`).
In addition, the result of each predicate is an intension.
This expresses the fact that the set of people who left in one world
may be different than the set of people who left in a different world.
in the way that the intensionality monad makes most natural.
The intensional types are more complicated than the extensional
-types. Wouldn't it be nice to make the complicated types available
-for those expressions like attitude verbs that need to worry about
+types. Wouldn't it be nice to make the complicated types available
+for expressions like attitude verbs that need to worry about
intensions, and keep the rest of the grammar as extensional as
-possible? This desire is parallel to our earlier desire to limit the
+possible? This desire is parallel to our earlier desire to limit the
concern about division by zero to the division function, and let the
other functions, like addition or multiplication, ignore
division-by-zero problems as much as possible.
So here's what we do:
-In OCaml, we'll use integers to model possible worlds. Characters (characters in the computational sense, i.e., letters like `'a'` and `'b'`, not Kaplanian characters) will model individuals, and OCaml booleans will serve for truth values:
+In OCaml, we'll use integers to model possible worlds. `char`s (characters in the computational sense, i.e., letters like `'a'` and `'b'`, not Kaplanian characters) will model individuals, and OCaml `bool`s will serve for truth values:
- type s = int;;
- type e = char;;
- type t = bool;;
+ type s = int
+ type e = char
+ type t = bool
- let ann = 'a';;
- let bill = 'b';;
- let cam = 'c';;
+ let ann = 'a'
+ let bill = 'b'
+ let cam = 'c'
- let left1 (x:e) = true;;
- let saw1 (x:e) (y:e) = y < x;;
+ let left1 (x : e) = true
+ let saw1 (y : e) (x : e) = x < y
- left1 ann;; (* true *)
- saw1 bill ann;; (* true *)
- saw1 ann bill;; (* false *)
+ left1 ann (* true *)
+ saw1 bill ann (* true *)
+ saw1 ann bill (* false *)
So here's our extensional system: everyone left, including Ann;
-and Ann saw Bill (`saw1 bill ann`), but Bill didn't see Ann. (Note that the word
+and Ann saw Bill (`saw1 bill ann`), but Bill didn't see Ann. (Note that the word
order we're using is VOS, verb-object-subject.)
-Now we add intensions. Because different people leave in different
+Now we add intensions. Because different people leave in different
worlds, the meaning of *leave* must depend on the world in which it is
being evaluated:
- let left (x:e) (w:s) = match (x, w) with ('c', 2) -> false | _ -> true;;
- left ann 1;; (* true: Ann left in world 1 *)
- left cam 2;; (* false: Cam didn't leave in world 2 *)
+ let left (x : e) (w : s) = match (x, w) with ('c', 2) -> false | _ -> true
+ left ann 1 (* true: Ann left in world 1 *)
+ left cam 2 (* false: Cam didn't leave in world 2 *)
This new definition says that everyone always left, except that
in world 2, Cam didn't leave.
Note that although this general *left* is sensitive to world of
evaluation, it does not have the fully intensionalized type given in
-the chart above, which was `(s->e)->s->t`. This is because
+the chart above, which was `(s->e)->s->t`. This is because
*left* does not exploit the additional resolving power provided by
-making the subject an individual concept. In semantics jargon, we say
-that *leave* is extensional with respect to its first argument.
+making the subject an individual concept. In semantics jargon, we say
+that *leave* is extensional with respect to its first argument.
-Therefore we will adopt the general strategy of defining predicates
+We will adopt the general strategy of defining predicates
in a way that they take arguments of the lowest type that will allow
-us to make all the distinctions the predicate requires. When it comes
+us to make all the distinctions the predicate requires. When it comes
time to combine this predicate with monadic arguments, we'll have to
make use of various lifting predicates.
Likewise, although *see* depends on the world of evaluation, it is
extensional in both of its syntactic arguments:
- let saw x y w = (w < 2) && (y < x);;
- saw bill ann 1;; (* true: Ann saw Bill in world 1 *)
- saw bill ann 2;; (* false: no one saw anyone in world 2 *)
+ let saw (y : e) (x : e) (w : s) = (w < 2) && (x < y)
+ saw bill ann 1 (* true: Ann saw Bill in world 1 *)
+ saw bill ann 2 (* false: no one saw anyone in world 2 *)
This (again, partially) intensionalized version of *see* coincides
with the `saw1` function we defined above for world 1; in world 2, no
Now we are ready for the intensionality monad:
<pre>
-type 'a intension = s -> 'a;;
-let unit x = fun (w:s) -> x;;
-(* as before, bind can be written more compactly, but having
- it spelled out like this will be useful down the road *)
-let bind u f = fun (w:s) -> let a = u w in let u' = f a in u' w;;
+type 'a intension = s -> 'a
+let mid x = fun (w : s) -> x
+let (>>=) xx k = fun (w : s) -> let x = xx w in let yy = k x in yy w
+ (* or just `fun w -> k (xx w) w` *)
</pre>
-Then the individual concept `unit ann` is a rigid designator: a
+Then the individual concept `mid ann` is a rigid designator: a
constant function from worlds to individuals that returns `'a'` no
-matter which world is used as an argument. This is a typical kind of
-thing for a monad unit to do.
+matter which world is used as an argument. This is a typical kind of
+thing for a monadic `mid` to do.
Then combining a predicate like *left* which is extensional in its
-subject argument with an intensional subject like `unit ann` is simply bind
+subject argument with an intensional subject like `mid ann` is simply `mbind`
in action:
- bind (unit ann) left 1;; (* true: Ann left in world 1 *)
- bind (unit cam) left 2;; (* false: Cam didn't leave in world 2 *)
+ (mid ann >>= left) 1 (* true: Ann left in world 1 *)
+ (mid cam >>= left) 2 (* false: Cam didn't leave in world 2 *)
-As usual, bind takes a monad box containing Ann, extracts Ann, and
-feeds her to the extensional *left*. In linguistic terms, we take the
-individual concept `unit ann`, apply it to the world of evaluation in
+As usual, `>>=` takes a monadic box containing Ann, extracts Ann, and
+feeds her to the function *left*. In linguistic terms, we take the
+individual concept `mid ann`, apply it to the world of evaluation in
order to get hold of an individual (`'a'`), then feed that individual
-to the extensional predicate *left*.
+to the predicate *left*.
We can arrange for a transitive verb that is extensional in both of
its arguments to take intensional arguments:
- let lift2' f u v = bind u (fun x -> bind v (fun y -> f x y));;
+<pre>
+let map2' f xx yy = xx >>= (fun x -> yy >>= (fun y -> <span class=ul>f x y</span>))
+</pre>
-This is almost the same `lift2` predicate we defined in order to allow
-addition in our division monad example. The difference is that this
+This is almost the same `map2` predicate we defined in order to allow
+addition in our division monad example. The *difference* is that this
variant operates on verb meanings that take extensional arguments but
-returns an intensional result. Thus the original `lift2` predicate
-has `unit (f x y)` where we have just `f x y` here.
-
-The use of `bind` here to combine *left* with an individual concept,
-and the use of `lift2'` to combine *see* with two intensional
+returns an intensional result. Thus the original `map2` predicate
+has `mid (f x y)` where we have just <code><span class=ul>f x y</span></code> here.
+
+The use of `>>=` here to combine *left* with an individual concept,
+and the use of `map2'` to combine *see* with two intensional
arguments closely parallels the two of Montague's meaning postulates
(in PTQ) that express the relationship between extensional verbs and
their uses in intensional contexts.
<pre>
-lift2' saw (unit bill) (unit ann) 1;; (* true *)
-lift2' saw (unit bill) (unit ann) 2;; (* false *)
+map2' saw (mid bill) (mid ann) 1 (* true *)
+map2' saw (mid bill) (mid ann) 2 (* false *)
</pre>
-Ann did see bill in world 1, but Ann didn't see Bill in world 2.
+Ann did see Bill in world 1, but Ann didn't see Bill in world 2.
-Finally, we can define our intensional verb *thinks*. *Think* is
-intensional with respect to its sentential complement, though still extensional
-with respect to its subject. (As Montague noticed, almost all verbs
+Finally, we can define our intensional verb *thinks*. *Think* is
+intensional with respect to its sentential complement (it takes complements of type `s -> t`), though still extensional
+with respect to its subject (type `e`). (As Montague noticed, almost all verbs
in English are extensional with respect to their subject; a possible
-exception is "appear".)
+exception is *appear*.)
+
+ let thinks (p : s->t) (x : e) (w : s) =
+ match (x, p 2) with ('a', false) -> false | _ -> p w
- let thinks (p:s->t) (x:e) (w:s) =
- match (x, p 2) with ('a', false) -> false | _ -> p w;;
+In every world, Ann fails to believe any proposition that is false in world 2.
+Apparently, she stably thinks we may be in world 2. Otherwise, everyone
+believes a proposition iff that proposition is true in the world of evaluation.
-Ann disbelieves any proposition that is false in world 2. Apparently,
-she firmly believes we're in world 2. Everyone else believes a
-proposition iff that proposition is true in the world of evaluation.
+ (mid ann >>= thinks (mid bill >>= left)) 1 (* true *)
- bind (unit ann) (thinks (bind (unit bill) left)) 1;;
+So in world 1, Ann thinks that Bill left (because in worlds 1 and 2, Bill did leave).
-So in world 1, Ann thinks that Bill left (because in world 2, Bill did leave).
+But although Cam left in world 1:
- bind (unit ann) (thinks (bind (unit cam) left)) 1;;
+ (mid cam >>= left) 1 (* true *)
-But in world 1, Ann doesn't believe that Cam left (even though he
-did leave in world 1: `bind (unit cam) left 1 == true`). Ann's thoughts are hung up on
-what is happening in world 2, where Cam doesn't leave.
+he didn't leave in world 2, so Ann doesn't in world 1 believe that Cam left:
-*Small project*: add intersective ("red") and non-intersective
- adjectives ("good") to the fragment. The intersective adjectives
+ (mid ann >>= thinks (mid cam >>= left)) 1 (* false *)
+
+
+**Small project**: add intersective (*red*) and non-intersective
+ adjectives (*good*) to the fragment. The intersective adjectives
will be extensional with respect to the nominal they combine with
- (using bind), and the non-intersective adjectives will take
+ (using `mbind`), and the non-intersective adjectives will take
intensional arguments.
+**Connections with variable binding**: the rigid individual concepts generated by `mid ann` and the like correspond to the numerical constants, that don't interact with the environment in any way, in the variable binding examples we considered earlier on the page. If we had any non-contingent predicates that were wholly insensitive to intensional effects, they would be modeled using `map2` and would correspond to the operations like `map2 (+)` in the earlier examples. As it is, our predicates *lift* and *saw*, though only sensitive to the *extension* of their arguments, nonetheless *are* sensitive to the world of evaluation for their `bool` output. So these are somewhat akin, at the predicate level, to expressions like `var_n`, at the singular term level, in the variable bindings examples. Our predicate *thinks* shows yet a further kind of interaction with the intensional structures we introduced: namely, its output can depend upon evaluating its complement relative to other possible worlds. We didn't discuss analogues of this in the variable binding case, but they exist: namely, they are expressions like `let x = 2 in ...` and `forall x ...`, that have the effect of supplying their arguments with an environment or assignment function that is shifted from the one they are themselves being evaluated with.
+
notes: cascade, general env