X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week7.mdwn;h=351a6d953fac41901c96823c8bf31829b51e6733;hp=84a2af55dec222f8d8e21eb9bb51c4ea38667508;hb=53be231576f6b0d8b6452fceac662158eea1d710;hpb=a020d7de4c5b17ce19c0efbaf6760799a899cc4d

diff --git a/week7.mdwn b/week7.mdwn
index 84a2af55..351a6d95 100644
--- a/week7.mdwn
+++ b/week7.mdwn
@@ -1,17 +1,18 @@
 [[!toc]]
 
+
 Monads
 ------
 
 Start by (re)reading the discussion of monads in the lecture notes for
-week 6 [Towards Monads](http://lambda.jimpryor.net//week6/#index4h2).
+week 6 [[Towards Monads]].
 In those notes, we saw a way to separate thinking about error
 conditions (such as trying to divide by zero) from thinking about
 normal arithmetic computations.  We did this by making use of the
-option monad: in each place where we had something of type `int`, we
+`option` type: in each place where we had something of type `int`, we
 put instead something of type `int option`, which is a sum type
-consisting either of just an integer, or else some special value which
-we could interpret as signaling that something had gone wrong.
+consisting either of one choice with an `int` payload, or else a `None`
+choice which we interpret as  signaling that something has gone wrong.
 
 The goal was to make normal computing as convenient as possible: when
 we're adding or multiplying, we don't have to worry about generating
@@ -29,8 +30,8 @@ building a simple account of presupposition projection, we would have
 to do several things.  First, we would have to make use of the
 polymorphism of the `option` type.  In the arithmetic example, we only
 made use of `int option`s, but when we're composing natural language
-expression meanings, we'll need to use types like `N int`, `Det Int`,
-`VP int`, and so on.  But that works automatically, because we can use
+expression meanings, we'll need to use types like `N option`, `Det option`,
+`VP option`, and so on.  But that works automatically, because we can use
 any type for the `'a` in `'a option`.  Ultimately, we'd want to have a
 theory of accommodation, and a theory of the situations in which
 material within the sentence can satisfy presuppositions for other
@@ -38,167 +39,373 @@ material that otherwise would trigger a presupposition violation; but,
 not surprisingly, these refinements will require some more
 sophisticated techniques than the super-simple option monad.]
 
-So what exactly is a monad?  As usual, we're not going to be pedantic
-about it, but for our purposes, we can consider a monad to be a system
+So what exactly is a monad?  We can consider a monad to be a system
 that provides at least the following three elements:
 
-* A way to build a complex type from some basic type.  In the division
-  example, the polymorphism of the `'a option` type provides a way of
-  building an option out of any other type of object.  People often
-  use a container metaphor: if `x` has type `int option`, then `x` is
-  a box that (may) contain an integer.
+*	A complex type that's built around some more basic type. Usually
+	the complex type will be polymorphic, and so can apply to different basic types.
+	In our division example, the polymorphism of the `'a option` type
+	provides a way of building an option out of any other type of object.
+	People often use a container metaphor: if `u` has type `int option`,
+	then `u` is a box that (may) contain an integer.
+
+		type 'a option = None | Some of 'a;;
+
+*	A way to turn an ordinary value into a monadic value.  In OCaml, we
+	did this for any integer `x` by mapping it to
+	the option `Some x`.  In the general case, this operation is
+	known as `unit` or `return.` Both of those names are terrible. This
+	operation is only very loosely connected to the `unit` type we were
+	discussing earlier (whose value is written `()`). It's also only
+	very loosely connected to the "return" keyword in many other
+	programming languages like C. But these are the names that the literature
+	uses.
+
+	The unit/return operation is a way of lifting an ordinary object into
+	the monadic box you've defined, in the simplest way possible. You can think
+	of the singleton function as an example: it takes an ordinary object
+	and returns a set containing that object. In the example we've been
+	considering:
+
+		let unit x = Some x;;
+		val unit : 'a -> 'a option = <fun>
+
+	So `unit` is a way to put something inside of a monadic box. It's crucial
+	to the usefulness of monads that there will be monadic boxes that
+	aren't the result of that operation. In the option/maybe monad, for
+	instance, there's also the empty box `None`. In another (whimsical)
+	example, you might have, in addition to boxes merely containing integers,
+	special boxes that contain integers and also sing a song when they're opened. 
+
+	The unit/return operation will always be the simplest, conceptually
+	most straightforward way to lift an ordinary value into a monadic value
+	of the monadic type in question.
+
+*	Thirdly, an operation that's often called `bind`. This is another
+	unfortunate name: this operation is only very loosely connected to
+	what linguists usually mean by "binding." In our option/maybe monad, the
+	bind operation is:
+
+		let bind u f = match u with None -> None | Some x -> f x;;
+		val bind : 'a option -> ('a -> 'b option) -> 'b option = <fun>
+
+	Note the type: `bind` takes two arguments: first, a monadic box
+	(in this case, an `'a option`); and second, a function from
+	ordinary objects to monadic boxes. `bind` then returns a monadic
+	value: in this case, a `'b option` (you can start with, e.g., `int option`s
+	and end with `bool option`s).
+
+	Intuitively, the interpretation of what `bind` does is this:
+	the first argument is a monadic value `u`, which 
+	evaluates to a box that (maybe) contains some ordinary value, call it `x`.
+	Then the second argument uses `x` to compute a new monadic
+	value.  Conceptually, then, we have
+
+		let bind u f = (let x = unbox u in f x);;
+
+	The guts of the definition of the `bind` operation amount to
+	specifying how to unbox the monadic value `u`.  In the `bind`
+	opertor for the option monad, we unboxed the monadic value by
+	matching it with the pattern `Some x`---whenever `u`
+	happened to be a box containing an integer `x`, this allowed us to
+	get our hands on that `x` and feed it to `f`.
+
+	If the monadic box didn't contain any ordinary value,
+	we instead pass through the empty box unaltered.
+
+	In a more complicated case, like our whimsical "singing box" example
+	from before, if the monadic value happened to be a singing box
+	containing an integer `x`, then the `bind` operation would probably
+	be defined so as to make sure that the result of `f x` was also
+	a singing box. If `f` also wanted to insert a song, you'd have to decide
+	whether both songs would be carried through, or only one of them.
+
+	There is no single `bind` function that dictates how this must go.
+	For each new monadic type, this has to be worked out in an
+	useful way.
+
+So the "option/maybe monad" consists of the polymorphic `option` type, the
+`unit`/return function, and the `bind` function.  With the option monad, we can
+think of the "safe division" operation:
+
+	# let safe_divide num den = if 0 = den then None else Some (num/den);;
+	val safe_divide : int -> int -> int option = <fun>
 
-    `type 'a option = None | Some of 'a;;`
+as basically a function from two integers to an integer, except with
+this little bit of option plumbing on the side.
 
-* A way to turn an ordinary value into a monadic value.  In Ocaml, we
-  did this for any integer `n` by mapping it to
-  the option `Some n`.  To be official, we can define a function
-  called unit:
+A note on notation: Haskell uses the infix operator `>>=` to stand
+for `bind`. Chris really hates that symbol.  Following Wadler, he prefers to
+use an infix five-pointed star &#8902;, or on a keyboard, `*`. Jim on the other hand
+thinks `>>=` is what the literature uses and students won't be able to
+avoid it. Moreover, although &#8902; is OK (though not a convention that's been picked up), overloading the multiplication symbol invites its own confusion
+and Jim feels very uneasy about that. If not `>>=` then we should use
+some other unfamiliar infix symbol (but `>>=` already is such...)
 
-    `let unit x = Some x;;`
+In any case, the course leaders will work this out somehow. In the meantime,
+as you read around, wherever you see `u >>= f`, that means `bind u f`. Also,
+if you ever see this notation:
 
-    `val unit : 'a -> 'a option = <fun>`
+	do
+		x <- u
+		f x
 
-    So `unit` is a way to put something inside of a box.
+That's a Haskell shorthand for `u >>= (\x -> f x)`, that is, `bind u f`.
+Similarly:
 
-* A bind operation (note the type):
+	do
+		x <- u
+		y <- v
+		f x y
 
-     `let bind m f = match m with None -> None | Some n -> f n;;`
+is shorthand for `u >>= (\x -> v >>= (\y -> f x y))`, that is, `bind u (fun x
+-> bind v (fun y -> f x y))`. Those who did last week's homework may recognize
+this.
 
-     `val bind : 'a option -> ('a -> 'b option) -> 'b option = <fun>`
+(Note that the above "do" notation comes from Haskell. We're mentioning it here
+because you're likely to see it when reading about monads. It won't work in
+OCaml. In fact, the `<-` symbol already means something different in OCaml,
+having to do with mutable record fields. We'll be discussing mutation someday
+soon.)
 
-     `bind` takes two arguments (a monadic object and a function from
-     ordinary objects to monadic objects), and returns a monadic
-     object.
+As we proceed, we'll be seeing a variety of other monad systems. For example, another monad is the list monad. Here the monadic type is:
 
-     Intuitively, the interpretation of what `bind` does is like this:
-     the first argument computes a monadic object m, which will
-     evaluate to a box containing some ordinary value, call it `x`.
-     Then the second argument uses `x` to compute a new monadic
-     value.  Conceptually, then, we have
+	# type 'a list
 
-    `let bind m f = (let x = unwrap m in f x);;`
+The `unit`/return operation is:
 
-    The guts of the definition of the `bind` operation amount to
-    specifying how to unwrap the monadic object `m`.  In the bind
-    opertor for the option monad, we unwraped the option monad by
-    matching the monadic object `m` with `Some n`--whenever `m`
-    happend to be a box containing an integer `n`, this allowed us to
-    get our hands on that `n` and feed it to `f`.
+	# let unit x = [x];;
+	val unit : 'a -> 'a list = <fun>
 
-So the "option monad" consists of the polymorphic option type, the
-unit function, and the bind function.  With the option monad, we can
-think of the "safe division" operation
+That is, the simplest way to lift an `'a` into an `'a list` is just to make a
+singleton list of that `'a`. Finally, the `bind` operation is:
 
-<pre>
-# let divide num den = if den = 0 then None else Some (num/den);;
-val divide : int -> int -> int option = <fun>
-</pre>
+	# let bind u f = List.concat (List.map f u);;
+	val bind : 'a list -> ('a -> 'b list) -> 'b list = <fun>
+	
+What's going on here? Well, consider `List.map f u` first. This goes through all
+the members of the list `u`. There may be just a single member, if `u = unit x`
+for some `x`. Or on the other hand, there may be no members, or many members. In
+any case, we go through them in turn and feed them to `f`. Anything that gets fed
+to `f` will be an `'a`. `f` takes those values, and for each one, returns a `'b list`.
+For example, it might return a list of all that value's divisors. Then we'll
+have a bunch of `'b list`s. The surrounding `List.concat ( )` converts that bunch
+of `'b list`s into a single `'b list`:
 
-as basically a function from two integers to an integer, except with
-this little bit of option frill, or option plumbing, on the side.
+	# List.concat [[1]; [1;2]; [1;3]; [1;2;4]]
+	- : int list = [1; 1; 2; 1; 3; 1; 2; 4]
 
-A note on notation: Haskell uses the infix operator `>>=` to stand
-for `bind`.  I really hate that symbol.  Following Wadler, I prefer to
-infix five-pointed star, or on a keyboard, `*`.
+So now we've seen two monads: the option/maybe monad, and the list monad. For any
+monadic system, there has to be a specification of the complex monad type,
+which will be parameterized on some simpler type `'a`, and the `unit`/return
+operation, and the `bind` operation. These will be different for different
+monadic systems.
+
+Many monadic systems will also define special-purpose operations that only make
+sense for that system.
+
+Although the `unit` and `bind` operation are defined differently for different
+monadic systems, there are some general rules they always have to follow.
 
 
-The Monad laws
+The Monad Laws
 --------------
 
 Just like good robots, monads must obey three laws designed to prevent
 them from hurting the people that use them or themselves.
 
-*    Left identity: unit is a left identity for the bind operation.
-     That is, for all `f:'a -> 'a M`, where `'a M` is a monadic
-     object, we have `(unit x) * f == f x`.  For instance, `unit` is a
-     function of type `'a -> 'a option`, so we have
+*	**Left identity: unit is a left identity for the bind operation.**
+	That is, for all `f:'a -> 'a m`, where `'a m` is a monadic
+	type, we have `(unit x) * f == f x`.  For instance, `unit` is itself
+	a function of type `'a -> 'a m`, so we can use it for `f`:
 
-<pre>
-# let ( * ) m f = match m with None -> None | Some n -> f n;;
-val ( * ) : 'a option -> ('a -> 'b option) -> 'b option = <fun>
-# let unit x = Some x;;
-val unit : 'a -> 'a option = <fun>
-
-# unit 2;;
-- : int option = Some 2
-# unit 2 * unit;;
-- : int option = Some 2
-
-# divide 6 2;;
-- : int option = Some 3
-# unit 2 * divide 6;;
-- : int option = Some 3
-
-# divide 6 0;;
-- : int option = None
-# unit 0 * divide 6;;
-- : int option = None
-</pre>
+		# let ( * ) u f = match u with None -> None | Some x -> f x;;
+		val ( * ) : 'a option -> ('a -> 'b option) -> 'b option = <fun>
+		# let unit x = Some x;;
+		val unit : 'a -> 'a option = <fun>
+
+		# unit 2;;
+		- : int option = Some 2
+		# unit 2 * unit;;
+		- : int option = Some 2
 
-The parentheses is the magic for telling Ocaml that the
+		# divide 6 2;;
+		- : int option = Some 3
+		# unit 2 * divide 6;;
+		- : int option = Some 3
+
+		# divide 6 0;;
+		- : int option = None
+		# unit 0 * divide 6;;
+		- : int option = None
+
+The parentheses is the magic for telling OCaml that the
 function to be defined (in this case, the name of the function
 is `*`, pronounced "bind") is an infix operator, so we write
-`m * f` or `( * ) m f` instead of `* m f`.
+`u * f` or `( * ) u f` instead of `* u f`.
 
-*    Associativity: bind obeys a kind of associativity, like this:
+*	**Associativity: bind obeys a kind of associativity**. Like this:
 
-    `(m * f) * g == m * (fun x -> f x * g)`
+		(u * f) * g == u * (fun x -> f x * g)
 
-    If you don't understand why the lambda form is necessary (the "fun
-    x" part), you need to look again at the type of bind.
+	If you don't understand why the lambda form is necessary (the "fun
+	x" part), you need to look again at the type of `bind`.
 
-    Some examples of associativity in the option monad:
+	Some examples of associativity in the option monad:
 
-<pre>
-# Some 3 * unit * unit;; 
-- : int option = Some 3
-# Some 3 * (fun x -> unit x * unit);;
-- : int option = Some 3
-
-# Some 3 * divide 6 * divide 2;;
-- : int option = Some 1
-# Some 3 * (fun x -> divide 6 x * divide 2);;
-- : int option = Some 1
-
-# Some 3 * divide 2 * divide 6;;
-- : int option = None
-# Some 3 * (fun x -> divide 2 x * divide 6);;
-- : int option = None
-</pre>
+		# Some 3 * unit * unit;; 
+		- : int option = Some 3
+		# Some 3 * (fun x -> unit x * unit);;
+		- : int option = Some 3
+
+		# Some 3 * divide 6 * divide 2;;
+		- : int option = Some 1
+		# Some 3 * (fun x -> divide 6 x * divide 2);;
+		- : int option = Some 1
+
+		# Some 3 * divide 2 * divide 6;;
+		- : int option = None
+		# Some 3 * (fun x -> divide 2 x * divide 6);;
+		- : int option = None
 
 Of course, associativity must hold for arbitrary functions of
-type `'a -> M 'a`, where `M` is the monad type.  It's easy to
-convince yourself that the bind operation for the option monad
-obeys associativity by dividing the inputs into cases: if `m`
+type `'a -> 'a m`, where `m` is the monad type.  It's easy to
+convince yourself that the `bind` operation for the option monad
+obeys associativity by dividing the inputs into cases: if `u`
 matches `None`, both computations will result in `None`; if
-`m` matches `Some n`, and `f n` evalutes to `None`, then both
+`u` matches `Some x`, and `f x` evalutes to `None`, then both
 computations will again result in `None`; and if the value of
-`f n` matches `Some r`, then both computations will evaluate
-to `g r`.
+`f x` matches `Some y`, then both computations will evaluate
+to `g y`.
 
-*    Right identity: unit is a right identity for bind.  That is, 
-     `m * unit == m` for all monad objects `m`.  For instance,
+*	**Right identity: unit is a right identity for bind.**  That is, 
+	`u * unit == u` for all monad objects `u`.  For instance,
+
+		# Some 3 * unit;;
+		- : int option = Some 3
+		# None * unit;;
+		- : 'a option = None
 
-<pre>
-# Some 3 * unit;;
-- : int option = Some 3
-# None * unit;;
-- : 'a option = None
-</pre>
 
-Now, if you studied algebra, you'll remember that a *monoid* is an
+More details about monads
+-------------------------
+
+If you studied algebra, you'll remember that a *monoid* is an
 associative operation with a left and right identity.  For instance,
 the natural numbers along with multiplication form a monoid with 1
 serving as the left and right identity.  That is, temporarily using
-`*` to mean arithmetic multiplication, `1 * n == n == n * 1` for all
-`n`, and `(a * b) * c == a * (b * c)` for all `a`, `b`, and `c`.  As
+`*` to mean arithmetic multiplication, `1 * u == u == u * 1` for all
+`u`, and `(u * v) * w == u * (v * w)` for all `u`, `v`, and `w`.  As
 presented here, a monad is not exactly a monoid, because (unlike the
 arguments of a monoid operation) the two arguments of the bind are of
-different types.  But if we generalize bind so that both arguments are
-of type `'a -> M 'a`, then we get plain identity laws and
-associativity laws, and the monad laws are exactly like the monoid
-laws (see <http://www.haskell.org/haskellwiki/Monad_Laws>, near the bottom).
+different types.  But it's possible to make the connection between
+monads and monoids much closer. This is discussed in [Monads in Category
+Theory](/advanced_notes/monads_in_category_theory).
+See also <http://www.haskell.org/haskellwiki/Monad_Laws>.
+
+Here are some papers that introduced monads into functional programming:
+
+*	[Eugenio Moggi, Notions of Computation and Monads](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf): Information and Computation 93 (1).
+
+*	[Philip Wadler. Monads for Functional Programming](http://homepages.inf.ed.ac.uk/wadler/papers/marktoberdorf/baastad.pdf):
+in M. Broy, editor, *Marktoberdorf Summer School on Program Design Calculi*, Springer Verlag, NATO ASI Series F: Computer and systems sciences, Volume 118, August 1992. Also in J. Jeuring and E. Meijer, editors, *Advanced Functional Programming*, Springer Verlag, LNCS 925, 1995. Some errata fixed August 2001.
+	The use of monads to structure functional programs is described. Monads provide a convenient framework for simulating effects found in other languages, such as global state, exception handling, output, or non-determinism. Three case studies are looked at in detail: how monads ease the modification of a simple evaluator; how monads act as the basis of a datatype of arrays subject to in-place update; and how monads can be used to build parsers.
+
+*	[Philip Wadler. The essence of functional programming](http://homepages.inf.ed.ac.uk/wadler/papers/essence/essence.ps):
+invited talk, *19'th Symposium on Principles of Programming Languages*, ACM Press, Albuquerque, January 1992.
+	This paper explores the use monads to structure functional programs. No prior knowledge of monads or category theory is required.
+	Monads increase the ease with which programs may be modified. They can mimic the effect of impure features such as exceptions, state, and continuations; and also provide effects not easily achieved with such features. The types of a program reflect which effects occur.
+	The first section is an extended example of the use of monads. A simple interpreter is modified to support various extra features: error messages, state, output, and non-deterministic choice. The second section describes the relation between monads and continuation-passing style. The third section sketches how monads are used in a compiler for Haskell that is written in Haskell.
+
+There's a long list of monad tutorials on the [[Offsite Reading]] page. Skimming the titles makes me laugh.
+
+In the presentation we gave above---which follows the functional programming conventions---we took `unit`/return and `bind` as the primitive operations. From these a number of other general monad operations can be derived. It's also possible to take some of the others as primitive. The [Monads in Category
+Theory](/advanced_notes/monads_in_category_theory) notes do so, for example.
+
+Here are some of the specifics. You don't have to master these; they're collected here for your reference.
+
+You may sometimes see:
+
+	u >> v
+
+That just means:
+
+	u >>= fun _ -> v
+
+that is:
+
+	bind u (fun _ -> v)
+
+You could also do `bind u (fun x -> v)`; we use the `_` for the function argument to be explicit that that argument is never going to be used.
+
+The `lift` operation we asked you to define for last week's homework is a common operation. The second argument to `bind` converts `'a` values into `'b m` values---that is, into instances of the monadic type. What if we instead had a function that merely converts `'a` values into `'b` values, and we want to use it with our monadic type. Then we "lift" that function into an operation on the monad. For example:
+
+	# let even x = (x mod 2 = 0);;
+	val g : int -> bool = <fun>
+
+`even` has the type `int -> bool`. Now what if we want to convert it into an operation on the option/maybe monad?
+
+	# let lift g = fun u -> bind u (fun x -> Some (g x));;
+	val lift : ('a -> 'b) -> 'a option -> 'b option = <fun>
+
+`lift even` will now be a function from `int option`s to `bool option`s. We can
+also define a lift operation for binary functions:
+
+	# let lift2 g = fun u v -> bind u (fun x -> bind v (fun y -> Some (g x y)));;
+	val lift2 : ('a -> 'b -> 'c) -> 'a option -> 'b option -> 'c option = <fun>
+
+`lift (+)` will now be a function from `int option`s  and `int option`s to `int option`s. This should look familiar to those who did the homework.
+
+The `lift` operation (just `lift`, not `lift2`) is sometimes also called the `map` operation. (In Haskell, they say `fmap` or `<$>`.) And indeed when we're working with the list monad, `lift f` is exactly `List.map f`!
+
+Wherever we have a well-defined monad, we can define a lift/map operation for that monad. The examples above used `Some (g x)` and so on; in the general case we'd use `unit (g x)`, using the specific `unit` operation for the monad we're working with.
+
+In general, any lift/map operation can be relied on to satisfy these laws:
+
+	* lift id = id
+	* lift (compose f g) = compose (lift f) (lift g)
+
+where `id` is `fun x -> x` and `compose f g` is `fun x -> f (g x)`. If you think about the special case of the map operation on lists, this should make sense. `List.map id lst` should give you back `lst` again. And you'd expect these
+two computations to give the same result:
+
+	List.map (fun x -> f (g x)) lst
+	List.map f (List.map g lst)
+
+Another general monad operation is called `ap` in Haskell---short for "apply." (They also use `<*>`, but who can remember that?) This works like this:
+
+	ap [f] [x; y] = [f x; f y]
+	ap (Some f) (Some x) = Some (f x)
+
+and so on. Here are the laws that any `ap` operation can be relied on to satisfy:
+
+	ap (unit id) u = u
+	ap (ap (ap (return compose) u) v) w = ap u (ap v w)
+	ap (unit f) (unit x) = unit (f x)
+	ap u (unit x) = ap (unit (fun f -> f x)) u
+
+Another general monad operation is called `join`. This is the operation that takes you from an iterated monad to a single monad. Remember when we were explaining the `bind` operation for the list monad, there was a step where
+we went from:
+
+	[[1]; [1;2]; [1;3]; [1;2;4]]
+
+to:
+
+	[1; 1; 2; 1; 3; 1; 2; 4]
+
+That is the `join` operation.
+
+All of these operations can be defined in terms of `bind` and `unit`; or alternatively, some of them can be taken as primitive and `bind` can be defined in terms of them. Here are various interdefinitions:
+
+	lift f u = ap (unit f) u
+	lift2 f u v = ap (lift f u) v = ap (ap (unit f) u) v
+	ap u v = lift2 id u v
+	lift f u = u >>= compose unit f
+	lift f u = ap (unit f) u
+	join m2 = m2 >>= id
+	u >>= f = join (lift f u)
+	u >> v = u >>= (fun _ -> v)
+	u >> v = lift2 (fun _ -> id) u v
+
 
 
 Monad outlook
@@ -210,10 +417,22 @@ which will enable us to model mutation: variables whose values appear
 to change as the computation progresses.  Later, we will study the
 Continuation monad.
 
+In the meantime, we'll look at several linguistic applications for monads, based
+on what's called the *reader monad*.
+
+
+The reader monad
+----------------
+
+Introduce
+
+Heim and Kratzer's "Predicate Abstraction Rule"
+
+
+
 The intensionality monad
 ------------------------
-
-In the meantime, we'll see a linguistic application for monads:
+...
 intensional function application.  In Shan (2001) [Monads for natural
 language semantics](http://arxiv.org/abs/cs/0205026v1), Ken shows that
 making expressions sensitive to the world of evaluation is
@@ -225,9 +444,9 @@ intensionality](http://parles.upf.es/glif/pub/sub11/individual/bena_wint.pdf),
 though without explicitly using monads.
 
 All of the code in the discussion below can be found here: [[intensionality-monad.ml]].
-To run it, download the file, start Ocaml, and say 
+To run it, download the file, start OCaml, and say 
 
-    # #use "intensionality-monad.ml";;
+	# #use "intensionality-monad.ml";;
 
 Note the extra `#` attached to the directive `use`.
 
@@ -283,24 +502,22 @@ as possible.
 
 So here's what we do:
 
-In Ocaml, we'll use integers to model possible worlds:
+In OCaml, we'll use integers to model possible worlds:
 
-    type s = int;;
-    type e = char;;
-    type t = bool;;
+	type s = int;;
+	type e = char;;
+	type t = bool;;
 
 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.
+OCaml booleans will serve for truth values.
 
-<pre>
-type 'a intension = s -> 'a;;
-let unit x (w:s) = x;;
+	type 'a intension = s -> 'a;;
+	let unit x (w:s) = x;;
 
-let ann = unit 'a';;
-let bill = unit 'b';;
-let cam = unit 'c';;
-</pre>
+	let ann = unit 'a';;
+	let bill = unit 'b';;
+	let cam = unit 'c';;
 
 In our monad, the intension of an extensional type `'a` is `s -> 'a`,
 a function from worlds to extensions.  Our unit will be the constant
@@ -313,18 +530,16 @@ as an argument.
 
 Let's test compliance with the left identity law:
 
-<pre>
-# let bind m f (w:s) = f (m w) w;;
-val bind : (s -> 'a) -> ('a -> s -> 'b) -> s -> 'b = <fun>
-# bind (unit 'a') unit 1;;
-- : char = 'a'
-</pre>
+	# let bind u f (w:s) = f (u w) w;;
+	val bind : (s -> 'a) -> ('a -> s -> 'b) -> s -> 'b = <fun>
+	# bind (unit 'a') unit 1;;
+	- : char = 'a'
 
 We'll assume that this and the other laws always hold.
 
 We now build up some extensional meanings:
 
-    let left w x = match (w,x) with (2,'c') -> false | _ -> true;;
+	let left w x = match (w,x) with (2,'c') -> false | _ -> true;;
 
 This function says that everyone always left, except for Cam in world
 2 (i.e., `left 2 'c' == false`).
@@ -333,31 +548,29 @@ Then the way to evaluate an extensional sentence is to determine the
 extension of the verb phrase, and then apply that extension to the
 extension of the subject:
 
-<pre>
-let extapp fn arg w = fn w (arg w);;
+	let extapp fn arg w = fn w (arg w);;
 
-extapp left ann 1;;
-# - : bool = true
+	extapp left ann 1;;
+	# - : bool = true
 
-extapp left cam 2;;
-# - : bool = false
-</pre>
+	extapp left cam 2;;
+	# - : bool = false
 
 `extapp` stands for "extensional function application".
 So Ann left in world 1, but Cam didn't leave in world 2.
 
 A transitive predicate:
 
-    let saw w x y = (w < 2) && (y < x);;
-    extapp (extapp saw bill) ann 1;; (* true *)
-    extapp (extapp saw bill) ann 2;; (* false *)
+	let saw w x y = (w < 2) && (y < x);;
+	extapp (extapp saw bill) ann 1;; (* true *)
+	extapp (extapp saw bill) ann 2;; (* false *)
 
 In world 1, Ann saw Bill and Cam, and Bill saw Cam.  No one saw anyone
 in world two.
 
 Good.  Now for intensions:
 
-    let intapp fn arg w = fn w arg;;
+	let intapp fn arg w = fn w arg;;
 
 The only difference between intensional application and extensional
 application is that we don't feed the evaluation world to the argument.
@@ -372,12 +585,10 @@ Just as we used bind to define a version of addition that interacted
 with the option monad, we now use bind to intensionalize an
 extensional verb:
 
-<pre>
-let lift pred w arg = bind arg (fun x w -> pred w x) w;;
+	let lift pred w arg = bind arg (fun x w -> pred w x) w;;
 
-intapp (lift left) ann 1;; (* true: Ann still left in world 1 *)
-intapp (lift left) cam 2;; (* false: Cam still didn't leave in world 2 *)
-</pre>
+	intapp (lift left) ann 1;; (* true: Ann still left in world 1 *)
+	intapp (lift left) cam 2;; (* false: Cam still didn't leave in world 2 *)
 
 Because `bind` unwraps the intensionality of the argument, when the
 lifted "left" receives an individual concept (e.g., `unit 'a'`) as
@@ -388,12 +599,10 @@ third meaning postulate.)
 
 Likewise for extensional transitive predicates like "saw":
 
-<pre>
-let lift2 pred w arg1 arg2 = 
-  bind arg1 (fun x -> bind arg2 (fun y w -> pred w x y)) w;;
-intapp (intapp (lift2 saw) bill) ann 1;;  (* true: Ann saw Bill in world 1 *)
-intapp (intapp (lift2 saw) bill) ann 2;;  (* false: No one saw anyone in world 2 *)
-</pre>
+	let lift2 pred w arg1 arg2 = 
+	  bind arg1 (fun x -> bind arg2 (fun y w -> pred w x y)) w;;
+	intapp (intapp (lift2 saw) bill) ann 1;;  (* true: Ann saw Bill in world 1 *)
+	intapp (intapp (lift2 saw) bill) ann 2;;  (* false: No one saw anyone in world 2 *)
 
 Crucially, an intensional predicate does not use `bind` to consume its
 arguments.  Attitude verbs like "thought" are intensional with respect
@@ -401,22 +610,18 @@ to their sentential complement, but extensional with respect to their
 subject (as Montague noticed, almost all verbs in English are
 extensional with respect to their subject; a possible exception is "appear"):
 
-<pre>
-let think (w:s) (p:s->t) (x:e) = 
-  match (x, p 2) with ('a', false) -> false | _ -> p w;;
-</pre>
+	let think (w:s) (p:s->t) (x:e) = 
+	  match (x, p 2) with ('a', false) -> false | _ -> p w;;
 
 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.
 
-<pre>
-intapp (lift (intapp think
-                     (intapp (lift left)
-                             (unit 'b'))))
-       (unit 'a') 
-1;; (* true *)
-</pre>
+	intapp (lift (intapp think
+						 (intapp (lift left)
+								 (unit 'b'))))
+		   (unit 'a') 
+	1;; (* true *)
 
 So in world 1, Ann thinks that Bill left (because in world 2, Bill did leave).
 
@@ -424,13 +629,11 @@ The `lift` is there because "think Bill left" is extensional wrt its
 subject.  The important bit is that "think" takes the intension of
 "Bill left" as its first argument.
 
-<pre>
-intapp (lift (intapp think
-                     (intapp (lift left)
-                             (unit 'c'))))
-       (unit 'a') 
-1;; (* false *)
-</pre>
+	intapp (lift (intapp think
+						 (intapp (lift left)
+								 (unit 'c'))))
+		   (unit 'a') 
+	1;; (* false *)
 
 But even in world 1, Ann doesn't believe that Cam left (even though he
 did: `intapp (lift left) cam 1 == true`).  Ann's thoughts are hung up
@@ -445,8 +648,8 @@ on what is happening in world 2, where Cam doesn't leave.
 Finally, note that within an intensional grammar, extensional funtion
 application is essentially just bind:
 
-<pre>
-# let swap f x y = f y x;;
-# bind cam (swap left) 2;;
-- : bool = false
-</pre>
+	# let swap f x y = f y x;;
+	# bind cam (swap left) 2;;
+	- : bool = false
+
+