Start by (re)reading the discussion of monads in the lecture notes for
week 6 [Towards Monads](http://lambda.jimpryor.net//week6/#index4h2).
-In those notes, we saw a way to separate thining about error
+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 monad: 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.
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
any new errors, so we do want to think about the difference between
-ints and int options. We tried to accomplish this by defining a
-`bind` operator, which enabled us to peel away the option husk to get
+`int`s and `int option`s. We tried to accomplish this by defining a
+`bind` operator, which enabled us to peel away the `option` husk to get
at the delicious integer inside. There was also a homework problem
-which made this even more convenient by mapping any bindary operation
+which made this even more convenient by mapping any binary operation
on plain integers into a lifted operation that understands how to deal
-with int options in a sensible way.
+with `int option`s in a sensible way.
[Linguitics note: Dividing by zero is supposed to feel like a kind of
presupposition failure. If we wanted to adapt this approach to
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 options, but when we're composing natural language
+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
any type for the `'a` in `'a option`. Ultimately, we'd want to have a
not surprisingly, these refinements will require some more
sophisticated techniques than the super-simple option monad.]
-So what examctly is a monad? As usual, we're not going to be pedantic
+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
that provides at least the following three elements:
`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 n by mapping an arbitrary integer `n` to
+ did this for any integer `n` by mapping it to
the option `Some n`. To be official, we can define a function
called unit:
happend to be a box containing an integer `n`, this allowed us to
get our hands on that `n` and feed it to `f`.
-So the "Option monad" consists of the polymorphic option type, the
-unit function, and the bind function.
+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
-A note on notation: some people use the infix operator `>==` to stand
+<pre>
+# let divide num den = if den = 0 then None else Some (num/den);;
+val divide : int -> int -> int option = <fun>
+</pre>
+
+as basically a function from two integers to an integer, except with
+this little bit of option frill, or option plumbing, on the side.
+
+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, `*`.
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>
- 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`.
+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`.
* Associativity: bind obeys a kind of associativity, like this:
- (m * f) * g == m * (fun x -> f x * g)
+ `(m * f) * g == m * (fun x -> f x * g)`
- If you don't understand why the lambda form is necessary, you need
- to look again at the type of bind. This is important.
+ 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.
- For an illustration of associativity in the option monad:
+ Some examples of associativity in the option monad:
<pre>
-Some 3 * unit * unit;;
+# Some 3 * unit * unit;;
- : int option = Some 3
-Some 3 * (fun x -> unit x * unit);;
+# 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>
- 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`
- matches `None`, both computations will result in `None`; if
- `m` matches `Some n`, and `f n` 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`.
+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`
+matches `None`, both computations will result in `None`; if
+`m` matches `Some n`, and `f n` 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`.
* Right identity: unit is a right identity for bind. That is,
`m * unit == m` for all monad objects `m`. For instance,
<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
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>).
+laws (see <http://www.haskell.org/haskellwiki/Monad_Laws>, near the bottom).
Monad outlook
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 `# #use
-"intensionality-monad.ml";;`.
+To run it, download the file, start Ocaml, and say
+
+ # #use "intensionality-monad.ml";;
+
+Note the extra `#` attached to the directive `use`.
Here's the idea: since people can have different attitudes towards
different propositions that happen to have the same truth value, we
-can't have sentences denoting simple truth values. Then if John
+can't have sentences denoting simple truth values. If we did, then if John
believed that the earth was round, it would force him to believe
Fermat's last theorem holds, since both propositions are equally true.
The traditional solution is to allow sentences to denote a function
</pre>
This system is modeled on the way Montague arranged his grammar.
-(There are significant simplifications: for instance, determiner
+There are significant simplifications: for instance, determiner
phrases are thought of as corresponding to individuals rather than to
-generalized quantifiers.) If you're curious about the initial `s`'s
+generalized quantifiers. If you're curious about the initial `s`'s
in the extensional types, they're there because the behavior of these
expressions depends on which world they're evaluated at. If you are
in a situation in which you can hold the evaluation world constant,
-you can further simplify the extensional types. (Usually, the
+you can further simplify the extensional types. Usually, the
dependence of the extension of an expression on the evaluation world
is hidden in a superscript, or built into the lexical interpretation
-function.)
+function.
The main difference between the intensional types and the extensional
types is that in the intensional types, the arguments are functions
those 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 concern about division by
-zero to the division function, and let the other functions ignore
-division-by-zero problems as much as possible.
+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: