```+let div' (x:int) (y:int) =
+  match y with
+	  0 -> None
+    | _ -> Some (x / y);;
+
+(*
+val div' : int -> int -> int option = fun
+# div' 12 2;;
+- : int option = Some 6
+# div' 12 0;;
+- : int option = None
+# div' (div' 12 2) 3;;
+Characters 4-14:
+  div' (div' 12 2) 3;;
+        ^^^^^^^^^^
+Error: This expression has type int option
+       but an expression was expected of type int
+*)
+```
-* A bind operation (note the type): +This starts off well: dividing 12 by 2, no problem; dividing 12 by 0, +just the behavior we were hoping for. But we want to be able to use +the output of the safe-division function as input for further division +operations. So we have to jack up the types of the inputs: - `let bind m f = match m with None -> None | Some n -> f n;;` +
```+let div' (u:int option) (v:int option) =
+  match u with
+	  None -> None
+	| Some x -> (match v with
+				  Some 0 -> None
+				| Some y -> Some (x / y));;
+
+(*
+val div' : int option -> int option -> int option =
+# div' (Some 12) (Some 2);;
+- : int option = Some 6
+# div' (Some 12) (Some 0);;
+- : int option = None
+# div' (div' (Some 12) (Some 0)) (Some 3);;
+- : int option = None
+*)
+```
+ +Beautiful, just what we need: now we can try to divide by anything we +want, without fear that we're going to trigger any system errors. - `val bind : 'a option -> ('a -> 'b option) -> 'b option = ` +I prefer to line up the `match` alternatives by using OCaml's +built-in tuple type: - `bind` takes two arguments (a monadic object and a function from - ordinary objects to monadic objects), and returns a monadic - object. +
```+let div' (u:int option) (v:int option) =
+  match (u, v) with
+	  (None, _) -> None
+    | (_, None) -> None
+    | (_, Some 0) -> None
+	| (Some x, Some y) -> Some (x / y);;
+```
- 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 +So far so good. But what if we want to combine division with +other arithmetic operations? We need to make those other operations +aware of the possibility that one of their arguments has triggered a +presupposition failure: - `let bind m f = (let x = unwrap m in f x);;` +
```+let add' (u:int option) (v:int option) =
+  match (u, v) with
+	  (None, _) -> None
+    | (_, None) -> None
+    | (Some x, Some y) -> Some (x + y);;
+
+(*
+val add' : int option -> int option -> int option =
+# add' (Some 12) (Some 4);;
+- : int option = Some 16
+# add' (div' (Some 12) (Some 0)) (Some 4);;
+- : int option = None
+*)
+```
- 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`. +This works, but is somewhat disappointing: the `add'` operation +doesn't trigger any presupposition of its own, so it is a shame that +it needs to be adjusted because someone else might make trouble. -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 +But we can automate the adjustment. The standard way in OCaml, +Haskell, etc., is to define a `bind` operator (the name `bind` is not +well chosen to resonate with linguists, but what can you do). To continue our mnemonic association, we'll put a `'` after the name "bind" as well.
```-# let divide num den = if den = 0 then None else Some (num/den);;
-val divide : int -> int -> int option =
+let bind' (u: int option) (f: int -> (int option)) =
+  match u with
+	  None -> None
+    | Some x -> f x;;
+
+let add' (u: int option) (v: int option)  =
+  bind' u (fun x -> bind' v (fun y -> Some (x + y)));;
+
+let div' (u: int option) (v: int option) =
+  bind' u (fun x -> bind' v (fun y -> if (0 = y) then None else Some (x / y)));;
+
+(*
+#  div' (div' (Some 12) (Some 2)) (Some 3);;
+- : int option = Some 2
+#  div' (div' (Some 12) (Some 0)) (Some 3);;
+- : int option = None
+# add' (div' (Some 12) (Some 0)) (Some 3);;
+- : int option = None
+*)
```
```-# let ( * ) m f = match m with None -> None | Some n -> f n;;
-val ( * ) : 'a option -> ('a -> 'b option) -> 'b option =
-# let unit x = Some x;;
-val unit : 'a -> 'a option =
+		# let unit x = Some x;;
+		val unit : 'a -> 'a option =
+		# let ( >>= ) u f = match u with None -> None | Some x -> f x;;
+		val ( >>= ) : 'a option -> ('a -> 'b option) -> 'b option =

-# unit 2;;
-- : int option = Some 2
-# unit 2 * unit;;
-- : int option = Some 2
+	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
+	`u >>= f` or equivalently `( >>= ) u f` instead of `>>= u
+	f`.

-# divide 6 2;;
-- : int option = Some 3
-# unit 2 * divide 6;;
-- : int option = Some 3
+		# unit 2;;
+		- : int option = Some 2
+		# unit 2 >>= unit;;
+		- : int option = Some 2

-# divide 6 0;;
-- : int option = None
-# unit 0 * divide 6;;
-- : int option = None
-```
+ Now, for a less trivial instance of a function from `int`s to `int option`s: -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`. + # let divide x y = if 0 = y then None else Some (x/y);; + val divide : int -> int -> int option = + # divide 6 2;; + - : int option = Some 3 + # unit 2 >>= divide 6;; + - : int option = Some 3 -* Associativity: bind obeys a kind of associativity, like this: + # divide 6 0;; + - : int option = None + # unit 0 >>= divide 6;; + - : int option = None - `(m * f) * g == m * (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. +* **Associativity: bind obeys a kind of associativity**. Like this: - Some examples of associativity in the option monad: + (u >>= f) >>= g == u >>= (fun x -> f x >>= g) -
```-# Some 3 * unit * unit;;
-- : int option = Some 3
-# Some 3 * (fun x -> unit x * unit);;
-- : int option = Some 3
+	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 3 * divide 6 * divide 2;;
-- : int option = Some 1
-# Some 3 * (fun x -> divide 6 x * divide 2);;
-- : int option = Some 1
+	Some examples of associativity in the option monad (bear in
+	mind that in the Ocaml implementation of integer division, 2/3
+	evaluates to zero, throwing away the remainder):

-# Some 3 * divide 2 * divide 6;;
-- : int option = None
-# Some 3 * (fun x -> divide 2 x * divide 6);;
-- : int option = None
-```
+ # 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` +Of course, associativity must hold for *arbitrary* functions of +type `'a -> 'b 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
-```
```-Extensional types                 Intensional types       Examples
--------------------------------------------------------------------
-
-S         s->t                    s->t                    John left
-DP        s->e                    s->e                    John
-VP        s->e->t                 s->(s->e)->t            left
-Vt        s->e->e->t              s->(s->e)->(s->e)->t    saw
-Vs        s->t->e->t              s->(s->t)->(s->e)->t    thought
-```
+That just means: -This system is modeled on the way Montague arranged his grammar. -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 -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 -dependence of the extension of an expression on the evaluation world -is hidden in a superscript, or built into the lexical interpretation -function. - -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 intensional 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). - -The intenstional types are more complicated than the intensional -types. Wouldn't it be nice to keep the complicated types to just -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, 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: - - 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. + u >>= fun _ -> v -
```-type 'a intension = s -> 'a;;
-let unit x (w:s) = x;;
+that is:

-let ann = unit 'a';;
-let bill = unit 'b';;
-let cam = unit 'c';;
-```
+ bind u (fun _ -> v) -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 -function (an instance of the K combinator) that returns the same -individual at each world. +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. -Then `ann = unit 'a'` is a rigid designator: a constant function from -worlds to individuals that returns `'a'` no matter which world is used -as an argument. +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's test compliance with the left identity law: + # let even x = (x mod 2 = 0);; + val g : int -> bool = -
```-# let bind m f (w:s) = f (m w) w;;
-val bind : (s -> 'a) -> ('a -> s -> 'b) -> s -> 'b =
-# bind (unit 'a') unit 1;;
-- : char = 'a'
-```
+`even` has the type `int -> bool`. Now what if we want to convert it into an operation on the option/maybe monad? -We'll assume that this and the other laws always hold. + # let lift g = fun u -> bind u (fun x -> Some (g x));; + val lift : ('a -> 'b) -> 'a option -> 'b option = -We now build up some extensional meanings: +`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 left w x = match (w,x) with (2,'c') -> false | _ -> true;; + # 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 = -This function says that everyone always left, except for Cam in world -2 (i.e., `left 2 'c' == false`). +`lift2 (+)` 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. -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: +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`! -
```-let extapp fn arg w = fn w (arg w);;
+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.

-extapp left ann 1;;
-# - : bool = true
+In general, any lift/map operation can be relied on to satisfy these laws:

-extapp left cam 2;;
-# - : bool = false
-```
+ * lift id = id + * lift (compose f g) = compose (lift f) (lift g) -`extapp` stands for "extensional function application". -So Ann left in world 1, but Cam didn't leave in world 2. +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: -A transitive predicate: + List.map (fun x -> f (g x)) lst + List.map f (List.map g lst) - let saw w x y = (w < 2) && (y < x);; - extapp (extapp saw bill) ann 1;; (* true *) - extapp (extapp saw bill) ann 2;; (* false *) +Another general monad operation is called `ap` in Haskell---short for "apply." (They also use `<*>`, but who can remember that?) This works like this: -In world 1, Ann saw Bill and Cam, and Bill saw Cam. No one saw anyone -in world two. + ap [f] [x; y] = [f x; f y] + ap (Some f) (Some x) = Some (f x) -Good. Now for intensions: +and so on. Here are the laws that any `ap` operation can be relied on to satisfy: - let intapp fn arg w = fn w arg;; + ap (unit id) u = u + ap (ap (ap (unit 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 -The only difference between intensional application and extensional -application is that we don't feed the evaluation world to the argument. -(See Montague's rules of (intensional) functional application, T4 -- T10.) -In other words, instead of taking an extension as an argument, -Montague's predicates take a full-blown intension. +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: -But for so-called extensional predicates like "left" and "saw", -the extra power is not used. We'd like to define intensional versions -of these predicates that depend only on their extensional essence. -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: + [[1]; [1;2]; [1;3]; [1;2;4]] -
```-let lift pred w arg = bind arg (fun x w -> pred w x) w;;
+to:

-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 *)
-```
+ [1; 1; 2; 1; 3; 1; 2; 4] -Because `bind` unwraps the intensionality of the argument, when the -lifted "left" receives an individual concept (e.g., `unit 'a'`) as -argument, it's the extension of the individual concept (i.e., `'a'`) -that gets fed to the basic extensional version of "left". (For those -of you who know Montague's PTQ, this use of bind captures Montague's -third meaning postulate.) +That is the `join` operation. -Likewise for extensional transitive predicates like "saw": - -
```-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 -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"): - -
```-let think (w:s) (p:s->t) (x:e) =
-  match (x, p 2) with ('a', false) -> false | _ -> p w;;
-```
+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: -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. + lift f u = u >>= compose unit f + lift f u = ap (unit f) u + lift2 f u v = u >>= (fun x -> v >>= (fun y -> unit (f x y))) + lift2 f u v = ap (lift f u) v = ap (ap (unit f) u) v + ap u v = u >>= (fun f -> lift f v) + ap u v = lift2 id u v + join m2 = m2 >>= id + u >>= f = join (lift f u) + u >> v = u >>= (fun _ -> v) + u >> v = lift2 (fun _ -> id) u v -
```-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). -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. +Monad outlook +------------- -
```-intapp (lift (intapp think
-                     (intapp (lift left)
-                             (unit 'c'))))
-       (unit 'a')
-1;; (* false *)
-```
```-# let swap f x y = f y x;;