X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week7.mdwn;h=abb6cfe82690c24556155b8dcf0b730bddcc2535;hp=c81ff1253ceca9ee8cb724163cee5e139bcfe234;hb=7d27bb2510dd466a8c8135d83b3dc0150ce93bef;hpb=1713e01a3a0982e0f8fc68ed93035cea6ca8f46e diff --git a/week7.mdwn b/week7.mdwn index c81ff125..abb6cfe8 100644 --- a/week7.mdwn +++ b/week7.mdwn @@ -1,28 +1,155 @@ [[!toc]] -Monads ------- - -Start by (re)reading the discussion of monads in the lecture notes for -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` 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 one choice with an `int` payload, or else a `None` -choice which we interpret as signaling that something has gone wrong. +Towards Monads: Safe division +----------------------------- -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 -`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 binary operation -on plain integers into a lifted operation that understands how to deal -with `int option`s in a sensible way. +[This section used to be near the end of the lecture notes for week 6] + +We begin by reasoning about what should happen when someone tries to +divide by zero. This will lead us to a general programming technique +called a *monad*, which we'll see in many guises in the weeks to come. + +Integer division presupposes that its second argument +(the divisor) is not zero, upon pain of presupposition failure. +Here's what my OCaml interpreter says: + + # 12/0;; + Exception: Division_by_zero. + +So we want to explicitly allow for the possibility that +division will return something other than a number. +We'll use OCaml's `option` type, which works like this: + + # type 'a option = None | Some of 'a;; + # None;; + - : 'a option = None + # Some 3;; + - : int option = Some 3 + +So if a division is normal, we return some number, but if the divisor is +zero, we return `None`. As a mnemonic aid, we'll append a `'` to the end of our new divide function. + +
+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
+*)
+
+ +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 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. + +I prefer to line up the `match` alternatives by using OCaml's +built-in tuple type: + +
+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);;
+
+ +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 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
+*)
+
+ +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. + +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 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
+*)
+
+ +Compare the new definitions of `add'` and `div'` closely: the definition +for `add'` shows what it looks like to equip an ordinary operation to +survive in dangerous presupposition-filled world. Note that the new +definition of `add'` does not need to test whether its arguments are +None objects or real numbers---those details are hidden inside of the +`bind'` function. + +The definition of `div'` shows exactly what extra needs to be said in +order to trigger the no-division-by-zero presupposition. [Linguitics note: Dividing by zero is supposed to feel like a kind of presupposition failure. If we wanted to adapt this approach to @@ -39,6 +166,29 @@ 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.] + +Monads in General +----------------- + +We've just seen 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` +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 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 +any new errors, so we would rather not think about the difference +between `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 defining a +`lift` operator that mapped any binary operation on plain integers +into a lifted operation that understands how to deal with `int +option`s in a sensible way. + So what exactly is a monad? We can consider a monad to be a system that provides at least the following three elements: @@ -59,7 +209,12 @@ that provides at least the following three elements: 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. + uses. [The rationale for "unit" comes from the monad laws + (see below), where the unit function serves as an identity, + just like the unit number (i.e., 1) serves as the identity + object for multiplication. The rationale for "return" comes + from a misguided desire to resonate with C programmers and + other imperative types.] 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 @@ -81,7 +236,7 @@ that provides at least the following three elements: 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 +* Thirdly, an operation that's often called `bind`. As we said before, 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: @@ -119,6 +274,8 @@ that provides at least the following three elements: 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. + (Are you beginning to realize how wierd and wonderful monads + can be?) 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 @@ -128,17 +285,11 @@ So the "option/maybe monad" consists of the polymorphic `option` type, the `unit`/return function, and the `bind` function. -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 ⋆, 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 ⋆ 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...) +A note on notation: Haskell uses the infix operator `>>=` to stand for +`bind`: wherever you see `u >>= f`, that means `bind u f`. +Wadler uses ⋆, but that hasn't been widely adopted (unfortunately). -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: +Also, if you ever see this notation: do x <- u @@ -152,9 +303,14 @@ Similarly: y <- v f x y -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 last expression. +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 last expression. You can think of the +notation like this: take the singing box `u` and evaluate it (which +includes listening to the song). Take the int contained in the +singing box (the end result of evaluting `u`) and bind the variable +`x` to that int. So `x <- u` means "Sing me up an int, which I'll call +`x`". (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 @@ -209,64 +365,69 @@ 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 - type, we have `(unit x) * f == f x`. For instance, `unit` is itself + That is, for all `f:'a -> 'b m`, where `'b 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`: # 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 = + # let ( >>= ) u f = match u with None -> None | Some x -> f x;; + val ( >>= ) : 'a option -> ('a -> 'b option) -> 'b option = 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 `( * ) u f` instead of `* u f`. Now: + is `>>=`, pronounced "bind") is an infix operator, so we write + `u >>= f` or equivalently `( >>= ) u f` instead of `>>= u + f`. # unit 2;; - : int option = Some 2 - # unit 2 * unit;; + # unit 2 >>= unit;; - : int option = Some 2 + Now, for a less trivial instance of a function from `int`s to `int option`s: + # 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;; + # unit 2 >>= divide 6;; - : int option = Some 3 # divide 6 0;; - : int option = None - # unit 0 * divide 6;; + # unit 0 >>= divide 6;; - : int option = None * **Associativity: bind obeys a kind of associativity**. Like this: - (u * f) * g == u * (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 (bear in + mind that in the Ocaml implementation of integer division, 2/3 + evaluates to zero, throwing away the remainder): - # 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;; + # Some 3 >>= divide 6 >>= divide 2;; - : int option = Some 1 - # Some 3 * (fun x -> divide 6 x * divide 2);; + # Some 3 >>= (fun x -> divide 6 x >>= divide 2);; - : int option = Some 1 - # Some 3 * divide 2 * divide 6;; + # Some 3 >>= divide 2 >>= divide 6;; - : int option = None - # Some 3 * (fun x -> divide 2 x * divide 6);; + # Some 3 >>= (fun x -> divide 2 x >>= divide 6);; - : int option = None Of course, associativity must hold for *arbitrary* functions of -type `'a -> 'a m`, where `m` is the monad type. It's easy to +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 @@ -276,11 +437,11 @@ computations will again result in `None`; and if the value of to `g y`. * **Right identity: unit is a right identity for bind.** That is, - `u * unit == u` for all monad objects `u`. For instance, + `u >>= unit == u` for all monad objects `u`. For instance, - # Some 3 * unit;; + # Some 3 >>= unit;; - : int option = Some 3 - # None * unit;; + # None >>= unit;; - : 'a option = None @@ -290,23 +451,18 @@ 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 * u == u == u * 1` for all +serving as the left and right identity. That is, `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 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). +Theory](/advanced_topics/monads_in_category_theory). See also . 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) 1991. - -* [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. - +* [Eugenio Moggi, Notions of Computation and Monads](http://www.disi.unige.it/person/MoggiE/ftp/ic91.pdf): Information and Computation 93 (1) 1991. Would be very difficult reading for members of this seminar. However, the following two papers should be accessible. * [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. @@ -314,12 +470,19 @@ invited talk, *19'th Symposium on Principles of Programming Languages*, ACM Pres 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.--> -* [Daniel Friedman. A Schemer's View of Monads](/schemersviewofmonads.ps): from but hosted the link above is to a local copy. +* [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. + -There's a long list of monad tutorials on the [[Offsite Reading]] page. Skimming the titles makes me laugh. + +There's a long list of monad tutorials on the [[Offsite Reading]] page. (Skimming the titles is somewhat amusing.) If you are confused by monads, make use of these resources. Read around until you find a tutorial pitched at a level that's helpful for you. 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. +Theory](/advanced_topics/monads_in_category_theory) notes do so, for example. Here are some of the other general monad operations. You don't have to master these; they're collected here for your reference. @@ -337,7 +500,7 @@ that is: 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: +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 = @@ -412,244 +575,15 @@ Monad outlook ------------- We're going to be using monads for a number of different things in the -weeks to come. The first main application will be the State monad, +weeks to come. One major application will be the State monad, 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 ------------------------- -... -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 -conceptually the same thing as making use of a *reader monad* (which -we'll see again soon). This technique was beautifully re-invented -by Ben-Avi and Winter (2007) in their paper [A modular -approach to -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 - - # #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. 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 -from worlds to truth values, what Montague called an intension. -So if `s` is the type of possible worlds, we have the following -situation: - - -
-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
-
- -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. - - type 'a intension = s -> 'a;; - let unit x (w:s) = x;; - - 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 -function (an instance of the K combinator) that returns the same -individual at each world. - -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. - -Let's test compliance with the left identity law: - - # let bind u f (w:s) = f (u w) w;; - val bind : (s -> 'a) -> ('a -> s -> 'b) -> s -> 'b = - # 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;; - -This function says that everyone always left, except for Cam in world -2 (i.e., `left 2 'c' == false`). - -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: - - let extapp fn arg w = fn w (arg w);; - - extapp left ann 1;; - # - : bool = true - - 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 *) - -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;; - -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. - -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: - - 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 *) - -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.) - -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;; - -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. - - 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. - - 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 -on what is happening in world 2, where Cam doesn't leave. - -*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 - intensional arguments. - -Finally, note that within an intensional grammar, extensional funtion -application is essentially just bind: +But first, we'll look at several linguistic applications for monads, based +on what's called the *Reader monad*. - # let swap f x y = f y x;; - # bind cam (swap left) 2;; - - : bool = false +##[[Reader monad for Variable Binding]]## +##[[Reader monad for Intensionality]]##