> Topics: Types, Polymorphism, Unit and Bottom, Dividing by Zero/[[Towards Monads]]
-(1 Nov) Lecture notes for Week7; Assignment6.
+(1 Nov) Lecture notes for [[Week7]]; Assignment6.
-> Topics: Monads
+> Topics: Monads; [[Reader Monad]]; [[Intensionality Monad]]
[[Upcoming topics]]
--- /dev/null
+The intensionality monad
+------------------------
+In the meantime, we'll look at several linguistic applications for monads, based
+on
+
+what's called the *reader 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:
+
+
+<pre>
+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
+</pre>
+
+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 = <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;;
+
+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:
+
+ # let swap f x y = f y x;;
+ # bind cam (swap left) 2;;
+ - : bool = false
+
+
--- /dev/null
+Introduce
+
+Heim and Kratzer's "Predicate Abstraction Rule"
+
+
+
In the meantime, we'll look at several linguistic applications for monads, based
on what's called the *reader monad*.
+##[[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:
-
-
-<pre>
-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
-</pre>
-
-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 = <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;;
-
-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:
-
- # let swap f x y = f y x;;
- # bind cam (swap left) 2;;
- - : bool = false
-
+##[[Intensionality monad]]##