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
+
+In the meantime, we'll look at several linguistic applications for
+monads, based on what's called the *reader monad*, starting with
+intensional function application.
+
+First, the familiar linguistic problem:
+
+ Bill left.
+ Cam left.
+ Ann believes [Bill left].
+ Ann believes [Cam left].
+
+We want an analysis on which all four of these sentences can be true
+simultaneously. If sentences denoted simple truth values or booleans,
+we have a problem: if the sentences *Bill left* and *Cam left* are
+both true, they denote the same object, and Ann's beliefs can't
+distinguish between them.
+
+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*. 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.
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:
+The traditional solution to the problem sketched above 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
+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
+S t s->t John left
+DP e s->e John
+VP e->t (s->e)->s->t left
+Vt e->e->t (s->e)->(s->e)->s->t saw
+Vs t->e->t (s->t)->(s->e)->s->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.
+generalized quantifiers.
The main difference between the intensional types and the extensional
types is that in the intensional types, the arguments are functions
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).
+In addition, the result of each predicate is an intension.
+This expresses the fact that the set of people who left in one world
+may be different than the set of people who left in a different world.
+Normally, the dependence of the extension of a predicate to the world
+of evaluation is hidden inside of an evaluation coordinate, or built
+into the the lexical meaning function, but we've made it explicit here
+in the way that the intensionality monad makes most natural.
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.
+types. Wouldn't it be nice to make the complicated types available
+for those expressions like 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:
`'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'
+<pre>
+let ann = 'a';;
+let bill = 'b';;
+let cam = 'c';;
-We'll assume that this and the other laws always hold.
+let left1 (x:e) = true;;
+let saw1 (x:e) (y:e) = y < x;;
-We now build up some extensional meanings:
+left1 ann;;
+saw1 bill ann;; (* true *)
+saw1 ann bill;; (* false *)
+</pre>
- let left w x = match (w,x) with (2,'c') -> false | _ -> true;;
+So here's our extensional system: everyone left, including Ann;
+and Ann saw Bill, but Bill didn't see Ann. (Note that Ocaml word
+order is VOS, verb-object-subject.)
-This function says that everyone always left, except for Cam in world
-2 (i.e., `left 2 'c' == false`).
+Now we add intensions. Because different people leave in different
+worlds, the meaning of *leave* must depend on the world in which it is
+being evaluated:
-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 left (x:e) (w:s) = match (x, w) with ('c', 2) -> false | _ -> true;;
- let extapp fn arg w = fn w (arg w);;
+This new definition says that everyone always left, except that
+in world 2, Cam didn't leave.
- extapp left ann 1;;
- # - : bool = true
+ let saw x y w = (w < 2) && (y < x);;
+ saw bill ann 1;; (* true: Ann saw Bill in world 1 *)
+ saw bill ann 2;; (* false: no one saw anyone in world 2 *)
- extapp left cam 2;;
- # - : bool = false
+Along similar lines, this general version of *see* coincides with the
+`saw1` function we defined above for world 1; in world 2, no one saw anyone.
-`extapp` stands for "extensional function application".
-So Ann left in world 1, but Cam didn't leave in world 2.
+Just to keep things straight, let's get the facts of the world set:
-A transitive predicate:
+<pre>
+ World 1: Everyone left.
+ Ann saw Bill, Ann saw Cam, Bill saw Cam, no one else saw anyone.
+ World 2: Ann left, Bill left, Cam didn't leave.
+ No one saw anyone.
+</pre>
- let saw w x y = (w < 2) && (y < x);;
- extapp (extapp saw bill) ann 1;; (* true *)
- extapp (extapp saw bill) ann 2;; (* false *)
+Now we are ready for the intensionality monad:
-In world 1, Ann saw Bill and Cam, and Bill saw Cam. No one saw anyone
-in world two.
+<pre>
+type 'a intension = s -> 'a;;
+let unit x (w:s) = x;;
+let bind m f (w:s) = f (m w) w;;
+</pre>
-Good. Now for intensions:
+Then the individual concept `unit ann` is a rigid designator: a
+constant function from worlds to individuals that returns `'a'` no
+matter which world is used as an argument. This is a typical kind of
+thing for a monad unit to do.
- let intapp fn arg w = fn w arg;;
+Then combining a prediction like *left* which is extensional in its
+subject argument with a monadic subject like `unit ann` is simply bind
+in action:
-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.
+ bind (unit ann) left 1;; (* true: Ann left in world 1 *)
+ bind (unit cam) left 2;; (* false: Cam didn't leave in world 2 *)
-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:
+As usual, bind takes a monad box containing Ann, extracts Ann, and
+feeds her to the extensional *left*. In linguistic terms, we take the
+individual concept `unit ann`, apply it to the world of evaluation in
+order to get hold of an individual (`'a'`), then feed that individual
+to the extensional predicate *left*.
- let lift pred w arg = bind arg (fun x w -> pred w x) w;;
+We can arrange for an extensional transitive verb to take intensional
+arguments:
- 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 *)
+ let lift f u v = bind u (fun x -> bind v (fun y -> f x y));;
-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.)
+This is the exact same lift predicate we defined in order to allow
+addition in our division monad example.
-Likewise for extensional transitive predicates like "saw":
+<pre>
+lift saw (unit bill) (unit ann) 1;; (* true *)
+lift saw (unit bill) (unit ann) 2;; (* false *)
+</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 *)
+Ann did see bill in world 1, but Ann didn't see Bill 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"):
+Finally, we can define our intensional verb *thinks*. *Think* is
+intensional with respect to its sentential complement, but extensional
+with respect to its 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;;
+ let thinks (p:s->t) (x:e) (w:s) =
+ 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 *)
+ bind (unit ann) (thinks (bind (unit bill) left)) 1;;
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 *)
+ bind (unit ann) (thinks (bind (unit cam) left)) 1;;
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.
+did: `bind (unit cam) left 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
-
-