reader: tweaking markup

[lambda.git] / intensionality_monad.mdwn

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