+++ /dev/null
-Now we'll look at using monads to do intensional function application.
-This is just another application of the Reader monad, not a new monad.
-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 the 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.
-
-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`.
-
-First, the familiar linguistic problem:
-
- Bill left.
- Cam left.
- Ann believes [Bill left].
- Ann believes [Cam left].
-
-We want an analysis on which the first three sentences can be true at
-the same time that the last sentence is false. 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.
-
-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
--------------------------------------------------------------------
-
-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.
-
-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 so-called "individual 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 intensional types are more complicated than the extensional
-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:
-
-In OCaml, we'll use integers to model possible worlds. 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 s = int;;
- type e = char;;
- type t = bool;;
-
- let ann = 'a';;
- let bill = 'b';;
- let cam = 'c';;
-
- let left1 (x:e) = true;;
- let saw1 (x:e) (y:e) = y < x;;
-
- left1 ann;; (* true *)
- saw1 bill ann;; (* true *)
- saw1 ann bill;; (* false *)
-
-So here's our extensional system: everyone left, including Ann;
-and Ann saw Bill (`saw1 bill ann`), but Bill didn't see Ann. (Note that the word
-order we're using is VOS, verb-object-subject.)
-
-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:
-
- let left (x:e) (w:s) = match (x, w) with ('c', 2) -> false | _ -> true;;
-
-This new definition says that everyone always left, except that
-in world 2, Cam didn't leave.
-
- 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 *)
-
-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.
-
-Just to keep things straight, let's get the facts of the world set:
-
-<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>
-
-Now we are ready for the intensionality monad:
-
-<pre>
-type 'a intension = s -> 'a;;
-let unit x = fun (w:s) -> x;;
-let bind u f = fun (w:s) -> f (u w) w;;
-</pre>
-
-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.
-
-Then combining a predicate like *left* which is extensional in its
-subject argument with an intensional subject like `unit ann` is simply bind
-in action:
-
- bind (unit ann) left 1;; (* true: Ann left in world 1 *)
- bind (unit cam) left 2;; (* false: Cam didn't leave in world 2 *)
-
-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*.
-
-We can arrange for an extensional transitive verb to take intensional
-arguments:
-
- let lift2 f u v = bind u (fun x -> bind v (fun y -> f x y));;
-
-This is the exact same lift predicate we defined in order to allow
-addition in our division monad example.
-
-<pre>
-lift2 saw (unit bill) (unit ann) 1;; (* true *)
-lift2 saw (unit bill) (unit ann) 2;; (* false *)
-</pre>
-
-Ann did see bill in world 1, but Ann didn't see Bill in world 2.
-
-Finally, we can define our intensional verb *thinks*. *Think* is
-intensional with respect to its sentential complement, though still 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 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.
-
- 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).
-
- bind (unit ann) (thinks (bind (unit cam) left)) 1;;
-
-But in world 1, Ann doesn't believe that Cam left (even though he
-did leave in world 1: `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.
-
-