-The intensionality monad
-------------------------
-
-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
+Now we'll look at using monads to do intensional function application.
+This really 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.
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
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
+take 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
+(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.
+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 make the complicated types available
thing for a monad unit to do.
Then combining a prediction like *left* which is extensional in its
-subject argument with a monadic subject like `unit ann` is simply bind
+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 *)