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[lambda.git] / topics / week8_reader_monad.mdwn

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[[!toc levels=2]]

The goal for these notes is to introduce the Reader Monad, and present
two linguistic applications: binding and intensionality. Along the
way, we'll continue to think through issues related to order, and a
related notion of flow of information.

At this point, we've seen monads in general, and three examples of
monads: the Identity monad (invisible boxes), the Option/Maybe monad (option
types), and the List monad.

We've also seen an application of the Option/Maybe monad to safe division.
The starting point was to allow the division function to return an `int
option` instead of an `int`. If we divide `6` by `2`, we get the answer `Some
3`. But if we divide `6` by `0`, we get the answer `None`.

The next step was to adjust the other arithmetic functions to teach
them what to do if they received `None` instead of a boxed integer.
This meant changing the type of their input from `int`s to `int option`s.
But we didn't need to do this piecemeal; rather, we "lift"ed the
ordinary arithmetic operations into the monad using the various tools
provided by the monad.

We'll go over this lifting operation in detail in the next section.

## Tracing the effect of `safe_div` on a larger computation

So let's see how this works in terms of a specific computation.

<!--
\tree ((((+) (1)) (((*) (((/) (6)) (2))) (4))))
-->
<pre>
(+ 1 (* (/ 6 2) 4)) in tree format:

 ___________
 |         |
_|__    ___|___
|  |    |     |
+  1  __|___  4
      |    |
      *  __|___
         |    |
        _|__  2
        |  |
        /  6
</pre>

No matter what order we evaluate it in, this computation
should eventually reduce to `13`. Given a specific reduction strategy,
we can watch the order in which the computation proceeds. Following
on the evaluator developed during the previous homework, let's
adopt the following reduction strategy:

> In order to reduce an expression of the form (head arg), do the following in order:

> 1. Reduce `head` to `head'`
> 2. Reduce `arg` to `arg'`.
> 3. If `(head' arg')` is a redex, reduce it.

There are many details left unspecified here, but this will be enough
for today. The order in which the computation unfolds will be

> 1.  Reduce head `(+ 1)` to itself
> 2.  Reduce arg `((* ((/ 6) 2)) 4)`
>     1.  Reduce head `(* ((/ 6) 2))`
>         1.  Reduce head `*` to itself
>         2.  Reduce arg `((/ 6) 2)`
>             1.  Reduce head `(/ 6)` to itself
>             2.  Reduce arg `2` to itself
>             3.  Reduce `((/ 6) 2)` to `3`
>         3.  Reduce `(* 3)` to itself
>     2.  Reduce arg `4` to itself
>     3.  Reduce `((* 3) 4)` to `12`
> 3.  Reduce `((+ 1) 12)` to `13`

This reduction pattern follows the structure of the original
expression exactly, at each node moving first to the left branch,
processing the left branch, then moving to the right branch, and
finally processing the results of the two subcomputation. (This is
called a depth-first postorder traversal of the tree.)

[diagram with arrows traversing the tree]

It will be helpful to see how the types change as we make adjustments.

    type num = int
    type contents = Num of num   | Op2 of (num -> num -> num)
    type tree = Leaf of contents | Branch of tree * tree

Never mind that these types will allow us to construct silly
arithmetric trees such as `+ *` or `2 3`. Note that during the
reduction sequence, the result of reduction was in every case a
well-formed subtree. So the process of reduction could be animated by
replacing subtrees with the result of reduction on that subtree, until
the entire tree is replaced by a single integer (namely, `13`).

Now we replace the number `2` with `0`:

<!--
\tree ((((+) (1)) (((*) (((/) (6)) (0))) (4))))
-->
<pre>
 ___________
 |         |
_|__    ___|___
|  |    |     |
+  1  __|___  4
      |    |
      *  __|___
         |    |
        _|__  0
        |  |
        /  6
</pre>

When we reduce, we get quite a ways into the computation before things
break down:

> 1.  Reduce head `(+ 1)` to itself
> 2.  Reduce arg `((* ((/ 6) 0)) 4)`
>     1.  Reduce head `(* ((/ 6) 0))`
>         1.  Reduce head `*` to itself
>         2.  Reduce arg `((/ 6) 0)`
>             1.  Reduce head `(/ 6)` to itself
>             2.  Reduce arg `0` to itself
>             3.  Reduce `((/ 6) 0)` to ACKKKK

This is where we replace `/` with `safe_div`. `safe_div` returns not an `int`, but an `int option`.  If the division goes
through, the result is `Some n`, where `n` is the integer result.
But if the division attempts to divide by zero, the result is `None`.

We could try changing the type of the arithmetic operators from `int
-> int -> int` to `int -> int -> int option`; but since we now have to
anticipate the possibility that *any* argument might involve division by
zero inside of it, it would be better to prepare for the possibility
that any subcomputation might return `None` here. So our operators should have
the type `int option -> int option -> int option`. Let's bring that about
by just changing the type `num` from `int` to `int option`, leaving everying else the same:

    type num = int option
    type contents = Num of num   | Op2 of (num -> num -> num)
    type tree = Leaf of contents | Branch of tree * tree

The only difference is that instead of defining our numbers to be
simple integers, they are now `int option`s; and so `Op` is an operator
over `int option`s.

At this point, we bring in the monadic machinery. In particular, here
is the `⇧` and the `map2` function from the notes on safe division:

    ⇧ (x : 'a) = Some x

    map2 (f : 'a -> 'b -> 'c) (xx : 'a option) (yy : 'b option) =
      match xx with
      | None -> None
      | Some x ->
          (match yy with
          | None -> None
          | Some y -> Some (f x y))

Then we lift the entire computation into the monad by applying `⇧` to
the integers, and by applying `map2` to the operators. Only, we will replace `/` with `safe_div`, defined as follows:

    safe_div (xx : 'a option) (yy : 'b option) =
      match xx with
      | None -> None
      | Some x ->
          (match yy with
          | None -> None
          | Some 0 -> None
          | Some y -> Some ((/) x y))

<!--
\tree ((((map2 +) (⇧1)) (((map2 *) (((map2 /) (⇧6)) (⇧0))) (⇧4))))
-->
<pre>
     ___________________
     |                 |
  ___|____         ____|_____
  |      |         |        |
map2 +  ⇧1    _____|_____  ⇧4
              |         |
            map2 *  ____|____
                    |       |
                 ___|____  ⇧0
                 |      |
             safe_div  ⇧6
</pre>

With these adjustments, the faulty computation now completes smoothly:

> 1.  Reduce head `((map2 +) ⇧1)`
> 2.  Reduce arg `(((map2 *) ((safe_div ⇧6) ⇧0)) ⇧4)`
>     1.  Reduce head `((map2 *) ((safe_div ⇧6) ⇧0))`
>         1.  Reduce head `(map2 *)`
>         2.  Reduce arg `((safe_div ⇧6) ⇧0)`
>             1.  Reduce head `(safe_div ⇧6)`
>             2.  Reduce arg `⇧0`
>             3.  Reduce `((safe_div ⇧6) ⇧0)` to `None`
>         3.  Reduce `((map2 *) None)`
>     2.  Reduce arg `⇧4`
>     3.  Reduce `(((map2 *) None) ⇧4)` to `None`
> 3.  Reduce `(((map2 +) ⇧1) None)` to `None`

As soon as we try to divide by `0`, `safe_div` returns `None`.
Thanks to the details of `map2`, the fact that `None` has been returned
by one of the arguments of a `map2`-ed operator guarantees that the
`map2`-ed operator will pass on the `None` as its result. So the
result of each enclosing computation will be `None`, up to the root
of the tree.

It is unfortunate that we need to continue the computation after
encountering our first `None`. We know immediately at the result of
the entire computation will be `None`, yet we continue to compute
subresults and combinations. It would be more efficient to simply
jump to the top as soon as `None` is encoutered. Let's call that
strategy Abort. We'll arrive at an `Abort` operator later in the semester.

So at this point, we can see how the Option/Maybe monad provides
plumbing that allows subcomputations to send information from one part
of the computation to another. In this case, the `safe_div` function
can send the information that division by zero has been attempted
upwards to the rest of the computation. If you think of the plumbing
as threaded through the tree in depth-first, postorder traversal, then
`safe_div` drops `None` into the plumbing half way through the
computation, and that `None` travels through the rest of the plumbing
till it comes out of the result faucet at the top of the tree.


## Information flowing in the other direction: top to bottom

We can think of this application as facilitating information flow.
In the `safe_div` example, a subcomputation created a message that
propagated upwards to the larger computation:

<pre>
                 message: Division by zero occurred!
                      ^
 ___________          |
 |         |          |
_|__    ___|___       |
|  |    |     |       |
+  1  __|___  4       |
      |    |          |
      *  __|___  -----|
         |    |
        _|__  0
        |  |
        /  6
</pre>

(The message was implemented by `None`.)

We might want to reverse the direction of information flow, making 
information available at the top of the computation available to the
subcomputations: 

<pre>
                    [λn]  
 ___________          |
 |         |          |
_|__    ___|___       |
|  |    |     |       |
+  1  __|___  4       |
      |    |          |
      *  __|___       |
         |    |       |
        _|__  n  <----|
        |  |
        /  6
</pre>

We've seen exactly this sort of configuration before: it's exactly
what we have when a lambda binds a variable that occurs in a deeply
embedded position. Whatever the value of the argument that the lambda
form combines with, that is what will be substituted in for free
occurrences of that variable within the body of the lambda.


## Binding

So our next step is to add a (primitive) version of binding to our
computation. We'll allow for just one binding dependency for now, and
then generalize later.

Binding is independent of the safe division, so we'll return to a
situation in which the Option/Maybe monad hasn't been added. One of the nice
properties of programming with monads is that it is possible to add or
subtract layers according to the needs of the moment. Since we need
simplicity, we'll set the Option/Maybe monad aside for now.

So we'll go back to the point where the num type is simple `int`, not
`int option`s.

    type num = int

And we won't be using the `map2` or `⇧` from the Option/Maybe monad anymore.

As you might guess, the technique we'll use to arrive at binding will
be to use the Reader monad, defined here in terms of `mid`/`⇧` and `mbind`/`>>=`:

<pre>
type <u>α</u> = int -> α
⇧x = \n. x
xx >>= k = \n. k (xx n) n
map  f xx = \n. f (xx n)
ff ¢ xx = \n. (ff n) (xx n)
map2 f xx yy = map f xx ¢ yy = \n. f (xx n) (yy n)
</pre>

A boxed type in this monad will be a function from an integer to an
value in the original type. The `mid`/`⇧` function lifts a value `x` to
a function that expects to receive an integer, but throws away the
integer and returns `x` instead (most values in the computation don't
depend on the input integer).

The `mbind`/`>>=` function in this monad takes a monadic value `xx`, a function
`k` taking non-monadic values into the monad, and returns a function
that expects an integer `n`. It feeds `n` to `xx`, which delivers a
result in the orginal type, which is fed in turn to `k`. `k` returns
a monadic value, which upon being fed an integer, also delivers a result
in the orginal type.

The `map`, `¢`, and `map2` functions corresponding to this `mbind` are given
above. They may look familiar --- we'll comment on this in a moment.

Lifing the computation into the monad, we have the following adjusted
types:

    type num = int -> int

That is, `num` is once again replaced with the type of a boxed `int`.
When we were dealing with the Option/Maybe monad, a boxed `int` had type `int
option`. In this monad, a boxed int has type `int -> int`.

<!--
\tree ((((map2 +) (⇧1)) (((map2 *) (((map2 /) (⇧6)) (x))) (⇧4))))
-->
<pre>
     __________________
     |                |
  ___|____        ____|_____
  |      |        |        |
map2 +  ⇧1    ____|_____  ⇧4
              |        |
            map2 *  ___|____
                    |      |
                 ___|____  x
                 |      |
               map2 /  ⇧6
</pre>

It remains only to decide how the variable `n` will access the input value supplied
at the top of the tree. The input value is supposed to be the
value that gets used for the variable `n`. Like every leaf in the tree
in argument position, the code we want in order to represent the
variable will have the type of a boxed `int`, namely, `int -> int`. So
we have the following:

    var_n = fun (n : int) -> n

That is, variables in this system denote the identity function!

The result of evaluating this tree will be a boxed integer: a function
from any integer `n` to `(+ 1 (* (/ 6 n)) 4)`.

Take a look at the definition of the Reader monad again. The
`mid` takes some object `x` and returns `\n. x`. In other words,
`⇧x = Kx`, for our familiar combinator **K**, so `⇧ = K`. Likewise, the `map` operation
is just function composition, and the `mapply`/`¢` operation is our friend the **S** combinator. `map2 f xx yy` for the Reader monad is `(f ○ xx) ¢ yy` or `S (f ○ xx) yy`.

We've seen this before as a strategy for translating a binding
construct into a set of combinators. To remind you, here is a part of
the general scheme for translating a lambda abstract into Combinatory
Logic. The translation function `[.]` translates a lambda term into a
term in Combinatory Logic:

    [(MN)] = ([M] [N])
    [\a.a] = I
    [\a.M] = K[M]       (assuming a not free in M)
    [\a.(MN)] = S[\a.M][\a.N]

The reason we can make do with this subset of the full translation
scheme is that we're making the simplifying assumption that there is
at most a single lambda involved. So once again we have the identity
function **I** as the translation of the bound variable, **K** as the function
governing expressions that don't contain an instance of the bound
variable, **S** as an operation that manages `¢`, that is, the applicative combination of complex expressions.


## Jacobson's Variable Free Semantics as a Reader Monad

We've designed the presentation above to make it as easy as possible
to show that Jacobson's Variable Free Semantics (e.g., Jacobson 1999,
[Towards a Variable-Free
Semantics](http://www.springerlink.com/content/j706674r4w217jj5/))
implements a Reader monad.

More specifically, it will turn out that Jacobson's geach combinator
*g* is exactly our `map` operator, and her binding combinator *z* is
exactly our `mbind` (though with the arguments reversed).

Jacobson's system contains two main combinators, *g* and *z*. She
calls *g* the Geach rule, and *z* performs binding. Here is a typical
computation. This implementation is based closely on email from Simon
Charlow, with beta reduction as performed by the on-line evaluator:

<pre>
; Jacobson's analysis of "Everyone_i thinks he_i left"
let g = \f xx. \n. f (xx n) in ; or `f ○ xx`
let z = \k xx. \n. k (xx n) n in ; or `S (flip k) xx`, or `Reader.(>>=) xx k`
let he = \n. n in
let everyone = \P. FORALL i (P i) in

everyone (z thinks (g left he))

~~>  FORALL i (thinks (left i) i)
</pre>

Two things to notice: First, pronouns once again denote identity
functions, just as we saw in the reader monad in the previous section.

Second, *g* plays the role of transmitting a binding dependency for an
embedded constituent to a containing constituent.

The basic recipe in Jacobson's system is that you transmit the
dependence of a pronoun upwards through the tree using *g* until just
before you are about to combine with the binder, when you finish off
with *z*. Here is an example with a longer chain of *g*'s:

<pre>
; "Everyone_i thinks that Bill said he_i left"

everyone (z thinks (g (T bill) (g said (g left he))))
; or `everyone (thinks =<< T bill ○ said ○ left ○ I)`

~~> FORALL i (thinks (said (left i) bill) i)
</pre>

If you compare Jacobson's values for *g* and *z* to the functions in
the reader monad given above, you'll see that Jacobson's *g*
combinator is exactly our `map` operator. Furthermore, Jacobson's *z*
combinator is identical to our Reader `>>=`, except with the order of the
arguments reversed (i.e., `(z k xx) == (xx >>= k)`).

(The `T` combinator in the derivations above is given by `T x <~~> \f. f x`;
it handles situations in which English word order reverses
the usual function/argument order. `T` is what Curry and Steedman call this
combinator. Jacobson calls it "lift", but it shouldn't be confused with the
`mid` and `map` operations that lift values into the Reader monad we're focusing
on here. It does lift values into a *different* monad, that we'll consider in
a few weeks.)

In other words, 
Jacobson's variable-free semantics is essentially a Reader monad.

One of the peculiar aspects of Jacobson's system is that binding is
accomplished not by applying *z* to the element that will (in some
pre-theoretic sense) bind the pronoun, here, *everyone*, but rather by
applying *z* instead to the predicate that will take *everyone* as an
argument, here, *thinks*.


## The Reader monad for intensionality

Now we'll look at using the Reader monad to do 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 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: [[code/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 that Bill left.
    Ann believes that 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 compared to Montague: for
instance, determiner phrases are thought of here 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 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. `char`s (characters in the computational sense, i.e., letters like `'a'` and `'b'`, not Kaplanian characters) will model individuals, and OCaml `bool`s 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 (y : e) (x : e) = x < y 

        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
    left ann 1 (* true: Ann left in world 1 *)
    left cam 2 (* false: Cam didn't leave in world 2 *) 

This new definition says that everyone always left, except that 
in world 2, Cam didn't leave.

Note that although this general *left* is sensitive to world of
evaluation, it does not have the fully intensionalized type given in
the chart above, which was `(s->e)->s->t`. This is because
*left* does not exploit the additional resolving power provided by
making the subject an individual concept. In semantics jargon, we say
that *leave* is extensional with respect to its first argument.

We will adopt the general strategy of defining predicates
in a way that they take arguments of the lowest type that will allow
us to make all the distinctions the predicate requires. When it comes
time to combine this predicate with monadic arguments, we'll have to
make use of various lifting predicates.

Likewise, although *see* depends on the world of evaluation, it is
extensional in both of its syntactic arguments:

    let saw (y : e) (x : e) (w : s) = (w < 2) && (x < y)
    saw bill ann 1 (* true: Ann saw Bill in world 1 *)
    saw bill ann 2 (* false: no one saw anyone in world 2 *)

This (again, partially) intensionalized 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 review the facts:

<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 mid x = fun (w : s) -> x
let (>>=) xx k = fun (w : s) -> let x = xx w in let yy = k x in yy w
                 (* or just `fun w -> k (xx w) w` *)
</pre>

Then the individual concept `mid 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 monadic `mid` to do.

Then combining a predicate like *left* which is extensional in its
subject argument with an intensional subject like `mid ann` is simply `mbind`
in action:

    (mid ann >>= left) 1  (* true: Ann left in world 1 *)
    (mid cam >>= left) 2  (* false: Cam didn't leave in world 2 *)

As usual, `>>=` takes a monadic box containing Ann, extracts Ann, and
feeds her to the function *left*. In linguistic terms, we take the
individual concept `mid ann`, apply it to the world of evaluation in
order to get hold of an individual (`'a'`), then feed that individual
to the predicate *left*.

We can arrange for a transitive verb that is extensional in both of
its arguments to take intensional arguments:

<pre>
let map2' f xx yy = xx >>= (fun x -> yy >>= (fun y -> <span class=ul>f x y</span>))
</pre>

This is almost the same `map2` predicate we defined in order to allow
addition in our division monad example. The *difference* is that this
variant operates on verb meanings that take extensional arguments but
returns an intensional result. Thus the original `map2` predicate
has `mid (f x y)` where we have just <code><span class=ul>f x y</span></code> here.

The use of `>>=` here to combine *left* with an individual concept,
and the use of `map2'` to combine *see* with two intensional
arguments closely parallels the two of Montague's meaning postulates
(in PTQ) that express the relationship between extensional verbs and
their uses in intensional contexts.

<pre>
map2' saw (mid bill) (mid ann) 1  (* true *)
map2' saw (mid bill) (mid 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 (it takes complements of type `s -> t`), though still extensional
with respect to its subject (type `e`). (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

In every world, Ann fails to believe any proposition that is false in world 2.
Apparently, she stably thinks we may be in world 2. Otherwise, everyone
believes a proposition iff that proposition is true in the world of evaluation.

    (mid ann >>= thinks (mid bill >>= left)) 1  (* true *)

So in world 1, Ann thinks that Bill left (because in worlds 1 and 2, Bill did leave).

But although Cam left in world 1:

    (mid cam >>= left) 1  (* true *)

he didn't leave in world 2, so Ann doesn't in world 1 believe that Cam left:

    (mid ann >>= thinks (mid cam >>= left)) 1  (* false *)


**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 `mbind`), and the non-intersective adjectives will take
 intensional arguments.

**Connections with variable binding**: the rigid individual concepts generated by `mid ann` and the like correspond to the numerical constants, that don't interact with the environment in any way, in the variable binding examples we considered earlier on the page. If we had any non-contingent predicates that were wholly insensitive to intensional effects, they would be modeled using `map2` and would correspond to the operations like `map2 (+)` in the earlier examples. As it is, our predicates *lift* and *saw*, though only sensitive to the *extension* of their arguments, nonetheless *are* sensitive to the world of evaluation for their `bool` output. So these are somewhat akin, at the predicate level, to expressions like `var_n`, at the singular term level, in the variable bindings examples. Our predicate *thinks* shows yet a further kind of interaction with the intensional structures we introduced: namely, its output can depend upon evaluating its complement relative to other possible worlds. We didn't discuss analogues of this in the variable binding case, but they exist: namely, they are expressions like `let x = 2 in ...` and `forall x ...`, that have the effect of supplying their arguments with an environment or assignment function that is shifted from the one they are themselves being evaluated with.

notes: cascade, general env