`u >>=`_{set} \[[∃x]] >>=_{set} \[[Px]]

What does \[[∃x]] need to be here? Here's what they say, on the top of p. 13:
> Suppose an information state `s` is updated with the sentence ∃xPx. Possibilities in `s` in which no entity has the property P will be eliminated.
We can defer that to a later step, where we do `... >>= \[[Px]]`.
> The referent system of the remaining possibilities will be extended with a new peg, which is associated with `x`. And for each old possibility `i` in `s`, there will be just as many extensions `i[x/d]` in the new state `s'` and there are entities `d` which in the possible world of `i` have the property P.
Deferring the "property P" part, this corresponds to:
```
u updated with \[[∃x]] ≡
let extend_one = fun one_dpm ->
fun truth_value ->
if truth_value = false
then empty_set
else List.map (fun d -> new_peg_and_assign 'x' d) domain
in bind_set u extend_one
```

where `new_peg_and_assign` is the operation we defined in [hint 3](/hints/assignment_7_hint_3):
let new_peg_and_assign (var_to_bind : char) (d : entity) =
fun ((r, h) : assignment * store) ->
(* first we calculate an unused index *)
let newindex = List.length h
(* next we store d at h[newindex], which is at the very end of h *)
(* the following line achieves that in a simple but inefficient way *)
in let h' = List.append h [d]
(* next we assign 'x' to location newindex *)
in let r' = fun v ->
if v = var_to_bind then newindex else r v
(* the reason for returning true as an initial element should now be apparent *)
in (true, r',h')
What's going on here? For each `bool dpm` in `u` that wraps a `true`, we collect `dpm`s that are the result of extending their input `(r, h)` by allocating a new peg for entity `d`, for each `d` in our whole domain of entities, and binding the variable `x` to the index of that peg. For `bool dpm`s in `u` that wrap `false`, we just discard them. We could if we wanted instead return `unit_set (unit_dpm false)`.
A later step can then filter out all the `dpm`s according to which the
entity `d` we did that with doesn't have property P.
So if we just call the function `extend_one` defined above \[[∃x]], then `u` updated with \[[∃x]] updated with \[[Px]] is just:
```
u >>= \[[∃x]] >>= \[[Px]]
```

or, being explicit about which "bind" operation we're representing here with `>>=`, that is:
```
bind_set (bind_set u \[[∃x]]) \[[Px]]
```

* Can you figure out how to handle \[[not φ]] on your own? If not, here are some [more hints](/hints/assignment_7_hint_6).