2 * How shall we handle \[[∃x]]. As we said, GS&V really tell us how to interpret \[[∃xPx]], but what they say about this breaks naturally into two pieces, such that we can represent the update of our starting set `u` with \[[∃xPx]] as:
4 <pre><code>u >>=<sub>set</sub> \[[∃x]] >>=<sub>set</sub> \[[Px]]
7 What does \[[∃x]] need to be here? Here's what they say, on the top of p. 13:
9 > 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.
11 We can defer that to a later step, where we do `... >>= \[[Px]]`.
13 > 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.
15 Deferring the "property P" part, this corresponds to:
17 <pre><code>u updated with \[[∃x]] ≡
18 let extend_one = fun one_dpm ->
20 if truth_value = false
22 else List.map (fun d -> new_peg_and_assign 'x' d) domain
23 in bind_set u extend_one
26 where `new_peg_and_assign` is the operation we defined in [hint 3](/hints/assignment_7_hint_3):
28 let new_peg_and_assign (var_to_bind : char) (d : entity) =
29 fun ((r, h) : assignment * store) ->
30 (* first we calculate an unused index *)
31 let newindex = List.length h
32 (* next we store d at h[newindex], which is at the very end of h *)
33 (* the following line achieves that in a simple but inefficient way *)
34 in let h' = List.append h [d]
35 (* next we assign 'x' to location newindex *)
37 if v = var_to_bind then newindex else r v
38 (* the reason for returning true as an initial element should now be apparent *)
41 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)`.
43 A later step can then filter out all the `dpm`s according to which the
44 entity `d` we did that with doesn't have property P.
46 So if we just call the function `extend_one` defined above \[[∃x]], then `u` updated with \[[∃x]] updated with \[[Px]] is just:
48 <pre><code>u >>= \[[∃x]] >>= \[[Px]]
51 or, being explicit about which "bind" operation we're representing here with `>>=`, that is:
53 <pre><code>bind_set (bind_set u \[[∃x]]) \[[Px]]
56 * Can you figure out how to handle \[[not φ]] on your own? If not, here are some [more hints](/hints/assignment_7_hint_6).