-* 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 `s` with \[[∃xPx]] as:
+* 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:
- <pre><code>s >>= \[[∃x]] >>= \[[Px]]
+ <pre><code>u >>=<sub>set</sub> \[[∃x]] >>=<sub>set</sub> \[[Px]]
</code></pre>
What does \[[∃x]] need to be here? Here's what they say, on the top of p. 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.
- Deferring the "property P" part, this says:
+ Deferring the "property P" part, this corresponds to:
- <pre><code>s updated with \[[∃x]] ≡
- s >>= (fun (r, h) -> List.map (fun d -> newpeg_and_bind 'x' d) domain)
+ <pre><code>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
</code></pre>
+
+ 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')
- That is, for each pair `(r, h)` in `s`, we collect the result of extending `(r, h)` by allocating a new peg for entity `d`, for each `d` in our whole domain of entities (here designated `domain`), and binding the variable `x` to the index of that peg.
+ 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 possibilities in which the entity `d` we did that with doesn't have property P.
+ 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 `(fun (r, h) -> ...)` above \[[∃x]], then `s` updated with \[[∃x]] updated with \[[Px]] is just:
+ So if we just call the function `extend_one` defined above \[[∃x]], then `u` updated with \[[∃x]] updated with \[[Px]] is just:
- <pre><code>s >>= \[[∃x]] >>= \[[Px]]
+ <pre><code>u >>= \[[∃x]] >>= \[[Px]]
</code></pre>
or, being explicit about which "bind" operation we're representing here with `>>=`, that is:
- <pre><code>bind_set (bind_set s \[[∃x]]) \[[Px]]
+ <pre><code>bind_set (bind_set u \[[∃x]]) \[[Px]]
</code></pre>
-* In def 3.1 on p. 14, GS&V define `s` updated with \[[not φ]] as:
-
- > { i &elem; s | i does not subsist in s[φ] }
-
- where `i` *subsists* in <code>s[φ]</code> if there are any `i'` that *extend* `i` in <code>s[φ]</code>.
-
- Here's how we can represent that:
-
- <pre><code>bind_set s (fun (r, h) ->
- let u = unit_set (r, h)
- in let descendents = u >>= \[[φ]]
- in if descendents = empty_set then u else empty_set
- </code></pre>
-
-
+* Can you figure out how to handle \[[not φ]] on your own? If not, here are some [more hints](/hints/assignment_7_hint_6).