- or as it's written using Haskell's infix notation for bind:
-
- s >>= \[[Qx]]
-
-* Now 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:
-
- <pre><code>s >>= \[[∃x]] >>= \[[Px]]
- </code></pre>
-
- 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 object 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 objects `d` which in the possible world of `i` have the property P.
-
- Deferring the "property P" part, this says:
-
- <pre><code>s updated with \[[∃x]] ≡
- s >>= (fun (r, g) -> List.map (fun d -> newpeg_and_bind 'x' d) domain)
- </code></pre>
-
- That is, for each pair `(r, g)` in `s`, we collect the result of extending `(r, g)` by allocating a new peg for object `d`, for each `d` in our whole domain of objects (here designated `domain`), and binding the variable `x` to the index of that peg.
-
- A later step can then filter out all the possibilities in which the object `d` we did that with doesn't have property P.
-
- So if we just call the function `(fun (r, g) -> ...)` above \[[∃x]], then `s` updated with \[[∃x]] updated with \[[Px]] is just:
-
- <pre><code>s >>= \[[∃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]]
- </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, g) ->
- let u = unit_set (r, g)
- in let descendents = u >>= \[[φ]]
- in if descendents = empty_set then u else empty_set
- </code></pre>
+ (* we want to return a function that we can bind to a bool dpm *)
+ let new_peg_and_assign (var_to_bind : char) (d : entity) : bool -> bool dpm =
+ fun truth_value ->
+ fun (r, h) ->
+ (* first we calculate an unused index *)
+ let new_index = List.length h
+ (* next we store d at h[new_index], 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 new_index *)
+ in let r' = fun var ->
+ if var = var_to_bind then new_index else r var
+ (* we pass through the same truth_value that we started with *)
+ in (truth_value, r', h');;