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]]`. GS&V continue:
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'` as 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 : bool dpm) ->
19 List.map (fun d -> bind_dpm one_dpm (new_peg_and_assign 'x' d)) domain
20 in bind_set u extend_one
23 where `new_peg_and_assign` is the operation we defined in [hint 3](/hints/assignment_7_hint_3):
25 let new_peg_and_assign (var_to_bind : char) (d : entity) =
26 (* we want to return a function that we can bind to a bool dpm *)
27 fun (truth_value : bool) ->
28 fun ((r, h) : assignment * store) ->
29 (* first we calculate an unused index *)
30 let new_index = List.length h
31 (* next we store d at h[new_index], which is at the very end of h *)
32 (* the following line achieves that in a simple but inefficient way *)
33 in let h' = List.append h [d]
34 (* next we assign 'x' to location new_index *)
35 in let r' = fun var ->
36 if var = var_to_bind then new_index else r var
37 (* we pass through the same truth_value that we started with *)
38 in (truth_value, r', h')
40 What's going on in this representation of `u` updated with \[[∃x]]? For each `bool dpm` in `u`, we collect `dpm`s that are the result of passing through their `bool`, but 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.
42 A later step can then filter out all the `dpm`s where the entity `d` we did that with doesn't have property P.
44 So if we just call the function `extend_one` defined above \[[∃x]], then `u` updated with \[[∃x]] updated with \[[Px]] is just:
46 <pre><code>u >>= \[[∃x]] >>= \[[Px]]
49 or, being explicit about which "bind" operation we're representing here with `>>=`, that is:
51 <pre><code>bind_set (bind_set u \[[∃x]]) \[[Px]]
54 * Let's compare this to what \[[∃xPx]] would look like on a non-dynamic semantics, for example, where we use a simple reader monad to implement variable binding. Reminding ourselves, we'd be working in a framework like this. (Here we implement environments or assignments as functions from variables to entities, instead of as lists of pairs of variables and entities. An assignment `r` here is what `fun c -> List.assoc c r` would have been in [week6](
55 /reader_monad_for_variable_binding).)
57 type assignment = char -> entity;;
58 type 'a reader = assignment -> 'a;;
60 let unit_reader (x : 'a) = fun r -> x;;
62 let bind_reader (u : 'a reader) (f : 'a -> 'b reader) =
68 let getx = fun r -> r 'x';;
70 let lift (predicate : entity -> bool) =
73 let obj = entity_reader r
74 in unit_reader (predicate obj)
76 `lift predicate` converts a function of type `entity -> bool` into one of type `entity reader -> bool reader`. The meaning of \[[Qx]] would then be:
78 <pre><code>\[[Q]] ≡ lift q
80 \[[Qx]] ≡ \[[Q]] \[[x]] ≡
83 in unit_reader (q obj)
86 Recall also how we defined \[[lambda x]], or as [we called it before](/reader_monad_for_variable_binding), \\[[who(x)]]:
88 let shift (var_to_bind : char) (clause : bool reader) =
89 (* we return a lifted predicate, that is a entity reader -> bool reader *)
91 fun (r : assignment) ->
92 let new_value = entity_reader r
93 (* remember here we're implementing assignments as functions rather than as lists of pairs *)
94 in let r' = fun var -> if var = var_to_bind then new_value else r var
97 Now, how would we implement quantifiers in this setting? I'll assume we have a function `exists` of type `(entity -> bool) -> bool`. That is, it accepts a predicate as argument and returns `true` if any element in the domain satisfies that predicate. We could implement the reader-monad version of that like this:
99 fun (lifted_predicate : entity reader -> bool reader) ->
100 fun r -> exists (fun (obj : entity) ->
101 lifted_predicate (unit_reader obj) r)
103 That would be the meaning of \[[∃]], which we'd use like this:
105 <pre><code>\[[∃]] \[[Q]]
110 <pre><code>\[[∃]] ( \[[lambda x]] \[[Qx]] )
113 If we wanted to compose \[[∃]] with \[[lambda x]], we'd get:
115 let shift var_to_bind clause =
116 fun entity_reader r ->
117 let new_value = entity_reader r
118 in let r' = fun var -> if var = var_to_bind then new_value else r var
120 in let lifted_exists =
121 fun lifted_predicate ->
122 fun r -> exists (fun obj -> lifted_predicate (unit_reader obj) r)
123 in fun bool_reader -> lifted_exists (shift 'x' bool_reader)
125 which we can simplify as:
128 fun entity_reader r ->
129 let new_value = entity_reader r
130 in let r' = fun var -> if var = 'x' then new_value else r var
132 in let lifted_exists =
133 fun lifted_predicate ->
134 fun r -> exists (fun obj -> lifted_predicate (unit_reader obj) r)
135 in fun bool_reader -> lifted_exists (shifted bool_reader)
139 fun entity_reader r ->
140 let new_value = entity_reader r
141 in let r' = fun var -> if var = 'x' then new_value else r var
143 in fun r -> exists (fun obj -> shifted' (unit_reader obj) r)
146 let shifted'' r obj =
147 let new_value = (unit_reader obj) r
148 in let r' = fun var -> if var = 'x' then new_value else r var
150 in fun r -> exists (fun obj -> shifted'' r obj)
153 let shifted'' r obj =
155 in let r' = fun var -> if var = 'x' then new_value else r var
157 in fun r -> exists (shifted'' r)
160 let shifted'' r new_value =
161 let r' = fun var -> if var = 'x' then new_value else r var
163 in fun r -> exists (shifted'' r)
165 This gives us a value for \[[∃x]], which we use like this:
167 <pre><code>\[[∃x]] ( \[[Qx]] )
170 Contrast the way we use \[[∃x]] in GS&V's system. Here we don't have a function that takes \[[Qx]] as an argument. Instead we have a operation that gets bound in a discourse chain:
172 <pre><code>u >>= \[[∃x]] >>= \[[Qx]]
175 The crucial difference in GS&V's system is that the distinctive effect of the \[[∃x]]---to allocate new pegs in the store and associate variable `x` with the objects stored there---doesn't last only while interpreting clauses supplied as arguments to \[[∃x]]. Instead, it persists through the discourse, possibly affecting the interpretation of claims outside the logical scope of the quantifier. This is how we'll able to interpret claims like:
177 > If ∃x (man x and ∃y y is wife of x) then (x kisses y).
180 * Can you figure out how to handle \[[not φ]] and the other connectives? If not, here are some [more hints](/hints/assignment_7_hint_6). But try to get as far as you can on your own.