2 * How shall we handle \[[∃x]]? As we said, GS&V really tell us how to interpret \[[∃xPx]], but for our purposes, what they say about this can be broken naturally into two pieces, such that we represent the update of our starting set `u` with \[[∃xPx]] as:
4 <pre><code>u >>= \[[∃x]] >>= \[[Px]]
7 (Extra credit: how does the discussion on pp. 25-29 of GS&V bear on the possibility of this simplification?)
9 What does \[[∃x]] need to be here? Here's what they say, on the top of p. 13:
11 > 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.
13 We can defer that to a later step, where we do `... >>= \[[Px]]`. GS&V continue:
15 > 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.
17 Deferring the "property P" part, this corresponds to:
19 <pre><code>u updated with \[[∃x]] ≡
20 let extend one_dpm (d : entity) =
21 bind_dpm one_dpm (new_peg_and_assign 'x' d)
22 in bind_set u (fun one_dpm -> List.map (fun d -> extend one_dpm d) domain)
25 where `new_peg_and_assign` is the operation we defined in [hint 3](/hints/assignment_7_hint_3):
27 let new_peg_and_assign (var_to_bind : char) (d : entity) : bool -> bool dpm =
30 (* first we calculate an unused index *)
31 let new_index = List.length h
32 (* next we store d at h[new_index], 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 new_index *)
36 in let r' = fun var ->
37 if var = var_to_bind then new_index else r var
38 (* we pass through the same truth_value that we started with *)
39 in (truth_value, r', h');;
41 What's going on in this proposed representation of \[[∃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. A later step can then filter out all the `dpm`s where the entity `d` we did that with doesn't have property P. (Again, consult GS&V pp. 25-9 for extra credit.)
43 If we call the function `(fun one_dom -> List.map ...)` defined above \[[∃x]], then `u` updated with \[[∃x]] updated with \[[Px]] is just:
45 <pre><code>u >>= \[[∃x]] >>= \[[Px]]
48 or, being explicit about which "bind" operation we're representing here with `>>=`, that is:
50 <pre><code>bind_set (bind_set u \[[∃x]]) \[[Px]]
53 * 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](
54 /reader_monad_for_variable_binding).)
56 type assignment = char -> entity;;
57 type 'a reader = assignment -> 'a;;
59 let unit_reader (value : 'a) : 'a reader = fun r -> value;;
61 let bind_reader (u : 'a reader) (f : 'a -> 'b reader) : 'b reader =
67 Here the type of a sentential clause is:
69 type clause = bool reader;;
71 Here are meanings for singular terms and predicates:
73 let getx : entity reader = fun r -> r 'x';;
75 type lifted_unary = entity reader -> bool reader;;
77 let lift (predicate : entity -> bool) : lifted_unary =
80 let obj = entity_reader r
81 in unit_reader (predicate obj)
83 The meaning of \[[Qx]] would then be:
85 <pre><code>\[[Q]] ≡ lift q
87 \[[Qx]] ≡ \[[Q]] \[[x]] ≡
90 in unit_reader (q obj)
93 Recall also how we defined \[[lambda x]], or as [we called it before](/reader_monad_for_variable_binding), \\[[who(x)]]:
95 let shift (var_to_bind : char) (clause : clause) : lifted_unary =
98 let new_value = entity_reader r
99 (* remember here we're implementing assignments as functions rather than as lists of pairs *)
100 in let r' = fun var -> if var = var_to_bind then new_value else r var
103 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:
105 fun (lifted_predicate : lifted_unary) ->
106 fun r -> exists (fun (obj : entity) ->
107 lifted_predicate (unit_reader obj) r)
109 That would be the meaning of \[[∃]], which we'd use like this:
111 <pre><code>\[[∃]] \[[Q]]
116 <pre><code>\[[∃]] ( \[[lambda x]] \[[Qx]] )
119 If we wanted to compose \[[∃]] with \[[lambda x]], we'd get:
121 let shift var_to_bind clause =
122 fun entity_reader r ->
123 let new_value = entity_reader r
124 in let r' = fun var -> if var = var_to_bind then new_value else r var
126 in let lifted_exists =
127 fun lifted_predicate ->
128 fun r -> exists (fun obj -> lifted_predicate (unit_reader obj) r)
129 in fun bool_reader -> lifted_exists (shift 'x' bool_reader)
131 which we can simplify to:
135 fun entity_reader r ->
136 let new_value = entity_reader r
137 in let r' = fun var -> if var = 'x' then new_value else r var
139 in let lifted_exists =
140 fun lifted_predicate ->
141 fun r -> exists (fun obj -> lifted_predicate (unit_reader obj) r)
142 in fun bool_reader -> lifted_exists (shifted bool_reader)
146 fun entity_reader r ->
147 let new_value = entity_reader r
148 in let r' = fun var -> if var = 'x' then new_value else r var
150 in fun r -> exists (fun obj -> shifted' (unit_reader obj) r)
153 let shifted'' r obj =
154 let new_value = (unit_reader obj) r
155 in let r' = fun var -> if var = 'x' then new_value else r var
157 in fun r -> exists (fun obj -> shifted'' r obj)
160 let shifted'' r obj =
162 in let r' = fun var -> if var = 'x' then new_value else r var
164 in fun r -> exists (shifted'' r)
168 let shifted r new_value =
169 let r' = fun var -> if var = 'x' then new_value else r var
171 in fun r -> exists (shifted r)
173 This gives us a value for \[[∃x]], which we use like this:
175 <pre><code>\[[∃x]] ( \[[Qx]] )
178 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:
180 <pre><code>u >>= \[[∃x]] >>= \[[Qx]]
183 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 some 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:
185 > If ∃x (man x and ∃y y is wife of x) then (x kisses y).
187 See the discussion on pp. 24-5 of GS&V.
190 * 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.