* 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:
u >>= \[[∃x]] >>= \[[Px]]

(Extra credit: how does the discussion on pp. 25-29 of GS&V bear on the possibility of this simplification?) 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 entity has the property P will be eliminated. We can defer that to a later step, where we do `... >>= \[[Px]]`. GS&V continue: > 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. Deferring the "property P" part, this corresponds to:
u updated with \[[∃x]] ≡
let extend one_dpm (d : entity) =
bind_dpm one_dpm (new_peg_and_assign 'x' d)
in bind_set u (fun one_dpm -> List.map (fun d -> extend one_dpm d) domain)

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) : 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');; 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.) If we call the function `(fun one_dom -> List.map ...)` defined above \[[∃x]], then `u` updated with \[[∃x]] updated with \[[Px]] is just:
u >>= \[[∃x]] >>= \[[Px]]

or, being explicit about which "bind" operation we're representing here with `>>=`, that is:
bind_set (bind_set u \[[∃x]]) \[[Px]]

* 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 [week7]( /reader_monad_for_variable_binding).) type assignment = char -> entity;; type 'a reader = assignment -> 'a;; let unit_reader (value : 'a) : 'a reader = fun r -> value;; let bind_reader (u : 'a reader) (f : 'a -> 'b reader) : 'b reader = fun r -> let a = u r in let u' = f a in u' r;; Here the type of a sentential clause is: type clause = bool reader;; Here are meanings for singular terms and predicates: let getx : entity reader = fun r -> r 'x';; type lifted_unary = entity reader -> bool reader;; let lift (predicate : entity -> bool) : lifted_unary = fun entity_reader -> fun r -> let obj = entity_reader r in unit_reader (predicate obj) The meaning of \[[Qx]] would then be:
\[[Q]] ≡ lift q
\[[x]] ≡ getx
\[[Qx]] ≡ \[[Q]] \[[x]] ≡
fun r ->
let obj = getx r

Recall also how we defined \[[lambda x]], or as [we called it before](/reader_monad_for_variable_binding), \\[[who(x)]]: let shift (var_to_bind : char) (clause : clause) : lifted_unary = fun entity_reader -> fun r -> let new_value = entity_reader r (* remember here we're implementing assignments as functions rather than as lists of pairs *) in let r' = fun var -> if var = var_to_bind then new_value else r var in clause r' 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: fun (lifted_predicate : lifted_unary) -> fun r -> exists (fun (obj : entity) -> lifted_predicate (unit_reader obj) r) That would be the meaning of \[[∃]], which we'd use like this:
\[[∃]] ( \[[Q]] )

or this:
\[[∃]] ( \[[lambda x]] \[[Qx]] )

If we wanted to compose \[[∃]] with \[[lambda x]], we'd get: let shift var_to_bind clause = fun entity_reader r -> let new_value = entity_reader r in let r' = fun var -> if var = var_to_bind then new_value else r var in clause r' in let lifted_exists = fun lifted_predicate -> fun r -> exists (fun obj -> lifted_predicate (unit_reader obj) r) in fun bool_reader -> lifted_exists (shift 'x' bool_reader) which we can simplify to: fun bool_reader -> let shifted r new_value = let r' = fun var -> if var = 'x' then new_value else r var in bool_reader r' in fun r -> exists (shifted r) This gives us a value for \[[∃x]], which we use like this:
\[[∃x]] ( \[[Qx]] )

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:
u >>= \[[∃x]] >>= \[[Qx]]

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: > If ∃x (man x and ∃y y is wife of x) then (x kisses y). See the discussion on pp. 24-5 of GS&V. * 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.