* 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>u >>=<sub>set</sub> \[[∃x]] >>=<sub>set</sub> \[[Px]]
+ <pre><code>u >>= \[[∃x]] >>= \[[Px]]
</code></pre>
What does \[[∃x]] need to be here? Here's what they say, on the top of p. 13:
Deferring the "property P" part, this corresponds to:
<pre><code>u updated with \[[∃x]] ≡
- let extend_one = fun (one_dpm : bool dpm) ->
+ let extend_one : clause = fun one_dpm ->
List.map (fun d -> bind_dpm one_dpm (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) =
- (* we want to return a function that we can bind to a bool dpm *)
- fun (truth_value : bool) ->
- fun ((r, h) : assignment * store) ->
+ 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 *)
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')
+ in (truth_value, r', h');;
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.
- 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.
+ A later step can then filter out all the `dpm`s where the entity `d` we did that with doesn't have property P.
So if we just call the function `extend_one` defined above \[[∃x]], then `u` updated with \[[∃x]] updated with \[[Px]] is just:
type assignment = char -> entity;;
type 'a reader = assignment -> 'a;;
- let unit_reader (x : 'a) = fun r -> x;;
+ let unit_reader (value : 'a) : 'a reader = fun r -> value;;
- let bind_reader (u : 'a reader) (f : 'a -> 'b reader) =
+ 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;;
- let getx = fun r -> r 'x';;
+ Here the type of a sentential clause is:
+
+ type clause = bool reader;;
+
+ Here are meanings for singular terms and predicates:
- let lift (predicate : entity -> bool) =
+ 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)
- `lift predicate` converts a function of type `entity -> bool` into one of type `entity reader -> bool reader`. The meaning of \[[Qx]] would then be:
+ The meaning of \[[Qx]] would then be:
<pre><code>\[[Q]] ≡ lift q
\[[x]] ≡ getx
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 : bool reader) =
- (* we return a lifted predicate, that is a entity reader -> bool reader *)
+ let shift (var_to_bind : char) (clause : clause) : lifted_unary =
fun entity_reader ->
- fun (r : assignment) ->
+ 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
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 : entity reader -> bool reader) ->
- fun r -> exists (fun (obj : entity) -> lifted_predicate (unit_reader obj) r)
+ 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:
This gives us a value for \[[∃x]], which we use like this:
- <pre><code>\[[∃x]]<sub>reader</sub> ( \[[Qx]] )
+ <pre><code>\[[∃x]] ( \[[Qx]] )
</code></pre>
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:
<pre><code>u >>= \[[∃x]] >>= \[[Qx]]
</code></pre>
- 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're able to interpret claims like:
+ 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:
- > If ∃y (farmer y and ∃x y owns x) then (y beats x).
+ > If ∃x (man x and ∃y y is wife of x) then (x kisses y).
* 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.