* At the top of p. 13 (this is in between defs 2.8 and 2.9), GS&V give two examples, one for \[[∃xPx]] and the other for \[[Qx]]. In fact it will be most natural to break \[[∃xPx]] into two pieces, \[[∃x]] and \[[Px]]. But first we need to get clear on expressions like \[[Qx]].
-* GS&V say that the effect of updating an information state `s` with the meaning of "Qx" should be to eliminate possibilities in which the entity associated with the peg associated with the variable `x` does not have the property Q. In other words, if we let `q` be the function from entities to `bool`s that gives the extension of "Q", then `s` updated with \[[Qx]] should be `s` filtered by the function `fun (r, h) -> let obj = List.nth h (r 'x') in q obj`. When `... q obj` evaluates to `true`, that `(r, h)` pair is retained, else it is discarded.
+* GS&V say that the effect of updating an information state `s` with the meaning of "Qx" should be to eliminate possibilities in which the entity associated with the peg associated with the variable `x` does not have the property Q. In other words, if we let `q` be the function from entities to `bool`s that gives the extension of Q, then `s` updated with \[[Qx]] should be `s` filtered by the function `fun (r, h) -> let obj = List.nth h (r 'x') in q obj`. When `... q obj` evaluates to `true`, that `(r, h)` pair is retained, else it is discarded.
- OK, we face two questions then. First, how do we carry this over to our present framework, where we're working with sets of `dpm`s instead of sets of discourse possibilities? And second, how do we decompose the behavior here ascribed to \[[Qx]] into some meaning for "Q" and a different meaning for "x"?
+ OK, we face two questions then. First, how do we carry this over to our present framework, where we're working with sets of `dpm`s instead of sets of discourse possibilities? And second, how do we decompose the behavior here attributed to \[[Qx]] into some meaning for "Q" and a different meaning for "x"?
* Answering the first question: we assume we've got some `bool dpm set` to start with. I won't call this `s` because that's what GS&V use for sets of discourse possibilities, and we don't want to confuse discourse possibilities with `dpm`s. Instead I'll call it `u`. Now what we want to do with `u` is to map each `dpm` it gives us to one that results in `(true, r, h)` only when the entity that `r` and `h` associate with variable `x` has the property Q. As above, I'll assume Q's extension is given by a function `q` from entities to `bool`s.
u >>= \[[Qx]]
-* Now our second question: how do we decompose the behavior here ascribed to \[[Qx]] into some meaning for "Q" and a different meaning for "x"?
+* Now our second question: how do we decompose the behavior here attributed to \[[Qx]] into some meaning for "Q" and a different meaning for "x"?
Well, we already know that \[[x]] will be a kind of computation that takes an assignment function `r` and store `h` as input. It will look up the entity that those two together associate with the variable `x`. So we can treat \[[x]] as an `entity dpm`. We don't worry here about sets of `dpm`s; we'll leave that to our predicates to interface with. We'll just make \[[x]] be a single `entity dpm`. So what we want is:
Can you persuade yourself that these are equivalent?)
-* Reviewing: now we've determined how to define \[[Q]] and \[[x]] such that \[[Qx]] can be the result of applying the function \[[Q]] to the `entity dpm` \[[x]]. And \[[Qx]] in turn is now a function that takes a `bool dpm` as input and returns a `bool dpm set` as output. We compose this with a `bool dpm set` we already have on hand:
+* Reviewing: now we've determined how to define \[[Q]] and \[[x]] such that \[[Qx]] can be the result of applying the function \[[Q]] to the `entity dpm` \[[x]]. And \[[Qx]] in turn is now a function that takes a `bool dpm` as input and returns a `bool dpm set` as output. We monadically bind this operaration to whatever `bool dpm set` we already have on hand:
bind_set u \[[Qx]]
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:
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 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.
let antecedent = fun one_dpm -> exists 'x' one_dpm >>= lift_predicate male getx >>= exists 'y' >>= lift_predicate2 wife_of getx gety;;
(* "if a man x has a wife y, x kisses y" *)
- run (initial_set >>= if_op antecedent lift_predicate2 kisses getx gety);;
+ run (initial_set >>= if_op antecedent (lift_predicate2 kisses getx gety));;
(* Bob has wife Carol, and kisses her; and Ted has wife Alice and kisses her; so this is true! *)
(* "if a man x has a wife y, x misses y" *)
- run (initial_set >>= if_op antecedent lift_predicate2 misses getx gety);;
+ run (initial_set >>= if_op antecedent (lift_predicate2 misses getx gety));;
(* Bob has wife Carol, and misses her; but Ted misses only Carol, not his wife Alice; so this is false! *)