> 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]]`.
+ 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'` and there are entities `d` which in the possible world of `i` have the property P.
+ > 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:
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 not a function that we can bind to a bool dpm *)
- fun (truth_value : bool) : bool dpm ->
+ (* we want to return a function that we can bind to a bool dpm *)
+ fun (truth_value : bool) ->
fun ((r, h) : assignment * store) ->
(* first we calculate an unused index *)
let new_index = List.length h
`lift predicate` converts a function of type `entity -> bool` into one of type `entity reader -> bool reader`. The meaning of \[[Qx]] would then be:
<pre><code>\[[Q]] ≡ lift q
- \[[x]] & equiv; getx
+ \[[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) entity_reader (v : 'a reader) =
- fun (r : assignment) ->
- 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 v r'
+ let shift (var_to_bind : char) (clause : bool reader) =
+ (* we return a lifted predicate, that is a entity reader -> bool reader *)
+ fun entity_reader ->
+ fun (r : assignment) ->
+ 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 : entity reader -> bool reader) : bool reader ->
+ fun (lifted_predicate : entity reader -> bool reader) ->
fun r -> exists (fun (obj : entity) -> lifted_predicate (unit_reader obj) r)
That would be the meaning of \[[∃]], which we'd use like this:
If we wanted to compose \[[∃]] with \[[lambda x]], we'd get:
- let shift var_to_bind entity_reader v =
- fun r ->
+ 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 v r'
+ 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' getx bool_reader)
+ in fun bool_reader -> lifted_exists (shift 'x' bool_reader)
- which we can simplify to:
+ which we can simplify as:
- let shifted v =
- fun r ->
- let new_value = r 'x'
+ let shifted clause =
+ fun entity_reader r ->
+ let new_value = entity_reader r
in let r' = fun var -> if var = 'x' then new_value else r var
- in v r'
+ 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 (shifted bool_reader)
- and simplifying further:
+ fun bool_reader ->
+ let shifted' =
+ fun entity_reader r ->
+ let new_value = entity_reader r
+ in let r' = fun var -> if var = 'x' then new_value else r var
+ in bool_reader r'
+ in fun r -> exists (fun obj -> shifted' (unit_reader obj) r)
fun bool_reader ->
- let shifted v =
- fun r ->
- let new_value = r 'x'
+ let shifted'' r obj =
+ let new_value = (unit_reader obj) r
in let r' = fun var -> if var = 'x' then new_value else r var
- in v r'
- let lifted_predicate = shifted bool_reader
- in fun r -> exists (fun obj -> lifted_predicate (unit_reader obj) r)
+ in bool_reader r'
+ in fun r -> exists (fun obj -> shifted'' r obj)
fun bool_reader ->
- let lifted_predicate = fun r ->
- let new_value = r 'x'
+ let shifted'' r obj =
+ let new_value = obj
in let r' = fun var -> if var = 'x' then new_value else r var
in bool_reader r'
- in fun r -> exists (fun obj -> lifted_predicate (unit_reader obj) r)
+ in fun r -> exists (shifted'' r)
+
+ 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:
+
+ <pre><code>\[[∃x]]<sub>reader</sub> ( \[[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:
+
+ > If ∃y (farmer y and ∃x y owns x) then (y beats x).
* Can you figure out how to handle \[[not φ]] on your own? If not, here are some [more hints](/hints/assignment_7_hint_6). But try to get as far as you can on your own.