X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=hints%2Fassignment_7_hint_5.mdwn;h=af3c947692681dca251dab87ff4967948d6caf11;hp=1b2ee6a9ceb116bfa936a44652b166934f96b475;hb=56875febe11ea5c63e753b64c546a7a45f28e343;hpb=be5eb752358a5067486efb6d515411551025f1e1

diff --git a/hints/assignment_7_hint_5.mdwn b/hints/assignment_7_hint_5.mdwn
index 1b2ee6a9..af3c9476 100644
--- a/hints/assignment_7_hint_5.mdwn
+++ b/hints/assignment_7_hint_5.mdwn
@@ -8,9 +8,9 @@
 
 	>	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:
 
@@ -23,8 +23,8 @@
 	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
@@ -76,7 +76,7 @@
 	`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]] &equiv; lift q
-	\[[x]] & equiv; getx
+	\[[x]] &equiv; getx
 	\[[Qx]] &equiv; \[[Q]] \[[x]] &equiv;
 		fun r ->
 			let obj = getx r
@@ -85,16 +85,18 @@
 
 	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 \[[&exist;]], which we'd use like this:
@@ -109,45 +111,69 @@
 
  	If we wanted to compose \[[&exist;]] 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 \[[&exist;x]], which we use like this:
+
+	<pre><code>\[[&exist;x]]<sub>reader</sub> ( \[[Qx]] )
+	</code></pre>
+
+	Contrast the way we use \[[&exist;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 >>= \[[&exist;x]] >>= \[[Qx]]
+	</code></pre>
+
+	The crucial difference in GS&V's system is that the distinctive effect of the \[[&exist;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 \[[&exist;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 &exist;y (farmer y and &exist;x y owns x) then (y beats x).
 
 
 *	Can you figure out how to handle \[[not &phi;]] 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.