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diff --git a/topics/_week15_continuation_applications.mdwn b/topics/_week15_continuation_applications.mdwn
index ec0cb5d6..d13efa7f 100644
--- a/topics/_week15_continuation_applications.mdwn
+++ b/topics/_week15_continuation_applications.mdwn
@@ -104,11 +104,16 @@ Raising closely resembles the reduction rule for shift:
Quantifier Raising: given a sentence [... [QDP] ...], build a new
sentence [QDP (\x.[... [x] ...])].
+Here, QDP is a scope-taking quantificational DP.
+
Just to emphasize the similarity between QR and shift, we can use QR
to provide insight into the tree task that mystified us earlier.
+
+
.
__|___
| |
@@ -119,11 +124,16 @@ a __|___
d _|__
| |
S e
+
-First we QR the lower shift operator
+First we QR the lower shift operator, replacing it with a variable and
+abstracting over that variable.
+
+
.
___|___
| |
@@ -138,11 +148,15 @@ S ___|___
d _|__
| |
x e
+
Next, we QR the upper shift operator
+
+
.
___|___
| |
@@ -161,10 +175,11 @@ S ___|____
d _|__
| |
x e
+
We then evaluate, using the same value for the shift operator proposed before:
- shift = \k.k(k "")
+ S = shift = \k.k(k "")
It will be easiest to begin evaluating this tree with the lower shift
operator (we get the same result if we start with the upper one).
@@ -172,8 +187,11 @@ The relevant value for k is (\x.a(y(d(x e)))). Then k "" is
a(y(d(""(e)))), and k(k "") is a(y(d((a(y(d(""(e)))))(e)))). In tree
form:
+
+
.
___|___
| |
@@ -196,13 +214,17 @@ S ___|____
d __|__
| |
"" e
+
Repeating the process for the upper shift operator replaces each
occurrence of y with a copy of the whole tree.
+
+
.
|
______|______
@@ -235,11 +257,13 @@ a ___|____ | |
d __|__
| |
"" e
+
-The yield of this tree (the sequence of leaf nodes) is aadadeedaadadeedee.
+The yield of this tree (the sequence of leaf nodes) is
+aadadeedaadadeedee, which is the expected output of the double-shifted tree.
-Exercise: the result is different, by the way, if the QR occurs in a
-different order.
+Exercise: the result is different, by the way, if the QR occurs in the
+opposite order.
Three lessons:
@@ -248,7 +272,9 @@ Three lessons:
dramatic increase in power and complexity.
* Operators that
- compose multiple copies of a context can be hard to understand.
+ compose multiple copies of a context can be hard to understand
+ (though keep this in mind when we see the continuations-based
+ analysis of coordination, which involves context doubling).
* When considering two-sided, tree-based continuation operators,
quantifier raising is a good tool for visualizing (defunctionalizing)
@@ -285,15 +311,14 @@ manipulates a list of information. It is natural to imagine
separating a box into two regions, the payload and the hidden scratch
space:
- _______________ _______________ _______________
- | [x->2, y->3] | | [x->2, y->3] | | [x->2, y->3] |
- ------------------- ------------------ ------------------
+
+ _______________ _______________ _______________
+ | [x->2, y->3] | | [x->2, y->3] | | [x->2, y->3] |
+ ------------------- ------------------ ------------------
| | ¢ | | = | |
- | +2 | | y | | 5 |
- |______________| |______________| |______________|
-
-
-(Imagine the + operation has been lifted into the Reader monad too.)
+ | +2 | | y | | 5 |
+ |______________| |______________| |______________|
+
For people who are familiar with Discourse Representation Theory (Kamp
1981, Kamp and Reyle 1993), this separation of boxes into payload and
@@ -309,14 +334,18 @@ We won't keep the outer box, but we will keep the horizontal line
dividing main effects from side-effects.
Tower convention for types:
+
γ | β
(α -> β) -> γ can be equivalently written -----
α
+
Tower convention for values:
+
g[]
\k.g[k(x)] can be equivalently written ---
x
+
If \k.g[k(x)] has type (α -> β) -> γ, then k has type (α -> β).
@@ -329,12 +358,15 @@ individuals) and S (the type of truth values).
Then in the spirit of monadic thinking, we'll have a way of lifting an
arbitrary value into the tower system:
- [] γ|β
- LIFT (x:α) = \k.kx : (α -> β) -> γ == --- : ---
- x α
+ [] β|β
+ LIFT (x:α) = \k.kx : (α -> β) -> β == -- : ---
+ x α
Obviously, LIFT is exactly the midentity (the unit) for the continuation monad.
-The name comes from Partee's 1987 theory of type-shifters for
+Notice that LIFT requires the result type of the continuation argument
+and the result type of the overall expression to match (here, both are β).
+
+The name LIFT comes from Partee's 1987 theory of type-shifters for
determiner phrases. Importantly, LIFT applied to an
individual-denoting expression yields the generalized quantifier
proposed by Montague as the denotation for proper names:
@@ -347,6 +379,14 @@ So if the proper name *John* denotes the individual j, LIFT(j) is the
generalized quantifier that maps each property k of type DP -> S to true
just in case kj is true.
+Crucially for the discussion here, LIFT does not apply only to DPs, as
+in Montague and Partee, but to any expression whatsoever. For
+instance, here is LIFT applied to a lexical verb phrase:
+
+ [] S|S
+ LIFT (left:DP\S) = \k.kx : (DP\S -> S) -> S == ---- : ---
+ left DP
+
Once we have expressions of type (α -> β) -> γ, we'll need to combine
them. We'll use the ¢ operator from the continuation monad:
@@ -373,6 +413,16 @@ operator:
Not a monad (Wadler); would be if the types were
Neverthless, obeys the monad laws.
+Oh, one more thing: because natural language allows the functor to be
+on the left or on the right, we replace the type arrow -> with a
+left-leaning version \ and a right-leaning version, as follows:
+
+ α/β β = α
+ β β\α = α
+
+This means we need two versions of ¢ too (see Barker and Shan 2014
+chapter 1 for full details).
+
This is (almost) all we need to get some significant linguistic work
done.