XGitUrl: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week10_gsv.mdwn;h=1cd25a800c4c03b671bdfcb37cc09fe044699913;hp=79b82a265d5ef809c0b99baefc977cd5dca511cd;hb=6e3c775751114918d3c5fb7be4708dc0c555315f;hpb=43e2eb7bdded470e524b75291f842e34cc0c0432
diff git a/topics/_week10_gsv.mdwn b/topics/_week10_gsv.mdwn
index 79b82a26..1cd25a80 100644
 a/topics/_week10_gsv.mdwn
+++ b/topics/_week10_gsv.mdwn
@@ 9,7 +9,15 @@
GSV are interested in developing and establishing a reasonable theory
of discourse update. One way of looking at this paper is like this:
 GSV = GS + V
+ GSV = GS + V, where
+
+ GS = Dynamic theories of binding of Groenendijk and Stokhof, e.g.,
+ Dynamic Predicate Logic L&P 1991: dynamic binding, donkey anaphora
+ Dynamic Montague Grammar 1990: generalized quantifiers, discourse referents
+
+ V = a dynamic theory of epistemic modality, e.g.,
+ Veltman, Frank. "Data semantics."
+ In Truth, Interpretation and Information, Foris, Dordrecht (1984): 4363.
That is, Groenendijk and Stokhof have a wellknown theory of dynamic
semantics, and Veltman has a wellknown theory of epistemic modality,
@@ 21,88 +29,42 @@ view and from a practical engineering point of view. On the
theoretical level, these scholars are proposing a strategy for
managing the connection between variables and the objects they
designate in way that is flexible enough to be useful for describing
natural language. The main way they attempt to do this is by
inserting an extra level in between the variable and the object:
instead of having an assignment function that maps variables directly
onto objects, GSV provide *pegs*: variables map onto pegs, and pegs
map onto objects. We'll discuss in considerable detail what pegs
allow us to do, since it is highly relevant to one of the main
applications of the course, namely, reference and coreference.
+natural language.
What are pegs? The term harks back to a paper by Landman called `Pegs
and Alecs'. There pegs are simply hooks for hanging properties on.
Pegs are supposed to be as anonymous as possible. Think of hanging
your coat on a physical peg: you don't care which peg it is, only that
there are enough pegs for everyone's coat to hang from. Likewise, for
the pegs of GSV, all that matters is that there are enough of them.
(Incidentally, there is nothing in Gronendijk and Stokhof's original
DPL paper that corresponds naturally to pegs; but in their Dynamic
Montague Grammar paper, pegs serve a purpose similar to discourse
referents there, though the connection is not simple.)

On an engineering level, the fact that GSV are combining anaphora and
bound quantification with epistemic quantification means that they are
gluing together related but distinct subsystems into a single
fragment. These subsystems naturally cleave into separate layers in a
way that is obscured in the paper. We will argue in detail that
reengineering GSV using monads will lead to a cleaner system that
does all of the same theoretical work.

Empirical targets: on the anaphoric side, GSV want to

On the epistemic side, GSV aim to account for asymmetries such as

 It might be raining. It's not raining.
 #It's not raining. It might be raining.

## Basics

There are a lot of formal details in the paper in advance of the
empirical discussion. Here are the ones that matter:

 type var = string
 type peg = int
 type refsys = var > peg
 type ent = Alice  Bob  Carl
 type assignment = peg > ent

So in order to get from a variable to an object, we have to compose a
refsys `r` with an assignment `g`. For instance, we might have
r (g ("x")) = Alice.

 type pred = string
 type world = pred > ent > bool
 type pegcount = int
 type poss = world * pegcount * refsys * assignment
 type infostate = [poss]
+## Basics of GSV's fragment
Worlds in general settle all matters of fact in the world. In
particular, they determine the extensions of predicates and relations.
In this discussion, we'll (crudely) approximate worlds by making them
a function from predicates such as "man" to a function mapping each
entity to a boolean.
+The fragment in this paper is unusually elegant. We'll present it on
+its own terms, with the exception that we will not use pegs. See the
+digression below concerning pegs for an explanation. After presenting
+the paper, we'll reengineering the fragment using explicit monads.
As we'll see, indefinites as a side effect increase the number of pegs
by one. GSV assume that we can determine what integer the next unused
peg corresponds to by examining the range of the refsys function.
We'll make things easy on ourselves by simply tracking the total
number of used pegs in a counter called `pegcount`.
+In this fragment, points of evaluation are not just worlds, but a pair
+of a world and an assginment function. This is familiar from Heim's
+1983 File Change Semantics. We'll follow GSV and call a
+worldassignment pair a "possibility". Then a context is a set (an
+"information state") is a set of possiblities. Infostates
+simultaneously track both information about the world (which possible
+worlds are live possibilities?) as well as information about the
+discourse (which objects to the variables refer to?).
So information states track both facts about the world (e.g., which
objects count as a man), and facts about the discourse (e.g., how many
pegs have been used).
+Worlds in general settle all matters of fact in the world. In
+particular, they determine the extensions of predicates and relations.
The formal language the fragment interprets is Predicate Calculus with
equality, existential and universal quantification, and one unary
modality (box and diamond, corresponding to epistemic necessity and
epistemic possibility).
+equality, existential and universal quantification, along with one
+unary modality (box and diamond, corresponding to epistemic necessity
+and epistemic possibility).
+
+An implementation in OCaml is available [[herecode/gsv.ml]]; consult
+that code for details of syntax, types, and values. [[An implementation
+in Haskellcode/gsv.hs]] is available as well, if you prefer.
Terms in this language are either individuals such as Alice or Bob, or
else variables. So in general, the referent of a term can depend on a
possibility:
ref(i, t) = t if t is an individual, and
 g(r(t)) if t is a variable, where i = (w,n,r,g)
+ g(t) if t is a variable, where i = (w,g)
Here are the main clauses for update (their definition 3.1).
@@ 111,31 +73,28 @@ state `s` with the information in Ï) as `s[Ï]`.
s[P(t)] = {i in s  w(P)(ref(i,t))}
So `man(x)` is the set of live possibilities `i = (w,r,g)` in s such that
+So `man(x)` is the set of live possibilities `i = (w,g)` in s such that
the set of men in `w` given by `w(man)` maps the object referred to by
`x`, namely, `r(g("x"))`, to `true`. That is, update with "man(x)"
+`x`, namely, `g("x")`, to `true`. That is, update with "man(x)"
discards all possibilities in which "x" fails to refer to a man.
 s[t1 = t2] = {i in s  ref(i,t1) = ref(i,t2)}
+ s[t1 = t2] = {i in s  ref(i,t1) == ref(i,t2)}
s[Ï and Ï] = s[Ï][Ï]
When updating with a conjunction, first update with the left conjunct,
then update with the right conjunct.
Existential quantification requires adding a new peg to the set of
discourse referents.

 s[âxÏ] = {(w, n+1, r[x>n], g[n>a])  (w,n,r,g) in s and a in ent}[Ï]
+Existential quantification is somewhat intricate.
Here's the recipe: for every possibility (w,n,r,g) in s, and for every
entity a in the domain of discourse, construct a new possibility with
the same world w, an incrementd peg count n+1, and a new r and g
adjusted in such a way that the variable x refers to the object a.
+ s[âxÏ] = Union {{(w, g[x>a])  (w,g) in s}[Ï]  a in ent}
Note that this recipe does not examine Ï. This means that this
analysis treats the formula prefix `âx` as if it were a meaningful
constituent independent of Ï.
+Here's the recipe: given a starting infostate s, choose an object a
+from the domain of discourse. Construct a modified infostate s' by
+adjusting the assignment function of each possibility so as to map the variable x to a.
+Then update s' with Ï. Finally, take the union over the results of
+doing this for every object a in the domain of discourse. If you're
+unsure about this, examine the [[codecode/gsv.ml]].
Negation is natural enough:
@@ 146,60 +105,28 @@ possibility i returns the empty information state, then not Ï is true
with respect to i.
In GSV, disjunction, the conditional, and the universals are defined
in terms of negation and the other connectives.

Exercise: assume that there are two entities in the domain of
discourse, Alice and Bob. Assume that Alice is a woman, and Bob is a
man. Show the following computations, where `i = (w,n,r,g)`:

 1. {i}[âx.person(x)]

 = {(w,n+1,r[x>n],g[n>a]),(w,n+1,r[x>n],g[n>b])}[person(x)]
 = {(w,n+1,r[x>n],g[n>a]),(w,n+1,r[x>n],g[n>b])}
+in terms of negation and the other connectives (see fact 3.2).
 2. {i}[âx.man(x)]
+Exercise: assume that there are three entities in the domain of
+discourse, Alice, Bob, and Carl. Assume that Alice is a woman, and
+Bob and Carl are men.
 = {(w,n+1,r[x>n],g[n>a]),(w,n+1,r[x>n],g[n>b])}[person(x)]
 = {(w,n+1,r[x>n],g[n>b])}
+Compute the following:
+ 1. {(w,g)}[âx.man(x)]
 3. {i}[âxây.person(x) and person(y)]
+ = {(w,g[n>a])}[man(x)] ++ {(w,g[n>b])}[man(x)]
+ ++ {(w,g[n>c])}[man(x)]
+ = {} ++ {(w,g[n>b])} ++ {(w,g[n>c])}
+ = {(w,g[n>a]),(w,g[n>b]),(w,g[n>c])}
+  Bob and Carl are men
 = {(w,n+1,r[x>n],g[n>a]),(w,n+1,r[x>n],g[n>b])}[ây.person(x) and person(y)]
 = {(w, n+2, r[x>n][y>n+1], g[n>a][n+1>a]),
 (w, n+2, r[x>n][y>n+1], g[n>a][n+1>b]),
 (w, n+2, r[x>n][y>n+1], g[n>b][n+1>a]),
 (w, n+2, r[x>n][y>n+1], g[n>b][n+1>b])
 }[person(x) and person(y)]
 = {(w, n+2, r[x>n][y>n+1], g[n>a][n+1>a]),
 (w, n+2, r[x>n][y>n+1], g[n>a][n+1>b]),
 (w, n+2, r[x>n][y>n+1], g[n>b][n+1>a]),
 (w, n+2, r[x>n][y>n+1], g[n>b][n+1>b])
 }
+ 2. {(w,g)}[âx.woman(x)]
+ 3. {(w,g)}[âxây.man(x) and man(y)]
+ 4. {(w,n,r,g)}[âxây.x=y]
 4. {i}[âxây.x=x]
+Running the [[codecode/gsv.ml]] gives the answers.
 = {(w, n+2, r[x>n][y>n+1], g[n>a][n+1>a]),
 (w, n+2, r[x>n][y>n+1], g[n>a][n+1>b]),
 (w, n+2, r[x>n][y>n+1], g[n>b][n+1>a]),
 (w, n+2, r[x>n][y>n+1], g[n>b][n+1>b])
 }[âxây.x=x]
 = {(w, n+2, r[x>n][y>n+1], g[n>a][n+1>a]),
 (w, n+2, r[x>n][y>n+1], g[n>a][n+1>b]),
 (w, n+2, r[x>n][y>n+1], g[n>b][n+1>a]),
 (w, n+2, r[x>n][y>n+1], g[n>b][n+1>b])
 }

 5. {i}[âxây.x=y]

 = {(w, n+2, r[x>n][y>n+1], g[n>a][n+1>a]),
 (w, n+2, r[x>n][y>n+1], g[n>a][n+1>b]),
 (w, n+2, r[x>n][y>n+1], g[n>b][n+1>a]),
 (w, n+2, r[x>n][y>n+1], g[n>b][n+1>b])
 }[âxây.x=y]
 = {(w, n+2, r[x>n][y>n+1], g[n>a][n+1>a]),
 (w, n+2, r[x>n][y>n+1], g[n>b][n+1>b])
 }
## Order and modality
@@ 209,9 +136,10 @@ The final remaining update rule concerns modality:
This is a peculiar rule: a possibility `i` will survive update just in
case something is true of the information state `s` as a whole. That
means that either every `i` in `s` will survive, or none of them will. The
criterion is that updating `s` with the information in Ï does not
produce the contradictory information state (i.e., `{}`).
+means that either every `i` in `s` will survive, or none of them will.
+The criterion is that updating `s` with the information in the
+prejacent Ï does not produce the contradictory information state
+(i.e., `{}`).
So let's explore what this means. GSV offer a contrast between two
discourses that differ only in the order in which the updates occur.
@@ 223,9 +151,9 @@ order shows that the system is ordersensitive.
According to GSV, the combination of these sentences in this order is
`inconsistent', and they mark the second sentence with the star of
ungrammaticality. We'll say instead that the discourse is
gramamtical, leave the exact word to use for its intuitive effect up
for grabs. What is important for our purposes is to get clear on how
the fragment behaves with respect to these sentences.
+gramamtical, leave the exact way to think about its intuitive status
+up for grabs. What is important for our purposes is to get clear on
+how the fragment behaves with respect to these sentences.
We'll start with an infostate containing two possibilities. In one
possibility, Alice is hungry (call this possibility "hungry"); in the
@@ 266,25 +194,10 @@ In contrast, consider the sentences in the opposite order:
We'll start with the same two possibilities.

= {hungry, full}[Alice might be hungry][Alice isn't hungry]
= {hungry, full}[Alice isn't hungry]
= {full}
Update with *Alice might be hungry* depends on the result of updating
with the prejacent, *Alice is hungry*. Here's the side calculation:

 {hungry, full}[Alice is hungry]
 = {hungry}

Since this update is nonempty, all of the original possibilities
survive update with *Alice might be hungry*. By now it should be
obvious that update with a *might* sentence either has no effect, or
produces an empty information state. The net result is that we can
then go on to update with *Alice isn't hungry*, yielding an updated
information state that contains only possibilities in which Alice
isn't hungry.

GSV comment that a single speaker couldn't possibly be in a position
to utter the discourse in (2). The reason is that in order for the
speaker to appropriately assert that Alice isn't hungry, that speaker
@@ 300,7 +213,8 @@ speaker assumes that as far as the listener knows, Alice might be
hungry, they can utter the discourse in (2). Here's a variant that
makes this thought more vivid:
 3. Based on public evidence, Alice might be hungry. But in fact she's not hungry.
+ 3. Based on public evidence, Alice might be hungry.
+ But in fact I have private knowledge that she's not hungry.
The main point to appreciate here is that the update behavior of the
discourses depends on the order in which the updates due to the
@@ 313,9 +227,368 @@ concerning negation.
5. Alice is hungry. (So of course) Alice might be hungry.
Both of these discourses lead to the same update effect: all and only
those possibilites in which Alice is hungry survive. If you think
that asserting *might* requires that the prejacent be undecided, you
will have to consider an update rule for the diamond on which update
with the prejacent and its negation must both be nonempty.
+those possibilites in which Alice is hungry survive. You might think
+that asserting *might* requires that the prejacent be not only
+possible, but undecided. If you like this idea, you can easily write
+an update rule for the diamond on which update with the prejacent and
+its negation must both be nonempty.
+
+## Order and binding
+
+The GSV fragment differs from the DPL and the DMG dynamic semantics in
+important details. Nevertheless, it says something highly similar to
+DPL about anaphora, binding, quantificational binding, and donkey
+anaphora (at least, when modality is absent, as we'll discuss below).
+
+In particular, continuing the theme of orderbased asymmetries,
+
+ 6. A man^x entered. He_x sat.
+ 7. He_x sat. A man^x entered.
+
+These discourses differ only in the order of the sentences. Yet the
+first allows for coreference between the indefinite and the pronoun,
+where the second discourse does not. In order to demonstrate, we'll
+need an information state whose refsys is defined for at least one
+variable.
+
+ 8. {(w,g[x>b])}
+
+This infostate contains a refsys and an assignment that maps the
+variable x to Bob. Here are the facts in world w:
+
+ extension w "enter" a = false
+ extension w "enter" b = true
+ extension w "enter" c = true
+
+ extension w "sit" a = true
+ extension w "sit" b = true
+ extension w "sit" c = false
+
+We can now consider the discourses in (6) and (7) (after magically
+converting them to the Predicate Calculus):
+
+ 9. Someone^x entered. He_x sat.
+
+ {(w,g[x>b])}[âx.enter(x)][sit(x)]
+
+ = ( {(w,g[x>b][x>a])}[enter(x)]
+ ++ {(w,g[x>b][x>b])}[enter(x)]
+ ++ {(w,g[x>b][x>c])}[enter(x)])[sit(x)]
+
+  "enter(x)" filters out the possibility in which x refers
+  to Alice, since Alice didn't enter
+
+ = ( {}
+ ++ {(w,g[x>b][x>b])}
+ ++ {(w,g[x>b][x>c])})[sit(x)]
+
+  "sit(x)" filters out the possibility in which x refers
+  to Carl, since Carl didn't sit
+
+ = {(w,g[x>b][x>b])}
+
+One of the key facts here is that even though the existential has
+scope only over the first sentence, in effect it binds the pronoun in
+the following clause. This is characteristic of dynamic theories in
+the style of Groenendijk and Stokhof, including DPL and DMG.
+
+The outcome is different if the order of the sentences is reversed.
+
+ 10. He_x sat. Someone^x entered.
+
+ {(w,g[x>b])}[sit(x)][âx.enter(x)]
+
+  evaluating `sit(x)` rules out nothing, since (coincidentally)
+  x refers to Bob, and Bob is a sitter
+
+ = {(w,g[x>b])}[âx.enter(x)]
+
+  Just as before, the existential adds a new peg and assigns
+  it to each object
+
+ = {(w,g[x>b][x>a])}[enter(x)]
+ ++ {(w,g[x>b][x>b])}[enter(x)]
+ ++ {(w,g[x>b][x>c])}[enter(x)]
+
+  enter(x) eliminates all those possibilities in which x did
+  not enter
+
+ = {} ++ {(w,g[x>b][x>b])}
+ ++ {(w,g[x>b][x>c])}
+
+ = {(w,g[x>b][x>b]), (w,g[x>b][x>c])}
+
+The result is different than before. Before, there was only one
+possibility: that x refered to the only person who both entered and
+sat. Here, there remain two possibilities: that x refers to Bob, or
+that x refers to Carl. This makes predictions about the
+interpretation of continuations of the dialogs:
+
+ 11. A man^x entered. He_x sat. He_x spoke.
+ 12. He_x sat. A man^x entered. He_x spoke.
+
+The construal of (11) as marked entails that the person who spoke also
+entered and sat. The construal of (12) guarantees only that the
+person who spoke also entered. There is no guarantee that the person
+who spoke sat.
+Intuitively, there is a strong impression in (12) that the person who
+entered and spoke not only should not be identified as the person who
+sat, he should be different from the person who sat. Some dynamic
+systems, such as Heim's File Change Semantics, guarantee nonidentity.
+That is not guaranteed by the GSV fragment. If you wanted to add this
+as a refinement to the fragment, you could require that the
+existential only considers object in the domain that are not in the
+range of the starting assignment function.
+
+As usual with dynamic semantics, a point of pride is the ability to
+give a good account of donkey anaphora, as in
+
+ 13. If a woman entered, she sat.
+
+See the paper for details.
+
+## Interactions of binding with modality
+
+At this point, we have a fragment that handles modality, and that
+handles indefinites and pronouns. It it only interesting to combine
+these two elements if they interact in nontrivial ways. This is
+exactly what GSV argue.
+
+The discussion of indefinites in the previous section established the
+following dynamic equivalence:
+
+ (âx.enter(x)) and (sit(x)) â¡ âx (enter(x) and sit(x))
+
+In words, existentials take effective scope over subsequent clauses.
+
+The presence of modal possibility, however, disrupts this
+generalization. GSV illustrate this with the following story.
+
+ The Broken Vase:
+ There are three children: Alice, Bob, and Carl.
+ One of them broke a vase.
+ Alice is known to be innocent.
+ Someone is hiding in the closet.
+
+ (âx.closet(x)) and (âguilty(x)) â¡/â¡ âx (closet(x) and âguilty(x))
+
+To see this, we'll start with the left hand side. We'll need at least
+two worlds.
+
+ in closet guilty
+  
+ w: a true a false
+ b false b true
+ c true c false
+
+ w': a false a false
+ b false b false
+ c true c true
+
+GSV say that (âx.closet(x)) and (âguilty(x)) is true if there is at
+least one possibility in which a person in the closet is guilty. In
+this scenario, world w' is the verifying world: Carl is in the closet,
+and he's guilty. It remains possible that there are closet hiders who
+are not guilty in any world. Alice fits this bill: she's in the
+closet in world w', but she is not guilty in any world.
+
+Let's see how this works out in detail.
+
+ 14. Someone^x is in the closet. He_x might be guilty.
+
+ {(w,g), (w',g}[âx.closet(x)][âguilty(x)]
+
+  existential introduces new peg
+
+ = ( {(w,g[x>a])}[closet(x)]
+ ++ {(w,g[x>b])}[closet(x)]
+ ++ {(w,g[x>c])}[closet(x)]
+ ++ {(w',g[x>a])}[closet(x)]
+ ++ {(w',g[x>b])}[closet(x)]
+ ++ {(w',g[x>c])}[closet(x)])[âguilty(x)]
+
+  only possibilities in which x is in the closet survive
+  the first update
+
+ = {(w,g[x>a]), (w',g[x>c])}[âguilty(x)]
+
+  Is there any possibility in which x is guilty?
+  yes: for x = Carl, in world w' Carl broke the vase
+  that's enough for the possiblity modal to allow the entire
+  infostate to pass through unmodified.
+
+ = {(w,g[x>a]),(w',g[x>c])}
+
+Now we consider the second half:
+
+ 15. Someone^x is in the closet who_x might be guilty.
+
+ {(w,g), (w',g)}[âx(closet(x) & âguilty(x))]
+
+ = {(w,g[x>a])}[closet(x)][âguilty(x)]
+ ++ {(w,g[x>b])}[closet(x)][âguilty(x)]
+ ++ {(w,g[x>c])}[closet(x)][âguilty(x)]
+ ++ {(w',g[x>a])}[closet(x)][âguilty(x)]
+ ++ {(w',g[x>b])}[closet(x)][âguilty(x)]
+ ++ {(w',g[x>c])}[closet(x)][âguilty(x)]
+
+  filter out possibilities in which x is not in the closet
+  and filter out possibilities in which x is not guilty
+  the only person who was guilty in the closet was Carl in
+  world w'
+
+ = {(w',g[x>c])}
+
+The result is different. Fewer possibilities remain.
+We have elminated both possible worlds and possible discourses.
+So the second formula is more informative.
+
+One of main conclusions of GSV is that in the presence of modality,
+the hallmark of dynamic treatmentsthat existentials bind outside of
+their syntactic scopeneeds to refined into a more nuanced understanding.
+Binding still occurs, but the extent of the syntactic scope of an existential
+has a detectable effect on truth conditions.
+
+As we discovered in class, there is considerable work to be done to
+decide which expressions in natural language (if any) are capable of
+expressing which of the two translations into the GSV fragment. We
+can certainly grasp the truth conditions, but that is not the same
+thing as discovering that there are natural language sentences that
+express one or the other or both.
+
+
+## Binding, modality, and identity
+
+The fragment correctly predicts the following contrast:
+
+ 16. Someone^x entered. He_x might be Bob. He_x might not be Bob.
+ (âx.enter(x)) & âx=b & ânot(x=b)
+  This discourse requires a possibility in which Bob entered
+  and another possibility in which someone who is not Bob entered
+
+ 17. Someone^x entered who might be Bob and who might not be Bob.
+ âx (enter(x) & âx=b & ânot(x=b))
+  This is a contradition: there is no single person who might be Bob
+  and who simultaneously might be someone else
+
+These formulas are expressing extensional, dereish intuitions. If we
+add individual concepts to the fragment, the ability to express
+fancier claims would come along.
+
+## GSV's "Identifiers"
+
+Let Î± be a term which differs from x. Then Î± is an identifier if the
+following formula is supported by every information state:
+
+ âx(â(x=Î±) > (x=Î±))
+
+The idea is that Î± is an identifier just in case there is only one
+object that it can refer to. Here is what GSV say:
+
+ A term is an identifier per se if no mattter what the information
+ state is, it cannot fail to decie what the denotation of the term is.
+
+## Digression on pegs
+
+One of the more salient aspects of the technical part of the paper is
+that GSV insert an extra level in between the variable and the object:
+instead of having an assignment function that maps variables directly
+onto objects, GSV provide *pegs*: variables map onto pegs, and pegs
+map onto objects. It happens that pegs play no role in the paper
+whatsoever. We'll demonstrate this by providing a faithful
+implementation of the paper that does not use pegs at all.
+
+Nevertheless, it makes sense to pause here to discuss pegs briefly,
+since this technique is highly relevant to one of the main
+applications of the course, namely, reference and coreference.
+What are pegs? The term harks back to a 1986 paper by Fred Landman
+called `Pegs and Alecs'. Pegs are simply hooks for hanging properties
+on. Pegs are supposed to be as anonymous as possible. Think of
+hanging your coat on a physical peg: you don't care which peg it is,
+only that there are enough pegs for everyone's coat to hang from.
+Likewise, for the pegs of GSV, all that matters is that there are
+enough of them. (Incidentally, there is nothing in Gronendijk and
+Stokhof's original DPL paper that corresponds naturally to pegs; but
+in their Dynamic Montague Grammar paper, pegs serve a purpose similar
+to discourse referents there, though the connection is not simple.)
+
+Pegs can be highly useful for exploring puzzles of reference and
+coreference.
+
+ Standard assignment function System with Pegs (drefs)
+  
+ Variable Object Var Peg Object
+     
+ x > a x > 0 > a
+ y / y /
+ z > b z > 1 > a
+
+A standard assignment function can map two different variables onto
+the same object. In the diagram, x and y are both mapped onto the
+object a. With discourse referents in view, we can have two different
+flavors of coreference. Just as with ordinary assignment functions,
+variables can be mapped onto pegs (discourse referents) that are in
+turn mapped onto the same object. In the diagram, x is mapped onto
+the peg 0, which in turn is mapped onto the object a, and z is mapped
+onto a discourse referent that is mapped onto a. On a deeper level,
+we can suppose that y is mapped onto the same discourse referent as
+x. With a system like this, we are free to reassign the discourse
+referent associated with z to a different object, in which case x and
+z will no longer refer to the same object. But there is no way to
+change the object associated with x without necessarily changing the
+object associated with y. They are coreferent in a deeper, less
+accidental sense.
+
+GSV could make use of this expressive power. But they don't. In
+fact, their system is careful designed to guarantee that every
+variable is assigned a discourse referent distinct from all previous
+discourse referents.
+
+The addition of pegs tracks an active discussion in the dynamic
+literature around the time of publication of the paper. Groenendijk
+and Stokhof (Two theories of dynamic semantics, 1989) noted that it
+was possible in DPL for information to be "lost".
+
+ 18. (âx.P(x)) & (âx.Q(x)) & R(x)
+
+If the two existentials happen to bind the same variable (here, "x"),
+then the second existential occludes the first. That is, at the point
+at which we evalute R(x), all of the assignment functions will be
+mapping the variable "x" to objects that have property Q. The
+information that there exist objects with property P has been lost.
+If you want your dynamic system to be eliminativeor in more general
+terms, if you want the amount of information embodied by an updated
+information state to be monotonically increasingthen this is a
+problem.
+
+A syntactic solution is to require that the variable bound
+by an existential to be chosen fresh.
+
+Vermeulen, Cees FM. "Merging without mystery or: Variables in dynamics
+semantics." Journal of Philosophical Logic 24.4 (1995): 405450 offers
+a different approach, one based on *referent systems*. GSV's pegs are
+a referent system. In the pegs system, when (18) is processed, the
+information that there is an object that has property P is maintained
+in the information state. Curiously, however, there is still no way
+to refer to that object, at least, not with a variable, since there is
+no variable that is associated with the peg that points to the
+relevant object. So the information is present, but not accessible.
+
+That does not mean that there aren't other expression types that are
+able to latch onto peg. An intriguing suggestion based on an example
+in Vermeulen is that "former" might be able to provide access to a
+hidden peg:
+
+ 19. Someone entered. Someone spoke. The former was a woman.
+
+Presumably we want *the former* to be able to pick out the person who
+entered, whether or not the two existentials bind the same variable or
+not. If we allow "former" to latch onto the second most recently
+established peg, no matter whether there is a variable still pointing
+to that peg, the desired effect is achieved.
+
+But none of this is relevant for any of the explanations or analyses
+provide by the GSV fragment, and it is considerably simpler to see
+what their fragment is about if we leave referent systems out of it.