-<!-- λ ∃ Λ ∀ ≡ α β γ ρ ω φ ψ Ω ○ μ η δ ζ ξ ⋆ ★ • ∙ ● ⚫ 𝟎 𝟏 𝟐 𝟘 𝟙 𝟚 𝟬 𝟭 𝟮 ⇧ (U+2e17) ¢ -->
+<!-- λ â\97\8a â\89 â\88\83 Î\9b â\88\80 â\89¡ α β γ Ï\81 Ï\89 Ï\86 Ï\88 Ω â\97\8b μ η δ ζ ξ â\8b\86 â\98\85 â\80¢ â\88\99 â\97\8f â\9a« ð\9d\9f\8e ð\9d\9f\8f ð\9d\9f\90 ð\9d\9f\98 ð\9d\9f\99 ð\9d\9f\9a ð\9d\9f¬ ð\9d\9f ð\9d\9f® â\87§ (U+2e17) ¢ -->
[[!toc levels=2]]
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): 43-63.
That is, Groenendijk and Stokhof have a well-known theory of dynamic
semantics, and Veltman has a well-known theory of epistemic modality,
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
-re-engineering 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 re-engineering 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
+world-assignment 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 [[here|code/gsv.ml]]; consult
+that code for details of syntax, types, and values.
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).
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[φ and ψ] = s[φ][ψ]
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[∃xφ] = {(w, n+1, r[x->n], g[n->a]) | (w,n,r,g) in s and a in ent}[φ]
+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[∃xφ] = Union {{(w, g[x->a]) | (w,g) in s}[φ] | a in ent}
-Note that this recipe does not examine φ. This means that this
-analysis treats the formula prefix `∃x` as if it were a meaningful
-constituent independent of φ.
+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 φ. Finally, take the union over the results of
+doing this for every object a in the domain of discourse.
Negation is natural enough:
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}[∃x.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])}
-
- 2. {i}[∃x.man(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->b])}
-
-
- 3. {i}[∃x∃y.person(x) and person(y)]
-
- = {(w,n+1,r[x->n],g[n->a]),(w,n+1,r[x->n],g[n->b])}[∃y.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])
- }
-
- 4. {i}[∃x∃y.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])
- }[∃x∃y.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}[∃x∃y.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])
- }[∃x∃y.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])
- }
+in terms of negation and the other connectives (see fact 3.2).
+
+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.
+
+Compute the following:
+
+ 1. {(w,g)}[∃x.man(x)]
+
+ = {(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
+
+ 2. {(w,g)}[∃x.woman(x)]
+ 3. {(w,g)}[∃x∃y.man(x) and man(y)]
+ 4. {(w,n,r,g)}[∃x∃y.x=y]
+
+Running the [[code|code/gsv.ml]] gives the answers.
+
+
+## Order and modality
+
+The final remaining update rule concerns modality:
+
+ s[◊φ] = {i in s | s[φ] ≠ {}}
+
+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 the
+prejacent φ 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.
+The fact that the predictions of the fragment differ depending on
+order shows that the system is order-sensitive.
+
+ 1. Alice isn't hungry. #Alice might be hungry.
+
+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 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
+other, she is not (call it "full").
+
+ {hungry, full}[Alice isn't hungry][Alice might be hungry]
+ = {full}[Alice might be hungry]
+ = {}
+
+As usual in dynamic theories, a sequence of sentences is treated as if
+the sentence were conjoined. This is the same thing as updating with
+the first sentence, then updating with the second sentence.
+Update with *Alice isn't hungry* eliminates the possibility in which
+Alice is hungry, leaving only the possibility in which she is full.
+Subsequent update with *Alice might be hungry* depends on the result
+of updating with the prejacent, *Alice is hungry*. Let's do that side
+calculation:
+
+ {full}[Alice is hungry]
+ = {}
+
+Because the only possibility in the information state is one in which
+Alice is not hungry, update with *Alice is hungry* results in an empty
+information state. That means that update with *Alice might be
+hungry* will also be empty, as indicated above.
+
+In order for update with *Alice might be hungry* to be non-empty,
+there must be at least one possibility in the input state in which
+Alice is hungry. That is what epistemic might means in this fragment:
+the prejacent must be possible. But update with *Alice isn't hungry*
+eliminates all possibilities in which Alice is hungry. So the
+prediction of the fragment is that update with the sequence in (1)
+will always produce an empty information state.
+
+In contrast, consider the sentences in the opposite order:
+
+ 2. Alice might be hungry. Alice isn't hungry.
+
+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}
+
+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
+would have to possess knowledge (or sufficient justification,
+depending on your theory of the norms for assertion) that Alice isn't
+hungry. But if they know that Alice isn't hungry, they couldn't
+appropriately assert *Alice might be hungry*, based on the predictions
+of the fragment.
+
+Another view is that it can be acceptable to assert a sentence if it
+is supported by the information in the common ground. So if the
+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 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
+individual sentence occur.
+
+Note, incidentally, that there is an asymmetry in the fragment
+concerning negation.
+
+ 4. Alice might be hungry. Alice *is* hungry.
+ 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. 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 non-empty.
+
+## 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 order-based 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])}[∃x.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)][∃x.enter(x)]
+
+ -- evaluating `sit(x)` rules out nothing, since (coincidentally)
+ -- x refers to Bob, and Bob is a sitter
+
+ = {(w,g[x->b])}[∃x.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 non-identity.
+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 non-trivial ways. This is
+exactly what GSV argue.
+
+The discussion of indefinites in the previous section established the
+following dynamic equivalence:
+
+ (∃x.enter(x)) and (sit(x)) ≡ ∃x (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.
+
+ (∃x.closet(x)) and (◊guilty(x)) ≡/≡ ∃x (closet(x) and ◊guilty(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 (∃x.closet(x)) and (◊guilty(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}[∃x.closet(x)][◊guilty(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)])[◊guilty(x)]
+
+ -- only possibilities in which x is in the closet survive
+ -- the first update
+
+ = {(w,g[x->a]), (w',g[x->c])}[◊guilty(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)}[∃x(closet(x) & ◊guilty(x))]
+
+ = {(w,g[x->a])}[closet(x)][◊guilty(x)]
+ ++ {(w,g[x->b])}[closet(x)][◊guilty(x)]
+ ++ {(w,g[x->c])}[closet(x)][◊guilty(x)]
+ ++ {(w',g[x->a])}[closet(x)][◊guilty(x)]
+ ++ {(w',g[x->b])}[closet(x)][◊guilty(x)]
+ ++ {(w',g[x->c])}[closet(x)][◊guilty(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.
+
+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.
+ (∃x.enter(x)) & ◊x=b & ◊not(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.
+ ∃x (enter(x) & ◊x=b & ◊not(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, de-reish intuitions. If we
+add individual concepts to the fragment, the ability to express
+fancier claims would come along.
+
+## GSV's "Identifiers"
+
+Let α be a term which differs from x. Then α is an identifier if the
+following formula is supported by every information state:
+
+ ∀x(◊(x=α) --> (x=α))
+
+The idea is that α 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.
+
+End of digression on pegs.