+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.