the fragment behaves with respect to these sentences.
We'll start with an infostate containing two possibilities. In one
-possibility (w1), Alice is hungry; in the other (w2), she is not.
+possibility, Alice is hungry (call this possibility "hungry"); in the
+other, she is not (call it "full").
- = {(w1,n,r,g), (w2,n,r,g)}[Alice isn't hungry][Alice might be hungry]
- = {(w2,n,r,g)}[Alice might be hungry]
+ {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 (w1), leaving only the possibility containing w2.
+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:
- {(w2,n,r,g)}[Alice is hungry]
+ {full}[Alice is hungry]
= {}
Because the only possibility in the information state is one in which
We'll start with the same two possibilities.
- = {(w1,n,r,g), (w2,n,r,g)}[Alice might be hungry][Alice isn't hungry]
- = {(w1,n,r,g), (w2,n,r,g)}[Alice isn't hungry]
- = {(w2,n,r,g)}
+ = {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:
- {(w1,n,r,g), (w2,n,r,g)}[Alice is hungry]
- = {(w1,n,r,g)}
+ {hungry, full}[Alice is hungry]
+ = {hungry}
Since this update is non-empty, all of the original possibilities
survive update with *Alice might be hungry*. By now it should be
will have to consider an update rule for the diamond on which update
with the prejacent and its negation must both be non-empty.
+## Binding
+
+The GSV fragment differs from the DPL and the DMG dynamic semantics in
+important details. Nevertheless, it has more or less the same things
+to say about anaphora, binding, quantificational binding, and donkey
+anaphora.
+
+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,1,r[x->0],g[0->b])}
+
+This infostate contains a refsys and an assignment that maps the
+variable x to Bob. Here are the facts in world w:
+
+ w "enter" a = false
+ w "enter" b = true
+ w "enter" c = true
+ w "sit" a = true
+ w "sit" b = true
+ 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,1,r[x->0],g[0->b])}[∃x.enter(x)][sit(x)]
+
+ -- the existential adds a new peg and assigns it to each
+ -- entity in turn
+
+ = {(w,2,r[x->0][x->1],g[0->b][1->a]),
+ (w,2,r[x->0][x->1],g[0->b][1->b]),
+ (w,2,r[x->0][x->1],g[0->b][1->c])}[enter(x)][sit(x)]
+
+ -- "enter(x)" filters out the possibility in which x refers
+ -- to Alice, since Alice didn't enter
+
+ = {(w,2,r[x->0][x->1],g[0->b][1->b]),
+ (w,2,r[x->0][x->1],g[0->b][1->c])}[sit(x)]
+
+ -- "sit(x)" filters out the possibility in which x refers
+ -- to Carl, since Carl didn't sit
+
+ = {(w,2,r[x->0][x->1],g[0->b][1->b])}
+
+Note that `r[x->0][x->1]` maps `x` to 1---the outermost adjustment is
+the operative one. In other words, `r[x->0][x->1] == (r[x->0])[x->1]`.
+
+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.
+
+ 10. He_x sat. Someone^x entered.
+
+ {(w,1,r[x->0],g[0->b])}[sit(x)][∃x.enter(x)]
+
+ -- evaluating `sit(x)` rules out nothing, since (coincidentally)
+ -- x refers to Bob, and Bob is a sitter
+
+ = {(w,1,r[x->0],g[0->b])}[∃x.enter(x)]
+
+ -- Just as before, the existential adds a new peg and assigns
+ -- it to each object
+
+ = {(w,2,r[x->0][x->1],g[0->b][1->a]),
+ (w,2,r[x->0][x->1],g[0->b][1->b]),
+ (w,2,r[x->0][x->1],g[0->b][1->c])}[enter(x)]
+
+ -- enter(x) eliminates all those possibilities in which x did
+ -- not enter
+
+ = {(w,2,r[x->0][x->1],g[0->b][1->b]),
+ (w,2,r[x->0][x->1],g[0->b][1->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. The GSV guarantees that
+the indefinite introduces a novel peg, but there is no requirement
+that the peg refers to a novel object. If you wanted to add this as a
+refinement to the fragment, you could required that whenever a new peg
+gets added, it must be mapped onto an object that is not in the range
+of the original 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 in effect take scope over subsequent clauses.
+
+The presence of modal possibility, however, disrupts this
+generalization:
+
+ (∃x.enter(x)) and (◊sit(x)) ≡/≡ ∃x (enter(x) and ◊sit(x))
+
+To see this, we'll start with the left hand side.
+
+ 14. Someone^x entered. He_x might sit.
+
+ {(w,1,r[x->0],g[0->b])}[∃x.enter(x)][◊sit(x)]
+
+ -- same computation up to the point of the modal
+
+ = {(w,2,r[x->0][x->1],g[0->b][1->b]),
+ (w,2,r[x->0][x->1],g[0->b][1->c])}[◊sit(x)]
+
+ -- modal returns all or none, depending on whether the
+ -- prejacent is consistent with the starting infostate.
+ -- since there is one choice for x who sat, returns all:
+
+ = {(w,2,r[x->0][x->1],g[0->b][1->b]),
+ (w,2,r[x->0][x->1],g[0->b][1->c])}
+
+To paraphrase, the requirements are that there must be a person who
+entered, and it might be possible that that person sat. But this is
+not metaphysical possibility: we're not choosing a person an wondering
+whether that person sat. If that's what we had in mind, we'd go off
+to a bunch of non-actual possible worlds and see what is happening
+there. Instead, this is supposed to be epistemic possibility. The
+paraphrase should be something like: there must be a person who
+entered, and for all we know, that person might have sat.
+
+The peculiar thing is that the uncertainty has nothing to do with the
+facts of the world, but only with the fact about the discourse: it's
+uncertainty about which object the pronoun refers to. GSV work hard
+to make this interpretation plausible. Here's their story:
+
+ There are three kids. One of them breaks a vase. One is known to
+ be innocent. There are sounds coming out of the closet.
+
+ 15. Someone^x is in the closet. He_x might be guilty.
+
+You have enough information to know that someone is in the closet.
+You use the pronoun to refer to the person in the closet, and assert
+that, for all you know, that person might be guilty. The fragment
+gives you guaranteed coreference---it's whoever is in the closet who
+might be guilty---in the presence of uncertainty about who the pronoun
+refers to.
+
+Now we consider the second half:
+
+ 14. Someone^x entered who_x might sit.
+
+ {(w,1,r[x->0],g[0->b])}[∃x.(enter(x) & ◊sit(x)]
+
+ = {(w,2,r[x->0][x->1],g[0->a]),
+ (w,2,r[x->0][x->1],g[0->b]),
+ (w,2,r[x->0][x->1],g[0->c])}[enter(x)][◊sit(x)]
+
+ -- recall that Alice didn't enter, so
+
+ = {(w,2,r[x->0][x->1],g[0->b]),
+ (w,2,r[x->0][x->1],g[0->c])}[◊sit(x)]