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[[!toc levels=2]]
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:
+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])
+ }
+
+## 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 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 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.
+
+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 (w1), leaving only the possibility containing w2.
+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}
+
+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 non-empty, 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
+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 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. 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 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)]
- 1. {}[∃x.person(x)]
- 2. {}[∃x.man(x)]
- 3. {}[∃x∃y.person(x) and person(y)]
- 4. {}[∃x∃y.x=x]
- 5. {}[∃x∃y.x=y]