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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, 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}
+
+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.
+
- 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]