of discourse update. One way of looking at this paper is like this:
GSV = GS + V
+
+ GS = Dynamic Predicate Logic L&P 1991: dynamic binding, donkey anaphora
+ Dynamic Montague Grammar 1990: generalized quantifiers, discourse referents
+
+ V = epistemic modality
That is, Groenendijk and Stokhof have a well-known theory of dynamic
semantics, and Veltman has a well-known theory of epistemic modality,
## Basics
There are a lot of formal details in the paper in advance of the
-empirical discussion. Here are the ones that matter:
+empirical discussion. Here are the ones that matter for our purposes:
type var = string
type peg = int
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.
+r (g ("x")) = Alice. A question to keep in mind as we proceed is why
+the mapping from variables to objects has been articulated into two
+functions. Why not map variables directly to objects? (We'll return
+to this question later.)
type pred = string
type world = pred -> ent -> bool
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, n+1, r[x->n], g[n->a]) | (w,n,r,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
+adding a peg to each possibility in s and adjusting the refsys and the
+assignment in order to map the variable x to a. Then update s' with
+φ, and collect 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.
+in terms of negation and the other connectives (see fact 3.2).
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, where `i = (w,n,r,g)`:
+man.
+
+We're using `++` here to mean set union.
- 1. {i}[∃x.person(x)]
+ 1. {(w,n,r,g)}[∃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])}[person(x)] ++ {(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])}
= {(w,n+1,r[x->n],g[n->a]),(w,n+1,r[x->n],g[n->b])}
+ -- both a and b are people
- 2. {i}[∃x.man(x)]
+ 2. {(w,n,r,g)}[∃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->a])}[man(x)] ++ {(w,n+1,r[x->n],g[n->b])}[man(x)]
+ = {} ++ {(w,n+1,r[x->n],g[n->b])}
= {(w,n+1,r[x->n],g[n->b])}
+ -- only b is a man
+ 3. {(w,n,r,g)}[∃x∃y.person(x) and person(y)]
- 3. {i}[∃x∃y.person(x) and person(y)]
+ = {(w,n+1,r[x->n],g[n->a])}[∃y.person(x) and person(y)]
+ ++ {(w,n+1,r[x->n],g[n->b])}[∃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])
- }
+ = ( {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->a])}[person(x)][person(y)]
+ ++ {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->b])}[person(x)][person(y)])
+ ++ ( {(w, n+2, r[x->n][y->n+1], g[n->b][n+1->a])}[person(x)][person(y)]
+ ++ {(w, n+2, r[x->n][y->n+1], g[n->b][n+1->b])}[person(x)][person(y)])
- 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])}
- = {(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])
- }
+ = {(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]
+ -- there are four ways of assigning x and y to people
+
+
+ 4. {(w,n,r,g)}[∃x∃y.x=y]
+
+ = ( {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->a])}[x=y]
+ ++ {(w, n+2, r[x->n][y->n+1], g[n->a][n+1->b])}[x=y]
+ ++ ( {(w, n+2, r[x->n][y->n+1], g[n->b][n+1->a])}[x=y]
+ ++ {(w, n+2, r[x->n][y->n+1], g[n->b][n+1->b])}[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])}
= {(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])
- }
+ (w, n+2, r[x->n][y->n+1], g[n->b][n+1->b])}
+
+ -- two ways to assign x and y to the same value
## Order and modality
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:
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])}[enter(x)]
+ ++ {(w,2,r[x->0][x->1],g[0->b][1->b])}[enter(x)]
+ ++ {(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.
+
+The outcome is different if the order of the sentences is reversed.
+
+ 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])}[enter(x)]
+ ++ {(w,2,r[x->0][x->1],g[0->b][1->b])}[enter(x)]
+ ++ {(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])}
+
+ = {(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 require 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 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 sons, Bob, Carl, and Dave.
+ One of them broke a vase.
+ Bob 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: b false b false
+ c false c false
+ d true d true
+
+ w': b false b false
+ c true c false
+ d false d true
+
+GSV observe 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. It remains possible
+that there are closet hiders who are not guilty in any world. Carl
+fits this bill: he's in the closet in world w', but he 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,0,r,g), (w',0,r,g}[∃x.closet(x)][◊guilty(x)]
+
+ -- existential introduces new peg
+
+ = ( {(w,1,r[x->0],g[0->b])}[closet(x)]
+ ++ {(w,1,r[x->0],g[0->c])}[closet(x)]
+ ++ {(w,1,r[x->0],g[0->d])}[closet(x)]
+ ++ {(w',1,r[x->0],g[0->b])}[closet(x)]
+ ++ {(w',1,r[x->0],g[0->c])}[closet(x)]
+ ++ {(w',1,r[x->0],g[0->d])}[closet(x)])[◊guilty(x)]
+
+ -- only possibilities in which x is in the closet survive
+
+ = {(w,1,r[x->0],g[0->d]),
+ (w',1,r[x->0],g[0->c])}[◊guilty(x)]
+
+ -- Is there any possibility in which x is guilty?
+ -- yes: for x = Dave, in world w Dave broke the vase
+
+ = {(w,1,r[x->0],g[0->d]),
+ (w',1,r[x->0],g[0->c])}
+
+Now we consider the second half:
+
+ 14. Someone^x is in the closet who_x might be guilty.
+
+ {(w,0,r,g), (w',0,r,g)}[∃x(closet(x) & ◊guilty(x))]
+
+ -- existential introduces new peg
+
+ = {(w,1,r[x->0],g[0->b])}[closet(x)][◊guilty(x)]
+ ++ {(w,1,r[x->0],g[0->c])}[closet(x)][◊guilty(x)]
+ ++ {(w,1,r[x->0],g[0->d])}[closet(x)][◊guilty(x)]
+ ++ {(w',1,r[x->0],g[0->b])}[closet(x)][◊guilty(x)]
+ ++ {(w',1,r[x->0],g[0->c])}[closet(x)][◊guilty(x)]
+ ++ {(w',1,r[x->0],g[0->d])}[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 Dave in
+ -- world 1
+
+ = {(w,1,r[x->0],g[0->d])}
+
+The result is different, and more informative.
+
+## Binding, modality, and identity
+
+The fragment correctly predicts the following contrast:
+
+ 15. 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
+
+ 16. 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.
+
+### 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.
+
+## Why articulate the mapping from variables to objects into two parts?
+
+In the current system, variables are associated with values in two
+steps.
+
+ Variables Pegs Entities
+ --------- r ---- g --------
+ x --> 0 --> a
+ y --> 1 --> b
+ z --> 2 --> c
+
+Here, r is a refsys mapping variables to pegs, and g is an assignment
+function mapping pegs to entities.
+
+Assignment functions are free to map different pegs to the same
+entity:
+
+ Variables Pegs Entities
+ --------- r ---- g --------
+ x --> 0 --> a
+ y --> 1 --> a
+ z --> 2 --> c
+
+But this is possible with ordinary assignment functions as well.
+
+It is possible to imagine a refsys that maps more than one variable to
+the same peg. But the fragment is designed to prevent that from ever
+happening: the only way to associate a variable with a peg is by
+evaluating an existential quantifier, and the existential quantifier
+always introduces a fresh, unused peg.
+
+So what does the bipartite system do that ordinary assignment
+functions can't do?