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:
w "enter" a = false
w "enter" b = true
w "enter" c = true
+
w "sit" a = true
w "sit" b = true
w "sit" c = false
-- 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)]
+ = ( {(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)]
+ = ( {}
+ ++ {(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])}
+ = {(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 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.
-- 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)]
+ = {(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->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
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
+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.
(∃x.enter(x)) and (sit(x)) ≡ ∃x (enter(x) and sit(x))
-In words, existentials in effect take scope over subsequent clauses.
+In words, existentials take effective scope over subsequent clauses.
The presence of modal possibility, however, disrupts this
-generalization:
+generalization. GSV illustrate this with the following story.
- (∃x.enter(x)) and (◊sit(x)) ≡/≡ ∃x (enter(x) and ◊sit(x))
+ 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.
-To see this, we'll start with the left hand side.
+ (∃x.closet(x)) and (◊guilty(x)) ≡/≡ ∃x (closet(x) and ◊guilty(x))
- 14. Someone^x entered. He_x might sit.
+To see this, we'll start with the left hand side. We'll need at least
+two worlds.
- {(w,1,r[x->0],g[0->b])}[∃x.enter(x)][◊sit(x)]
+ in closet guilty
+ --------------- ---------------
+ w: b false b false
+ c false c false
+ d true d true
- -- same computation up to the point of the modal
+ w': b false b false
+ c true c false
+ d false d true
- = {(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)]
+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.
- -- 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:
+Let's see how this works out in detail.
- = {(w,2,r[x->0][x->1],g[0->b][1->b]),
- (w,2,r[x->0][x->1],g[0->b][1->c])}
+ 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)]
-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.
+ -- existential introduces new peg
-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:
+ = ( {(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)]
- There are three kids. One of them breaks a vase. One is known to
- be innocent. There are sounds coming out of the closet.
+ -- only possibilities in which x is in the closet survive
- 15. Someone^x is in the closet. He_x might be guilty.
+ = {(w,1,r[x->0],g[0->d]),
+ (w',1,r[x->0],g[0->c])}[◊guilty(x)]
-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.
+ -- 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 entered who_x might sit.
+ 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:
- {(w,1,r[x->0],g[0->b])}[∃x.(enter(x) & ◊sit(x)]
+ Variables Pegs Entities
+ --------- r ---- g --------
+ x --> 0 --> a
+ y --> 1 --> a
+ z --> 2 --> c
- = {(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)]
+But this is possible with ordinary assignment functions as well.
- -- recall that Alice didn't enter, so
+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.
- = {(w,2,r[x->0][x->1],g[0->b]),
- (w,2,r[x->0][x->1],g[0->c])}[◊sit(x)]
+So what does the bipartite system do that ordinary assignment
+functions can't do?