Extended application: Groenendijk, Stokhof and Veltman's Coreference and Modality

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

What are pegs?  A paper by Landman called Pegs and Alecs'.

It might be raining.  It's not raining.
#It's not raining.  It might be raining.


Two-part assignment functions

type var = string
type peg = int
type refsys = var -> peg
type ent = Alice | Bob | Carl
type assignment = peg -> ent

type pred = string
type world = pred -> ent -> bool
type pegcount = int
type poss = world * pegcount * refsys * assignment
type infostate = [poss]


So information states track both facts about the world (e.g., which
objects count as a man), and facts about the discourse (e.g., how many
pegs have been used).

ref(i, t) = t if t is an individual, and
g(r(t)) if t is a variable, where i = (w,n,r,g)

s[P(t)] = {i in s | w(P)(ref(i,t))}

s[t1 = t2] = {i in s | ref(i,t1) = ref(i,t2)}

s[φ and ψ] = s[φ][ψ]

s[∃xφ] = Union {{(w, n+1, r[x->n], g[n->a]) | (w,n,r,g) in s}[φ] | a in ent}

s[neg φ] =  {i | {i}[φ] = {}}

1. {(w,n,r,g)}[∃x.person(x)]
2. {(w,n,r,g)}[∃x.man(x)]
3. {(w,n,r,g)}[∃x∃y.person(x) and person(y)]
4. {(w,n,r,g)}[∃x∃y.x=y]


Order and modality

s[◊φ] = {i in s | s[φ] ≠ {}}

1. Alice isn't hungry.  #Alice might be hungry.

{hungry, full}[Alice isn't hungry][Alice might be hungry]
= {full}[Alice might be hungry]
= {}

2. Alice might be hungry.  Alice isn't hungry.

= {hungry, full}[Alice might be hungry][Alice isn't hungry]
= {hungry, full}[Alice isn't hungry]
= {full}


GSV: a single speaker couldn't possibly be in a position to utter (2).

3. Based on public evidence, Alice might be hungry.  But in fact she's not hungry.

4. Alice might be hungry.  Alice *is* hungry.
5. Alice is hungry.  (So of course) Alice might be hungry.


consider: update with the prejacent and its negation must both be non-empty.

Order and binding

6. A man^x entered.  He_x sat.
7. He_x sat.  A man^x entered.

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

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])}


existential in effect binds the pronoun in the following clause

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])}

11. A man^x entered.  He_x sat.  He_x spoke.
12. He_x sat.  A man^x entered.  He_x spoke.


consider: new peg requires object not in the domain of the assignment fn

13. If a woman entered, she sat.


Interactions of binding with modality

(∃x.enter(x)) and (sit(x)) ≡ ∃x (enter(x) and sit(x))


but (∃x.closet(x)) and (◊guilty(x)) ≡/≡ ∃x (closet(x) and ◊guilty(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.

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.

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])}


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


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 have a two-part assignment function?

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

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