[[!toc levels=2]]
# Doing things with monads
## Extended application: Groenendijk, Stokhof and Veltman's *Coreference and Modality*
GSV are interested in developing and establishing a reasonable theory
of discourse update. One way of looking at this paper is like this:
GSV = GS + V
That is, Groenendijk and Stokhof have a well-known theory of dynamic
semantics, and Veltman has a well-known theory of epistemic modality,
and this fragment brings both of those strands together into a single
system.
We will be interested in this paper both from a theoretical point of
view and from a practical engineering point of view. On the
theoretical level, these scholars are proposing a strategy for
managing the connection between variables and the objects they
designate in way that is flexible enough to be useful for describing
natural language. The main way they attempt to do this is by
inserting an extra level in between the variable and the object:
instead of having an assignment function that maps variables directly
onto objects, GSV provide *pegs*: variables map onto pegs, and pegs
map onto objects. We'll discuss in considerable detail what pegs
allow us to do, since it is highly relevant to one of the main
applications of the course, namely, reference and coreference.
What are pegs? The term harks back to a paper by Landman called `Pegs
and Alecs'. There pegs are simply hooks for hanging properties on.
Pegs are supposed to be as anonymous as possible. Think of hanging
your coat on a physical peg: you don't care which peg it is, only that
there are enough pegs for everyone's coat to hang from. Likewise, for
the pegs of GSV, all that matters is that there are enough of them.
(Incidentally, there is nothing in Gronendijk and Stokhof's original
DPL paper that corresponds naturally to pegs; but in their Dynamic
Montague Grammar paper, pegs serve a purpose similar to discourse
referents there, though the connection is not simple.)
On an engineering level, the fact that GSV are combining anaphora and
bound quantification with epistemic quantification means that they are
gluing together related but distinct subsystems into a single
fragment. These subsystems naturally cleave into separate layers in a
way that is obscured in the paper. We will argue in detail that
re-engineering GSV using monads will lead to a cleaner system that
does all of the same theoretical work.
Empirical targets: on the anaphoric side, GSV want to
On the epistemic side, GSV aim to account for asymmetries such as
It might be raining. It's not raining.
#It's not raining. It might be raining.
## Basics
There are a lot of formal details in the paper in advance of the
empirical discussion. Here are the ones that matter:
type var = string
type peg = int
type refsys = var -> peg
type ent = Alice | Bob | Carl
type assignment = peg -> ent
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.
type pred = string
type world = pred -> ent -> bool
type pegcount = int
type poss = world * pegcount * refsys * assignment
type infostate = [poss]
Worlds in general settle all matters of fact in the world. In
particular, they determine the extensions of predicates and relations.
In this discussion, we'll (crudely) approximate worlds by making them
a function from predicates such as "man" to a function mapping each
entity to a boolean.
As we'll see, indefinites as a side effect increase the number of pegs
by one. GSV assume that we can determine what integer the next unused
peg corresponds to by examining the range of the refsys function.
We'll make things easy on ourselves by simply tracking the total
number of used pegs in a counter called `pegcount`.
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).
The formal language the fragment interprets is Predicate Calculus with
equality, existential and universal quantification, and one unary
modality (box and diamond, corresponding to epistemic necessity and
epistemic possibility).
Terms in this language are either individuals such as Alice or Bob, or
else variables. So in general, the referent of a term can depend on a
possibility:
ref(i, t) = t if t is an individual, and
g(r(t)) if t is a variable, where i = (w,n,r,g)
Here are the main clauses for update (their definition 3.1).
Following GSV, we'll write `update(s, φ)` (the update of information
state `s` with the information in φ) as `s[φ]`.
s[P(t)] = {i in s | w(P)(ref(i,t))}
So `man(x)` is the set of live possibilities `i = (w,r,g)` in s such that
the set of men in `w` given by `w(man)` maps the object referred to by
`x`, namely, `r(g("x"))`, to `true`. That is, update with "man(x)"
discards all possibilities in which "x" fails to refer to a man.
s[t1 = t2] = {i in s | ref(i,t1) = ref(i,t2)}
s[φ and ψ] = s[φ][ψ]
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]
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.
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.
Negation is natural enough:
s[neg φ] = {i | {i}[φ] = {}}
If updating φ with the information state that contains only the
possibility i returns the empty information state, then not φ is true
with respect to i.
In GSV, disjunction, the conditional, and the universals are defined
in terms of negation and the other connectives.
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)`:
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 (w1), Alice is hungry; in the other (w2), she is not.
= {(w1,n,r,g), (w2,n,r,g)}[Alice isn't hungry][Alice might be hungry]
= {(w2,n,r,g)}[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:
{(w2,n,r,g)}[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.
= {(w1,n,r,g), (w2,n,r,g)}[Alice might be hungry][Alice isn't hungry]
= {(w1,n,r,g), (w2,n,r,g)}[Alice isn't hungry]
= {(w2,n,r,g)}
Update with *Alice might be hungry* depends on the result of updating
with the prejacent, *Alice is hungry*. Here's the side calculation:
{(w1,n,r,g), (w2,n,r,g)}[Alice is hungry]
= {(w1,n,r,g)}
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.