Doing things with monads (an 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, where

GS = Dynamic theories of binding of Groenendijk and Stokhof, e.g., Dynamic Predicate Logic L&P 1991: dynamic binding, donkey anaphora Dynamic Montague Grammar 1990: generalized quantifiers and discourse referents

V = a dynamic theory of epistemic modality, e.g., Veltman, Frank. "Data semantics." In Truth, Interpretation and Information, Foris, Dordrecht (1984): 43-63, or Veltman, Frank. "Defaults in update semantics." Journal of philosophical logic 25.3 (1996): 221-261.

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. The key result, as we'll discuss, is that adding modality to dynamic semantics creates some unexpected and fascinating interactions.

Basics of GSV's fragment

The fragment in this paper is unusually elegant. We'll present it on its own terms, with the exception that we will not use GSV's "pegs". See the discussion below below concerning pegs for an explanation. After presenting the paper, we'll re-engineer the fragment using explicit monads.

In this fragment, points of evaluation are not just worlds, but pairs consisting of a world and an assginment function. This conception of an evaluation point is familiar from Heim's 1983 File Change Semantics. Following GSV, we'll call a world-assignment pair a "possibility", and so a context (an "information state") will be set of possiblities. As GSV emphasize, infostates simultaneously track information about the world (which possible worlds are live possibilities?) as well as information about the discourse (which objects to the variables refer to?).

The formal language the fragment interprets is the Predicate Calculus with equality, existential and universal quantification, and one unary modality, interpreted as epistemic possibility.

An implementation in OCaml is available here; consult that code for details of syntax, types, and values. An implementation in Haskell is available as well, if you prefer.

(We've also written a fuller and separate implementation using the OCaml Juli8 Monad libraries. This fuller implementation demonstrates a graduated collection of semantics, so that you can see how we start from a monadic implementation of classic variable binding, and step by step approach the system presented by GSV. In the fullness of time, these two bodies of code will be merged. But for the time being, we just present them both for your edification.)

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(t) if t is a variable, where i = (w,g)

Immediately following are the recipes for context update (GSV's 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)] = {(w,g) in s | extension w P (ref((w,g),t))}

So man(x) is the set of live possibilities (w,g) in s such that the set of men in w given by extension w "man" maps the object referred to by x, namely, 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 is somewhat intricate.

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

Here's the recipe: given a starting infostate s, choose an object a from the domain of discourse. Construct a modified infostate s' by adjusting the assignment function of each possibility so as to map the variable x to a. Then update s' with φ. Finally, take the union over the results of doing this for every object a in the domain of discourse. If you're unsure about exactly what this recipe does, examine the implementations linked above.

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 (see fact 3.2).

Exercise: assume that there are three entities in the domain of discourse, Alice, Bob, and Carl. Assume that Alice is a woman, and Bob and Carl are men.

Compute the following:

1. {(w,g)}[∃]

   = {(w,g[x->a])}[man(x)] ++ {(w,g[x->b])}[man(x)] 
                           ++ {(w,g[x->c])}[man(x)] 
   = {} ++ {(w,g[x->b])} ++ {(w,g[x->c])}
   = {(w,g[x->a]),(w,g[x->b]),(w,g[x->c])}
   -- Bob and Carl are men

2. {(w,g)}[∃x.woman(x)]
3. {(w,g)}[∃x∃ and man(y)]
4. {(w,g)}[∃x∃y.x=y]

Running the code gives the answers.

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 the prejacent 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 way to think about its intuitive status 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: there must be a possibility in the starting infostate that is consistent with the prejacent. 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}

This is a very different result: the two sentences are consistent, and do not guarantee an empty output infostate.

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 I have private knowledge that) 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 sentences are processed.

Note, incidentally, that the treatment of modality contains an asymmetry related to 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. So negating an assertion rules out the possibility, but asserting the non-negated version does not.

You might think that asserting might requires that the prejacent be not merely possible, but undecided. If you like this idea, you can easily write an update rule for the diamond on which update with the prejacent and its negation must both be non-empty.

Order and binding

The GSV fragment differs from the DPL and the DMG dynamic semantics in important details. Nevertheless, it is highly similar to DPL with respect to anaphora, binding, quantificational binding, and donkey anaphora (at least, until we add modality into the mix, as we will below).

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 how the fragment treats these discourses, we'll need an information state whose refsys is defined for at least one variable.

8. {(w,g[x->b])}

This infostate contains a refsys and an assignment that maps the variable x to Bob. Here are the facts in world w:

extension w "enter" a = false
extension w "enter" b = true
extension w "enter" c = true

extension w "sit" a = true
extension w "sit" b = true
extension 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,g[x->b][x->a])}[enter(x)]
      ++ {(w,g[x->b][x->b])}[enter(x)]
      ++ {(w,g[x->b][x->c])}[enter(x)])[sit(x)]

      -- "enter(x)" filters out the possibility in which x refers
      -- to Alice, since Alice didn't enter

   = (   {}
      ++ {(w,g[x->b][x->b])}
      ++ {(w,g[x->b][x->c])})[sit(x)]

      -- "sit(x)" filters out the possibility in which x refers
      -- to Carl, since Carl didn't sit

   =  {(w,g[x->b][x->b])}

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. 


     -- evaluating `sit(x)` rules out nothing, since (coincidentally)
     -- x refers to Bob, and Bob is a sitter

   = {(w,g[x->b])}[∃x.enter(x)]

     -- Just as before, the existential adds a new peg and assigns
     -- it to each object

   =    {(w,g[x->b][x->a])}[enter(x)]
     ++ {(w,g[x->b][x->b])}[enter(x)]
     ++ {(w,g[x->b][x->c])}[enter(x)]

     -- enter(x) eliminates all those possibilities in which x did
     -- not enter

   = {} ++ {(w,g[x->b][x->b])}
        ++ {(w,g[x->b][x->c])}

   = {(w,g[x->b][x->b]), (w,g[x->b][x->c])}

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. If you wanted to add this as a refinement to the fragment, you could require that the existential only considers object in the domain that are not in the range of the starting 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 can bind pronouns in subsequent clauses even if they don't take syntactic scope over those clauses.

The presence of modal possibility, however, disrupts this generalization. GSV illustrate this with the following story.

The Broken Vase:
There are three children: Alice, Bob, and Carl.
One of them broke a vase.  
Alice 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:  a  true          a  false
    b  false         b  true
    c  false         c  false

w': a  false         a  false
    b  false         b  false
    c  true          c  true

GSV say 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: Carl is in the closet, and he's guilty. It remains possible that there are closet hiders who are not guilty in any world. Alice fits this bill: she's in the closet in world w, but she is not guilty in any world.

Let's see how this works out in detail.

14. Someone^x is in the closet.  They_x might be guilty.

     {(w,g), (w',g}[∃x.closet(x)][◊guilty(x)]

     -- existential introduces new peg

   = (   {(w,g[x->a]), (w',g[x->a])}[closet(x)]
      ++ {(w,g[x->b]), (w',g[x->b])}[closet(x)]
      ++ {(w,g[x->c]), (w',g[x->c])}[closet(x)]

     -- only possibilities in which x is in the closet survive
     -- the first update

   = {(w,g[x->a]), (w',g[x->c])}[◊guilty(x)]

     -- Is there any possibility in which x is guilty?
     -- yes: for x = Carl, in world w' Carl broke the vase
     -- that's enough for the possiblity modal to allow the entire
     -- infostate to pass through unmodified.

   = {(w,g[x->a]),(w',g[x->c])}

Now we consider the second half:

15. Someone^x is in the closet who_x might be guilty.

     {(w,g), (w',g)}[∃x(closet(x) & ◊guilty(x))]

   =    {(w,g[x->a]), (w',g[x->a])}[closet(x)][◊guilty(x)]
     ++ {(w,g[x->b]), (w',g[x->b])}[closet(x)][◊guilty(x)]
     ++ {(w,g[x->c]), (w',g[x->c])}[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 Carl in
      -- world w'

   = {(w',g[x->c])}

The result is different. Fewer possibilities remain. We have eliminated one of the possible worlds (w is ruled out), and we have eliminated one of the possible discourses (x cannot refer to Alice). So the second formula is more informative.

One of main conclusions of GSV is that in the presence of modality, the hallmark of dynamic treatments--that existentials bind outside of their syntactic scope--needs to refined into a more nuanced understanding. Binding still occurs, but the extent of the syntactic scope of an existential has a detectable effect on truth conditions.

As we discovered in class, there is considerable work to be done to decide which expressions in natural language (if any) are capable of expressing which of the two translations into the GSV fragment. We can certainly grasp the two distinct sets of truth conditions, but that is not the same thing as discovering that there are natural language sentences that conventionally express one or the other or both.

Binding, modality, and identity

The fragment correctly predicts the following contrast:

16. 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

17. 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-re-ish intuitions. If we add individual concepts to the fragment, the ability to express fancier claims would come along.

GSV's "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.

About the pegs

One of the more salient aspects of the technical part of the paper is that GSV insert 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. It happens that pegs play no role in the paper whatsoever. We've demonstrated this by providing a faithful implementation of the paper that does not use pegs at all.

Nevertheless, it makes sense to pause here to discuss pegs briefly, since this technique 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 1986 paper by Fred Landman called `Pegs and Alecs'. 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.)

Pegs can be highly useful for exploring puzzles of reference and coreference.

Standard assignment function    System with Pegs (drefs)
----------------------------    ------------------------
 Variable      Object           Var      Peg      Object
---------      -------          ---      ---      ------
    x     -->    a               x   -->  0   -->   a
    y     -/                     y   -/   
    z     -->    b               z   -->  1   -->   a

A standard assignment function can map two different variables onto the same object. In the diagram, x and y are both mapped onto the object a. With discourse referents in view, we can have two different flavors of coreference. Just as with ordinary assignment functions, variables can be mapped onto pegs (discourse referents) that are in turn mapped onto the same object. In the diagram, x is mapped onto the peg 0, which in turn is mapped onto the object a, and z is mapped onto a discourse referent that is mapped onto a. On a deeper level, we can suppose that y is mapped onto the same discourse referent as x. With a system like this, we are free to reassign the discourse referent associated with z to a different object, in which case x and z will no longer refer to the same object. But there is no way to change the object associated with x without necessarily changing the object associated with y. They are coreferent in a deeper, less accidental sense.

GSV could make use of this expressive power. But they don't. In fact, their system is careful designed to guarantee that every variable is assigned a discourse referent distinct from all previous discourse referents.

The addition of pegs tracks an active discussion in the dynamic literature around the time of publication of the paper. Groenendijk and Stokhof (Two theories of dynamic semantics, 1989) noted that it was possible in DPL for information to be "lost".

18. (∃x.P(x)) & (∃x.Q(x)) & R(x)

If the two existentials happen to bind the same variable (here, "x"), then the second existential occludes the first. That is, at the point at which we evalute R(x), all of the assignment functions will be mapping the variable "x" to objects that have property Q. The information that there exist objects with property P has been lost. If you want your dynamic system to be eliminative---or in more general terms, if you want the amount of information embodied by an updated information state to be monotonically increasing---then this is a problem.

A syntactic solution is to require that the variable bound by an existential to be chosen fresh.

Vermeulen, Cees FM. "Merging without mystery or: Variables in dynamics semantics." Journal of Philosophical Logic 24.4 (1995): 405-450 offers a different approach, one based on referent systems. GSV's pegs are a referent system. In the pegs system, when (18) is processed, the information that there is an object that has property P is maintained in the information state. Curiously, however, there is still no way to refer to that object, at least, not with a variable, since there is no variable that is associated with the peg that points to the relevant object. So the information is present, but not accessible.

That does not mean that there aren't other expression types besides pronouns or variables that might be able to latch onto pegs. An intriguing suggestion based on an example in Vermeulen is that "former" might be able to provide access to a hidden peg:

19. Someone entered.  Someone spoke.  The former was a woman.

Presumably we want the former to be able to pick out the person who entered, whether or not the two existentials bind the same variable or not. If we allow "former" to latch onto the second most recently established peg, no matter whether there is a variable still pointing to that peg, the desired effect is achieved.

But none of this is relevant for any of the explanations or analyses provided by the GSV fragment, and it is considerably simpler to see what their fragment is about if we leave referent systems out of it.