X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week10_gsv.mdwn;h=4ff29764874b83c02a7bef3918c29319e58e14d5;hp=b956d6c85b81786e8d87ef926b665cbdb3bbd627;hb=291555207a30b07dd33b3d493ca863bc64ea9edb;hpb=6a178df76a1175c4f89887087e426a5cceee0bf3 diff --git a/topics/_week10_gsv.mdwn b/topics/_week10_gsv.mdwn index b956d6c8..4ff29764 100644 --- a/topics/_week10_gsv.mdwn +++ b/topics/_week10_gsv.mdwn @@ -9,133 +9,90 @@ 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 + 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. - -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). +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|code/gsv.ml]]; consult +that code for details of syntax, types, and values. [[An implementation +in Haskell|code/gsv.hs]] is available as well, if you prefer. 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). + ref (i,t) = t if t is an individual, and + g(t) if t is a variable, where i = (w,g) -Following GSV, we'll write `update(s, φ)` (the update of information -state `s` with the information in φ) as `s[φ]`. +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)] = {i in s | w(P)(ref(i,t))} + s[P(t)] = {(w,g) in s | extension w P (ref((w,g),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. +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[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] +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, g[x->a]) | (w,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 +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: @@ -146,60 +103,28 @@ 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. +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)`: +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. - 1. {i}[∃x.person(x)] +Compute the following: - = {(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])} + 1. {(w,g)}[∃x.man(x)] - 2. {i}[∃x.man(x)] + = {(w,g[n->a])}[man(x)] ++ {(w,g[n->b])}[man(x)] + ++ {(w,g[n->c])}[man(x)] + = {} ++ {(w,g[n->b])} ++ {(w,g[n->c])} + = {(w,g[n->a]),(w,g[n->b]),(w,g[n->c])} + -- Bob and Carl are men - = {(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])} + 2. {(w,g)}[∃x.woman(x)] + 3. {(w,g)}[∃x∃y.man(x) and man(y)] + 4. {(w,n,r,g)}[∃x∃y.x=y] +Running the [[code|code/gsv.ml]] gives the answers. - 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 @@ -209,9 +134,10 @@ The final remaining update rule concerns modality: 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., `{}`). +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. @@ -223,9 +149,9 @@ order shows that the system is order-sensitive. 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. +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 @@ -239,7 +165,7 @@ 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. +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: @@ -255,7 +181,8 @@ 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* +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. @@ -266,24 +193,12 @@ In contrast, consider the sentences in the opposite order: 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} -Update with *Alice might be hungry* depends on the result of updating -with the prejacent, *Alice is hungry*. Here's the side calculation: - - {hungry, full}[Alice is hungry] - = {hungry} - -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. +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 @@ -300,22 +215,390 @@ 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. + 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 updates due to the -individual sentence occur. +discourses depends on the order in which the sentences are processed. -Note, incidentally, that there is an asymmetry in the fragment -concerning negation. +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. 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. +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.enter(x)][sit(x)] + + = ( {(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. + + {(w,g[x->b])}[sit(x)][∃x.enter(x)] + + -- 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 true 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])}[closet(x)] + ++ {(w,g[x->b])}[closet(x)] + ++ {(w,g[x->c])}[closet(x)] + ++ {(w',g[x->a])}[closet(x)] + ++ {(w',g[x->b])}[closet(x)] + ++ {(w',g[x->c])}[closet(x)])[◊guilty(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])}[closet(x)][◊guilty(x)] + ++ {(w,g[x->b])}[closet(x)][◊guilty(x)] + ++ {(w,g[x->c])}[closet(x)][◊guilty(x)] + ++ {(w',g[x->a])}[closet(x)][◊guilty(x)] + ++ {(w',g[x->b])}[closet(x)][◊guilty(x)] + ++ {(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 one of +the possible worlds (w is ruled out), and we have ruled out 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.