X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week10_gsv.mdwn;h=d61759426301cd2e6c0beabc97d61ed5f7c5537f;hp=3fd046c95548f61761fb4d4073a9cc0d5dec3b81;hb=9e249131827f4f17862110e1dddf34b8895de613;hpb=f747612ad27e9facc7c92c04e4c6981b40a561ad diff --git a/topics/_week10_gsv.mdwn b/topics/_week10_gsv.mdwn index 3fd046c9..d6175942 100644 --- a/topics/_week10_gsv.mdwn +++ b/topics/_week10_gsv.mdwn @@ -1,4 +1,4 @@ - + [[!toc levels=2]] @@ -10,6 +10,11 @@ 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 + + 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, @@ -58,7 +63,7 @@ On the epistemic side, GSV aim to account for asymmetries such as ## 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 @@ -68,7 +73,10 @@ empirical discussion. Here are the ones that matter: 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 @@ -123,19 +131,16 @@ discards all possibilities in which "x" fails to refer to a man. 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: @@ -146,14 +151,464 @@ 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: +man. + +We're using `++` here to mean set union. + + 1. {(w,n,r,g)}[∃x.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. {(w,n,r,g)}[∃x.man(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)] + + = {(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+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)]) + + = {(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])} + + -- 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->b][n+1->b])} + + -- two ways to assign x and y to the same value + +## 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, 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: +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. + + + = {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. + +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. + +## Binding + +The GSV fragment differs from the DPL and the DMG dynamic semantics in +important details. Nevertheless, it has more or less the same things +to say about anaphora, binding, quantificational binding, and donkey +anaphora. + +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, we'll +need an information state whose refsys is defined for at least one +variable. + + 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 + +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,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])} + +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 outcome is different if the order of the sentences is reversed. + + 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])} + +The result is different than before. 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. 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 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. + +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 take effective scope over subsequent clauses. + +The presence of modal possibility, however, disrupts this +generalization. GSV illustrate this with the following story. + + 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. + + (∃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: 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. + +Let's see how this works out in detail. + + 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])} + +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: + + Variables Pegs Entities + --------- r ---- g -------- + x --> 0 --> a + y --> 1 --> a + z --> 2 --> c + +But this is possible with ordinary assignment functions as well. + +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. + +So what does the bipartite system do that ordinary assignment +functions can't do? - 1. {}[∃x.person(x)] - 2. {}[∃x.man(x)] - 3. {}[∃x∃y.person(x) and person(y)] - 4. {}[∃x∃y.x=x] - 5. {}[∃x∃y.x=y]