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=ca85c305b7501dba7536acfc9ac7d185310a0529;hb=9e249131827f4f17862110e1dddf34b8895de613;hpb=a2294b4df1fd5cab018bf234ab107f05ce54bc3e diff --git a/topics/_week10_gsv.mdwn b/topics/_week10_gsv.mdwn index ca85c305..d6175942 100644 --- a/topics/_week10_gsv.mdwn +++ b/topics/_week10_gsv.mdwn @@ -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: @@ -146,60 +151,65 @@ 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)`: +man. + +We're using `++` here to mean set union. - 1. {i}[∃x.person(x)] + 1. {(w,n,r,g)}[∃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])}[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. {i}[∃x.man(x)] + 2. {(w,n,r,g)}[∃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->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)] - 3. {i}[∃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+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]) - } + = ( {(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)]) - 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])} - = {(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]) - } + = {(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] + -- 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->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]) - } + (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 @@ -228,22 +238,23 @@ 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. +possibility, Alice is hungry (call this possibility "hungry"); in the +other, she is not (call it "full"). - = {(w1,n,r,g), (w2,n,r,g)}[Alice isn't hungry][Alice might be hungry] - = {(w2,n,r,g)}[Alice might be hungry] + {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 (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: - {(w2,n,r,g)}[Alice is hungry] + {full}[Alice is hungry] = {} Because the only possibility in the information state is one in which @@ -266,15 +277,15 @@ In contrast, consider the sentences in the opposite order: 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)} + = {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: - {(w1,n,r,g), (w2,n,r,g)}[Alice is hungry] - = {(w1,n,r,g)} + {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 @@ -317,4 +328,287 @@ 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?