From: Chris Date: Wed, 8 Apr 2015 20:48:09 +0000 (-0400) Subject: unstable state of gsv discussion X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=f4895699b6d5a76840479ef2149f055bfa02b471;ds=sidebyside unstable state of gsv discussion --- diff --git a/topics/_week10_gsv.mdwn b/topics/_week10_gsv.mdwn index b956d6c8..f408648c 100644 --- a/topics/_week10_gsv.mdwn +++ b/topics/_week10_gsv.mdwn @@ -318,4 +318,198 @@ 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]), + (w,2,r[x->0][x->1],g[0->b][1->b]), + (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. + + 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]), + (w,2,r[x->0][x->1],g[0->b][1->b]), + (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])} + +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 required 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 in effect take scope over subsequent clauses. + +The presence of modal possibility, however, disrupts this +generalization: + + (∃x.enter(x)) and (◊sit(x)) ≡/≡ ∃x (enter(x) and ◊sit(x)) + +To see this, we'll start with the left hand side. + + 14. Someone^x entered. He_x might sit. + + {(w,1,r[x->0],g[0->b])}[∃x.enter(x)][◊sit(x)] + + -- same computation up to the point of the modal + + = {(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)] + + -- modal returns all or none, depending on whether the + -- prejacent is consistent with the starting infostate. + -- since there is one choice for x who sat, returns all: + + = {(w,2,r[x->0][x->1],g[0->b][1->b]), + (w,2,r[x->0][x->1],g[0->b][1->c])} + +To paraphrase, the requirements are that there must be a person who +entered, and it might be possible that that person sat. But this is +not metaphysical possibility: we're not choosing a person an wondering +whether that person sat. If that's what we had in mind, we'd go off +to a bunch of non-actual possible worlds and see what is happening +there. Instead, this is supposed to be epistemic possibility. The +paraphrase should be something like: there must be a person who +entered, and for all we know, that person might have sat. + +The peculiar thing is that the uncertainty has nothing to do with the +facts of the world, but only with the fact about the discourse: it's +uncertainty about which object the pronoun refers to. GSV work hard +to make this interpretation plausible. Here's their story: + + There are three kids. One of them breaks a vase. One is known to + be innocent. There are sounds coming out of the closet. + + 15. Someone^x is in the closet. He_x might be guilty. + +You have enough information to know that someone is in the closet. +You use the pronoun to refer to the person in the closet, and assert +that, for all you know, that person might be guilty. The fragment +gives you guaranteed coreference---it's whoever is in the closet who +might be guilty---in the presence of uncertainty about who the pronoun +refers to. + +Now we consider the second half: + + 14. Someone^x entered who_x might sit. + + {(w,1,r[x->0],g[0->b])}[∃x.(enter(x) & ◊sit(x)] + + = {(w,2,r[x->0][x->1],g[0->a]), + (w,2,r[x->0][x->1],g[0->b]), + (w,2,r[x->0][x->1],g[0->c])}[enter(x)][◊sit(x)] + + -- recall that Alice didn't enter, so + + = {(w,2,r[x->0][x->1],g[0->b]), + (w,2,r[x->0][x->1],g[0->c])}[◊sit(x)]