From: Chris Date: Wed, 8 Apr 2015 16:49:53 +0000 (-0400) Subject: edits X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=15cf1a049393b9087a14cccf4dd83dc1ebc76825;ds=inline edits --- diff --git a/topics/_week10_gsv.mdwn b/topics/_week10_gsv.mdwn index e971fe93..ca85c305 100644 --- a/topics/_week10_gsv.mdwn +++ b/topics/_week10_gsv.mdwn @@ -1,4 +1,4 @@ - + [[!toc levels=2]] @@ -201,3 +201,120 @@ man. Show the following computations, where `i = (w,n,r,g)`: (w, n+2, r[x->n][y->n+1], g[n->b][n+1->b]) } +## 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 (w1), Alice is hungry; in the other (w2), she is not. + + = {(w1,n,r,g), (w2,n,r,g)}[Alice isn't hungry][Alice might be hungry] + = {(w2,n,r,g)}[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. +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] + = {} + +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. + + + = {(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)} + +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)} + +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. + +