From 247d01832209e40ef4dffa3746fded9f74052674 Mon Sep 17 00:00:00 2001 From: Chris Date: Sun, 1 Feb 2015 14:42:11 -0500 Subject: [PATCH] added discussion of order, with F&#; adjusted index and topics file --- topics/week1_order.mdwn | 246 ++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 246 insertions(+) create mode 100644 topics/week1_order.mdwn diff --git a/topics/week1_order.mdwn b/topics/week1_order.mdwn new file mode 100644 index 00000000..ad0db9a9 --- /dev/null +++ b/topics/week1_order.mdwn @@ -0,0 +1,246 @@ +## Major theme: order ## + +Notes from part of the first lecture on one of the larger themes of +the course, namely, order. + +### Order in programming languages, order in natural languages ### + +In programming languages, order matters. Consider the following +program fragment: + + x := 2 + x := 1 + print x + +We're using ":=" to mean "takes on the value of" (not "is equal to"). +This is a fragment written in an imperative style. When this program +is executed, three things should happen: the value of the variable `x` +should be set to 1, the value of `x` should be set to 2, and the value +of `x` should be printed. But in addition, these things should happen +in a specific order, namely, the order in which the commands are +written. Compare: + + x := 1 + x := 2 + print x + +In this fragment, the same things happen, but they happen in a +different order. One way to see this is to note that the expected +behavior is different. The first program will print the number 1, +and the second program will print the number 2. + +A similar point is familiar from discussions in the lingusitics +literature concerning discourse anaphora. + + 1. a. A woman arrived. + b. She spoke. + c. "a woman" == "she": OK + + 2. a. She spoke. + b. A woman arrived. + c. "a woman" == "she": nearly impossible + +In the discourse fragment in (1), two events are described: an arrival +event and a speaking event. It is easy to interpret the discourse as +describing a situation in which the same woman who entered spoke. In +contrast, in discourse (2), it is much more difficult---in fact, +barring time travel, nearly impossible---to interpret the situation as +describing two events involving a single person. + +The standard explanation is that the use of an indefinite such as "a +woman" creates a new discourse referent, which a pronoun such as "she" +can refer back to under appropriate circumstances. In the discourse +in (1), the indefinite occurs first, and the pronoun in the second +sentence is able to access the discourse referent created by the +indefinite. In the discourse in (2), the pronoun occurs first. Since +the definite has not yet had a chance to create its discourse +referent, the pronoun has nothing local to latch onto, and must take +its value independently of the resources provided by the discourse. + +We'll discuss a number of specific analyses that will seek to capture +the contrast between (1) and (2) later in the course. + +Note that the analogy we are making between the program fragments and +the discourse fragments suggests that it makes sense to think of +natural language meaning as if it were a computer program. We are +going to take this analogy very seriously indeed: we will suggest that +natural language meanings are isomorphic to computer programs. A +closely related version of this claim is the Curry-Howard isomorphism, +which establishes a parallel correspondence between logical +derivations and programs. + +One consequence of this correspondence is that it makes sense to think +of interpreting an expression in natural language in the same terms as +we think of interpreting a program: they are "evaluated" or "run". + +### Dynamic versus static ### + +There is a major long-standing debate in the fields of linguistics and +the philoosphy of language about whether it is right to think of +natural language meanings as being dynamic in this way. The +alternative, to oversimplify, is to think of natural language (well, +the fragment of natural language consisting of declarative sentences) +as expressing propositions, which we can treat for the moment as +denoting truth values. The the denotation of a sentence like (1a) +will be `true` just in case a woman arrived, and `false` just in case +no woman arrived. On this kind of view, the obvious asymmetry between +the discourse in (1) versus the discourse in (2) is supposed to be the +result of the ways in which people tend to react to sentences as they +exchange information. That is, it's a fact about the psychology of +belief revision, and not part of the meaning of the sentences. In the +terminology of the debate, we can call our view, that sentences +express programs, a *dynamic* view, and the notion that sentence +meaning is truth conditions and nothing else a *static* view. (See +recent work of Yalcin and Rothschild for a recent version of the +static view, with pointers into the literature.) + +One of the things that makes the dynamic/static debate so interesting +is that it is not always easy to tell whether a system ought to be +classified as dynamic or as static just by looking at its formal +properties. On the one hand, it is well-known (see work of Yalcin and +Rothschild, or Groenendijk and Stokhof) that grammars taking the form +of a dynamic update system can be reformulated as static grammars if +certain conditions are met. (We'll explore this point in more detail once +we have more experience with dynamic grammars.) So being expressed in +the form of a dynamic recipe is no guarantee that the grammar is +essentially dynamic. + +On the other hand, it is much less appreciated that supposedly static +grammars can nevertheless express analyses that have dynamic +intuitions embedded deeply within them. To see this, consider +Classical Logic, the paradigm example of a static system. Classical +theorems are timeless, in the sense that conclusions are independent +of the order of the premises. Here is the meaning of one of the +logical connectives of classical logic, expressed in the form of a +standard truth table: + + A B A&B + ----------- + T T T + T F F + F T F + F F F + +This table says that if either of the conjuncts is false, the +conjunction as a whole will be false. + +But now consider a small but crucial extension of classical logic. +Instead of limiting values to `true` and `false`, we'll allow one +additional value: `undefined`, which we'll write as `#`. To motivate +this extension, think of sentences whose presuppositions are not +satisfied. + + 3. The earth is round. (true) + 4. The sun is green. (false) + 5. The King of France is bald. (undefined) + +The usual attitude towards sentences like (5), which presupposes the +existence of a specific object that does not in fact exist, is that +they are neither true nor false. Certainly (5) is not true, and +saying that it is false appears to commit you to believing that its +negation is true, which is not a commitment that everyone is willing +to make. + +Given that a partial-function approach to presupposition failure is +coherent, let's consider one way to extend classical conjunction: + + p q p&q + -------------- + a. T T T + b. T F F + c. F T F + d. F F F + ------------- + e. # # # The King of France is bald and the King of France is bald. + f. T # # The earth is round and the King of France is bald. + g. # T # The King of France is bald and the earth is round. + ------------- + h. F # F + i. # F # + +The truth table begins just as before (lines (a) through (d)): when +both conjuncts are defined, the value of the conjunction as a whole +conincides with classical conjunction. In lines (e) and (f), we +imagine conjoining a true proposition with an undefined one. In order +for a conjoined sentence to be true, both conjuncts must be true; and +if one of the conjuncts is undefined, there is no way that requirement +can be met. If both conjuncts are undefined, as in (g), then of +course the conjunction as a whole will be undefined. + +So far, so good. Nothing so far undermines the static view. But now +consider the two remaining possibilities, one by one, starting with +line (h). Here is a concrete sentence fitting the pattern addressed +by line (h), `F&#`: + + 6. The sun is green and the King of France is bald. + +Since a conjoined sentence is true only if both conjuncts are true, +(6) cannot possibly evaulate to true: the left conjunct is false, and +that settles the matter. It doesn't matter whether the second +conjunct is undefined---any rational and alert listener should be +prepared to commit to the falsity of the conjunction as soon as she +realizes that the first conjunct is false. In fact, she can simply +stop listening as soon as she hears "The sun is green and...". No +matter whether the second conjunct is well defined, the conjunction as +a whole must be false. + + 7. The King of France is bald and the sun is green. + +Concerning line (i), in a similar spirit, if the first conjunct is not +defined, by the time the first conjunct has been heard, a rational and +alert listener should be prepared to commit to the judgment that the +presuppositions of the sentence are impossible to satisfy. No matter +how the rest of the conjoined sentence continues, it will presuppose +that France has a King. Therefore is rational to judge the conjoined +sentence as a whole to be undefined. + +Comparing lines (h) and (i) in the truth table, there is an asymmetry: +the outcome depends on the order of the conjuncts. The truth table +embodies the following processing strategy: if the status of the first +conjunct reliably determines some aspect of the status of the +conjunction as a whole, let the value of the left conjunct control the +outcome. + +To be sure, it would also be coherent to choose a fully symmetric +truth table by replacing line (h) with one that maps `F&#` to `#`, or +by replacing line (i) with one that maps `#&F` to `F`. With respect +to natural language, of course, which truth table is a better match +for a given natural language is an empirical question, and not one +that can be settled by logical argument. If native speakers behave as +if sentences with the form in (i) are false, then that is the way a +truth table describing that language ought to look. + +But it is sufficient for our point here for the truth table as given +to merely be coherent. Consider a language with a conjunction as +defined in the table. Are the semantics for this language dynamic or +static? Well, there is no explicit notion of *before* or *after* in the +processing of a complex sentence. But the truth table has sensitivity +to order baked into its truth conditions. In that sense, it is +dynamic in spirit. + +We should mention that in a series of papers, Schlenker defends a +static view of presupposition, given detailed consideration to +situations very much like the ones we discuss here involving possible +continuations of a sentence, reasoning about the conclusions that a +rational listener would come to based on partial knowledge of the +sentence. + +An open-ended question: + +The asymmetry in the table arises from the asymmetry in the status +of true versus false with respect to conjunction, not from any +asymmetry between being undefined and true versus false. That is, +knowing that one conjunct is false is enough to constrain the value of +the conjunction as a whole, whereas knowing that one conjunct is true +leaves the final outcome completely open. This asymmetry is evident +from the orginal classical truth table. To what extent can the same +point be made using only the classical truth table? That is, is it +possible to argue that classical logic has some order sensitivity +baked into it? It may be worthwhile thinking about material +implication: afer all, in the material implication `T --> X`, +the value of the implication as a whole depends on the value of `X`, +but in the material implication `F --> ?`, the outcome is `T` no +matter what the value of `X` turns out to be. + +[to be added: citation details; reasoning about order sensitivity in +an order-independent way] -- 2.11.0