--- /dev/null
+## 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]