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 and 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 and 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 and #:

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 and # to #, or by replacing line (i) with one that maps # and 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.)

The preceding discussion has endeavored to bring out some similarities between the kind of order-dependence in our three-valued truth-table, and the kinds of order-dependence exhibited by "dynamic semantics." But there are also of course substantial differences between them, and these are also, perhaps even more interesting. Over the course of this semester we hope to clarify and help you to think more carefully about both the similarities and the differences.