Language and information
Lecture 1. A formal theory of syntax
1.3. The partial order constraint
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Now, the first constraint creates sentence structure. It is a partial order of words, that is, roughly, an ordering in which some words are higher or lower on some scale than others, while some are neither higher nor lower than others. The partial order holds between word-occurrences in utterances. It determines all sentences, but is at work only in a subset, from which, however, all other sentences can be derived. Grammatical relations can be defined in terms of it. We can see the partial order most clearly in very short sentences. Now I'll give just a very few examples. This will be very hard to follow and boring, but just to explain what the partial order is.
We consider a few words, for instance, the words that can occur with sleep. In certain sentences, there are certain words will occur with sleep: man, child, even tree sleeps in winter, earth sleeps under a blanket of snow, and a few others. Some others are very, very rare, but they can be said. A stone sleeps for 5 billion years … or something of this sort. Some of them are more common than others, we'll come to that. But in any case, there's a certain type of word—a certain set of words, actually, though we're not defining what this set is now—which occurs with sleep. I can show you words that do not occur with sleep, in short sentences, like, for instance, sleep sleeps, or walking sleeps, or because sleeps. I mean, those are English words, but you can't do anything with it, as you can with a rock sleeps or has slept for billions of years. So sleep occurs with one word of a certain set. Let's call this, temporarily, the 'argument set' for sleep.
Let's take wear. Wear will be found to take two words of the same set: a man wears a coat, a tree wears bark, or something of this sort. There are various things one can say, some of them are somewhat crazy, but what you cannot say that sleeping wears running, or something of this sort, or again that because was coming or something like this. So we will say that [wear] has two—that the argument classes of [wear] are two words of that same set, of earth and so forth: the earth wears a blanket of snow also.
Now, let's take another word: to assert. Assert occurs with one word of this class that we already know—a man asserts, a child asserts, even a tumor asserts something, even a painting, I'm told, asserts something. And that also has another thing at the end, a man asserts something. That something, however, is not a coat or a man—you cannot say a man asserts a coat or something of this sort, it has be a man asserts sleep. Well, he doesn't assert sleep, he asserts a tree sleeps. But that is because a man asserts sleeping, and the sleeping requires a tree. So that a man asserts that a tree sleeps has man and sleep as the arguments of assert and tree as the argument of sleep.
At this point, let me try to say the general statement about this. Because I should say that there are—of course there are many other words—it turns out that all of the argument classes that we need for all the words are only either the set to which man, earth, tree, and so forth belong—namely, that's the nouns, not all nouns, concrete nouns—or else, things that we would call verbs—though it's a class that also includes adjectives—like sleep, and wear, and kill, and so forth. These two classes—we can describe the arguments of all words of the language as some combination: one of this class, two words out of these classes, one word of this class and one word of the other class, and so forth. So that you'll see that the material is not terribly hard, even though the number of combinations that we have to make is very large.
Now, the partial order relation I have been describing here is as follows: For each word, whether it's man or sleep or coat or whatever, there are zero or more classes of words which will be called its argument, such that the given word will not occur in a sentence unless one word—any word—of each argument class is present, and in the stated position next to it. So, sleep wants one word of this noun class, wear wants two words of the noun class, assert wants one word of the noun class and one word of this so-called verb class, and so on.
Now, every word, for instance man or tree has zero arguments, because I said zero or more word classes, and therefore it's possible for a word to have no arguments. If it has zero argument, we call them zero-level words. Its argument is null, it's nothing. If it has a non-zero argument—that is, all other words—we call them operators on their arguments, on the words that it requires.
Now this relation is a dependence. That is to say, man depends on nothing, in this organization; sleep depends on one thing that depends on nothing, and so on. This is dependence with each word in the sentence requiring the presence of certain other words in the sentence.
Now, at this point I want to introduce—first of all, let me say something about the partial order already, even at this point. It is of course a constraint on word combination, because it says that with the argument position next to a given operator, the frequency of certain words, those which are not in the argument class, is zero. So it already constrains a large number of words not occurring in a certain position. Each satisfaction of this partial order, let's say each occurrence of an operator with its arguments, is a sentence. So we already have sentences. These already are sentences. A man sleeps is a sentence. The partial ordering also has a meaning. As we will see next week, each operator is being said about its argument. The meaning of the presence of the operator is 'aboutness'. The operator has its own meaning—for instance sleep or wear—but it also has a grammatical meaning. It is not just a juxtaposition, it is said about. Why it has this meaning of 'about' will come up next week. So we have introduced a meaning which this thing brings in, which the partial order brings in to a sentence.
Now, this dependence relation, the requirement, has a very, very important property, which is this: if we ask what determines for each word which word class it in turn requires, we find that the required words are identified—the way we really know what the required words [are], whereas we didn't give the whole class, I just stated a few examples, saying man, tree, and so forth. The required words are identified by what they in turn require. In other words, a word does not require a particular list of words. As a matter of fact, it would be a very poor situation if it did, because we would then have to modify the list each time those come in and go out. A word is not identified by what list of words it goes with, it is identified by the requirement property of the words that it will go with. It will accept only words that have the requirement property. We'll see what this means in a moment. This means, for instance, that sleep requires one word that requires null. So you can't say sleep sleeps; you can say man sleeps. Wear requires two words which themselves require null. Assert, however, requires two words, one of which requires null, the other word has to require something. That is, a man asserts sleeping: he asserts sleeping about a tree, he asserts that a tree sleeps. But, while I'm saying it now sort of in a semantic way, but what I mean is, these are the combinations that make sentences, while the other combinations don't make sentences. Now, the relation that imposes the partial order therefore is not just dependence of words on a stated class of words, but dependence of a word on the dependence property of words. This is the kind of relation that can define a system without recourse to any externally defined elements. It has the essential property of a mathematical object.
Having come to this, it is worth noting that the language elements involved—namely words—have indeed no inherent property which has to be used for sentence construction. The sounds of words are not essential to their meaning or to their combinability, and from the viewpoint of the partial order then the word occurrences form a set of arbitrary elements closed under the dependence-on-dependence relation, with every combination which satisfies this relation being a sentence.
We will also see that this relation is sufficient as a basis for everything else that has to be found grammatically and also for authentification of word meanings.