Language and information
Lecture 2. Sublanguages
2.1. Subsets of sentences
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Today, we will speak about subsets of sentences and subsets of language, including the metalanguage of language, and about science sublanguages, and even science languages. It's a long list of things.
In the theory that was presented yesterday, there is a certain property which was presented yesterday, and that is that sentences are formed by the very first step of the theory. It isn't that there is a long process of development in this making of a sentence before we get to the first sentence; the first-level operators, when they act on a zero-level word, already produce an elementary sentence, like night falls, John sleeps, or whatever it may be. I'm disregarding the a and so forth, these are matters that can be handled otherwise. So the sentences are made immediately, by the very first step of syntax, if you wish. Thereafter, all further sentences are made by events on the elementary sentences. They're not made differently, they're made either by second-level operators acting on the sentences that have been made, like John's sleeping continued or something like that, or I know that John sleeps, or else they are made by reductions of these sentences. This means that from the moment that the first elementary sentences have been made, the first thing which has happened within syntax, we are dealing with relations among sentences. All further things are relations among sentences. Between an expanded sentence and the sentence which it contains, and from which it was expanded; or between a reduced sentence and the sentence from which it was reduced. And this is the whole of the rest of the syntax. Therefore it is really relations among sentences, beginning with the elementary sentences.
The result of this is that this theory forms subsets of sentences immediately, subsets determined by the relations. There are: elementary sentences, which we have seen; base sentences, which we discussed last time, which are sentences that are not reduced—they may be very, very long, and they have a great proliferation of operators, one on the other, even if it's only at the three levels that we saw, zero level words, first-level operators, and second-level operators, but these can act one on the other, especially the second-level operators can—and reductions of many kinds. And this describes all of language, and not merely, as I said last time, not merely by fiat, but by offering a structure which suffices for all the sentences.
Now, a subset of sentences which is closed under some operation, certain operations of the language as a whole, is what is called a sublanguage. That is, it would be called a sublanguage, that is how subsystems are generally defined. The elementary sentences, for instance, are not a sublanguage, because any operation on them will take them out of the elementary set. But the base subset of sentences is also a sublanguage. It is closed under and or under many other operations. The reduced sentences are also a [sublanguage]. If you take any two reduced sentences, and operate on them, with any conjunction, or some other ways of operating, you again get a reduced sentence, so it is part of the same closed set.
Now, I want to introduce for a moment a particularly important sublanguage, and this is the metalanguage. The metalanguage of a system, I remind you, is a set of sentences in which one can talk about the system, or describe the system. As we saw, natural language does not have an external metalanguage. You can either describe it with another language with the same structure, or you can describe it in itself. An English grammar, for instance, written in English. What, then, is the metalanguage of language? It is a certain set of sentences, let's say in English, in the same language, which talk about the language; they either are the grammar of it or the sentences which talk about the sentences of the language or its parts. This is the metalanguage, and it is a subset of English. So we now know that the metlanguage is not just English or in English, it is a subset of English, it is a sublanguage of English, for obvious reasons.
I want now to introduce quickly a little construction made on these metalanguages as an example of the utility of investigating the structure of sublanguages. And this is as follows. It'll be a chain of metalanguages, one on top of the other. If we consider the metalanguage of English, it will have sentences about English. The nouns, or the zero-level words of the metalanguage will be any expression of English which is being described, a sentence or a part of a sentence. The first-level operators will be words like is a sentence, is a word, is not a word, is next to, various words of that sort. And a second-level operator will be, for instance, is frequent, meaning the occurrence of a certain word next to another word is frequent, is highly likely, and so forth. All right, that is the metalanguage, that is the structure, the description of the metalanguage. This is already metalanguage 2. Metalanguage 2 describes metalanguage 1, which is a metalanguage of English. Metalanguage 2 also has a structure, which we will describe in metalanguage 3. The structure is a little bit different, because it doesn't deal any longer with the expressions of English, it deals with expressions in the metalanguage of English, or that is said in the metalanguage of English. It won't have is frequent as an operator, because, as we will see later, the frequency of word occurrence is relevant in figuring out the grammar of the language, but not in figuring out the grammar of the metalanguage. We will see this. So this is metalanguage 3. Well metalanguage 3, its structure is described in metalanguage 4. But at this point we see something. Metalanguage 4 is identical with metalanguage 3. The referents are different, of course, in respect that metalanguage 3 talks about metalanguage 2, metalanguage 4 talks about metalanguage 3, but the sentences are the same sentences, because nothing changes any longer in what is said, only the referent changes. Therefore, though everybody knows that there has to be an infinite regress in metalanguages, this is an infinite regress in the referents of metalanguages, and not in the structure. The structure of metalanguages as a general matter stops at three, the fourth metalanguage is the same as the third.
Now, naturally I cannot demonstrate this in detail, I'm only saying this.