In this section we introduce the notion of formal literature which is to be contrasted with the well-known notion of formal language. We shall mention some possible applications of this concept in constructing an epistemologically adequate system.
A formal literature is like a formal language with a history: we imagine that up to a certain time a certain sequence of sentences have been said. The literature then determines what sentences may be said next. The formal definition is as follows.
Let A be a set of potential sentences, for example, the set of all finite strings in some alphabet. Let Seq(A) be the set of finite sequences of elements of A and let be such that if and L(s), that is L(s)=true, and is an initial segment of then . The pair (A,L) is termed a literature. The interpretation is that may be said after provided . We shall also write and refer to as a string of the literature L.
From a literature L and a string we introduce the derived literature . Namely, if and only if , where denotes the concatenation of and .
We shall say that the language L is universal for the class of literatures if for every literature there is a string such that ; that is, if and only if .
We shall call a literature computable if its strings form a recursively enumerable set. It is easy to see that there is a computable literature U(C) that is universal with respect to the set C of computable literatures. Namely, let e be a computable literature and let c be the representation of the Gödel number of the recursively enumerable set of e as a string of elements of A. Then, we say if and only if .
It may be more convenient to describe natural languages as formal literatures than as formal languages: if we allow the definition of new terms and require that new terms be used in accordance with their definitions, then we have restrictions on sentences that depend on what sentences have previously been uttered. In a programming language, the restriction that an identifier not be used until it has been declared, and then only consistently with the declaration, is of this form.
Any natural language may be regarded as universal with respect to the set of natural languages in the approximate sense that we might define French in terms of English and then say `From now on we shall speak only French'.
All the above is purely syntactic. The applications we envisage to artificial intelligence come from a certain kind of interpreted literature. We are not able to describe precisely the class of literatures that may prove useful, only to sketch a class of examples.
Suppose we have an interpreted language such as first-order logic perhaps including some modal operators. We introduce three additional operators: , , and . We start with a list of sentences as hypotheses. A new sentence may be added to a string of sentences according to the following rules:
1. Any consequence of sentences of may be added.
2. If a sentence is consistent with , then may be added. Of course, this is a non-computable rule. It may be weakened to say that may be added provided can be shown to be consistent with by some particular proof procedure.
4. is a possible deduction.
5. If is a possible deduction then
is also a possible deduction.
The intended application to our formalism is as follows:
In part 3 we considered the example of one person telephoning another, and in this example we assumed that if p looks up q's phone-number in the book, he will know it, and if he dials the number he will come into conversation with q. It is not hard to think of possible exceptions to these statements such as:
1. The page with q's number may be torn out.
2. p may be blind.
3. Someone may have deliberately inked out q's number.
4. The telephone company may have made the entry incorrectly.
5. q may have got the telephone only recently.
6. The phone system may be out of order.
7. q may be incapacitated suddenly.
For each of these possibilities it is possible to add a term excluding the difficulty in question to the condition on the result of performing the action. But we can think of as many additional difficulties as we wish, so it is impractical to exclude each difficulty separately.
We hope to get out of this difficulty by writing such sentences as
We would then be able to deduce
provided there were no statements like
present in the system.
Many of the problems that give rise to the introduction of frames might be handled in a similar way.
The operators normally, consistent and probably are all modal and referentially opaque. We envisage systems in which and and therefore will arise. Such an event should give rise to a search for a contradiction.
We hereby warn the reader, if it is not already clear to him, that these ideas are very tentative and may prove useless, especially in their present form. However, the problem they are intended to deal with, namely the impossibility of naming every conceivable thing that may go wrong, is an important one for artificial intelligence, and some formalism has to be developed to deal with it.