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Formal Literatures


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 tex2html_wrap_inline1117 be such that if tex2html_wrap_inline1119 and L(s), that is L(s)=true, and tex2html_wrap_inline1125 is an initial segment of tex2html_wrap_inline831 then tex2html_wrap_inline1129 . The pair (A,L) is termed a literature. The interpretation is that tex2html_wrap_inline1133 may be said after tex2html_wrap_inline1135 provided tex2html_wrap_inline1137 . We shall also write tex2html_wrap_inline1139 and refer to tex2html_wrap_inline831 as a string of the literature L.

From a literature L and a string tex2html_wrap_inline1139 we introduce the derived literature tex2html_wrap_inline1149 . Namely, tex2html_wrap_inline1151 if and only if tex2html_wrap_inline1153 , where tex2html_wrap_inline1155 denotes the concatenation of tex2html_wrap_inline831 and tex2html_wrap_inline751 .

We shall say that the language L is universal for the class tex2html_wrap_inline1163 of literatures if for every literature tex2html_wrap_inline1165 there is a string tex2html_wrap_inline1167 such that tex2html_wrap_inline1169 ; that is, tex2html_wrap_inline1171 if and only if tex2html_wrap_inline1173 .

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 tex2html_wrap_inline1187 if and only if tex2html_wrap_inline1189 .

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: tex2html_wrap_inline1191 , tex2html_wrap_inline1193 , and tex2html_wrap_inline1195 . We start with a list of sentences as hypotheses. A new sentence may be added to a string tex2html_wrap_inline831 of sentences according to the following rules:

1. Any consequence of sentences of tex2html_wrap_inline831 may be added.

2. If a sentence tex2html_wrap_inline1163 is consistent with tex2html_wrap_inline831 , then tex2html_wrap_inline1205 may be added. Of course, this is a non-computable rule. It may be weakened to say that tex2html_wrap_inline1205 may be added provided tex2html_wrap_inline1163 can be shown to be consistent with tex2html_wrap_inline831 by some particular proof procedure.

3. tex2html_wrap_inline1213 .

4. tex2html_wrap_inline1215 is a possible deduction.

5. If tex2html_wrap_inline1217 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 tex2html_wrap_inline1247 and tex2html_wrap_inline1249 and therefore tex2html_wrap_inline1251 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.

next up previous
Next: Probabilities Up: REMARKS AND OPEN PROBLEMS Previous: The Frame Problem

John McCarthy
Mon Apr 29 19:20:41 PDT 1996