Series No. WP95/01 June, 1995 Editor: Dr. Matthew Lee

Reprinted April, 1996 Assoc. Editor : Robert Davison

There is a problem about explaining how systems that are true by definition can be informative about a world that is not. This problem is particularly relevant to the design of computerized information systems. A solution is offered based on the distinction between logical and factual truth.

Keywords: Information, Systems, Logic, Logical Truth

The real world mapping problem is the problem of explaining how does a system that is an invention of the human mind help us to understand a world that is not? It is self evident that some systems of rules and definition, such as those that constitute the game of chess, do not help use to understand anything beyond themselves, others, such as mathematics, do. The real world mapping problem is the problem of understanding which systems will help us understand the real world and which will not.

This paper will show that real world mapping is a particular problem for information systems. It will present a logical analysis of the problem and suggest a solution. The solution is in the form of a logical template which all information systems should conform to.

Next, we look at the background of the mapping problem. The first section explains how a adequate solution has escaped both the philosophy of mathematics and the philosophy of logic. Section two uses the example of Britain's Poll Tax to show how real world mapping is a problem in current information system design. In the third section the bizarre results of an information systems mapping failure are emphasized by a comparison with the fiction of Franz Kafka. The last section explains why systems maintenance is not usually a solution to the mapping problem.

In "Logic and Language", we seek to explain, in terms of philosophical logic, how languages can have reference to real world objects and events. The difference between universals and particulars and between logical and factual truth is described in the first section. The second section presents the arguments that both logical and factual truth are required in a system that maps on to the real world. The first part of the argument is that factual truth cannot be derived from logical truth; this argument comes from David Hume and is well known. The second part of the argument is that a system which includes factual truth also needs to include logical truth; this argument is, to the best of the author's knowledge, original. The third section explains the way in which logical truth is determined. Most existing computerized information systems have a logical form that comprises only logical universals and factual particulars, the fourth section explains why these cannot be informative about the real world.

Finally, a logical template for an information system is presented. This comprises three elements: logical universals, factual universals and factual particulars.

Einstein posed the mapping problem for mathematics: "How can it be that mathematics, being after all a product of human thought independent of existence, is so admirably adapted to the objects of reality?" An answer to this has not emerged and currently pure mathematicians, for example, often seem to be perplexed about why their subject is useful (Barrow, 1992, Chapter 1).

The term "the real world mapping problem" is a term of the present author's invention. There is no universally recognized name for the problem, instead the problem pops up in various places under various names and in various guises. In Hofstadter's "Godel, Esher, Bach" (1980) the term "isomorphism" is used to describe the situation when a system of rules and definitions corresponds to the real world. But Hofstadter's popular book does not explain how, why and when an isomorphism occurs. When one looks for the academic literature giving an account of this notion one finds that it does not exist. Hofstadter posses puzzles but does not provide the solutions.

One would expect that the subject would be addressed in the philosophy of mathematics. While much of this work bears upon the issue, a straight forward solution is rarely offered. Indeed, there are hardly any papers that confront the problem directly. The absence of adequate coverage in the philosophy of mathematics might be explained by a the problem being shunted sideways into the philosophy of logic. This move might be legitimate because there are number of arguments to the effect that mathematics can be reduced to logic (Haack, 1978). Given this the question now becomes "how does logic map on to the real world?". The problem here is that some authors would argue that logic does not map on to the real world nor is it intended to. The later sections of this paper will argue that the solution does lie in the area of logic but logic does not map on to the real world itself rather it is the role of logic to describe the mapping process.

There is an initial plausibility, and one that will be substantiated in a later section, that computerized information systems are, like mathematics, systems of rules and definition. As such they are an invention of the human mind and it needs to be explained how these systems, such as stock control systems, can help us to deal with real world objects and events such as the movement of stock in a warehouse.

The lack of an adequate theory is a more pressing problem for information system design than it is for mathematics. The early systems of arithmetic existed in 9000 BC (Barrow, 1992), Over millennia people have had the time to learn how to use mathematics even though this process of learning may have been driven by trial and error rather than theory. People have built up a tacit understanding of how, when and where mathematics is useful. Computerized information systems are simply too important in our daily lives and the time frame for their development too short to allow a tacit understanding to be built up by trial and error. Another problem is that information systems tend to define their own terms of success, their measures of performance tend to be internal rather than external. The result of this is that they can create their own surrogate reality in which the need for real world mapping is not even perceived.

That something very strange is happening with information systems is evident from cases that arose with the introduction of Britain's poll tax. Local government had previously been financed by rates, a tax levied by local government on the real estate in their area. Under the Thatcher administration this was changed to a tax on people living in the local area. Local governments brought in computer systems companies of their own choice to change their systems. Many of these companies based the new system on the old system of rates and seem to have overlooked, or at least failed to accommodate, the fact that people unlike buildings have a tendency to move.

The local authorities sent some poll tax bills, for the forthcoming year, to people who had recently moved from their area and who were not liable to pay. These bills were sent to their old address, in cases where there was no subsequent occupier these bills were not forwarded. When these people failed to pay, the local authorities placed the matter before the court. The courts sent a summons to the old address. When there was no response to the summons the court tried the people in absentia, found them guilty, and ordered a fine to be paid. When the fine was not paid a bailiff was called. Bailiffs were quite capable of locating most of the people who had moved because they were registered in the area they had moved to and were paying poll tax to the local authorities of that area. The result was that some people, who had moved from an address that was not subsequently occupied and who were paying poll tax to their new local authorities, first discovered that they had been tried, found guilty of an offence they had not committed, and fined was when a bailiff arrived at their doorstep threatening to seize their furniture and household goods. (This actually happened to the author).

This case is interesting not because it shows how incompetent information system design leads to a travesty of justice but because it shows how extensive the real world mapping problem has become and how readily people accept a situation that belongs in the sort of fiction written by Franz Kafka.

Kafka's novel "The Trial" (1983) tells a story about Joseph K. who is arrested for a crime that is never specified. During the course of the book it becomes doubtful that he has even been accused of anything. A trial is normally to determine whether the accused committed the crime referred to. In Kafka's story the trial has no reference. This is strikingly similar to the poll tax case where people were penalized for a crime that was never committed.

In spite of the lack of reference Kafka's story seems to make sense. It has an internal coherence despite the near paradox of some of the situations. The characters are not concerned with how the system maps on to the real world, it is as though the real world has ceased to matter. When K. considers the legitimacy of the proceedings in which he is involved he says "They are in fact only legal proceedings if I choose to recognize them as such". What we have here is a system in which belief alone is a sufficient justification of the system. In the poll tax cases it seems that the court determined that a crime had been committed just because the local authorities believed that a crime had been committed.

The reader might think that the poll tax problem would not have arisen if people leaving an area had notified the local authorities that they were leaving. However, this presumes that there was a mechanism to do so and that this information could be accommodated in the system. Problems in this area are the subject of another of Kafka's works. "The Castle" (1983) is a full length novel that describes how K., who has been hired as a land surveyor by the authorities of a principality, arrives and tries, unsuccessfully, to register his presence in the principality with authorities that called for his services.

Kafka's works are classics of twentieth century literature. They can be contrasted with George Orwell's 1984 (1984) or Animal Farm (1971). Orwell's vision of was of a evil dictatorship working through a centralized and efficient bureaucracy. Kafka's world is of nebulous systems populated by hapless individuals; a system which everyone assents to but which nobody, including the officials, can seem to understand. Kafka's works are generally considered to be a disturbing view of life and society. They are not, like Orwell's 1984, considered to be prophetic. Yet it is Kafka that seems to be becoming an appropriate model for modern information systems. The connection between Kafka and work in the area of information systems has already been noted by Probert (1991) who argues that some aspects of Soft Systems Methodology are Kafkaesque.

There is an argument to the effect that if an information system suffers from a lack of reference and fails to map on to the real world then this can be rectified by systems maintenance. It is the argument that the problems with systems like the poll tax would have been sorted out given enough time (the poll tax system never got the time because the scheme was abandoned). The fact that the poll tax system was trying to bill non-existent residents would have been noticed and the system modified to overcome the problem. With successive modifications the system would come in to line with reality and system requirements. Space constraints prevent a detailed discussion of this argument. However, there are a number of a priori reasons why this is inadequate and these will be briefly described.

1) The viability of the maintenance argument assumes that either a mistake does not matter much or that there is a manual check on the system. In many systems, such as safety critical systems, a mistake is just not acceptable. One might argue that mistakes in official systems such as the poll tax are not acceptable either. If the a manual check on the system is anything other than fortuitous, then the fact that the system might make an error in a certain area will have been anticipated. This will not help to deal with problem that have not been anticipated.

2) A maintenance solution assumes that the rate of change in the environment is slower than the rate of maintenance. Maintenance backlogs are a practical problem because manual maintenance is labour intensive and time consuming. For many systems the rate of environmental change will be faster than the rate of maintenance.

3) Alterations made to correct a lack of reference in one part of the system can have a knock on effect that causes a lack of reference to develop in another parts of the system.

4) People who use an information system have a tendency to believe the system. Like the characters in a Kafka novel if the system indicates that something does not exist they will tend to act as though it does not exist despite physical evidence to the contrary. In these circumstances the need to correct the system is never officially recognized.

Unlike robots, information systems produce "statements" and many of these are putative descriptions of objects or events outside the information system itself, as such they are tractable to analysis in terms of logic and the philosophy of language.

In order to understand how an information system that fails in reference can continue to operate it is necessary to make a number of logical distinctions. A distinction needs to be made between universals and particulars and between logical and factual truth. Universals are statements that refer to all members of a class and have the form: "all cats are animals", "no cat is a dog" etc. Particulars are statements that refer to particular members of a class and have the form: "Tiddles is a cat", "some cats like milk", "that is a cat" etc.

The term "logically true" is used to describe what is inevitably true in a contrived system. In formal systems there are axioms and rules of production, applying the rules of production to the axioms results in theorems all of which are logically true. Behind human conversation there are definitions and statements that follow from definitions by the law of non-self-contradiction, all of these are logically true. Language is, as Wittgenstein taught, a game; logically true statements are the rules of the game. The contradictory of a logically true statement is meaningless.

The term "factually true" is used to describe what is contingently true; that is, what is true but not logically true. It is used to describe true statements about objects and events in the real world. One could equate the real world with the physical world, however, this might involve an unwarranted assumption that the laws of physics govern everything that is not a contrived system. All factually true statements are contingent, this means that they are always open to falsification and that the contradictory of a factual statement is not meaningless.

An example might help to illustrate this distinction.

i) All panthers are black

ii) All crows are black

Statement i) is, in standard English, a logical truth. Standard English is what lexicographer try to capture with their dictionary definitions. English speakers play a language game and have agreed that one of the rules of this game is that "Panther" is the word for a black leopard. Therefore, the statement "some panthers are yellow" is self contradictory and meaningless. There are, of course, variants on standard English. Among a group of regular interlocutors the word "panthers" might stand for the members of a political movement and the word "yellow" might for stand for "cowardly". In this language game "some panthers are yellow" might be true. What counts as a logical truth depends upon the language game that is being played.

Statement ii) is, in standard English, a factual statement. Although "some crows are yellow" is false, it is not self contradictory and it is meaningful.

Logical truths are true because of the way we have set up our language or system. Factual truth are true because of the way the world, the world which is not entirely an invention of the human mind, is. Two important points must be recognized about the relations between logical and factual truth. The first comes from David Hume and states that a factual truth cannot be derived from a logical truth. One cannot infer a factual statement from a set of statements all of which are logically true. This is a well known principle that underlies all of modern physical science. When some people first become acquainted with these ideas they are apt to think that it is only factual truth that is important. This is a mistake.

The second important point is that any system that contains factual truth must also contain logical truth. A more formal way of expressing this is that any factually true statement implies some logically true statements. This is because if we say that "All crows are black" is factual then we must have criteria for identifying crows that is independent of their colour. Those criteria cannot be contingent. If they were not then it would always be possible to say of any yellow crow-like thing that it was not a crow. That is, if we come across a thing that is just like a crow except that it is yellow we could say on the basis of this experience "it is not true that all crows are black" or we could say "this yellow thing is not a crow". If it is always possible to say that things that are not black are not crows then "All crows are black" could not be falsified and, therefore, could not be factual. Logical truths form an indispensable glue that binds factual statements together.

Some people have found this argument rather hard to follow so it is worth making the point in greater detail. When we say a statement is falsifiable we mean that it can be shown to be false by a counter example. If every crow we have come across has been black then we will be justified in believing that all crows are black. We will be justified in believing this until we come across a crow that is not black. This might be "Edgar" a crow that is white. But there is a problem here. How do we know that Edgar is a crow? There must be certain attributes that Edgar has which will allow us to recognize him as a crow and obviously being black is not one of these attributes. Another way of putting this is that there are certain statements that will justify us in saying that Edgar is a crow and one of these statements cannot be "All crows are black".

Crows make a distinctive sound called a "caw". We might recognize Edgar as a crow on this basis. The conjunction of "All birds that caw are crows" and "Edgar is a bird that caws" will justify us in saying that Edgar is a crow. Now consider the following:

1) All crows are black

2) All birds that caw are crows

3) Edgar is a bird that caws and Edgar is white

Let us suppose that we know on the basis of observation and incontrovertibly that 3) is true. Given that 2) is also true we can deduce that 1) is false. However, if all we know for certain is that 3) is true all we know about 1) and 2) is that at least one of them must be false. But it could be 2) just as easily as 1). We could take it that all crows are black is, in fact, true in which case the fact that Edgar is a bird that caws and is white only goes to show that not all birds that caw are crows.

If 1) and 2) are both factual (i.e. contingent) statements there is no way to determine on the basis of 3) which is true and which is false; but we can determine on the basis of 3) that at least one must be false. We do not know which has been falsified and this is not a very satisfactory situation. It makes a nonsense of the idea of falsification because it makes no sense to say that certain statements are falsifiable if we do never know when to say they are false. Also if falsification is given up then we will also need to give up the ideas of contingent truth and the idea of factual truth as we have defined these in terms of falsification. Furthermore, if we tried to represent this sort of system in software we would have computability problems because the system is not decidable.

The problem is not resolved by adding more statements to the system. Suppose we add:

4) All British carrion eating birds are crows

5) Edgar is a white, British carrion eating bird that caws

These added to 1) and 2) still do not show that 1) is false. Given that 5) is true it is still possible for 1) to be true if 2) and 4) are false. That is 5) could be taken as showing that both 2) and 4) are false just as easily as it could be taken as showing that 1) is false. This situation will continue no matter how many factual (e.g. contingent) universals we add to the system.

The only way to make the system coherent and decidable is to introduce logical truth. If we take it that 2) is a logical rather than a factual truth the problem disappears. In this case being a bird that caws is a defining property of crows. From this it follows, by definition, that Edgar must be a crow. And from this it follows that not all crows are black.

It now needs to be explained how logical truth can be established. Logical truths are true by definition. Statements that are logically true are either full definitions, statement specifying a defining attribute or statements that follow from these by the rules of logic.

It is common sense that we as individuals can define our terms in any way we like. However, if our definitions are going to be part of an information system, that is a system used to communicate with others, then the other people involved will need to know about and accept them. Indeed the amount of communication between individuals could be measured in terms of the definitions they accept. There will only perfect communication between two interlocutors if they accept all the definitions in a universe of discourse.

In information system design, therefore, it is of vital importance that there is a consensus about the definitions in the system among the people using the system. How this consensus can be achieved using Soft Systems Methodology has been the subject of previous papers (Gregory 1993a ).

Although logically true statements are essential for factual statements they are not in themselves, nor in conjunction with factual particulars, informative about the real world. Suppose that Tiddles is a panther. Then it might seem that by i) one can infer that Tiddles is black and that this tells one something one did not already know. But this is an illusion. As i) is logically true all it does is say something about how a panther is defined. If Tiddles does not fit this definition then Tiddles is not a panther. If i) is logically true one will not be able to infer from it any real world facts, i.e. contingent facts, about Tiddles.

The configuration of most computerized information systems is such that they embody rules that are logically true and only rules that are logically true. The entry of data, which is the equivalent of the introduction of factual particulars, will not change or falsify any of these implicit rules. People expect these systems to be genuinely informative about real world events, however, this would appear to be logically impossible.

It needs to be explained, therefore, how it is that people appear to find them useful. The simple answer is that computer systems do not function in isolation but in the context of a wider system of belief. This is the system of beliefs adhered to by the human operators and users. Their beliefs encompass a plethora of universal factual truths and it is these that complement the logical truths returned by the machine in such a way that they can be mapped on to the real world and produce genuine information.

An information system should consist of elements corresponding to logically true universals, factually true universals and factually true particulars. They should work together as follows:

1) Factual particular: Tiddles hunts at night.

2) Factual universal: Only panthers hunt at night.

3) Factual particular: Tiddles is a panther.

4) Logical universal: All panthers are black.

5) Factual universal: All black things are the colour of coal.

6) Factual particular: Tiddles is the colour of coal.

Here the factual universals act as buffers between the factual particulars and the logical universal and allows 6) to be a genuinely informative inference. At the same time the logical universal is an indispensable part of the system. The users do not identify panthers on the basis of the definition in the computer system, i.e. 4), but on the basis of another criterion, i.e. 2), which they have formulated themselves and which they believe to be true as a matter of fact. These factual universals are inductive hypotheses and can be falsified as the world or knowledge of the world changes. For example it might be discovered that a certain tiger hunts at night. In this case 2) would be falsified and the user would have to formulate another factual universal as a criterion for the identification of panthers.

This is how a system of logical rules and definitions can be mapped on to the real world. The mapping will break down when the user fails to formulate factual universals relevant to the system. In this case the user will be accused of not understanding the system. However, it will also break down if the logical rules are such that there are no factual universals that could effect a mapping on to the real world. In these cases there will be no interpretation of the system that will correspond with observed events. If the system is not changed then it can continue to operate in a Kafka mode. It will not map on to the real world but it can have an indirect effect, through the actions of the users, on people in the real world.

The fundamental problem is that while the logical elements of the machine can be created by systems designers, the users system of beliefs which are essential for reference to the real world cannot. However, this can be overcome by designing information systems that include factual universals as well as logical universals and particulars. A method for doing this explicitly has already been developed and sample programs that include factual universals have been written (Gregory, 1993b).

The problem of mapping information systems on to the real world is not as difficult as explaining mathematics. It is mainly a problem because is not recognized as the logical problem that it is. The tools of systems analysis, such as data flow diagrams and entity-relationship models, have very limited logical power. They are not sufficient to produce a system that will map on to the real world. However, most information system designers think they are adequate. Many designers have come to think like the systems they design.

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