The Relationship Between Probability Theory and Fuzzy Set The- The-ory Ultimately we may think of bridging the gap between fuzzy

interval arithmetic and the calculus of random variables, i.e. embedding both into a unique setting.

| Didier Dubois and Henri Prade Articial Intelligence is philosophical explication turned into computer programs.

| Clark Glymour

We will now attempt to briey review various arguments on the topic of proba-bility theory versus²⁰ fuzzy set theory, and their general interrelationship. This is, after twenty-ve years, still a contentious topic. We have certainly not aimed for completeness (that would require far more space), but we have aimed to be rea-sonably representative. Although our preference should quickly become apparent anyway, let us explicitly state our opinions here: We feel that fuzzy set theory is of little value in engineering applications. Our reasons for this conclusion are sev-eral, but the main two are the philosophical basis (which we believe to be confused and wrong), and the practical ecacy (apparently close to zero, when reasonably compared with probability theory). Since we do not have space to fully develop our arguments, we do not expect to convert many people in the following text.

Nevertheless we feel the issue too important to pass over in silence.

A lot of the debate between proponents of probability and fuzzy set theory is essentially philosophical. Whilst the immediate reaction of the engineer is to avoid this (just get on with building something that works), it turns out that this is neither desirable nor possible. This is especially true given that one of the main application

20As many people have observed, the question is really less a matter of probability versus fuzzy set theory than a balanced comparison of their merits. The point is that the two approaches can be, and have been, combined in a number of dierent ways. Nevertheless, since it is the purpose of the present section to highlight the dierences, asking the question in the above form is reasonable.

areas for the methods we have been discussing is Articial Intelligence. We agree with Glymour who says that articial intelligenceis philosophy. He argues that

Since AI is philosophy, the philosophical theory a program implements should be explicit. Any claim that a program solves some well-studied problem,::: but doesn't say how, should be disbelieved 314, p.206].

We aim to show, inter alia,that the philosophical basis of fuzzy set theory is inad-equate for engineering and articial intelligence problems.

One of the diculties in critically discussing fuzzy set theory was explained by Cheeseman as follows:

Unfortunately this the comparison between fuzzy and non-fuzzy theories]

is not as easy as it sounds because the \fuzzy approach" is itself fuzzy

| there are fuzzy sets, fuzzy logic, possibility theory and various higher order generalisations of these (e.g.fuzzy numbers within fuzzy set theory).

This diversity complicates the task of critiquing the fuzzy approach 141, p.97].

Toth 825]has recently tried to clarify some of the distinctions, and to develop a more rigorous foundation for fuzzy set-theory. The reason we mention Cheeseman's complaint is that because of the variety of dierent views, it is dicult to know which to criticise: any criticism can be deected by changing ground slightly. In general we will refrain from attacking the most absurd and the weakest arguments in favour of fuzzy set theory, and we will concentrate on what appears to be the most useful material.²¹

To us, the two most convincing arguments are as follows.

1. The whole enterprise of fuzzy set theory is based on the \inherent imprecision of natural language." This is supposed to be an \uncertainty" of a completely dierent kind to the uncertainty of probability theory. It has its roots insorites type paradoxes 81,82,701]. Arbib 31] has presented a simple argument against this. He observes that although \people can certainly draw a `degree of tallness' curve if pushed to it,::: this does not show that our concept of tallness has such a form" 31, p.947]. He goes on to note that vague terms are normally context sensitive (a notion fuzzy set theory either ignores, or handles in a very poor manner), and that natural language is not \inherently imprecise," although it may be used imprecisely in some circumstances:

Perhaps the most distressing mistake of fuzzy set theorists is to be-lieve that a natural language like English isimprecise. The fact that

21We allow ourselves one irresistable exception, namely Goguen's argument for the \social nature of truth" 319]. Goguen, upon realising the diculties in actually determining grades of membership or degrees of truth, suggested that the notion of objective truth was not as useful as one based on social con-sensus: \This paper suggests we must abandon classical presuppositions about truth, and view assertions in their social context" 319, p.65]. Whilst this may appeal to totalitarian governments, it has little to recommend it otherwise!

many people use English badly is no proof of inherent imprecision 31, p.948].

In any case, the point at issue, for practical purposes, is the referent of a word (what the word describes), rather than the word itself. A concentration on linguistic aspects was the cause for severe diculties in a stream of twentieth century philosophy which followed Ludwig Wittgenstein (see 57]).

2. Our second argument is more appealing to the engineer: Fuzzy set theory based methods do not work. More precisely, it seems generally fair to say that fuzzy set theory has not been used to develop any methods for any problems that are demonstrably better than probabilistic or non-fuzzy methods. Although numerous applications have been reported, the \fuzziness" of the methods is not essential to any success they may have. Furthermore there has been very little hard-headed and honest comparison with non-fuzzy techniques. We will use the example of fuzzy control to illustrate this point below.

Regarding the general applicability of fuzzy set theory, Zeleny 903, p.302]has argued that apart from human decision making and judgement, \there areno other areas of application." We agree with this, but would even question the applicability to human decision making.²²

Fuzzy Control

An example of a suggested engineering application of fuzzy set theory is fuzzy con-trol. The idea of this, which seems to have been rst studied by Mamdami 538], is to develop automatic controllers for dynamic plants by using linguistic informa-tion obtained by quesinforma-tioning human operators of the plant. That is, one asks the operator how he controls the plant, and then incorporates these \fuzzy rules" into an automatic controller. It is suggested that this approach (which does still seem to show some promise) is suitable for highly non-linear plants which it is dicult to model explicitly. A survey of fuzzy control is given by Sugeno 788]. Fuzzy control is considered to be one of the most developed and \successful applications]of the theory of fuzzy sets" 899, p.421]: Sugeno argues that \Fuzzy control is without doubt one of the most exciting and promising elds in fuzzy engineering" 789].

Not only are the fuzzy controllers rarely compared with the classical designs, but when they are, it is only with the simplest PID (Proportional, Integral, Derivative) controllers and little advantage (if any) is claimed 181]. The main disadvantage of fuzzy control (and this is admitted by Sugeno in his survey 788, p.78]) is that there are no analytical tools to test the stability of these controllers. Furthermore, as has

22Some authors consider fuzzy set theory as a purely mathematical theory and suggest that it be judged on its mathematical merits. Whilst we admit that there has been some very interesting mathematical work (especially by Hohle, Lowen and others), we generally agree with MacLane's assessment (see 527]

and the papers following) that most of fuzzy mathematics is valueless. It is believed that there are far too many mistakes and that most of the results are trivial. (A cynic might say that the same argument applies to all modern mathematics, to which we would reply \perhaps, but it applies more so to fuzzy mathematics"). See also Johnstone's open letter to Ian Graham: \Fuzzy mathematics is NOT an excuse for fuzzy thinking" 414].