Fuzzy Thinking

Finally, in recent times some claims have been made for Fuzzy sets. Zadeh originally proposed Fuzzy Logic and Fuzzy sets as a model for imprecision in natural language. The idea was that whereas using classical logic we have that `Fred is a man', is interpreted as ` Fred is a member of the set of all men', the similar sounding proposition `Fred is a tall man' has a rather dubious interpretation as `Fred is a member of the set of tall men'. When is a man tall? Fuzzy set theory attempts to overcome the essential vagueness about how tall a tall man is by assigning a number between 0 and 1 to the proposition `Fred is a tall man', and hence assigns a number to the extent to which Fred is a member of the `Fuzzy set' of tall men. How this number is arrived at is never made clear. The vector space representation would simply measure his height.

Modelling objects by Fuzzy predicates certainly seems more useful than modelling them by the quasi-logical predicates of AI, and the area has grown in recent times. It is liable, however, to a serious criticism which has never been convincingly rebutted: everything fuzzy sets can do, statistics and probability theory can do, too. And statistics and probability theory have relatively well argued rationales, a huge body of successful applications and fairly solid foundations. In contrast, the so-called applications of Fuzzy Logic and Fuzzy Set theory are often just examples of fuzzy thinking. Giving a rather badly drawn oval shape as an example of something which has fuzzy membership in the set of ellipses, for example, suggests a misunderstanding of how descriptions in Science and Engineering work. They aren't there to give you a warm inner glow, they are there to be used. The question is, what can you do with such a statement? On all the evidence, not much. To see that these are issues in linguistics rather than science, consider the response of a Trobriand Islander to the same badly drawn oval shape. He might, just possibly, see it as a `fuzzy' coconut or a `fuzzy' necklace, but he wouldn't feel inclined to assign it as having some (hard to measure) degree of elementhood in the set of ellipses, if only because he hasn't met the set before. If you feel an urge to put a metric on the set of simple closed curves in the plane so as to measure the extent to which such things are ellipses, it is not too difficult to do, but it would be a mistake to imagine there is anything unique about your choice of how to do it, or that your urge requires expression just because you have it. I frequently have the urge to beat Fuzzy Set theorists over the head with a bottle, but I curb this impulse.

Early applications of these ideas to control have been unconvincing. An early but typical one consisted of replacing a thermostat system which monitored the temperature and fed power to a furnace so as to heat the place up if the temperature went low, and cool it down when it went high. The original controller measured the temperature continuously with a thermocouple, the `fuzzy' controller divided the temperatures into three intervals, `too hot', `too cold', and `OK'. Describing a controller which treats temperatures which fall into intervals as fuzzy is sharp practice. If the controller sometimes acted as though `too hot' started at 80^o and sometimes at 70^o, or if the thermocouple was subject to large random variations, there might be a case, but the world isn't fuzzy except possibly at the quantum level. What is fuzzy is language, and we are more interested in the world. So far as I am aware, the sudden surge of so called AI applications of fuzzy logic in Japanese refrigerators is composed of ideas as banal as the fuzzy controller, and is intended to sell more refrigerators to the unwashed. Science and Engineering are rather more difficult than that.

Later applications of `Fuzzy Control' appear to be more interesting. The fuzzy control of the inverted pendulum, and more impressively the double inverted pendulum (balancing two billiard cues, one on top of the other) are claimed to work, whereas classical control theory finds these problems difficult. There is some reason to believe that the fuzzy control people have a good idea in there somewhere, but it seems doubtful if it is coherently worked out. Engineers and Physicists have a tradition of coming up with working systems using a rationale that is full of holes and logical nonsense. This has often led to mathematical advances when the mathematicians woke up to the fact that there had to be something there, and that the fact that it couldn't be expressed in conventional ways meant that mathematicians had to do some work. This may be the case with `fuzzy control'; they may have some good ideas mixed up with some awful philosophy and scrofulous mathematics that would be well worth sorting out properly.

Multi-valued logics have been around since Post in the 1930's, and John Maynard Keynes wrote around the same time on Probability interpretations in a way similar to Zadeh's model. More recently Ed Jaynes has given some good reasons for believing that the assignment of numbers to statements to measure their believability must be done in accordance with Bayesian statistics. This is discussed in the Information Theory II course in the Master's Degree.

What it comes down to is that when we have uncertainty in the real world, the semantics of our models involve counting things that happen in different outcomes when we cannot distinguish between the inputs, as when we throw dice. This is what probability theory was invented for, and it seems to do a reasonable job. When we wish to treat of such notions as closeness or proximity, we have all the machinery of topological and metric spaces already to hand. It is pointless to reinvent it rather amateurishly when it has been done well once already. Reinventing wheels is sometimes justifiable, but reinventing statistics and topology, badly, is time and energy wasted.

Fuzzy set theory seems to have been based upon and be driven by a desire to translate vague sentences of natural language into a mathematical formalism. This seems like a great idea to those many people who cannot distinguish clearly between properties of the world and properties of the language we use to talk about it. Such people may genuinely believe that the world is fuzzy around the edges.

Possibly for them it is; a consequence, perhaps, of having smoked or sniffed illegal substances as undergraduates. They should have stuck to brandy, cigars and sex, like the rest of us.

Getting clear descriptions of what is going on in a nuclear power plant is not much helped by going and asking the man in the street what he thinks. His description will certainly be describable as `fuzzy'. But most engineers would swap it for some measurements any day of the week. The same holds for just about any complicated system, not just nuclear power plants.

It has been said that if Statistics, Probability and Metric Space concepts had been better taught to engineers, as ideas instead of recipes, the fact that Fuzzy set theory and Fuzzy logic are unnecessary would have been apparent, and the subject would never have been invented. It is imprudent to buy further into this fight, so I'll stop there, but the reader should know that there is a fight. And I daresay by this time the reflective reader will have worked out whose side I am on, allowing me to eschew the hypocrisy of pretending to be neutral.

Since writing the above, I have had some discussions with Jim Bezdek, who has a commitment to Fuzziness. Now Jim is an intelligent and thoughtful man, and his views merit serious consideration. Nor is he wholly alone in this.

Bezdek claims that the semantics of Fuzzy Sets and the semantics of Probability theory are quite different. He gives as an example the case where you stagger through the desert, on the point of dying from dehydration, and meet Jim who offers you a choice of two bottles. He describes the one bottle by a `potability' value of 0.95, and the other as a probability of being pure water of 0.95. The latter, he says, represents a gamble: 100 bottles were taken, 95 filled with pure water, five with salt solution in toxic concentration, and a choice made at random. The question is, would you rather take the 0.95 potability bottle, or the other?

Were it me staggering out of the desert, my first response would be to sample the probability bottle, and if water, drink it. My second response would be to threaten to beat Jim over the head until he told me what `potability' means. I know it means `drinkable', but what does the 0.95 mean? Jim might argue that the term is inherently vague and has no answer, but I am bigger than he is and would insist. Does it mean that he took 95 parts of pure water and 5 parts salt, mixed them up and filled the bottle from the mixture? Or maybe it was 95 parts water and 5 parts potassium cyanide. I need to know. More crucially, the sort of data I want is something along the lines of `if you do this sort of thing often, what fraction of people who took the 0.95 potable bottle survived to talk about it? That is, we reduce to counts of possible outcomes that I care about.

The important thing I care about is, will drinking the contents of the bottle shorten my life expectancy or increase it? To be told that it is 0.95 potable is to be told that I have a whacko on my hands who proposes to tease me with meaningless jargon, and I don't take kindly to this even when not dying of thirst. To be told that of the people who drank the same mixture in the past, all complained of mild indigestion but went on walking to the next delicatessen, is to be given some idea of what it means. To be told that 95 percent lived but the other five percent died is again to be given some sort of useful information. But to be confronted with the label and no idea of how to interpret it is merely frustrating. If I found the bottles with their inscrutable labels but no Jim to interrogate, I should sniff the contents carefully and sample them to get some idea of what to do. But I'd do that anyway, since who would believe a label in such a case? What I would do with the (sampled) contents would depend on just how thirsty I was. This is what most people would do, I daresay, and the fact that none of us would bother about the label much tells us something about labels.

This might be thought to miss the point. In making decisions we often take into account things we have been told, and we are often told them with a fair degree of vagueness. We do not know the particular usage of terms employed by the teller, and if a short man tells you to look out for a big bloke at the airport, and he turns out to be of only average height, then one would not perhaps be too surprised that the word `tall' means something different to a short man than it does to a tall one. But we might reasonably suppose that the teller has some reasonably precise definition in his head, even if he doesn't know what it is. We could parade a collection of people of varying heights before our source, and ask him to classify them, and we could conclude that `tall' means, to him, approximately `over five foot nine inches in height'. We don't have the luxury of conducting the experiment, so we have to guess what he means. This is plainly a problem with our ignorance of how someone else uses a term. Probability theory can handle this quite well. We use as data the way we have heard the word `tall' used in the past. This is how we handle much of the vagueness of natural language.

Sometimes vagueness is used to connote indifference, sometimes ignorance, sometimes because only crude approximations are necessary and vague is quick. Few people know their weight to three places of decimals in kilograms or pounds, and few care to. But there seems to be no necessity for asserting that there is some numerical degree of membership in the class of overweight people, and no obvious intepretation of such a number when given.

For these reasons, I cannot take seriously the philosophical foundations of fuzzy sets and fuzzy logic, although the latter makes a harmless exercise in mathematics.

Early in the Nineteenth century, Napoleon, who thought Mathematics important and rather fancied himself as being, spiritually, a Mathematician who had just somehow lapsed into being an Emperor, was talking to Laplace. Napoleon twitted Laplace about the latter's book, Mecanique Celeste, which dealt with the origins of the solar system. ``I see that you have made in your work no mention of le bon Dieu, monsieur de Laplace''. ``Sire'', replied Laplace, ``I have no need of that hypothesis''.

This is roughly my position on fuzzy logic, fuzzy sets and fuzzy thinking. I can get along better without them.