Neural Networks and Fuzzy Systems (textbook)

Neural Networks for Signal Processing (textbook)


XV Preface




1. Shades of Gray

. The Fuzzy Principle 18

Theme 1: Bivalence vs. Multivalence:

Simplicity vs. Accuracy 21

Theme 2: Precision Up, FuzzUp 34

Theme 3: Fuzzy Reasoning Raises

Machine IQ 38

Theme 4: Don't Confuse Science with Scientists 40

3. The Whole in the Part 44

The God of Maximum Probability 47

To Believe and Not Believe 48

The Probability Instinct 53

Subsethood: To Contain and Not Contain 55

The Zen of Gambling: The Whole in the Part 56



4. The Fuzzy Past 67

. Aristotle vs. the Buddha 69

The Sociology of Creeping Fuzziness 70

Aristotle and the Buddha in the West 72

The Buddha in the East and Today 75

6. What Is Truth? 79

The Philosophy of Truth: Truth as a Scorecard 80

Statements as Vehicles of Truth 81

Logical Truth and Factual Truth 82

Coherence and Correspondence 83

Hemingway's Challenge: Truth as Accuracy 85

7. The Ways of Paradox 92

Sorites Paradoxes: The Road from A to Not-A 94

Paradox at Endpoints, Resolution at Midpoints 97

Uncertain World 103

What the Uncertainty Principle Says 104

Another Mismatch: Linear Math, Nonlinear World 107

Pythagoras and the Uncertainty Principle 110


8. The Fuzzy Present 119

9. Fuzzy Sets 121

Numbers Are Fuzzy Too 123

Fits of Fuzzy Entropy 126

Max Black: Vague Sets 135

Lotfi Zadeh: Fuzzy Sets 140

From Fuzzy Sets to Systems 155

10. Fuzzy Systems 156

Knowledge as Rules 158

Rules as Patches 161

The FAT Theorem 167

Fuzzy Associative Memory: Fire All Rules 171

Fuzzy Decisions in Life: Fuzzy Weighted

Averages 176

Fuzzy Systems as Judges: Rules vs.

Principles 178

FAMs in Practice: Fuzzy Products 180

LIFE in Japan 190

Yamakawa in the South 197

11. AdaptiveFuzzySystems 201

The DIRO Brain Suck: Data In, Rules Out 202

Learning as Change: Neural Nets in Brains 205

Learning Rules


Learning Rules


Fuzzy Cognitive Maps: Fuzzy Pictures of the World 222


12. The Fuzzy Future 239

13. Life and Death 241

Life Lines 243

Life Curves 246

Take a Poll and Find Out 247

Death in Degrees 250

14. Ethics and the Social Contract 254

The Fuzzy Truth of Ethics 256

The Test of Morals 257

Right and Wrong: Hurrah and Boo 260

The Force of Law 262

The Fuzzy Labyrinth of Law 263

The Fuzzy Social Contract 264

15. Man and God 267

The Anthropic Principle: It Is So That I Am 270

A Fuzzy Answer: Suppose Nothing 272

Cosmic Chips and God 276

Man a Fuzzy Machine: The Kiss of Machine IQs 281

Glossary 287

Bibliography 299

Index 309


One day I learned that science was not true. I do not recall the day but I recall the moment. The God of the twentieth century was no longer God.

There was a mistake and everyone in science seemed to make it. They said that all things were true or false. They were not always sure which things were true and which were false. But they were sure that all the things were either true or false. They could say whether grass is green or whether atoms vibrate or whether the number of lakes in Maine is an even or odd number. The truth of these claims had the same truth as claims about math or logic. They were true all or none, white or black, 1 or 0.

In fact, they were matters of degree. All facts were matters of degree. The facts were always fuzzy or vague or inexact to some degree. Only math was black and white and it was just an artificial system of rules and symbols. Science treated the gray or fuzzy facts as if they were the black-white facts of math. Yet no one had put forth a single fact about the world that was 100 true or 100%o false. They just said they all were.

That was the mistake and with it came a new level of doubt. Scientists could err at the level of logic and math. And they could maintain that error with all the pomp and intolerance of a religious cult.

I pursued gray truth in my graduate training in math and electrical engineering and machine intelligence. At first I worked with symbols on abstract math theorems. This made the gray or fuzzy world view seem like a dry exercise in a math textbook. Then I started to teach the subject and to combine the math with real applications of the fuzzy ideas. Students were quick to learn how to paint gray pictures of a gray world. Some students built real fuzzy systems or software packages and some of them patented their ideas and some of them went off to sell their wares or to start their own companies. In Japan engineers designed the first fuzzy "smart" commercial products. Soon there would be fuzzy camcorders and washing machines and microwave ovens and carburetors and hundreds of other smart products.

The applications showed that the fuzzy world view extended beyond the journal paper and textbook and classroom. The rapid spread of fuzzy ideas in the Far East and the opposition to them in the West showed even more. The fuzzy world view was a world view. It extended as much to culture and philosophy as it did to science and math. It reached back to thinkers as diverse as Aristotle and the Buddha. It reached forward to how we argue over law and abortion and the nature of intelligence and on to how someday we might make our peace with machines that can beat us in any race.

This book is my statement of the fuzzy world view. At the core is the paradigm shift from the black and white to the gray-from bivalence to multivalence. I tell the tale with a mix of science and philosophy and history and with episodes from my own fuzzy experience. The point was not to write a text on fuzzy logic. I already did that and it takes too many equations. The point was to show the fuzzy world view at work in the mind and in the flesh. To do that you have to have lived the field and fought the fights. You have to have doubted the God of science and felt a little of Her wrath.

Into every tidy scheme for arranging the pattern of human life, it is necessary to inject a certain dose of anarchism.



Everything Is a Matter of Degree



So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality.



Fuzzy theory is wrong, wrong, and pernicious. What we need is more logical thinking, not less. The danger of fuzzy logic is that it will encourage the sort of imprecise thinking that has brought us so much trouble. Fuzzy logic is the cocaine of science.



"Fuzzification ' is a kind of scientific permissiveness. It tends to result in socially appealing slogans unaccompanied by the discipline of hard scientific work and patient observation. PROFESSOR RUDOLF KALMAN UNIVERSITY OF FLORIDA AT GAINESVILLE

Fuzziness is probability in disguise. I can design a controller with probability that could do the same thing that you could do with fuzzy logic.



Hold an apple in your hand. Is it an apple? Yes. The object in your hand belongs to the clumps of space-time we call the set of apples--all apples anywhere ever. Now take a bite, chew it, swallow it. Let your digestive tract take apart the apple's molecules. Is the object in your hand still an apple? Yes or no? Take another bite. Is the new object still an apple? Take another bite, and so on down to void.

The apple changes from thing to nonthing, to nothing. But where does it cross the line from apple to nonapple? When you hold half an apple in your hand, the apple is as much there as not. The half apple foils all-or-none descriptions. The half apple is a fuzzy apple, the gray between the black and the white. Fuzziness is grayness.

Seventeenth-century philosopher Rene Descartes pondered a plug of beeswax one evening in front of his fireplace. He thumped the wax and heard its dull sound, smelled its honey fragrance, felt its smooth surface and cool temperature, tried to look through its milky texture. Then he held the wax plug near the fire. The hard white plug softened, warmed, stretched, lost its odor, became clear, became liquid, flowed. Some of it dripped onto the hot brick hearth and sizzled and boiled away into the atmosphere.

Where did the wax go? When did it change from wax plug to non-wax plug? Where did its identity lie? In the plug? On the hearth? In between?

We face the same questions every day when we look in the mirror. The face and hair and teeth and skin have changed. They even change slightly, at the molecular level, as we look. We pass slowly from self to a new nonself as we grow old one cell at a time, a molecule at a time. Molecules assemble and disassemble an atom or two at a time. Atoms assemble and disassemble a quark at a time. Quarks, our current smallest units of matter, assemble and disassemble in ways we do not yet know. The division of matter may go all the way down to Leibniz's monads, infinitesimal but intelligent points of existence.

All around us things change their identities. The atoms that make up the universe swirl and collide and keep swirling and colliding Everything is in flux. Everything flows. The universe unfolds as a river runs. The cosmological fluid seems to obey Einstein7s laws of general relativity in the large and seems to obey the laws of quantum mechanics in the small and obeys we do not know what in between.

Things flow smoothly to nonthings. The atoms in our fingertips swirl into the atoms in the air. There are finger atoms and nonfinger atoms. And there are the atoms in between, the atoms to some degree both finger atoms and air atoms and to some degree neither. A rose is a rose is a nonrose when its molecules change. The finger shades into the hand, the hand shades into the wrist, the wrist into the arm. Earth's atmosphere shades into space. The mountain crumbles into a hill and in time crumbles into a plain. The growing human embryo passes into a living human being and a living brain decays into death.

We can put black-and-white labels on these things. But the labels will pass from accurate to inaccurate as the things change. Language ties a string between a word and the thing it stands for. When the thing changes to a nonthing, the string stretches or breaks or tangles with other strings. "House" stands for a house even after the house falls apart or burns. Our world of words soon looks like a fishing boat that drifts with thousands of tangled and broken lines.

We know things change. Science reveals a world of jagged edges and quantities that vary smoothly. More precision does not take the gray out of things--it pins down the gray. Medical advances have not made it easier to draw the line between life and not-life at birth or at death. If we described the Earth's atmosphere molecule by molecule, we would still find no line that divides the atmosphere from space. Detailed maps of the surface of Earth and Mars and the Moon do not tell us where hills end and mountains begin.

Yet in much of our science, math, logic, and culture we have assumed a world of blacks and whites that does not change. Every statement is true or false. Every molecule in the cosmos belongs to your finger or not. Every law, statute, and club rule applies to you or not. The digital computer, with its high-speed binary strings of ls and 0s, stands as the emblem of the black and white and its triumph over the scientific mind.

This faith in the black and the white, this bivalence, reaches back in the West to at least the ancient Greeks. Democritus reduced the universe to atoms and void. Plato filled his world with the pure forms of redness and rightness and triangularity. Aristotle took time off from training his pupil Alexander the Great to write down what he felt were the black-and-white laws of logic, laws that scientists and mathematicians still use to describe and discuss the gray universe.

Aristotle's binary logic came down to one law: A OR not-A. Either this or not this. The sky is blue or not blue. It can't be both blue and not blue. It can't be A AND not-A. Aristotle's "law" defined what was philosophically correct for over two thousand years.

The binary faith has always faced doubt. It has always led to its own critical response, a sort of logical and philosophical underground. The Buddha lived in India five centuries before Jesus and almost two centuries before Aristotle. The first step in his belief system was to break through the black-and-white world of words, pierce the bivalent veil and see the world as it is, see it filled with "contradictions," with things and not-things, with roses that are both red and not red, with A AND not-A.

You find this fuzzy or gray theme in Eastern belief systems old and new, from Lao-tze's Taoism to the modern Zen in Japan. Either-or versus contradiction. A OR not-A versus A AND not A. Aristotle versus the Buddha.

The Greeks called their dissenters "sophists." Today we call sophistry reasoning we find faulty or silly. When one day in his Academy Plato defined man as a featherless biped, the next day a sophist student walked into class and held out to Plato a plucked chicken. Zeno picked a grain of sand from a sand heap and asked whether the heap was still a heap. Zeno could never find that sand grain that changed the heap to a nonheap, that took it from A to not-A. As he picked out more sand grains he seemed to get a heap and a nonheap, A AND not-A. The liar from Crete said that all Cretans are liars and asked if he lied. If he lied, then he did not. And if he did not lie, then he did. He seemed to lie and not lie at the same time. Modern philosophers, like Descartes, have brooded over the nature of identity and have searched in vain for the common substance that passes from wax plug to non-wax plug. David Hume saw the self dissolve into a nonself bundle of sensations. Werner Heisenberg

showed physicists that not all scientific statements are true or false- Many, if not most, statements are indeterminate, uncertain, gray--fuzzy. Logician Bertrand Russell found the Cretan's liar paradox at the foundations of modern math. Mathematicians and philosophers have since tried to patch and gerrymander those black-and-white foundations to get rid of the gray paradoxes. But the paradoxes and the brooding about them remain.

I brooded about grayness too. It led me from philosophy to mathematics to electrical engineering. I picked up degrees along the way and in time ended up teaching electrical engineering at the University of Southern California, where I had started out as an undergraduate in music composition. I ran across Einstein's quote about math mismatching reality when I was 21 and sat in a USC philosophy classroom. It shocked me to see Einstein doubt the very math framework of black-and-white science that he had helped build: ''So far as the laws of mathematics refer to reality, they are not certain. And so far as they are certain, they do not refer to reality." So Einstein brooded about grayness too.

I skimmed Sir A. J. Ayer's anthology on logical positivism, the dominant philosophy of science in this century. Positivism demands evidence, factual or math evidence, as a security guard demands positive ID, not just your say-so. Logical positivism holds that if you cannot test or mathematically prove what you say, you have said nothing. Positivism works out well for scientists and mathematicians, since it allows only them to speak. Everyone else utters "meaningless" statements about the world and life and morals and beauty. Problems of God and metaphysics and goodness and value reduce to mere "pseudo-problems," questions asked by those whom language has misled, those who do not know what counts as answers. Logical positivist Moritz

Schlick ended one of his essays in the anthology on just this point:

Philosophical writers will long continue to discuss the old pseudo-questions But in the end they will no longer be listened to. They will come to resemble actors who continue to play for some time before noticing thal the audience has slowly departed. Then it will no longer be necessary to speak of "philosophical problems . "

Every philosopher you ask will attack logical positivism, either on details or on some general principle, but it remains the working philosophy of modern science, medicine, and engineering. Logical positivism hands the future to scientists. It also hands them much of the present.

Einstein's quote punched through the black-and-white world of science and math. It sounded the positivists' lament. I read the quote over and over and found myself slowly becoming a fuzzy logical positivist. The world of math does not fit the world it describes. The two worlds differ, one artificial and the other real, one neat and the other messy. It takes faith in language, a dose of make-believe, to make the two worlds match.

I called this the mismatch problem: The world is gray but science is black and white. We talk in zeroes and ones but the truth lies in between. Fuzzy world, nonfuzzy description. The statements of formal logic and cornputer programming are all true or all false, I or 0. But statements about the world differ.

Statements of fact are not all true or all false. Their truth lies between total truth and total falsehood, between 1 and 0. They are not bivalent but multivalent, gray, fuzzy. These statements are not just tentative, they are imprecise and vague. The logical statement "Two equals two" and the math statement "2 + 2 = 4'' are precise and 100%o true--true, as philosophers say, "in all possible universes," even though philosophers have seen only one. But that does not affect how atoms swirl or how universes expand or how a strawberry tastes or how a face slap feels. We can never prove 100%o true a scientific statement or claim of fact like "The moon shines" or "Grass is green" or "e = mc2." Fresh evidence may topple any scientific belief, and objects of belief differ only approximately from their opposites. The blade of green grass turns brown. In the next instant the moon may stop shining and burst into flames or fall into the Earth or break a cherished law of science and collapse into a black hole or a ball of cheese.

Laws of science are not laws at all. They are not laws in the sense of logical Laws like two plus two equals four. Logic does not legislate them. Laws of science state tendencies we have recently observed in our corner of the universe. The best you can say about them is so far, so good. In the next instant every "law" of science may change. Their truth is a matter of degree and is always up for grabs. Yet the language of science, the language of math and logic and computer programming, is black and white. It deals only with statements that are 100% true or 100% false. Math talk differs in kind from science talk. But scientists talk it anyway.

I thought scientists and philosophers would see the mismatch problem as the central philosophical problem of modern science. But they seemed to ignore it. Einstein called it out as a quotable quote, an ironic throw-away line in the midst of the business of science. A rare philosopher might cite Einstein's quote to bring out the positivis1's favorite split, between words and objects, logic and fact. But I found no one wrestling with the mismatch problem and for a reason: they assumed the match too.

Philosophers assumed the world was black and white, bivalent, just like the words and math they used to describe it. After all these years and all that training they still took orders from Aristotle and did not question them. In theory they could tell matters of logic from matters of fact. In practice they ignored this split and treated the messy matters of fact as if they were neat matters of logic. They did this for two reasons: First, it was easy. Second. it was habit. They used the same artificial language to talk about matters of logic and matters of fact. They described math and the world with the same black-white "symbolic logic" that Aristotle set in motion over two thousand years ago.

And I used it too. In gym class the toughest guys do the most pushups or run the fastest mile or punch hardest. In a modern philosophy class they find the shortest proofs to the theorems of symbolic logic. The same holds in science. The more math an author throws at a problem, the less her audience understands her and the more they respect her. Your skill at logic and math places you in the pecking order of science. I competed with classmates and professors to move up in that pecking order. I saw formal logic as the flame the philosophers kept. But Einstein's quote kept eating through my faith in logic and science.

At first my doubt had no focus. I had no alternative to the black-white formalisms of science and logic. What else could there be? To stray from logic and science struck me as the nonlogic and nonsense of Eastern mysticism, the full-lotus stuff a young scientist makes jokes about when he eats in a Chinese restaurant.

Even Einstein had no alternatives to bivalence. Instead he and the league of scientists added a new theory to the old theory of bivalence. They added the theory of probability, the mathematical theory of "chance" or "randomness." The idea is that every event has a number attached to it, the probability that the event will occur. The event might be the flip of a coin. There is a probability that the coin comes up heads and a probability that it comes up tails. The coin comes up heads or tails and the two probabilities add up to one. In general the probability that an event occurs and the probability that it does not occur add up to one. That's probability theory. Event numbers add up to one and the events are black and white, they occur all or none.

Probability did not alter or even challenge the black-white picture of the world. It just showed how to gamble on it and in it. Aristotle's law of A OR not-A always holds in probability. The new physicists saw probability wherever they looked. But Einstein did not feel comfortable with it. That is what he meant when he said that "God does not play dice.'' Quantum mechanics, the physics of subatomic events, suggested otherwise. The universe seemed nothing but probability.

I found comfort in the math of probability but not in the very idea of it. What is probability? What kind of thing is it? What does it look like? How do you measure it? How do you test a probability claim? I hold a coin in my hand and claim that it is "fair"--the coin has a 50% "chance" or "probability" of coming up heads. Then I flip the coin and it comes up heads. Does that confirm the claim? If it does, it also confirms the claim that the heads probability is 55% or 90% or even 100%. We could even count the head flip as evidence for the claim that the heads probability is 45% or 10% or even 0.0000000001%. Just the "luck" of the draw. A probability experiment can go either way and you do not know which way. If I hold the white chess pawn in my hand behind my back and ask you to guess which hand holds it, I know which way the experiment goes but you must guess, estimate, calculate odds. For you the probability that my right hand holds the pawn is "real" or makes sense. For me it is an illusion. I know the outcome with "certainty."

Probability evaporates with increased information. Information up, probability down. The laws of physics determine whether the coin comes up heads or tails. To a supersmart, Supersensitive, supercomputing being all probability experiments are illusions. So maybe there is no probability. Maybe there is something else, maybe something fuzzy, that we sometimes call probability but that exists in the nature of things and nonthings, or in the relations between them.

I looked for the answer as to how probability differed from fuzziness but could not find it, because at that time I did not know what fuzziness was. I had not seen the math of fuzziness. In time I wrote my Ph.D. dissertation on fuzzy math to help me see and that still was not enough. I wanted to draw a line in the math sand between fuzziness and probability. But in my heart I suspected one contained the other. Critics said that all the time: fuzziness is probability in disguise.

I suspected it was the other way around. Since the days of ancient Sumeria, men and women have used probability words to refer to complex patterns of behavior in the environment and in society--whether it will rain, whether the beer will spoil, whether the hunters will find a deer, whether the other village will attack, whether the wife will get pregnant. Modern scientists have carried on this tradition and they did not question it. Instead they put the probability talk in math and exalted it in all the sciences. That made me even more suspicious. The probability that the bowman's arrow hits the deer does not lie in the arrow or the deer. If it lies anywhere it lies in his mind, his brain state, or in ours. Math or no math, I wondered, how wise is it to build these brain states into the foundations of quantum mechanics, the theory of subatomic particles, into the most fundamental descriptions of the universe?

These were problems with the idea of probability. But there was still the first problem of applying probability to the real world: Probability did not solve the mismatch problem. It compounded the problem. It piled a new theory on top of the blackwhite theory of bivalence. And that came with a price: the world filled with the new phantom of "randomness," the concept mathematicians for years have tried to define.

Probability deals with blacks and whites--heads or tails, success or failure, in the box or out. It puts odds on these precise events. It does not remove their precision, their bivalent flavor Indeed scientists round off gray things to black or white things before they apply probability. The electron orbits the atomic nucleus or not. The cell cluster turns cancerous or not. The lion catches the gazelle or not. The customer waits in line or not. The cloud is nimbus or not. The star belongs to the galaxy or not. The universe is opened or closed. At the conceptual level probability theory has filled the universe with the undefined, unobservable gas of "randomness." In practice it has made scientists draw more, not fewer, lines between things and nonthings. Probability has turned modern science into a truth casino.

The practice began almost a half millennium ago when scientists worked out the first probability math from examples of gambling, from games of chance with artificially precise rules and cutoff lines. Two or three centuries later scientists applied probability to disease and death statistics of city populations to give mathematical rebirth to the insurance industry. Either you had the disease or not, were married or not, were over 20 or not, were over the poverty line or not, were dead or not. Today commanders launch attacks or not depending on kill probabilities. Probability has proved a powerful tool for social prediction and control. But I could not see how it softened the mismatch between logic and fact.

Say you park your car in a parking lot with 100 painted parking spaces. The probability approach assumes you park in one parking space and each space has some probability that you will park in it. All these parking-space probabilities add up to 100%o. If the parking lot is full, there is zero probability that you will park in it. If there is only one empty parking space, say the thirty-fourth space, you will park there with 1005% probability. If the parking lot is empty, and if we know nothing else about the parking lot, you have the same slim chance, 1%, of parking in any one of the parking spaces.

The probability approach assumes parking in a space is a neat and bivalent affair. You park in the space or not, all or none, in or out. A walk through a real parking lot shows otherwise. Cars crowd into narrow spaces and at angles. One car hogs a space and a half and sets a precedent for the cars that follow. To apply the probability model we have to round off and say one car per space.

Up close things are fuzzy Borders are inexact and things coexist with nonthings. You may park your car 90% in the thirty fourth space and 10% in the space to the right of it, the thirty fifth space. Then the statement "I parked in the thirty-fourth parking space" is not all true and the statement "I did not park in the thirty-fourth parking space" is not all false. To a large degree you parked in the thirty-fourth space and to a lesser degree you did not. To some degree you parked in all the spaces

But most of those were zero degrees. This claim is fuzzy and yet more accurate. It better approximates the "fact'' that you parked in the thirty-fourth parking space.

I found a different fuzzy example as I sat in the philosophy classroom. The professor asked a question. I do not remember the question but I remember resenting the suggestion that either you know the answer or not and, if you know it, you raise your hand and in due course state the correct answer. Children first meet this bivalent filter when they go to preschool or kindergarten. Know it or not. Hands up or down. Answer right or wrong. Put up or shut up. I felt I knew a partial answer to the question. I did not know the answer all or none with a probability to decide which. I had just been studying multivalued or fuzzy logic and so I thought it made sense to raise my hand only part way up to show the degree to which I knew the answer. The innovation failed and the professor called on me to answer all or none.

I have since used this trick on audiences to show them a "real" fuzzy set. How many of you are male? Raise your hands. Male hands go up and female hands stay down. That gives a set and it is not fuzzy. Aristotle's A OR not-A still holds. How many of you are female? Raise your hands. The reverse happens and again the audience splits into two black-white sets, males and nonmales or females and nonfemales.

Then comes a harder question: How many of you are satisfied with your jabs? The hands bob up and down and soon come to rest with most elbows bent. A confident few point their arms straight up or do not raise them at all. Most persons are in between. That defines one fuzzy set, the set of those satisfied with their jobs, the happily employed. Now hands down. How many of you are not satisfied with your jobs? Many of the same hands go up again and bob and come to rest with elbows bent. This defines another fuzzy set, the unhappily employed, the opposite or negation of the first fuzzy set. A AND not-A. Now to some degree the Buddha's law holds. Fuzzy logic is reasoning with fuzzy sets.

The job sets differ from the male-female sets. The set of males does not intersect the set of females. No one is both male and female (in most audiences). Everyone is either male or female: A OR not-A. But most people are both satisfied and not satisfied with their jobs: A AND not-A. Few are 100% satisfied or 100% unsatisfied.

The audience example shows the essence of fuzziness: fuzzy things resemble fuzzy nonthings. A resembles not-A. Fuzzy things have vague boundaries with their opposites, with nonthings. The more a thing resembles its opposite, the fuzzier it is. In the fuzziest case the thing equals its opposite: the glass of water half empty and half full, the liar from Crete who says all Cretans lie and who both lies and not-lies, the borderline customer as satisfied as she is not satisfied. Here yin equals or balances yang, as in the ancient Taoist symbol (Figure 1.1).

The yin-yang symbol is the emblem of fuzziness. It stands for a world of opposites, a world we often associate with Eastern mysticism. The yin-yang symbol adorns the flag of South Korea. In Southern California it signifies a surf club.

For much of my youth I wrestled with this apparent mysticism in a world that science had whitewashed and blackwashed. Scientists had rounded off gray things to white and black things and then forgot about the rounding off and saw only a world of whites and blacks. The world is so much simpler if you can always cut the universe in exactly two pieces--if A OR not-A always holds. Modern scientists and philosophers would put a 1 or a 0, TRUE or FALSE, next to every sentence in this book,

Instead of a truth fraction somewhere in between. The men and women of science beg the question of bivalence, assume the point at issue, climb the ladder of bivalence and forget they stand on it. The practice looks far more like religion than science- They turn their assumption of bivalence into an entrance exam and fail those who dissent, and they banish them with all the intimidation modern science can marshal: sloppy reasoning, not rigorous enough, unscientific measurement, untrained eye, poor experimental design, won't fit in a computer, commonsensical, folk psychology, would know better if you knew more math.

I had lost my faith in establishment science and found myself in a type of reverse atheism. I had passed the bivalent entrance exams but in my heart and head still failed them. I learned how to apply the rules of science but did not believe they were true. I learned how to manipulate probability but did not believe it existed.

- Most of all the black-and-white world of science struck me as unreasonable, as when a zealous prosecutor or judge applies the letter and not the spirit of the law and you end up in jail if you spit on the sidewalk or deduct a nonbusiness dinner on your tax forms or mail-order the wrong magazine. Language, especially the math language of science, creates artificial boundaries between black and white. Reason or common sense smoothes them out. Reason works with grays.

I looked for an alternative that could challenge bivalent science on its own terms. If science rests on math, so should the alternative. Criticism fails without a working alternative Fuzzy logic provided that alternative. It had the same math flavor that probability had, it worked with percentages between 0% and 100%, but it described events happening to some degree, not whether "random'' events happened all or none. You paint one picture of the world if you say there is a 50% chance that an apple sits in the refrigerator. You paint a different picture if you say half an apple sits in the refrigerator. Same number, different worlds.

I pursued fuzziness through the channels of science and academia. i read and wrote papers on fuzziness, gave lectures and taught on it and videotaped new courses and seminars on it, sneaked it into the probability courses I taught at USC, helped organize conferences on it in the United States and Japan, wrote a textbook on it. I had to know if fuzziness existed. I was like a theist who had to know if God exists. If so, I would be a priest. If not, I would join the ranks of the opposition as an atheist zealot.

I looked for fuzziness and found it in a family of new math theorems, all housed in the geometry of a Rubik's cube, as we shall discuss. The math was so easy I could not believe someone else, everyone else, had not seen it. But soon I saw why even earlier fuzzy theorists might overlook this species of math or why they might see it as simple error if they looked at it at all. lt involved strange notions like the whole contained in the part, big things stuck inside smaller things. I could understand why Western scientists did not want to turn fuzziness loose in the pristine world of black-and-white absolutes that they had built up over centuries. I could understand their fear of contradiction, their manic reactions to thing and nonthing, to A AND not-A.

I could understand the cultural prejudice and emotional reactions, all dressed up in the technical language and mannerisms of science, but I could not forgive them. Fuzziness solved age old paradoxes of Western thought and opened new doors through mathematical infinity as it reduced black-and-white math to a special case of gray.

Most of all fuzziness made machines smarter. It increased the machine IQ of dozens of products in consumer electronics and manufacturing: cameras, camcorders, TVs, microwave ovens, washing machines? vacuum sweepers, transmissions, engine control, subway control. But it increased machine IQ in the land of A AND not-A, in the Far East, in Japan, where in the early 1990s fuzzy logic took hold inside and on TV sets, with even newscasters and politicians debating the meaning of fuzziness. Surely, I thought, money talks to scientists, since money drives everything in science and academia. But Western scientists and engineers only threw stones and we-could-do-it-toos at news of the fuzzy commercial successes in Japan. Earlier they had attacked fuzzy theory as lacking applications. Now they attacked the applications as lacking theory.

While Western scientists and engineers ignored or attacked fuzzy logic, their Eastern counterparts eagerly applied it and launched the long-awaited era of commercial machine intelligence. I often found myself reviled by Western scientists, especially the senior ones, including those in my own engineering department at USC--the grayer the hair, the more the reasoning seemed to be black and white. But in Japan I signed autographs and chaired conferences and waved at TV cameras. By the time we fuzzy theorists threw the first U.S. fuzzy conference in Austin, Texas, in June 1991 (at MCC or the Microelectronics and Computer Technology Corporation), the Japanese had already passed the $1 billion mark in annual sales of fuzzy products and taken another leap forward in their world leadership of consumer electronics and high-tech engineering and manufacturing. Cultural preferences come with costs.

Here is an IQ test for skeptics of fuzzy ideas: Can you explain how a fuzzy chip works? In the world of technology that question puts people into two nonfuzzy classes, the knows and the know-nots. Information does not respect rank or possessions or wrinkles. Logic helps people manage that information. Fuzzy logic helps machines manage it.

I don't think any human brain works with Aristotle's syllogisms or with computer precision. It's messier than that. The days of symbolic reasoning in "artificial intelligence" computer programs are over. They got unplugged with Hal the computer in the 1968 movie 2001: A Space Odyssey. When Arnold Schwarzenegger's cyborg in Terminator II tells us it can learn new behavior because "My CPU is a neural-net processor, a learning computer," it does not mean Artistotle in a box. As we shall see, Aristotle ends up not in a box but at the corners of a fuzzylogic cube, the rare moments of black and white in a world of gray.

If our reasoning has logic, it's fuzzy at best. We have only one decision rule: 1 do it if it feels right. The formal logic we first learn in tenth-grade geometry class has little to do with it. That's why we made it to tenth grade.

Fuzzy logic begins where Western logic ends.