Encefalus

Lately I have started dabbling with fuzzy logic and fuzzy set theory. You’ll probably start wondering what these are. I won’t try to give you a definition, but rather I’ll give you a few examples to help you understand yourself.

I don’t like the mathematical way that concepts like fuzzy logic are first presented. A mathematician will probably first present you some topics on set theory. However, I prefer the way that an engineer thinks about them. That is, instead of basing their validity on axioms, base it on usefulness. If a tool provides results, then it is useful. Therefore, I will provide you with examples in order to realize why fuzzy logic might not be another theoritical abstraction, but something really connected to the way our brains work.

Not this kind of fuzzy

You’re talking on the phone with a friend and you are trying to arrange what will you do this evening. Your friend tells you he has some work to do and so, he will be out somewhat late.

What does "late" mean?

For some people "late" might mean after 11 p.m. For others it might mean after 10 p.m. And for some people it might mean 1 a.m.

Let’s define late as after 11 p.m. If your friend goes out to have a few beers at 10:55 p.m., would you call him a liar? Of course not. If he did that at 10:30? At 10? At 9? At 7:15p.m.?

This is like the Sorites Paradox.I especially like the "baldness" variation. That is, let’s say that a man loses one hair from his head. Would you call him bald? Deffinitely not. But if he lost 2? How about a 100,or a 1000? Maybe 10.000? What is the exact threshold of "baldness"? Does a number X exists so that everything above X is not-bald and everything below X is non-bald?

Britney Spears: bald or not?

As you see, regular "crisp" logic, can many times create such paradoxes. Most things in life don’t have clearly-cut boundaries. Most things in life are not black and white. This includes both descriptions of physical objects and sizes ("large","small","tall"), as well as abstract notions, such as democracy. How democratic can a democratic state be? It’s something difficult-to-impossible to define.

You might think that these boundaries cannot be simulated adequately mathematically.  However, this is wrong.

All this changed by Lotfi Asker Zadeh,the man who created fuzzy set theory. In traditional logic an element is mapped to a set via a membership function that can give the value 0 (doesn’t belong to the set) or 1(it is part of the set). In fuzzy logic, all membership values are defined in a continuum in the domain [-1,1]. So, element X can have membership 0.7 to the set A.

Lotfi Asker Zadeh

Let’s think about it for a while in real-life. An orange might be 0.7 sweet. A man might have a membership of 0.6 to the set of tall people. This means that an orange is sweet, without being the sweetest thing you can eat, while a man might be slightly above average, but not really tall.

Fuzzy sets are inherent in the way we think and we interact with the world. People are constantly using them without even realizing it. Very few of the notions we use at our everyday lives are crisp. Yet, we get along fine. This may come as a revelation to many people who think that we cannot use a notion if it is not properly defined. I had many conversations where the other person tried to prove that an argument is invalid because a notion could not be described fully and have its boundaries clearly drawn. Yet, you don’t really need to know what peace, nature, or friendship are in order to move around in your everyday life. You can have many friends without really being able to give the definition of  "friendship".

Fuzzy the way you think is, young padawan

Your brain might make complex philosophical arguments of why you cannot define love of friendship, but you socialize nevertheless. This is the important point here. While fuzzy logic might not seem so "logical" to many people at first, it is, nevertheless, working. And this is what proves it right. Some people like to think like mathematicians and convert such a strict way of thinking to other areas of science and life. That is, we start from a set of axioms and reach a set of conclusions. However, life on Earth has evolved through millions of years without using axioms or complex arguments.

Our everyday lives are much fuzzier than we might think at first. Indeed, it could be said that it’s almost impossible to predict most important events that will happen in the future, even if we know that they will happen for sure. We can’t predict our death or the death of others. We can’t predict economic collapses and wars. We can’t predict all those Black Swans Nassim Nicholas Taleb has been talking about. How can we tread these dangerous grounds every day, yet come out victorious? It’s simple. We don’t really try to predict what’s going to happen. We just estimate what’s going to happen. Nevertheless, this proves just to be enough.

I believe that the greatest scientific revolution of the 20th century was the introduction of vagueness and ambiguity in science, through probability theory, statistics, neural networks (in artificial intelligence) and other theories or approaches. The 21st century will be the century where these theories from simple tools, will become the mainstream. Our world is inherently ambiguous; it is inherently fuzzy. Yet, our role as scientists is to try and analyze it. Fuzzy logic is just another tool in the toolbox, but a very good one indeed

This entry was posted on Sunday, January 3rd, 2010 at 12:11 pm and is filed under Cognitive, Complexity, Philosophical. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.