A new logic and more realistic numbers: Fuzzy Logic

Mathematicians have always relied on the conventional mathematical logic, which is the same logic studied and developed by the philosophers through centuries. The classical logic depends on some basic fundamentals, but the most important ones are:

1- Each statement must be either true or false, and there is no third option.

2- A statement cannot be both true and false at the same time.

For instance: The statement “Jordan is an Arabic country” is true, but the statement “Earth and Mars are in different solar systems” is false.

Similarly, we say that the mathematical statement “1=2” is false, whereas “2>1” is true.

Each of these statements must be one of two options: true or false, and there is no compromise or any other options.

The problem emerges when we try to judge a statement such as “Zaid is tall”. Here, we cannot say whether this piece of information is true or false firmly. Zaid might be tall, according to the average of people’s heights in some country, but short in another. Moreover, one can be “tall a little bit”, while another can be “taller”, and some people are really “towering”. Therefore, classifying people as “tall” or “not tall” is not realistic.

The same discussion can be held for the statement “The forest is large”, since a forest can be relatively large in comparison with the surrounding forests, but its area may form only a small proportion of the forests’ areas in another continent. Even in the same continent, there is a “large” forest, and a “larger” one. So, I cannot deal with such concept with the true-or-false logic.

Many other examples, such as “The ruler is just”, “The landscape is beautiful” or “Amr is a lovely guy”, indicate that there are some concepts in our lives that we cannot deal with using the conventional logic.

So, must the judgment be relative? The word “relative” may not be the correct word. Actually, these concepts are called “fuzzy”. To illustrate this word better, let’s talk about some mathematical problems that cannot be handled well using the known logic.

If I told you that “I want the profits of the company to be 7 thousand”, and you wanted to represent this number on the real line, then you can simply draw the line, put a point at the number 7 (ignoring the thousand for convenience), and go for this goal.

However, what if I told you that “I want the profits of the company to be about 7 thousand”? Here, I want to give you a kind of flexibility, since there is no need for the profits to be exactly 7 thousand. They can be slightly more or less, but I definitely don’t want them to be neither much less nor much more because -for example- I would have to pay more taxes. Can you now represent the needed profits on the real line similarly? Of course not! You cannot put one single point, and not even a number of points.

You may ask: “Why don’t we take an open interval around the number 7 into account, and consider every point in this interval to be “about 7” and the other points to be “not about 7”? This way, we can make our goal to reach a profit of one of the numbers inside the interval. Isn’t this a good solution?”

Here, you are dealing with the problem using the usual logic: Either “about 7” or “not about 7”. It’s OK. I will show you the problem, but let me first ask you: “How long will you choose the interval?”. And let’s suppose that we agreed to set the interval to be (6,8) as represented in the previous graph.

Now, I have some questions for you:

First: Isn’t the number 6.8 “about 7” more than the number 6.2? But by using the interval, you accepted both of them with the same degree.

Second: Why would I accept the number 7.99 to be “about 7”, and refuse the number 8.01?! Aren’t they so close to each other? Is this slight difference that worthy to take some important decisions in the company?

Third: Is it right to say that the number 9 is “not about 7” just as the number 1000?! Is this logical? Isn't the number 1000 much further from 7 in comparison to the number 9?

Although your assumption is quite right from a mathematical point of view, and you definitely can choose the interval you want and continue your work, you may get results, but are they realistic?

The three questions above, which are totally logical questions that our minds will accept immediately, imply that there is a gap between mathematics and our intuition. But how can we represent this vague number mathematically and solve these problems?

If you keep thinking with the classical logic -to be or not to be-, you won’t find a way. This is due to that the known logic is unable to find a realistic representation of the problems that have unclear boundaries. Don’t be afraid, someone found a way!

It is not strange if you couldn’t think of a way to solve the problem; an accepted mathematical representation was not found until 1965 CE through a research paper named “Fuzzy Sets” written by Zadeh, the founder of Fuzzy Sets. This new fuzzy way of thinking is still considered as a new discipline of mathematics, and there is a big number of the universities around the world that do not teach it.

Zadeh considered some words that we use in our daily life, such as “about”, “almost”, “more than” and “much less than”, as fuzzy words. And if we wanted to be more realistic, we can’t deal with such words as we do with the deterministic concepts and words.

In fact, our lives and thoughts are built on fuzzy concepts and stuff, there is not an only one correct way to do something, and there isn’t -in general- a concept that is not controversial. There is always a proportion of both right and wrong about everything.

Now, to understand the meaning of the “fuzziness” more, let’s think of a simple way to solve the previous problem about representing the number “about 7” on the real line, without taking too much care of its correct mathematical representation.

Let’s imagine a foggy area surrounding the number 7, with a high density at 7, that fades away as we move far from it.

test_1.png

We can think of the fog’s density as a criterion that tells us how much the number is “about 7”. This way, if we assume that the density of the fog is 100% at exactly 7, we can choose another real number and measure the degree of density at the corresponding point, then determine how close it is to 7. Here, there is no right nor wrong, the answer becomes a percentage that has an infinite number of choices.

In other words, if I told you: “I wanted the profits of the company to be about 7 thousand, but actually I profited 6 thousand. Is this good or bad?

You cannot answer with only “good” or “bad”, but you may say “It’s OK” or “It is kind of good”. My question was fuzzy, so your answer will definitely be the same. And now if you drew that foggy cloud, you would be able to get a number that illustrates how good my profits are.

One may ask: How should I determine the average in which the fog fades away as we go far from the number 7? Should it fade away quickly or slowly? The answer is: As you like, according to the situation and depending on the realistic data you have.

There is still a very important question here: How can I represent this cloud mathematically and calculate these percentages?

Actually, this is exactly what Zadeh did, and this is what we are going to discuss in the next article.

Follow us if you'd like to find out more about the Fuzzy Sets, Fuzzy Numbers and some of their applications. Is there a relation between the classical and the fuzzy logics?

You will also know the answer of a very common question: Is fuzziness the same as randomness? Why do people get confused about them? And what is the difference?

 

Marwa Tuffaha

marwat.math@gmail.com

 

Reference: Zadeh, L.A., 1965. Fuzzy sets. Information and control, 8(3), pp.338-353.