Professor in the Graduate School, Computer Science Division
Department of Electrical Engineering and Computer Sciences
University of California
Berkeley, CA 94720 -1776
Director, Berkeley Initiative in Soft Computing (BISC)

Address:
Computer Science Division
University of California
Berkeley, CA 94720-1776
zadeh@cs.berkeley.edu
Tel.(office): (510) 642-4959
Fax (office): (510) 642-1712
Tel.(home): (510) 526-2569
Fax (home): (510) 526-2433
Fax (home): (510) 526-5181

Before we can go and try to understand fuzzy logic, we first need to have a general understanding about logic. Let us start by trying to understand the basic concept behind logic. Well, according to the Dictionary of Mathematics, logic simply put, is the study of deductive arguments. The central concept in logic revolves around what is known as a valid argument that is made up of two parts a premise and a conclusion. In a valid argument, if the premise is true then the conclusion must also be true. This concept means that the conclusion logically follows from the premise. For example, if X is a man then X is mortal; Socrates is a man therefore Socrates must be mortal. It logically follows if A then B; A is true then B must be true and this is known as Modus ponens. According to the Handbook of Mathematics and computational science, this kind of logic is known as classical logic where a statement can only have two truth values (true=1 or false=0). Therefore, from this statement, we can clearly see that in classical logic a statement can be either true or false. For example, suppose X represents the set of even integers, now given a number y we can say that y is a member of X only if y is even. This concept is known as crisp logic. It is discrete and exact.

We can see how this kind of two-value logic or switching algebra can represent discrete concepts. However, what happened when we have a concept that is not so clearly defined. For example suppose, we want to represent the set of tall men, how do we go about doing that in term of logic? In classical logic, there is no confusion about what is true and what is false. We mutually agreed that X is an integer if and only if X is any of the positive or negative whole numbers. Therefore, if X is a fraction we know that X is not a member of the set of integers. However, we have no clearly mutually agreed value to denote a tall person. For example, a 4 feet person will say that a tall person is some one taller than 5 feet and again a 5 feet person may say that tall people are taller than 6 feet. So we can clearly see the ambiguity in this kind of statement. How do we get rid of this ambiguity? There is no definite answer for this question. How do we represent this concept in term of classical, crisp logic?

It is evident that we cannot represent a continuous changing concept in term of just two-truth value, true or false. In order to make up for that Zadeh introduces what is known as multi-valued logic whose algebraic operations are defined by mathematical operators. Multi-valued logic or fuzzy logic can now denote sets where members have truth-values from the closed interval [0,1]. In this case, a member with a truth-value of zero is absolutely false, which means it is not a member of the set. This concept is what is known as fuzzy set, that is a fuzzy set A which is a subset of a given fundamental set G whose elements are contained in A only to a certain degree (Handbook of Mathematics and computational science). Now using this concept of fuzzy logic we can remove the ambiguity that exists when we say that a person is tall. For example, in term of fuzzy logic when we say that a person is tall we mean that the person is a member of the set of tall people to a certain degree.

This is the concept of fuzzy set that was introduced in 1965 by Lotfi A. Zadeh. In fuzzy set, the members of the set are allowed to have membership value indicated by a value between 0 and 1. In fuzzy set we have many degrees of memberships. We can represent statement that are not entirely true and those that are not entirely false. This concept is very important if we want to write program that emulate humans reasoning, because human can work with incomplete data and uncertain data. We can formally state this fact like so:

U_{A (X):} X à [0,1]

This read as a membership function maps every element of the universe of discourse X to the interval [0,1]. In general, each member of a fuzzy set has a membership function. So now we can say that crisp set is a fuzzy set where its members can have a membership function of {0,1}.

Membership function y(x) in a classical set theory assumes only the values 0 and 1:

Membership function y(x) in fuzzy set assumes values be values between 0 and 1:

U_{A (X):} X à [0,1]

We can say that the fuzzy subset A of a set X is characterized by its membership function:

A = {U_{A(x)/x | x Î} X}

Using Fuzzy Methods, linguistic problems can be transformed to algorithmic mathematical expressions. Linguistic elements can be represented in fuzzy sets by means of characteristics or graph (Handbook of mathematics and computational science). In fuzzy linguistic, we can use linguistic variable and linguistic term to represent well defined linguistic statements. For example we can now explain term such as ‘hot’, ‘low’, ‘high’,’ medium’ and soon.

John W. Harris and Horst Stocker @1998 Springer-Verlag New York Inc.

Lefteri H. Tsoukalas and Robert E. Uhrig @1997 John Wiley & sons, Inc.

Edited by David Nelson @1989,1998 Penguin Books Ltd

Discrete and Combinatorial Mathematis 3ed

Ralph P. Crimaldi @1994 Addison-wesley Publishing Company Inc.

http://www.mathworks.com/products/fuzzylogic/zadeh.shtml

http://www.cs.berkeley.edu/People/Faculty/Homepages/zadeh.html