Tuesday, May 08, 2012

Lotfi Zadeh on Misconceptions about Fuzzy Logic

Lotfi A. Zadeh Mon, May 7, 2012 at 8:45 PM

Reply-To: bisc-group@lists.eecs.berkeley.edu
To: bisc-group@lists.eecs.berkeley.edu
Berkeley Initiative in Soft Computing (BISC)

Dear members of the BISC [Berkeley Initiative on Soft Computing] Group:

Sometimes labels are misleading. This applies to fuzzy logic. There are many kinds of logical
systems, some going back to antiquity and some of recent vintage. Among them are: 
Aristotelian logic, modal logic, deontic logic, multivalued logic, dynamic logic, probabilistic logic,
etc. A common misconception is that fuzzy logic is a member of this list. This is not the case.
Fuzzy logic is much broader than a logical system. Basically, fuzzy logic, FL, is a system of 
reasoning, modeling and computation. FL has four principal facets. First, the relational facet,
FLr. This facet is centered on fuzzy relations, fuzzy if-then rules, fuzzy systems analysis and 
fuzzy decision-analysis. Most applications of fuzzy logic relate to this facet. The most visible 
application area is fuzzy control. Second, the epistemic facet, FLe. This facet is concerned 
with knowledge representation, linguistic variables, possibility theory, search and natural 
languages. Third, the fuzzy-set-theoretic facet, FLs. This facet is focused on the theory of 
fuzzy sets. Fourth, the logical facet, FLl. In this facet, and only in this facet, fuzzy logic is 
viewed as a logical system. To differentiate between FL and FLl, FL and FLl are referred to 
as fuzzy logic in a wide sense and fuzzy logic in a narrow sense, respectively. Today, when 
we discuss fuzzy logic, it should be understood that we are talking about fuzzy logic in a wide 
sense, unless stated to the contrary. Equating FL to its logical facet, FLl, is a common 
misconception. This misconception is a source of a great deal of misunderstanding about 
what fuzzy logic is and what it has to offer.

In science and engineering, precision is respected and imprecision is not. What is widely
unrecognized is that in many important applications of fuzzy logic imprecision is deliberate.
The underlying rationale is the following. In most real-world problems there is some tolerance
for imprecision. Fuzzy logic exploits this tolerance for imprecision through the use of words in
place of numbers. Resulting in lower costs and greater simplicity. This is a key idea which
underlies Computing with Words (CWW). This idea is one of the most important features of
fuzzy logic, and it is unique to fuzzy logic.

In CWW, concepts and techniques drawn from the realm of natural languages play an
important role. Natural languages are intrinsically imprecise. In CWW, the accent is on
problem-solving rather than on axiomatics and precisely defined concepts. There is a rationale
for this attitude -- a rationale which is embodied in the Impossibility Principle. Briefly, the
Impossibility Principle states that as the complexity of a system increases, a point is reached
beyond which precision and relevance become incompatible. An example which I employed in
my earlier messages, March 23, 24 and April 4, 2011, is the taxicab problem. The taxicab
problem is a convenient platform for introduction of two basic concepts--the concepts of
p-validity (provable validity) and f-validity (fuzzy validity).

I hail a taxicab and ask the driver to take me from address A, where I am, to address B. There 
are two versions: (a) I ask the driver to take me to B the shortest way; and (b) I ask the driver 
to take me to B the fastest way. Abstractly, the street map is assumed to be a graph, and the 
problem is to move from node A to node B. Each link (block) is assumed to be associated with 
a constant, l, the length of the link, and a random variable, t, the traversal time. The traversal 
time, t, is assumed to depend on the time at which the taxicab enters the link.

Version (a) has a p-valid solution. The route that the driver chooses is an f-valid solution. 

Version (b) has an f-valid solution which is the route that the driver takes. Version (b) does not 
have a p-valid solution because we have no way of minimizing the sum of not-well-defined 
random variables. In summary, Version (a) is a tractable problem whereas Version (b) is an 
intractable problem.

An analogy is helpful. Assume that I want to reach the peak of a mountain. I start by driving a 

car toward the mountain. At some point, I cannot proceed further because of rough terrain. To 
proceed further, I use a mule. Eventually, I reach a point beyond which I have to proceed on 

Using a car in the first leg of my trip is analogous to the use of tools which are provided by 

traditional bivalent-logic-based mathematics. Classes are assumed to be crisp, that is, have 
sharp boundaries. Let us refer to the tools which I use as Modality 1. The second leg is 
analogous to the use of tools based on fuzzy logic. Classes are assumed to have unsharp 
boundaries which are precisely defined via membership functions. Broadly speaking, we 
employ what may be labeled fuzzy mathematics. Call it Modality 2. In the third leg, the 
machineries of traditional mathematics and fuzzy mathematics cease to be effective. Classes 
have unsharp boundaries which are not precisely defined. This is the world of everyday 
reasoning. What we employ may be viewed as quasi-mathematics--a kind of mathematics 
which I describe very briefly in my 2009 note on "Toward Extended Fuzzy Logic--A First Step," 
Fuzzy Sets and Systems 160, 3175-3181. Call this Modality 3. The taxicab problem, Version 
(b), falls within Modality 3.

Given a real-world problem, P, what modality does it fall into? The answer depends on how 

P is modeled. Idealization of an intractable problem may make it a tractable problem. This is 
common practice when we are faced with an intractable problem which we want to solve 
through the use of traditional mathematics.

    What I said above carries an important message. You should not assume that every problem 

that we are faced with falls into Modality 1, that is, can be solved through the use of traditional 
mathematics. Rigor and precision carry a price.



Lotfi A. Zadeh 
Professor Emeritus
Director, Berkeley Initiative in Soft Computing (BISC) 

729 Soda Hall #1776
Computer Science Division
Department of Electrical Engineering and Computer Sciences
University of California 
Berkeley, CA 94720-1776 
Tel.(office): (510) 642-4959 
Fax (office): (510) 642-1712 
Tel.(home): (510) 526-2569 
Fax (home): (510) 526-2433 
URL: http://www.cs.berkeley.edu/~zadeh/

BISC Homepage URLs
URL: http://zadeh.cs.berkeley.edu/


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