Fuzzy Rudolf Kalman UFlorida “Fuzziness is probability

logic was introduced in 1965 by Lofti A Zadeh in his paper “Fuzzy
Sets”. Zadeh and others continued to develop fuzzy logic at that time. The
idea of fuzzy sets and fuzzy logic were not accepted well within academic
circles because some of the underlying mathematics had not yet been explored. The
applications of fuzzy logic were slow to develop because of this, except in the
east. In Japan specifically fuzzy logic was fully accepted and implemented in
products simply because fuzzy logic worked, regardless of whether
mathematicians agreed or not. The success of many fuzzy logic based products in
Japan in the early 80s led to a revival in fuzzy logic in the US in the late

of the objections that faced fuzzy logic in its early days are shown below:

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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 UFlorida

is probability in disguise. I can design a controller with probability that
could do the same thing that you could do with fuzzy logic.” -Professor
Myron Tribus, on hearing of the fuzzy-logic control of the Sendai subway system
IEEE Institute, may 1988.

The theory of
fuzzy sets has advanced in a variety of ways and in many disciplines, since its
inception in 1965. Application of this theory can be found in artificial
intelligence, computer science, machine learning, robotics, pattern recognition
etc. Having capability to create algorithm which often can mimic the human
cognitive abilities like learning, reasoning, decision making and tolerance to
uncertainty due to vagueness, ambiguity, imprecision have paved the way of the
fuzzy system to be better suitable for the situation in which uncertainty
occurring in the system that cannot be accommodated for the rigorous
computation for modeling.

Let’s look at
the kind of problems in the history of mankind which paved the way for
development of fuzzy theory. The problems of uncertainty, imprecision and
vagueness have been discussed for many years. These problems have been major
topics in philosophical circles with much debate, in particular, about the
nature of vagueness and the ability of traditional Boolean logic to cope with
concepts and perceptions that are imprecise or vague. The Fuzzy Logic can be
considered a Multi-valued Logic, MVL. It is founded on, and is closely related
to-Fuzzy Sets Theory, and successfully applied on Fuzzy Systems. It may be
thought that fuzzy logic is quite recent and what has worked for a short time,
but its origins date back at least to the Greek philosophers and especially
Plato (428-347 B.C.). It even seems plausible to trace their origins in China
and India. It is because it seems that they were the first to consider that all
things need not be of a certain type or quit, but there are a stopover between.
That is, be the pioneers in considering that there may be varying degrees of
truth and falsehood. In case of colors, for example, between white and black
there is a whole infinite scale: the shades of gray. Some recent theorems show
that in principle fuzzy logic can be used to model any continuous system, be it
based in artificial intelligence or physics, or biology, or economics, etc.

development has advanced to a very high standard. The basic idea underlying all
these approaches is that of an intrinsic dichotomy between true and false. This
opposition implies the validity of two fundamental laws of classical logic: –
Principle of excluded middle: Every proposition is true or false, and there is
another possibility. – Principle of non-contradiction: No statement is true and
false simultaneously. This basic idea generates a series of paradoxes and
dissatisfaction that is based on the need to overcome this strict
truth-bivalence of classical logic. Accept that a proposition about a future
event is true or false becomes necessary or impossible, respectively, the event
expressed by the proposition. The solution proposed by Jan Lukasiewicz himself
in his classic 1920, is the acceptance of a logic with three truth values (or
three-valued), also called trivalent), which in addition to true and false,
accepts a value of indeterminate truth, which is ascribed a truth value or
grade of membership of 0.5. In the eighteenth century, David Hume (1711-1776)
and Immanuel Kant (1724-1804) were inquiring about such concepts. They
concluded that the reasoning is acquired through experiences throughout our
lives. Hume believed in the logic of common sense, and Kant thought that only
mathematicians could provide clear and precise definitions, both accepting that
there were conflicting principles that had no solution. In conclusion, both
were detecting conflicting principles within the so-called classical logic.
Then in the early twentieth century, the British philosopher and mathematician
Bertrand Russell reported the idea that classical logic inevitably leads to
contradictions. Also Charles Sanders Peirce (1839-1914) somewhat anticipated
this, but there are many who, like Bart Kosko, Bertrand Russell considered the
father of Fuzzy Logic. Because in the early twentieth century, the British
philosopher and mathematician Bertrand Russell (1872-1970) reported the idea
that classical logic inevitably leads to contradictions, making a study on the
vagaries of language, and concluding that the vagueness is precisely one
degree.   According to this theory, we
have a transfer function derived from the characteristic function usually
called the “membership function”, which runs from the universe of discourse, U,
until the unit closed interval of reals, which is 0, 1. Not so in the sets
“classic” or “crisp sets”, where the range of the function is reduced to a set
consisting of only two elements, namely was the {0, 1}. Therefore, fuzzy set
theory is a generalization of classical set theory. The theory of “vague sets”
(today, so-called Fuzzy Sets) proceeds from the quantum physicist and German
philosopher Max Black (1937) analyzed the problem of modeling “vagueness”. He
differs from Russell in that he proposes that traditional logic can be used by
representing vagueness at an appropriate level of detail and suggests that
Russell’s definition of vagueness confuses vagueness with generality. He
discusses vagueness of terms or symbols by using borderline cases where it is
unclear whether the term can be used to describe the case. When discussing
scientific measurement he points out “the indeterminacy which is characteristic
in vagueness is present also in all scientific measurement”. To the fuzzy logic
researcher of today these curves bear a strong resemblance to the membership
functions of (type-1)-fuzzy sets. At the beginning of its brainstorm, the
papers published by Lotfi A. Zadeh was not well received in the West, even in
many cases were bitterly dismissed by the more conservative elements of the
scientific community, as mentioned before. However, over time began to gain
enough supporters, which led to these theories were being extended again and
again, settling firmly among the most innovative scientists, and especially
among the best professionals, more than anywhere else, initially in Japan and
then South Korea, China and India. Europe and the States have been incorporated
into this new math, but more slowly. As a matter picturesque, if you will, but
true, we can tell that the now recognized by many as “the father of Fuzzy
Logic”, Lotfi A. Zadeh, in his time met with executives from IBM, which told
him that his “discovery” had no interest or no utility. Of course, it will be
considered a very clear model of intelligence and vision.

Some Important Land
marks in the Advancement of Fuzzy theory




Zadeh introduces the idea of Type-n fuzzy sets and, therefore,
Type-2 fuzzy sets. 1975. Zadeh presents the definition of Type n Fuzzy Sets.


Grattan-Guinness presents the notion of Set Values Fuzzy Sets
as well as some operations based on previous developments for many-valued algebras.


Atanassov presents the definition of Atanassov Intuitionistic
Fuzzy Sets


Dubois and Prade introduce the definition of Fuzzy Rough Sets


Zhang presents the definition of Bipolar Valued Fuzzy Sets of


Liang and Mendel introduce the idea of Interval Type-2 Fuzzy


Lee introduces a new concept with the name of bipolar-valued
Fuzzy Sets.


Smaradache introduces the concept of Nutrosophic sets.


Yager gives the idea of Pythagorean Fuzzy Sets.


Mesiarova-Zemankova et
al. present the concept of m-Polar Valued Fuzzy Sets.



analysis of system reliability often requires the use of subjective judgements,
uncertain data and approximate system models. By allowing imprecision and
approximate analysis fuzzy logic provides effective tools for characterizing
system reliability.  Indeed, the applications of fuzzy logic,
once thought to be an obscure mathematical curiosity, can be found in many
engineering and scientific works.

Fuzzy logic has been used in numerous
applications such as facial pattern recognition, air conditioners, washing
machines, vacuum cleaners, antiskid braking systems, transmission systems,
control of subway systems and unmanned helicopters, knowledge-based systems for
multi-objective optimization of power systems, weather forecasting systems,
models for new product pricing or project risk assessment, medical diagnosis
and treatment plans, and stock trading. Fuzzy logic has been successfully used
in numerous fields such as control systems engineering, image processing, power
engineering, industrial automation, robotics, consumer electronics, and
optimization. This branch of mathematics has instilled new life into scientific
fields that have been dormant for a long time. One of the most famous applications of fuzzy logic is that of the
Sendai Subway system in Sendai, Japan. This control of the Nanboku line,
developed by Hitachi, used a fuzzy controller to run the train all day long.
This made the line one of the smoothest running subway systems in the world and
increased efficiency as well as stopping time. This is also an example of the
earlier acceptance of fuzzy logic in the east since the subway went into
operation in 1988. In “Detection and
elimination of a potential fire in engine and battery compartments of hybrid
electric vehicles” by M. S. Dattathreya et al, the authors present a novel
fuzzy deterministic non-controller type (FDNCT) system and an FDNCT inference
algorithm (FIA). The FDNCT is used in an intelligent system for detecting and
eliminating potential fires in the engine and battery compartments of a hybrid
electric vehicle. They also present the simulation results of the comparison
between the FIA and singleton inference algorithms for detecting potential
fires and determining the actions for eliminating them. In “Comparison of
detection and classification algorithms using boolean and fuzzy techniques” by
R. Dixit and H. Singh, the authors compare various logic analysis methods and
present results for a hypothetical target classification scenario. They show
how preprocessing can reasonably preserve result confidence and compare the
results between Boolean, multi-quantization Boolean, and fuzzy techniques.

In conclusion, a word about the methodology of computing
with words (CWW) which is rooted in the concept of a linguistic variable. CWW
opens the door to construction of mathematical solutions of computational
problems which are stated in natural language. In coming years, CWW is likely
to play an increasingly important role in origination and development of
real-life applications of fuzzy logic.