**Fuzzy what ??!!**

Fuzzy logic is a superset of conventional(Boolean) logic that has been extended to handle the concept of partial truth- truth values between "completely true" and "completely false". As its name suggests, it is the logic underlying modes of reasoning which are approximate rather than exact. The importance of fuzzy logic derives from the fact that most modes of human reasoning and especially common sense reasoning are approximate in nature.

**Boolean vs. Fuzzy**

300 years B.C., the Greek philosopher, Aristotle came up with binary logic(0,1),
which is now the principle foundation of Mathematics. It came
down to one law: A or not-A, either this or not this. For example, a typical
rose is either red * or * not red. It cannot be red *and* not red.
Every statement or sentence is true or false or has the truth value 1 or 0. This is Aristotle's law of bivalence and was philosophically correct for over two
thousand years.

Two centuries before Aristotle, Buddha, had the belief which contradicted
the black-and-white world of worlds, which went beyond the bivalent cocoon and see the
world as it is, filled with contradictions, with things *and * not things. He stated that a rose, could be to a certain degree completely red, but at the same time
could also be at a certain degree not red. Meaning that it can be red * and
* not red at the same time. Conventional(Boolean) logic states
that a glass can be full *or *not full of water. However, suppose one were to fill
the glass only halfway. Then the glass can be half-full* and * half-not-full. Clearly,
this disprove's Aristotle's law of bivalence.
This concept of * certain degree* or multivalence
is the fundamental concept which propelled Zader Lofti of University
Berkely in the 1960's to introduce fuzzy logic. The essential characteristics
of fuzzy logic founded by him are as follows.

- In fuzzy logic, exact reasoning is viewed as a limiting case of approximate reasoning.
- In fuzzy logic everything is a matter of degree.
- Any logical system can be fuzzified
- In fuzzy logic, knowledge is interpreted as a collection of elastic or, equivalently , fuzzy constraint on a collection of variables
- Inference is viewed as a process of propagation of elastic constraints.

**Fuzzy Subset Theory**

There is a strong relationship between Boolean logic and the concept of a subset. Similarly there is a strong relationship between fuzzy logic and fuzzy subset theory.

*
young(x) = { 1, if age(x) <= 20.,*

*
(30-age(x))/10, if 20 < age(x) <= 30,*

Given this definition, here are some example values:

Person Age degree of youth -------------------------------------- Johan 10 1.00 Edwin 21 0.90 Parthiban 25 0.50 Arosha 26 0.40 Chin Wei 28 0.20 Rajkumar 83 0.00So given this definition, we'd say that the degree of truth of the statement "Parthiban is YOUNG" is 0.50.

Note: Membership functions almost never have as simple a shape as age(x). They will at least tend to be triangles pointing up, and they can be much more complex than that. Furthermore, membership functions so far is discussed as if they always are based on a single criterion, but this isn't always the case, although it is the most common case. One could, for example, want to have the membership function for YOUNG depend on both a person's age and their height (Arosha's short for his age). This is perfectly legitimate, and occasionally used in practice. It's referred to as a two-dimensional membership function. It's also possible to have even more criteria, or to have the membership function depend on elements from two completely different universes of discourse.

**Fuzzy Logic Operations**

It is clear what the statement * X is LOW *
means in fuzzy logic. But, how do we interpret a statement like

*
X is LOW and Y is HIGH or (not Z is MEDIUM)
*

The standard definitions in fuzzy logic as suggested by Lotfi are:

**References**

1. Eric Horstkotte on Fuzzy Logic

http://www.quadralay.com/www/Fuzzy/overview.html

Author: Peter Baur, Stephen Nouak, Roamn Winkler

2. Fuzzy Systems-A Tutorial

http://www.quadralay.com/www/Fuzzy/tutorial.html

Author: James F. Brule'

3. Fuzzy Thinking

Author: Bart Kosko

4. IEEE Transactions On Knowledge and

Data Engineering: Knowledege Representation in Fuzzy Logic.

Author: Zader Lofti