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# Fuzzy Logic

Fuzzy logic can't handle dependent variables correctly.

Fuzzy logic is a refomulation of probability and statisical inference.

## Why fuzzy logic offers nothing new

'The "degree of truth" is in fact a frequency probability -- that
of a competent speaker of the language using the fuzzy descriptor
in question to describe the candidate element which may be in
question, of the fuzzy set which is in some sense induced by the
fuzzy descriptor.

Seen in this way, the membership function is akin to a
likelihood function -- a semantic likelihood function in this
case -- induced by a (measurable, frequentist) uncertainty in the
use of fuzzy terms. In the same way that likelihood -- which
varies over parameter space, as distinct from the sample space
(from which come the data) to which it is related -- is distinct
from, though related to probability, the membership function over
a universe of discourse is not a probability distribution, but it
is related to the sample space of yes/no responses that would be
obtained when asking any speaker whether he/she would use a fuzzy
descriptor (eg. "tall") to describe any particular candidate
element (eg. height value) for a fuzzy subset consistent with the
descriptor in question. It is the collection of such response
probabilities, and related uncertainty of description (also
measurement in general) that determines a semantic likelihood
function (membership function) over the space of hypotheses --
now identified with the universe of discourse over which the
fuzzy subset is defined -- when a fuzzy descriptor is put to
actual use.'

S. F. Thomas, sci.stat.math > Fuzzy logic compared to probability
## Fuzzy Systems

A set of fuzzy rules that converts inputs to outputs.

## History of Fuzzy Theory

Fuzzy theory was first proposed by L. A. Zadeh in 1965. It was originally proposed to deal with the fuzziness found in expert knowledge.

## Fuzzy Logic

Fuzzy logic is a multivalued or "vague" logic where everything is a matter of degree, including truth and set membership. Because a degree of uncertainty is allowed, it is possible to represent imprecise notions such as "more or less true", "low", "fairly hot", "good", etc.

## Fuzzy Set Theory

A fuzzy set is one whose members belong to it to some degree. In fuzzy set theory, we require ordered pairs, which consist of a candidate for inclusion in the set and the degree of membership of the set. The second member of the ordered pair here is a real number in the interval [0, 1].
## Fuzzy Rules

A fuzzy rule is a conditional of the form "if X is A, then Y is B," where A and B are fuzzy sets. Fuzzy inference rules specify the relationships between fuzzy variables. A few fuzzy rules can encompass great complexity, compared to crisp rules.
## Applications of Fuzzy Systems

- Trading systems
- Portfolio management systems
- Fuzzy database retrieval
- Cameras and camcorders
- Cars
- Elevator banks
- Subway trains

## Papers

- DUBOIS, Didier, Henri PRADE and Philippe SMETS, Gradual properties vs. uncertainty : Fuzzy logic vs. possibilistic logic
- ELKAN, Charles The Paradoxical Success of Fuzzy Logic, 1993.
- FIORDALISO, Antonio, Combining forecasts: a fuzzy approach, 1997.
- FIORDALISO, Antonio, Analysis Improvement of Takagi??Sugeno Fuzzy Rules Using Convexity Constraints
- NGUYEN, Hung T., Masao MUKAIDONO and Vladik KREINOVICH, Probability of Implication, Logical Version of Bayes Theorem, and Fuzzy Logic Operations, 2002.

## Links