fuzzy logic

fuzzy logic

First order

fuzzy logic

is a new chapter of logic which originates from the notion of

fuzzy subset

proposed by L. A. Zadeh. From a semantical point of view, fuzzy logic is not different in nature from first-order multi-valued logic. Indeed in both the logics one refers to ”worlds with graded

properties

”. Instead, if we refer to the managment of the information on these worlds, and therefore to the

deduction

apparatus, fuzzy logic is a totally different and new topic. In fact it is based on the notion of approximate reasoning as suggested by Zadeh, Goguen, Pavelka and other authors. This means that if

F

denotes the set of

sentences

of the considered

first order language

, the available information (system of proper axioms) is represented by a fuzzy subet

s

:

F

→

[

0

,

1

]

of

formulas

. Such a fuzzy subset gives constraints on the possible truth degree of the formulas. Namely it says that, for every formula

α

, the truth degree of

α

is greater or equal to

s

⁢

(

α

)

. The managment of such an information is obtained by a deduction apparatus enabling us to define the fuzzy subset

D

⁢

(

s

)

:

F

→

[

0

,

1

]

of

logical consequences

of

s

. Again

D

⁢

(

s

)

is a constraint on the truth degree of the formulas but it is the best constraint we can obtain given

s

. We can define such an apparatus by fixing a suitable set of fuzzy

inference rules

and a suitable fuzzy subset of

logical axioms

. This gives a notion of proof

π

and a way to calculate the degree of

validity

V

⁢

a

⁢

l

⁢

i

⁢

d

⁢

(

π

,

s

)

of

π

given

s

. Then,

D

⁢

(

s

)

is obtained by setting

D

⁢

(

s

)

⁢

(

α

)

=

sup

⁡

{

V

⁢

a

⁢

l

⁢

i

⁢

d

⁢

(

π

,

s

)

:

π

⁢

is a proof of

⁢

α

}

.

Precise

definitions

and

completeness theorems

can be found in [7] and [8]. Notice that the so defined notion of approximate reasoning enables us to give an interesting solution of the famous heap

paradox

(see [2] and[3]).

An alternative and very important approach is obtained by introducing in the

language

propositional constants to denote truth values. In such a way it is possible to reduce the question of the deduction in fuzzy logic to the classical paradigm based on logical axioms and crisp inference rules (see the basic book of P. HÃÂ¡jek).

BIBLIOGRAPHY

1. Cignoli R., D Ottaviano I. M. L. and Mundici D.,

Algebraic

Foundations

of Many-Valued Reasoning

. Kluwer, Dordrecht, 1999.

2. Gerla G.,

Fuzzy logic: Mathematical tools for approximate reasoning

, Kluwer Academic Publishers, Dordrecht,2001.

3. Goguen J., The logic of inexact

concepts

,

Synthese

, vol. 19 (1968/69)

4. Gottwald S.,

A treatise on

many-valued logics

, Research Studies Press, Baldock 2000.

5. Gottwald, S., Mathematical fuzzy logic,

The Bulletin of

Symbolic Logic

, vol. 14, 2008, pp. 210-239.

6. HÃÂ¡jek P.,

Metamathematics

of fuzzy logic

. Kluwer 1998.

7. Novak V., Perfilieva I., Mockor J.,

Mathematical Principles of Fuzzy Logic

, Kluwer Academic Publ., 1999.

8. Pavelka J., On fuzzy logic I-III,

Zeitschrift f

u

¨

r Mathemathische Logik
und Grundlagen der Mathematik

, vol. 25 (1979), pp. 45–52; 119–134; 447–464.

9. Yager R. and Filev D.,

Essentials of Fuzzy Modeling and Control

(1994), ISBN 0-471-01761-2

10. Zimmermann H.,

Fuzzy Set Theory and its Applications

(2001), ISBN 0-7923-7435-5.

11. Zadeh L.A., Fuzzy Sets,

Information and Control

, 8 (1965) 338ÃÂ­-353.

12. Zadeh L. A., The concept of a linguistic

variable

and its application to approximate reasoning I-III,

Information Sciences

, vol. 8, 9(1975), pp. 199-275, 301-357, 43-80.

Title

fuzzy logic

Canonical

name

FuzzyLogic

Date of creation

2013-03-22 16:33:32

Last modified on

2013-03-22 16:33:32

Owner

ggerla (15808)

Last modified by

ggerla (15808)

Numerical id

53

Author

ggerla (15808)

Entry type

Topic

Classification

msc 03B52

Classification

msc 03B15

Classification

msc 03B10

Synonym

multi-valued logic

Related topic

FuzzySubset

Related topic

MinimalAndMaximalNumber

Related topic

FuzzyLogicsOfLivingSystems

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