## The Paradoxical Success of Fuzzy Logic (1993)

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Venue: | IEEE Expert |

Citations: | 70 - 1 self |

### BibTeX

@INPROCEEDINGS{Elkan93theparadoxical,

author = {Charles Elkan},

title = {The Paradoxical Success of Fuzzy Logic},

booktitle = {IEEE Expert},

year = {1993},

pages = {698--703}

}

### Years of Citing Articles

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### Abstract

Applications of fuzzy logic in heuristic control have been highly successful, but which aspects of fuzzy logic are essential to its practical usefulness? This paper shows that an apparently reasonable version of fuzzy logic collapses mathematically to two-valued logic. Moreover, there are few if any published reports of expert systems in real-world use that reason about uncertainty using fuzzy logic. It appears that the limitations of fuzzy logic have not been detrimental in control applications because current fuzzy controllers are far simpler than other knowledge-based systems. In the future, the technical limitations of fuzzy logic can be expected to become important in practice, and work on fuzzy controllers will also encounter several problems of scale already known for other knowledge-based systems. 1

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Citation Context ...re mentioned as a topic for future research by von Altrock et al. [1992]. Symbolic AI formalisms for representing systems whose behaviour depends on their history have been available since the 1960s [=-=McCarthy and Hayes, 1969-=-]. Neural networks with similar properties (called recurrent networks) have been available for several years [Watrous and Shastri, 1987; Elman, 1990], and have already been used in control application... |

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Citation Context ...and quicker settling by Burkhardt and Bonissone [1992] for example. The first demonstrations that fuzzy logic could be used in building heuristic controllers were published in the 1970s [Zadeh, 1973; =-=Mamdani, 1974-=-]. Work using fuzzy logic in heuristic control continued through the 1980s, and recently there has been an explosion of industrial interest in this area; for surveys see Yamakawa and Hirota [1989] and... |

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Citation Context ...In particular, the theorem remains true when negation is modeled by any operator in the Sugeno class [Sugeno, 1977], and when disjunction or conjunction are modeled by operators in the Yager classes [=-=Yager, 1980-=-]. Of course, the last postulate of Definition 1 is the most controversial one. In 1 Note however that Theorem 1 does not depend on any particular definition of implication in fuzzy logic. New definit... |

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Citation Context ...x makes A true} (8) where A is a Boolean proposition (a proposition that can only be true or false). It can be easily checked that for Boolean propositions A and B, we have n(A v B) = max(n(A), n(B)) =-=(9)-=- but that we only have the inequality n(A AB) 5 min(n(A), n(B)) (10) n the general case (equality holds when A ind B are logically independent). Indeed if !3 TA, n(A AB) = n(l) = 0, while nin(n(A), n(... |

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Citation Context ...ledge of the form “the more Xis A, the more Y is B,” such as, “the taller you are, the heavier you are.” This is captured by the implication defined by t(A -+ B) = 1 if t(A) 5 t(B) = 0 if t(A) > t(B) =-=(6)-=- This implication is the natural counterpart of Zadeh’s fuzzy set inclusion defined by the pointwise inequality of the membership functions.6 It is also directly associated with Equations 1-3, since A... |

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Citation Context ...our depends on their history have been available since the 1960s [McCarthy and Hayes, 1969]. Neural networks with similar properties (called recurrent networks) have been available for several years [=-=Watrous and Shastri, 1987-=-; Elman, 1990], and have already been used in control applications [Ku et al., 1992]. It remains to be seen whether research from a fuzzy logic perspective will provide new solutions to the fundamenta... |

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Citation Context ...trona1 Conference on Artificial Intelligence (AAA1 ’93), MIT Press, 1993, pp 698-703 Definition 1: Let A and B be arbitrary assertions. Then t(A A B) = min [ t(A), t(B)) t(A v B) = max { t(A), t(B)] t=-=(4)-=- = 1 - t(A) t(A) = t(B) if A and B are logically equivalent. Depending how the phrase “logically equivalent” is understood, Definition 1 yields different formal systems. A fuzzy logic system is intend... |

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Citation Context ...possibility. Given a [O,l]-valued possibility distribution n: describing an incomplete state of knowledge, Zadeh4 defines a so-called possibility measure n such that n(A) = sup( ~(x), x makes A true} =-=(8)-=- where A is a Boolean proposition (a proposition that can only be true or false). It can be easily checked that for Boolean propositions A and B, we have n(A v B) = max(n(A), n(B)) (9) but that we onl... |

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Citation Context ...etwork methods have been especially successful.22 What the tuning algorithms themselves have in common is that they are gradient-descent “hill-climbing” algorithms that learn by local 0ptimi~ation.l~ =-=(5)-=- By definition, fuzzy controllers use fuzzy logic operators. Typically, minimum and maximum are used, as are explicit possibility distributions (usually trapezoidal) and some fuzzy implication operato... |

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(Show Context)
Citation Context ...on mechanism between n rules with fuzzy condition parts and nonfuzzy conclusions of the form “if X is A, and Y is B, then Z = cl”, by computing the following output when X = x0 and Y = yo is observed =-=(7)-=- where K = min(pA,(xd, pei(yd), i = 1,n. Again, this kind of “inference” (which is widely used in fuzzy control) has nothing to do with uncertainty handling, since only an interpolation between typica... |

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2 |
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(Show Context)
Citation Context ...Zessity measure N is associated to n ac:ording to the relation (which can be viewed as a graded version of the relation between what is necessary and what is posiible in modal logic) N(A) = 1 - n(-A) =-=(11)-=- which states that A is all the more necessarily true as TA has a low possibility to be true. It entails and N(A A B) = min(N(A), N(B)) (12: N(A v B) 2 max(N(A), N(B)). (13: Equations 9, 1 I, and 12 s... |

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