# Fuzzy Logic

# FUZZY LOGIC

"Fuzzy logics" are multivalued logics intended to model human reasoning with certain types of imprecision. The field of fuzzy logic originated with a 1965 paper by Lotfi Zadeh, a professor of engineering at the University of California, Berkeley. It is significant that the inventor of fuzzy logic was neither a philosopher nor a linguist. Since 1965 research in fuzzy logic has always had an engineering and mathematical bent, while the philosophical foundations of fuzzy logic have always been under attack.

Many different formal systems have been proposed under the general name of fuzzy logic, but there is wide acceptance that the fundamental principles of fuzzy logic are**(1)** *t* (*A* ∧ *B* ) = min{*t* (*A* ),*t* (*B* )}**(2)** *t* (*A* ∨ *B* ) = max{*t* (*A* ),*t* (*B* )}**(3)** *t* (¬*A* ) = 1 − *t* (*A* ).

In these axioms *A* and *B* represent arbitrary propositions. The truth value of *A*, a real number between 0 and 1, is denoted *t* (*A* ). The first axiom above says that the truth value of *A* ∧ *B* is the lesser of the truth value of *A* and the truth value of *B.* The second and third axioms concerning disjunction and negation are to be understood similarly.

At the same time that Zadeh introduced fuzzy logic, he also introduced fuzzy set theory, a variant of naive set theory (i.e., everyday set theory as opposed to a foundational set theory such as the Zermelo-Fraenkel axioms) with the basic axioms**(1)** *μ* (*x* ∈ *P* ∩ *Q* ) = min{*μ* (*x* ∈ *P* ),*μ* (*x* ∈ *Q* )}**(2)** *μ* (*x* ∈ *P* ∪ *Q* ) = max{*μ* (*x* ∈ *P* ),*μ* (*x* ∈ *Q* )}**(3)** *μ* (*x* ∈ *P ^{c}* ) = 1 −

*μ*(

*x*∈

*P*).

Here

*μ*(

*x*∈

*P*) denotes the degree to which

*x*is a member of the set

*P.*Since 1965 many branches of mathematics have been generalized along fuzzy set theory lines.

There are two fundamental differences between fuzzy logics and conventional logics such as classical predicate calculus or modal logics. Although these differences are technical, they are of considerable philosophical significance. First, conventional logics (except intuitionistic logics) require for every proposition that either it or its negation be true, that is, that *t* (*A* ∨ ¬ *A* ) = 1 in fuzzy logic notation. In fuzzy logics this "law of the excluded middle" does not hold. Second, there is no consensus about a semantics for fuzzy logic that is well-defined independently of its proof theory, that is, the inferential axioms given above. In contrast, conventional logics have well-accepted semantics, for example Tarskian model theory for predicate calculus, and Kripkean possible worlds semantics for modal logics.

Fuzzy logics are claimed to be capable of representing the meanings of intrinsically imprecise natural language sentences, such as "Many Texans are rich," for which the law of excluded middle fails. There is disagreement as to whether fuzzy methods successfully represent the complexities of concepts such as "many" and "rich." What is clear is that the rules of fuzzy logic cannot be used for reasoning about frequentist or subjective types of uncertainty, whose properties are captured by standard probability theory. The central issue here is that the probability of a compound proposition such as *A* ∧ *B* is not a function just of the probabilities of the propositions *A* and *B* : The probability of *A* ∧ *B* also depends on the relationship between the propositions *A* and *B*, in particular on their independence or correlation.

The tolerance for ambiguity found in fuzzy logic, and specifically the rejection of the law of the excluded middle, is a revolutionary idea in mathematical logic. Some advocates of fuzzy logic claim that tolerance for ambiguity is also revolutionary philosophically, since Western philosophy, from Plato through René Descartes, has supposedly been an intrinsically dualistic tradition. According to this argument, fuzzy logic has been better received in Japan and other Asian countries than in the West because of the holistic, subtle nature of the Eastern intellectual tradition. Apart from the dualistic oversimplification of the distinction between "Western" and "Eastern" thought, this claim also ignores the continuous holistic tradition in European philosophical thought, from Zeno through Blaise Pascal to Martin Heidegger and Ludwig Wittgenstein.

There has been much artificial intelligence research on using fuzzy logic for representing real-world knowledge, and there has been some recent convergence between this work and parallel work by a distinct research community on knowledge representation using classical logics, nonmonotonic logics, and probability theory. So far this research has remained almost exclusively theoretical. In contrast, engineering work on using fuzzy logic for controlling complex machines heuristically has been highly successful in practice.

A fuzzy controller is a device, usually implemented as software for an embedded microprocessor, that continually monitors readings from sensors, and makes decisions about actuator settings. For example, a controller for the automatic transmission of a car monitors road speed, the position of the accelerator pedal, and other factors, and decides whether to shift gears down or up, or not to shift. The knowledge possessed by a fuzzy controller is typically represented as rules such as*μ* (*speed*, MODERATE) ∧ *μ* (*pedal*, FULL-DOWN) → *μ* (*shift*, DOWN)

Here *speed* and *pedal* are sensory readings, *shift* is a possible actuator setting, and MODERATE, FULL-DOWN, and DOWN are fuzzy sets. Through inference rules for the fuzzy connectives ∧ and →, the degree of membership of *speed* in MODERATE and of *pedal* in FULL-DOWN determines the desired degree of membership of *shift* in DOWN. Given a set of rules, a fuzzy controller continually computes the degree to which the antecedents of each rule are satisfied, and selects a conclusion that is the weighted average of the conclusion of each rule, where rules are weighted using these degrees.

Fuzzy controllers are widely used for two basic reasons. First, since the action chosen at each instant is typically the result of interpolating several rules, their behavior is smooth. Second, fuzzy controller rule sets are easy for humans to read and understand intuitively, hence easy to construct by trial and error.

** See also ** Artificial Intelligence; Descartes, René; Heidegger, Martin; Kripke, Saul; Logic, History of; Mathematics, Foundation of; Modal Logic; Model Theory; Pascal, Blaise; Plato; Probability and Chance; Proof Theory; Quantum Mechanics; Semantics; Set Theory; Tarski, Alfred; Wittgenstein, Ludwig Josef Johann; Zeno of Elea.

## Bibliography

Elkan, C. "The Paradoxical Success of Fuzzy Logic." *IEEE Expert* 6 (August 1994): 3–8.

Mamdani, E. H. "Application of Fuzzy Algorithms for Control of Simple Dynamic Plant." *Proceedings of the Institution of Electrical Engineers* 121 (1974): 1585–1588.

Zadeh, L. A. "Fuzzy Sets." *Information and Control* 8 (1965): 338–353.

Zimmermann, H.-J. *Fuzzy Set Theory—And Its Applications.* Boston: Kluwer Academic, 1991.

*Charles Elkan (1996)*

# fuzzy logic

**fuzzy logic ( fuzzy theory)** A branch of logic designed specifically for representing knowledge and human reasoning in such a way that it is amenable to processing by computer. Thus fuzzy logic is applicable to expert systems, knowledge engineering, and artificial intelligence.

The more traditional propositional and predicate logics do not allow for degrees of imprecision, indicated by words or phrases such as fairly, very, quite possibly. Instead of truth values such as true and false it is possible to introduce a multivalued logic consisting of, for example, the values true, not true, very true, not very true, more or less true, not very false, very false, not false, and false. Alternatively an interval such as [0,1] can be introduced and the degree of truth can be represented by some real number in this range. Predicates are then functions that map not into {true, false} but into these more general domains.

Fuzzy logic is concerned with the study of sets and predicates of this kind. There emerge such concepts as

*fuzzy sets*,

*fuzzy relationships*, and

*fuzzy quantifiers*.

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