by James F. Brule

**INTRODUCTION**

Fuzzy systems is an alternative to traditional notions of set membership and
logic that has its origins in ancient Greek philosophy, and applications at the
leading edge of Artificial Intelligence. Yet, despite its long-standing origins,
it is a relatively new field, and as such leaves much room for development. This
paper will present the foundations of fuzzy systems, along with some of the more
noteworthy objections to its use, with examples drawn from current research in
the field of Artificial Intelligence. Ultimately, it will be demonstrated that
the use of fuzzy systems makes a viable addition to the field of Artificial
Intelligence, and perhaps more generally to formal mathematics as a whole.

**THE PROBLEM: REAL-WORLD VAGUENESS**

Natural language abounds with vague and imprecise concepts, such as "Sally
is tall," or "It is very hot today." Such statements are
difficult to translate into more precise language without losing some of their
semantic value: for example, the statement "Sally's height is 152 cm."
does not explicitly state that she is tall, and the statement "Sally's
height is 1.2 standard deviations about the mean height for women of her age in
her culture" is fraught with difficulties: would a woman 1.1999999 standard
deviations above the mean be tall? Which culture does Sally belong to, and how
is membership in it defined?

While it might be argued that such vagueness is an obstacle to clarity of meaning, only the most staunch traditionalists would hold that there is no loss of richness of meaning when statements such as "Sally is tall" are discarded from a language. Yet this is just what happens when one tries to translate human language into classic logic. Such a loss is not noticed in the development of a payroll program, perhaps, but when one wants to allow for natural language queries, or "knowledge representation" in expert systems, the meanings lost are often those being searched for.

For example, when one is designing an expert system to mimic the diagnostic powers of a physician, one of the major tasks i to codify the physician's decision-making process. The designer soon learns that the physician's view of the world, despite her dependence upon precise, scientific tests and measurements, incorporates evaluations of symptoms, and relationships between them, in a "fuzzy," intuitive manner: deciding how much of a particular medication to administer will have as much to do with the physician's sense of the relative "strength" of the patient's symptoms as it will their height/weight ratio. While some of the decisions and calculations could be done using traditional logic, we will see how fuzzy systems affords a broader, richer field of data and the manipulation of that data than do more traditional methods.

**HISTORIC FUZZINESS**

The precision of mathematics owes its success in large part to the efforts of
Aristotle and the philosophers who preceded him. In their efforts to devise a
concise theory of logic, and later mathematics, the so-called "Laws of
Thought" were posited [7]. One of these, the "Law of the Excluded
Middle," states that every proposition must either be True or False. Even
when Parminedes proposed the first version of this law (around 400 B.C.) there
were strong and immediate objections: for example, Heraclitus proposed that
things could be simultaneously True and not True. It was Plato who laid the
foundation for what would become fuzzy logic, indicating that there was a third
region (beyond True and False) where these opposites "tumbled about."
Other, more modern philosophers echoed his sentiments, notably Hegel, Marx, and
Engels. But it was Lukasiewicz who first proposed a systematic alternative to
the bi-valued logic of Aristotle [8]. In the early 1900's, Lukasiewicz described
a three-valued logic, along with the mathematics to accompany it. The third
value he proposed can best be translated as the term "possible," and
he assigned it a numeric value between True and False. Eventually, he proposed
an entire notation and axiomatic system from which he hoped to derive modern
mathematics. Later, he explored four-valued logics, five-valued logics, and then
declared that in principle there was nothing to prevent the derivation of an
infinite-valued logic. Lukasiewicz felt that three- and infinite-valued logics
were the most intriguing, but he ultimately settled on a four-valued logic
because it seemed to be the most easily adaptable to Aristotelian logic.

Knuth proposed a three-valued logic similar to Lukasiewicz's, from which he speculated that mathematics would become even more elegant than in traditional bi-valued logic. His insight, apparently missed by Lukasiewicz, was to use the integral range [-1, 0, +1] rather than [0, 1, 2]. Nonetheless, this alternative failed to gain acceptance, and has passed into relative obscurity. It was not until relatively recently that the notion of an infinite-valued logic took hold. In 1965 Lotfi A. Zadeh published his seminal work "Fuzzy Sets" ([12], [13]) which described the mathematics of fuzzy set theory, and by extension fuzzy logic. This theory proposed making the membership function (or the values False and True) operate over the range of real numbers [0.0, 1.0]. New operations for the calculus of logic were proposed, and showed to be in principle at least a generalization of classic logic. It is this theory which we will now discuss.

**BASIC CONCEPTS**

The notion central to fuzzy systems is that truth values (in fuzzy logic) or
membership values (in fuzzy sets) are indicated by a value on the range [0.0,
1.0], with 0.0 representing absolute Falseness and 1.0 representing absolute
Truth. For example, let us take the statement: "Jane is old."

If Jane's age was 75, we might assign the statement the truth value of 0.80. The statement could be translated into set terminology as follows: "Jane is a member of the set of old people."

This statement would be rendered symbolically with fuzzy sets as: mOLD(Jane) = 0.80

where m is the membership function, operating in this case on the fuzzy set of old people, which returns a value between 0.0 and 1.0.

At this juncture it is important to point out the distinction between fuzzy systems and probability. Both operate over the same numeric range, and at first glance both have similar values: 0.0 representing False (or non- membership), and 1.0 representing True (or membership). However, there is a distinction to be made between the two statements: The probabilistic approach yields the natural-language statement, "There is an 80% chance that Jane is old," while the fuzzy terminology corresponds to "Jane's degree of membership within the set of old people is 0.80." The semantic difference is significant: the first view supposes that Jane is or is not old (still caught in the Law of the Excluded Middle); it is just that we only have an 80% chance of knowing which set she is in. By contrast, fuzzy terminology supposes that Jane is "more or less" old, or some other term corresponding to the value of 0.80. Further distinctions arising out of the operations will be noted below.

The next step in establishing a complete system of fuzzy logic is to define the operations of EMPTY, EQUAL, COMPLEMENT (NOT), CONTAINMENT, UNION (OR), and INTERSECTION (AND). Before we can do this rigorously, we must state some formal definitions:

- Definition 1: Let X be some set of objects, with elements noted as x. Thus, X = {x}.
- Definition 2: A fuzzy set A in X is characterized by a membership function mA(x) which maps each point in X onto the real interval [0.0, 1.0]. As mA(x) approaches 1.0, the "grade of membership" of x in A increases.
- Definition 3: A is EMPTY iff for all x, mA(x) = 0.0.
- Definition 4: A = B iff for all x: mA(x) = mB(x) [or, mA = mB].
- Definition 5: mA' = 1 - mA.
- Definition 6: A is CONTAINED in B iff mA <>
- Definition 7: C = A UNION B, where: mC(x) = MAX(mA(x), mB(x)).
- Definition 8: C = A INTERSECTION B where: mC(x) = MIN(mA(x), mB(x)).

It is important to note the last two operations, UNION (OR) and INTERSECTION (AND), which represent the clearest point of departure from a probabilistic theory for sets to fuzzy sets. Operationally, the differences are as follows:

For independent events, the probabilistic operation for AND is multiplication, which (it can be argued) is counterintuitive for fuzzy systems. For example, let us presume that x = Bob, S is the fuzzy set of smart people, and T is the fuzzy set of tall people. Then, if mS(x) = 0.90 and uT(x) = 0.90, the probabilistic result would be: mS(x) * mT(x) = 0.81

whereas the fuzzy result would be: MIN(uS(x), uT(x)) = 0.90

The probabilistic calculation yields a result that is lower than either of the two initial values, which when viewed as "the chance of knowing" makes good sense.

However, in fuzzy terms the two membership functions would read something like "Bob is very smart" and "Bob is very tall." If we presume for the sake of argument that "very" is a stronger term than "quite," and that we would correlate "quite" with the value 0.81, then the semantic difference becomes obvious. The probabilistic calculation would yield the statement

If Bob is very smart, and Bob is very tall, then Bob is a quite tall, smart
person.

The fuzzy calculation, however, would yield

If Bob is very smart, and Bob is very tall, then Bob is a very tall, smart
person.

Another problem arises as we incorporate more factors into our equations (such
as the fuzzy set of heavy people, etc.). We find that the ultimate result of a
series of AND's approaches 0.0, even if all factors are initially high. Fuzzy
theorists argue that this is wrong: that five factors of the value 0.90 (let us
say, "very") AND'ed together, should yield a value of 0.90 (again,
"very"), not 0.59 (perhaps equivalent to "somewhat").
Similarly, the probabilistic version of A OR B is (A+B - A*B), which approaches
1.0 as additional factors are considered. Fuzzy theorists argue that a sting of
low membership grades should not produce a high membership grade instead, the
limit of the resulting membership grade should be the strongest membership value
in the collection.

Other values have been established by other authors, as have other operations. Baldwin [1] proposes a set of truth value restrictions, such as "unrestricted" (mX = 1.0), "impossible" (mX = 0.0), etc. The skeptical observer will note that the assignment of values to linguistic meanings (such as 0.90 to "very") and vice versa, is a most imprecise operation. Fuzzy systems, it should be noted, lay no claim to establishing a formal procedure for assignments at this level; in fact, the only argument for a particular assignment is its intuitive strength. What fuzzy logic does propose is to establish a formal method of operating on these values, once the primitives have been established.

**HEDGES**

Another important feature of fuzzy systems is the ability to define
"hedges," or modifier of fuzzy values. These operations are provided
in an effort to maintain close ties to natural language, and to allow for the
generation of fuzzy statements through mathematical calculations. As such, the
initial definition of hedges and operations upon them will be quite a subjective
process and may vary from one project to another. Nonetheless, the system
ultimately derived operates with the same formality as classic logic.

The simplest example is in which one transforms the statement "Jane is old" to "Jane is very old." The hedge "very" is usually defined as follows: m"very"A(x) = mA(x)^2

Thus, if mOLD(Jane) = 0.8, then mVERYOLD(Jane) = 0.64.

Other common hedges are "more or less" [typically SQRT(mA(x))], "somewhat," "rather," "sort of," and so on. Again, their definition is entirely subjective, but their operation is consistent: they serve to transform membership/truth values in a systematic manner according to standard mathematical functions.

A more involved approach to hedges is best shown through the work of Wenstop [11] in his attempt to model organizational behavior. For his study, he constructed arrays of values for various terms, either as vectors or matrices. Each term and hedge was represented as a 7-element vector or 7x7 matrix. He ten intuitively assigned each element of every vector and matrix a value between 0.0 and 1.0, inclusive, in what he hoped was intuitively a consistent manner. For example, the term "high" was assigned the vector

0.0 0.0 0.1 0.3 0.7 1.0 1.0

and "low" was set equal to the reverse of "high," or

1.0 1.0 0.7 0.3 0.1 0.0 0.0

Wenstop was then able to combine groupings of fuzzy statements to create new
fuzzy statements, using the APL function of Max-Min matrix multiplication. These
values were then translated back into natural language statements, so as to
allow fuzzy statements as both input to and output from his simulator. For
example, when the program was asked to generate a label "lower than sortof
low," it returned "very low;" "(slightly higher) than
low" yielded "rather low," etc. The point of this example is to
note that algorithmic procedures can be devised which translate
"fuzzy" terminology into numeric values, perform reliable operations
upon those values, and then return natural language statements in a reliable
manner. Similar techniques have been adopted by others, primarily in the study
of fuzzy systems as applicable to linguistic approximation (e.g. [2], [3], [4]).
APL appears to be the language of choice, owing to its flexibility and power in
matrix operations.

**OBJECTIONS**

It would be remarkable if a theory as far-reaching as fuzzy systems did not
arouse some objections in the professional community. While there have been
generic complaints about the "fuzziness" of the process of assigning
values to linguistic terms, perhaps the most cogent criticisms come from Haack
[6]. A formal logician, Haack argues that there are only two areas in which
fuzzy logic could possibly be demonstrated to be "needed," and then
maintains that in each case it can be shown that fuzzy logic is not necessary.
The first area Haack defines is that of the nature of Truth and Falsity: if it
could be shown, she maintains, that these are fuzzy values and not discrete
ones, then a need for fuzzy logic would have been demonstrated. The other area
she identifies is that of fuzzy systems' utility: if it could be demonstrated
that generalizing classic logic to encompass fuzzy logic would aid in
calculations of a given sort, then again a need for fuzzy logic would exist.

In regards to the first statement, Haack argues that True and False are discrete terms. For example, "The sky is blue" is either true or false; any fuzziness to the statement arises from an imprecise definition of terms, not out of the nature of Truth. As far as fuzzy systems' utility is concerned, she maintains that no area of data manipulation is made easier through the introduction of fuzzy calculus; if anything, she says, the calculations become more complex. Therefore, she asserts, fuzzy logic is unnecessary. Fox [5] has responded to her objections, indicating that there are three areas in which fuzzy logic can be of benefit: as a "requisite" apparatus (to describe real-world relationships which are inherently fuzzy); as a "prescriptive" apparatus (because some data is fuzzy, and therefore requires a fuzzy calculus); and as a "descriptive" apparatus (because some inferencing systems are inherently fuzzy).

His most powerful arguments come, however, from the notion that fuzzy and classic logics need not be seen as competitive, but complementary. He argues that many of Haack's objections stem from a lack of semantic clarity, and that ultimately fuzzy statements may be translatable into phrases which classical logicians would find palatable. Lastly, Fox argues that despite the objections of classical logicians, fuzzy logic has found its way into the world of practical applications, and has proved very successful there. He maintains, pragmatically, that this is sufficient reason for continuing to develop the field.

**APPLICATIONS**

Areas in which fuzzy logic has been successfully applied are often quite
concrete. The first major commercial application was in the area of cement kiln
control, an operation which requires that an operator monitor four internal
states of the kiln, control four sets of operations, and dynamically manage 40
or 50 "rules of thumb" about their interrelationships, all with the
goal of controlling a highly complex set of chemical interactions. One such rule
is "If the oxygen percentage is rather high and the free-lime and kiln-
drive torque rate is normal, decrease the flow of gas and slightly reduce the
fuel rate" (see Zadeh [14]). A complete accounting of this very successful
system can be found in Umbers and King [10].

The objection has been raised that utilizing fuzzy systems in a dynamic control environment raises the likelihood of encountering difficult stability problems: since in control conditions the use of fuzzy systems can roughly correspond to using thresholds, there must be significant care taken to insure that oscillations do not develop in the "dead spaces" between threshold triggers. This seems to be an important area for future research.

Other applications which have benefited through the use of fuzzy systems theory have been information retrieval systems, a navigation system for automatic cars, a predicative fuzzy-logic controller for automatic operation of trains, laboratory water level controllers, controllers for robot arc-welders, feature-definition controllers for robot vision, graphics controllers for automated police sketchers, and more.

Expert systems have been the most obvious recipients of the benefits of fuzzy logic, since their domain is often inherently fuzzy. Examples of expert systems with fuzzy logic central to their control are decision-support systems, financial planners, diagnostic systems for determining soybean pathology, and a meteorological expert system in China for determining areas in which to establish rubber tree orchards [14]. Another area of application, akin to expert systems, is that of information retrieval [9].

**CONCLUSIONS**

Fuzzy systems, including fuzzy logic and fuzzy set theory, provide a rich and
meaningful addition to standard logic. The mathematics generated by these
theories is consistent, and fuzzy logic may be a generalization of classic
logic. The applications which may be generated from or adapted to fuzzy logic
are wide-ranging, and provide the opportunity for modeling of conditions which
are inherently imprecisely defined, despite the concerns of classical logicians.
Many systems may be modeled, simulated, and even replicated with the help of
fuzzy systems, not the least of which is human reasoning itself.

**REFERENCES**

[1] J.F. Baldwin, "Fuzzy logic and fuzzy reasoning," in Fuzzy
Reasoning and Its Applications, E.H. Mamdani and B.R. Gaines (eds.), London:
Academic Press, 1981.

[2] W. Bandler and L.J. Kohout, "Semantics of implication operators and
fuzzy relational products," in Fuzzy Reasoning and Its Applications, E.H.
Mamdani and B.R. Gaines (eds.), London: Academic Press, 1981.

[3] M. Eschbach and J. Cunnyngham, "The logic of fuzzy Bayesian
influence," paper presented at the International Fuzzy Systems Association
Symposium of Fuzzy information Processing in Artificial Intelligence and
Operational Research, Cambridge, England: 1984.

[4] F. Esragh and E.H. Mamdani, "A general approach to linguistic
approximation," in Fuzzy Reasoning and Its Applications, E.H. Mamdani and
B.R. Gaines (eds.), London: Academic Press, 1981.

[5] J. Fox, "Towards a reconciliation of fuzzy logic and standard
logic," Int. Jrnl. of Man-Mach. Stud., Vol. 15, 1981, pp. 213-220.

[6] S. Haack, "Do we need fuzzy logic?" Int. Jrnl. of Man-Mach. Stud.,
Vol. 11, 1979, pp.437-445.

[7] S. Korner, "Laws of thought," Encyclopedia of Philosophy, Vol. 4,
MacMillan, NY: 1967, pp. 414-417.

[8] C. Lejewski, "Jan Lukasiewicz," Encyclopedia of Philosophy, Vol.
5, MacMillan, NY: 1967, pp. 104-107.

[9] T. Radecki, "An evaluation of the fuzzy set theory approach to
information retrieval," in R. Trappl, N.V. Findler, and W. Horn, Progress
in Cybernetics and System Research, Vol. 11: Proceedings of a Symposium
Organized by the Austrian Society for Cybernetic Studies, Hemisphere Publ. Co.,
NY: 1982.

[10] I.G. Umbers and P.J. King, "An analysis of human decision-making in
cement kiln control and the implications for automation," Int. Jrnl. of
Man- Mach. Stud., Vol. 12, 1980, pp. 11-23.

[11] F. Wenstop, "Deductive verbal models of organizations," Int. Jrnl.
of Man-Mach. Stud., Vol. 8, 1976, pp. 293-311.

[12] L.A. Zadeh, "Fuzzy sets," Info. & Ctl., Vol. 8, 1965, pp.
338-353.

[13] L.A. Zadeh, "Fuzzy algorithms," Info. & Ctl., Vol. 12, 1968,
pp. 94- 102.

[14] L.A. Zadeh, "Making computers think like people," I.E.E.E.
Spectrum, 8/1984, pp. 26-32.

**REFERENCES RELATED TO DEFINITIONS OF
OPERATORS:**

Gougen, J.A. (1969) The logic of inexact concepts. Synthese, Vol. 19, pp
325-373.

Osherson, D.N., & Smith, E.E. (1981) On the adequacy of prototype theory as
a theory of concepts. Cognition. Vol. 9, pp. 35-38.

Osherson, D.N., & Smith, E.E. (1982) Gradedness and conceptual combination.
Cognition, Vol. 12, pp. 299-318.

Roth, E.M., & Mervis, C.B. (1983) Fuzzy set theory and class inclusion
relations in semantic categories. Journal of Verbal Learning and Verbal
Behavior, Vol. 22, pp. 509-525.

Zadeh, L.A. (1982) A note on prototype theory and fuzzy sets. Cognition, Vol.
12, pp. 291-297.

**BASIC REFERENCE ON PROTOTYPE THEORY
IN COGNITIVE PSYCHOLOGY:**

Mervis, C.B., & Rosch, E. (1981) Categorization of natural objects. Annual
Review of Psychology, Vol. 32, pp. 89-115.

**SELECTED REFERENCES ON FUZZY SET THEORY
GENERALLY & AI APPLICATIONS:**

Jain, R. Fuzzyism and real world problems. In P.P. Wang & S.K. Chang (Eds.),
Fuzzy Sets, New York: Plenum Press.

Zadeh, L.A. (1965) Fuzzy sets. Information and Control, Vol. 8, pp. 338-353.

Zadeh, L.A. (1978) PRUF - A meaning representation language for natural
languages. International Journal of Man-Machine Studies, Vol. 10, pp. 395-460.

Zadeh, L.A. (1983) The role of fuzzy logic in the management of uncertainty in
expert systems. Memorandum No. UCB/ERL M83/41, University of California,
Berkeley.