FUZZY LOGIC

"The model of fuzzy logic is the human mind" (Lofti Zadeh).

"The essence of fuzzy logic is that everything is a matter of degree, including the notion of membership" (Lofti Zadeh).

"Opposites are identical in nature, but different in degree" (The Kybalion).

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Fuzzy Logic
Fuzzy or fuzzy logic was first proposed by Lofti Zadeh (professor at UC Berkeley) in his famous 1965 paper entitled "fuzzy Sets" [Zadeh, 1965] which revolutionized traditional logical thinking. Its principles and characteristics are as follows:

• It is based on fuzzy, fuzzy concepts, which are of qualitative type, such as high, low, rich, poor, young, old, cold, hot, fast, slow, etc. The concepts "true" and "false" are also fuzzy. These fuzzy concepts are structured as qualitative variables and qualitative values. For example, the variable "age" and its possible qualitative values are: very young, young, mature, old, very old.
  
  Fuzzy or approximate is the opposite of exact, precise or crisp. According to Zadeh, fuzzy logic is the model of the human mind, so it is easy to understand and use.
  
  
• It is a generalization of classical, Boolean or bivalent logic. It is also a generalization of infinite−valent logic based on truth values between 0 (false) and 1 (true), where these values are complementary (add up to 1). In fuzzy logic, the values true (V) and false (F) are independent and may not add up to 1. A fuzzy concept or property may be both partially V and partially F. For example, a person may be neither tall nor short, but participate at the same time in both properties to some degree. Only when his height is above a certain value, he is completely tall and not short. And when it is below another certain value, it is considered completely short and not tall.
  
  
• The main fuzzy principle is that nothing is absolute, that "Everything is a matter of degree". A classic example often given is: "How many grains of sand make up a heap?". If we start from scratch, adding one grain at a time, at some point it could be graded as a heap. Once the heap is created, if we remove one grain, is it still a heap? The boundaries of the concept "heap" are fuzzy.
  
  
• It allows to solve the problem of logical paradoxes. The heap problem is an example of a paradox. Logical paradoxes appear with the bivalence of truth values (true and false) when a statement is both true and false. Paradoxes disappear when degrees of truth are considered.
  
  
• Like all polyvalent logics, fuzzy logic also breaks the dichotomy of Aristotelian logic and the two Aristotelian laws: 1) the law of non-contradiction: A cannot be both B and non-B;; 2) law of excluded third party (tertium non datur): A must be B or non-B. These two laws have formed the foundation of logic for 2000 years.
  
  
• It allows to represent imprecise and partial information, in a flexible and easily modifiable way. It also allows to easily represent knowledge for artificial intelligence applications. Knowledge is represented in a compact and modular way by a set of rules that are logical relations between qualitative variables, allowing qualitative or approximate reasoning. The rules are heuristic, rather than detailed, "brute force" rules. For example, in strategy games (such as chess), instead of considering all the possible moves in detail, knowledge can be represented in the form of general strategies, patterns, etc. This implies a higher level of knowledge, thus improving performance and effectiveness.
  
  
• It is very suitable for the so called "soft sciences", such as linguistics, psychology, sociology, economics, politics, etc. But it is also useful for the technical (computer science, electronics, artificial intelligence, etc.) and even for the theoretical (mathematics, logic, information theory, philosophy, etc.).

Logical operations

If we represent, in general, with C(x) the degree of truth of x with respect to a concept C, we have the following definitions:

• Contrary fuzzy logic: 1−C(x)
• Fuzzy logic sum:
  max(C(x), C(y))
• Fuzzy logic product:
  min(C(x), C(y))

The logical implication C(x)→C(y) admits many interpretations, there being different models, which take the name of their proponents. We highlight the following:

• Lucasiewicz implication.
  It is based on the equivalence p→q ≡ ≡ ¬p∨q of bivalent logic. Therefore, it is equivalent to max(1−C(x), C(y)).
  
  
• Gódel implication.
  C(x)→C(y) = 1 if C(x)≤C(y) otherwise.
  
  
• Sharp implication.
  C(x)→C(y) = 1 if C(x)≤C(y), 0 otherwise.
  
  
• Mamdani implication. C(x)→C(y) = min(C(x), C(y)).

Fuzzy sets
The idea behind a fuzzy set is very simple. In classical Cantor set theory, elements either belong or do not belong to sets. In fuzzy set theory, an element may partially belong to a set:

For any element x, its degree of belonging to a set C is defined by a function f(x). The value 1 of the function is interpreted as x belongs to C (x∈ C), and the value 0 as x does not belong to C (x∉C). This is the classical conception in set theory. But if this function takes values between 0 and 1, these values indicate the degree of set membership and C is a fuzzy set.

The formalization of fuzzy sets was made by Zadeh in his 1965 paper in which he created a generalized axiomatic system of set theory based on the concept of partial membership.


Formal definitions

• A fuzzy set A, in the universe of discourse X, is a set of ordered pairs (x, μ_A(x)), where μ_A(x) is the membership function of x to the set A.
  
  
• A fuzzy set A is a subset of a fuzzy set B if μ_A (x)≤ μ_B(x) para todo x∈A.
  
  
• The support of a fuzzy set A is the set of elements that have a degree of membership in A. If the support is ∅, it is called an empty fuzzy set. If the support is a single element, it is called singleton fuzzy.
  
  
• The height of a fuzzy set is the largest value of its membership function.
  
  
• The kernel of a fuzzy set is the set of elements that belong to the set with degree 1.
  
  
• The cardinality of a fuzzy set A, in the universe of discourse X, is:
  
  
  |A| = ∑A(x)
  The summation is for all elements x belonging to X, and A(x) being the degree of membership of x in A.
  
  
• The alpha−cut (α-cut) of a fuzzy set A is the set of elements having a degree of membership in A equal to or greater than α. A fuzzy set can be decomposed into a family of fuzzy sets by α-cuts.

Operations with fuzzy sets

If we represent, in general, with C(x) the degree of membership of x in C, we have the following definitions:

• Complementary set of a set A: A^C(x) = (1−A(x))
  
  
• Union of sets: (A∪B)(x) = max[A(x), B(x)]
  
  
• Set intersection: (A∩B)(x) = min[A(x), B(x)]

Fuzzy numbers

Zadeh proposed the concept of fuzzy number (fuzzy number) as a special case of fuzzy set. Just as fuzzy logic is an extension of Boolean logic, and fuzzy sets are an extension of traditional sets, fuzzy numbers are an extension of real numbers.

A fuzzy number is an extension of a real number in the sense that it refers not to a single value, but to a set of values, each with a corresponding "weight," factor, or degree of membership (a value between 0 and 1) to the real line. The set is convex, i.e., the set of values has no "gaps", they form a continuum, which can be the entire real line or an interval of it. In general, a fuzzy number is a finite or infinite segment of the real line, whose elements have a degree of membership in the real line.

An example of a fuzzy number on the entire real line is the following triangular number:

f(≤25) = 0
f(30) = 1
f(≥35) = 0
Intermediate values according to figure

A fuzzy number can be considered a function between the numbers of a segment of the real line and the numbers between 0 and 1. In the example, the degree of membership between 25 and 30 increases linearly between 0 and 1, and then decreases (also linearly) between 30 and 35 between 1 and 0.

A real number is a particular case of a fuzzy number: it is a segment reduced to a single point representing a real number with degree of membership 1.

Fuzzy arithmetic operations on fuzzy numbers are the same as operations with sets: complementary, sum, and product.


Linguistic variables

Zadeh introduced in 1973 the concept of "linguistic variable" in which he established an equivalence between variable and fuzzy set.

A numerical variable (e.g. "age") takes a concrete quantitative (numerical) value (e.g. age=43). A linguistic variable has a set of possible linguistic, qualitative values (or categories), e.g.,

{very young, young, young, middle-aged, old, very old}

The granularity of a linguistic variable is the number of qualitative values that are defined for it. In the example, the granularity of age is 5. A fine granularity indicates many qualitative values. A coarse granularity indicates few qualitative values.

There is a function that, for each age, assigns a numerical value (between 0 and 1) to each of these qualitative values, indicating the degree of membership in those categories. The result is a set. For example,

age=30 → {(0, very young), (0.4, young), (0.6, middle-aged), (0, old), (0, very old)}

The logical values "true" and "false" are qualitative values of the linguistic variable "truth".


Background information

• Plato was the first to question bivalent logic by stating, "There is a third region between the true and the false where opposites are presented together."
  
  
• Aristotle, a disciple of Plato, also questioned the principle of bivalence: "In some cases there is eventuality and of affirmation and negation neither is truer than the other" (De interpretatione).
  
  
• Charles Sanders Peirce disagreed with Cantor in characterizing things as belonging or not belonging to sets. He said that all things exist on a continuum.
  
  
• Vasiliev, in 1909, developed a trivalent logic and eliminated the principle of the excluded third of classical logic, but did not fully formalize it.
  
  
• Jan Lukasiewicz proposed in his 1920 paper "On three-valued logic", the idea that the logical dichotomy true−false should be extended with a third value interpreted as "indeterminate" or "uncertain", of value ½ (intermediate between 0 and 1), thus creating trivalent logic. Later, in 1970, he stated that any intermediate value between 0 (false) and 1 (true) could be assigned, which would represent the "degree of truth". Lukasiewicz is considered the creator of polyvalent logic and infinitovalent logic.
  
  
• Bertrand Russell believed that vagueness was a characteristic of human language, but not of reality. He published in 1973 the article "Vagueness" in which he wondered how many hairs a person has to lose to be considered bald.
  
  
• Finally, Lofty Zadeh, in 1965 −inspired by the idea of fuzzy membership of a set− proposed the idea of "fuzzy logic" (fuzzy logic), a logic of infinite values, associated to the values "true" and "false" (independent of each other) and to fuzzy concepts in general.

Fuzziness vs. probability

Blurredness and probability (or randomness) are distinct concepts. There are many conceptual and theoretical differences between the two. There are also similarities, the main one being that both describe uncertainties, with associated numbers between 0 and 1.

• Fuzzy logic and probability express different types of uncertainty. Randomness is a type of uncertainty, but it is not fuzzy. Fuzzy is a type of uncertainty, but it is not random.
  
  
• Randomness describes the uncertainty that a certain future event will occur, which may or may not occur. Fuzzy logic measures or describes the ambiguity or fuzziness of events that have already occurred, by assigning values (between 0 and 1) to the elements of a set.
  
  Whether or not an event occurs is randomness. To what degree it occurs is fuzziness. If all events are crisp, there is no fuzziness, there is only randomness. Blur is a property or attribute of an event.
  
  
• Probability is objective, and is a function of the available information. It can be calculated at the theoretical level and verified in practice. Fuzzy is subjective.
  
  In fuzzy logic, the sum of the degrees assigned to the different qualitative values or categories of a concept need not add up to 1. For example, a given height (e.g. 1.70) can be assigned a sum of 1. e. 1.70) can be associated with the values {0.1×tall, 0.3×short, 0.8×normal}, whose degrees do not add up to 1. And an irregular figure can be associated with {0.8×circle, 0.3×square}, whose degrees also do not add up to 1.
  
  
• With the increase of information (the possible cases), the probability becomes more precise and the fuzziness increases.

Soft Computing

According to Zadeh, fuzzy logic is not an isolated discipline, but is part of what he calls "Soft Computing". Its characteristics are:

• It studies and models complex phenomena that by traditional mathematical−analytical methods (numerical analysis, propositional logic, predicate logic, etc.) are insufficient or require a large consumption of resources to obtain a solution that is sometimes incomplete or imprecise. Complex systems appear in biology, social sciences, medicine, etc. Paradoxically, it uses simplified analytical models to capture complexity.
  
  
• Tolerates imprecision, uncertainty and partial truth. It uses fuzzy, approximate, probabilistic and heuristic reasoning. This model is that of the human mind. In this respect it differs from "hard" computing (such as, for example, operations research).
  
  
• It mainly encompasses fuzzy logic, neural networks and evolutionary computation (genetic algorithms, evolutionary algorithms). Other techniques it uses are: probabilistic reasoning, chaos theory, learning theory, rough set theory (rough set theory), group intelligence (swarm intelligence ), belief propagation networks, possibility theory, "soft" data analysis (soft data), special searches (tabu, reactive and harmonic), heuristics and meta−heuristics (evolutionary and relaxation), etc. It is called "relaxation" of a problem when a moment is reached when the problem is simplified due to the modification or weakening of some elements of the problem.
  
  A rough set (rough set) is a formal approximation of a conventional set by a pair of sets that provide the upper and lower approximation of the set. These two approximations can be either conventional or fuzzy sets.

The BISC program (The Berkeley Initiative in Soft Computing) is an initiative of Lofti Zadeh oriented to the study and development of Soft Computing.


Fuzzy Control Systems

The basic ideas of fuzzy logic control (fuzzy logic control) were provided by Zadeh [1972]. It deals with the design of control systems using a set of fuzzy rules, which have, in general, the form.

If x₁ is v₁ and x₂ is v₂2 ... , then y₁ is w₁, y₂ is w₂ ...

siendo x₁, x₂,... the qualitative input variables, and v₁, v₂,... the values corresponding to each of these variables. Similarly for the qualitative output variables and their values (y₁, y₂, . .. and w₁, w₂, ...). Expressions of the type "x_i is not v_i" and the operator "or" (logical disjunction) instead of "and" (logical conjunction) may also appear in the antecedent.

Each rule has a "weight" or importance, which is a numerical value.

The advantage of fuzzy rules is that they are conceptual in nature, so they are easy to interpret, since they represent knowledge at a qualitative level only. The reasoning is performed at a qualitative level. Once the qualitative results are obtained, they are transformed into quantitative ones.

The rules of a fuzzy model to be tested in practice. The rules are refined until a satisfactory model is finally obtained.

A simple illustrative example is a control system to regulate a fan. In this case there is only one input variable, the ambient temperature t, and one output variable w, the angular velocity (in rpm) of the fan. There are 3 qualitative values for the temperature: Cold, Warm and Hot, defined according to this graph:


The angular velocity w has 3 qualitative values associated with it: Minimum, Average and Maximum. In this case, each of these qualitative values is assigned a quantitative value: w₁, w₂, w₃, which correspond to the average or reference value of the angular velocity.

The fuzzy rules are, in this case, very simple:

R1. If t is Cold, then w is Minimum.
R2. If t is Temperate, then w is Average.
R3. If t is Hot, then w is Maximum.

In this case, the weights of the rules are all unity, i.e., they all have the same weight.

The operation of a fuzzy control system consists of 3 phases:

• Fuzzification of the input values.
  Also called fuzzy classification, it converts quantitative input values into qualitative ones through a fuzzifier or fuzzy classifier element.
  In this case, the input temperature t is converted into 3 fuzzy values: f₁×Fría, f₂×Templada y f₃×Caliente.
  
  
• Evaluation of fuzzy system rules.
  All rules are evaluated and rules that meet the conditions are triggered or activated. For example, if f₁=0, f₂=0.5 and f₃=0. 7, the rules R2 and R3 are triggered, obtaining the corresponding qualitative output values. In this case, Mean and Maximum.
  
  
• Defuzzification.
  From the weights of the triggered rules and the quantitative values associated with the qualitative output values, a function calculates the numerical output value of w, the angular velocity of the fan. This process is performed by a defuzzifier element, fuzzy declassifier.
  
  There are many defuzzification techniques, which combine in different ways these values to obtain the output. The most commonly used system is the "center of gravity" (or centroid) system. In the case of the fan,
  
  
  w = (f₁*w₁ + f₂*w₂ + f₃*w₃) / (f₁+f₂+f₃)
  
  En general, w = ∑f_iw_i/∑f_i
  (both summations go from i equal to 1 up to n)

Fuzzy quantizers

One of the most important aspects of fuzzy logic is that it allows to extend and formalize the concept of quantization, interpreted as fuzzy subset. This topic was also introduced by Zadeh [1983].

A crisp or precise quantifier is a concrete number. A fuzzy (or linguistic) quantifier allows to express a quantity in a qualitative, fuzzy, approximate way. It seems somewhat paradoxical that quantity and quality, which are different concepts, are united in this concept of fuzzy quantifier to express a quantity by means of a quality. But fuzzy quantifiers are essential for effective communication in natural language.

Fuzzy quantifiers can be of two types: absolute and relative.

• Absolute quantifier.
  An absolute quantifier expresses a quantity (qualitatively) over the total number of elements of a given set C. For example: between 5 and 10, more than 10, at least 2, about 8, many more than 100, etc.
  
  Absolute fuzzy quantifiers are defined as fuzzy sets on the positive real line, i.e., on the interval (0, ∞), i.e., as a fuzzy number defined on that interval.
  
  
• Relative quantizer.
  A relative (or proportional) quantifier also expresses a quantity over the total number of elements of a given set −as in the case of absolute quantifiers−, but expresses it by a proportion with respect to the whole set. For example, the majority, the minority, some, about half, almost none, at least a third, almost all, etc.
  
  Relative fuzzy quantifiers are defined as fuzzy sets on the segment [0, 1] of the real line, i.e., as a fuzzy number defined on that interval. To evaluate a relative quantifier, two quantities are needed: the number of elements of the original set C and the number of elements corresponding to the selected subset D . The result is also a number between 0 and 1.

Quantifiers are thus functions Q(φ) between φ and [0, 1], where φ is the number of elements (in the case of the absolute quantifier) or a proportion (in the case of the relative quantifier). If Q(φ) = 1, the quantifier is fully satisfied. If Q(φ) = 0, the quantifier is not satisfied at all. An intermediate value (between 0 and 1) indicates the degree to which the quantifier is satisfied, i.e., the degree of membership in the fuzzy set.

There are increasing quantifiers. For example:



And decreasing quantifiers. For example:



Unimodal quantifiers are those that have a membership function that is first increasing and then decreasing. For example (the first is absolute and the second is relative):



In mathematical logic there are only two quantifiers: the universal quantifier ("all elements", ∀) and the existential quantifier ("at least one element", ∃). They can be considered as particular cases of fuzzy quantifiers:




Bart Kosko: Fuzzy Thinking
Bart Kosko is a noted researcher and advocate of fuzzy logic, which he has related to the philosophy of mind. He has contributed greatly to popularizing it. His best known book is "Fuzzy Thinking" [Kosko, 1994]. Kosko is a total convert to the fuzzy doctrine, more radical even than Zadeh himself.

In the concept of truth, Kosko highlights the differences between Western philosophy versus Eastern philosophy, which he summarizes in the opposition between Buddha and Aristotle (Buddha died 100 years before Aristotle was born):

Aristotle	Buddha
A or non-A	A and non-A
All or nothing	A grade
Exact	Fuzzy
Discrete: 0 or 1	Continuous: between 0 and 1
Formal language	Natural language
Bit (binary digit)	Fit (fuzzy unit)

A bit is a binary digit (0 or 1). A fit is a degree, a value between 0 and 1.

Western philosophy, heir to Aristotle, accepted bivalence as a form of simplification of a complex reality. This philosophy has allowed remarkable advances in science, but it is an incomplete, partial philosophy. In contrast, Eastern philosophy accepts the union of opposites, the balance of the universal principles of yin and yang.

Kosko made several notable contributions to this topic, including: the concept of "fit" (the degree of membership or fuzziness), the geometric view of fuzzy sets, the clear conceptual demarcation between probability and fuzziness, fuzzy subsets, and fuzzy entropy.


The geometric view of fuzzy sets

If we have the set X={x₁, x₂}, its 2² = 4 subsets are: ∅, {x₁}, {x₂}, {x₁,x₂}. These subsets can be represented by bit strings (bivalent messages), where each bit (0 or 1) indicates non-belonging and belonging to the set X, respectively: (0, 0), (1, 0), (0, 1), (1, 1). In a two-dimensional Cartesian graphical system, these subsets are at the vertices of a square (see figure). In general, for a set of n elements, the 2ⁿ subsets are at the vertices in a hypercube of dimension n.



A fuzzy subset, e.g., A=(0.2, 0.6), is a point inside the square. The set of all fuzzy subsets are all points in the square except the 4 vertices, which correspond to crisp (non-fuzzy) subsets.

The complementary set of A is the complementary to 1 of its components: A^C = (0.8, 0.4), such that:

A∪A^C = (0.8, 0.6) ≠ X (the principle of the excluded third party is not fulfilled)
A∩A^C = (0.2, 0.4) ≠ ∅ (non-contradiction principle is not satisfied)

A subset A is fuzzy if and only if A∪A ^C ≠ X and A∩A^C ≠ ∅.

A subset A is crisp if and only if A∪A^C = X and A∩A^C = ∅.

The points corresponding to the sets A, A^C, A∪A^C and A∩A^C form the vertices of a square inside the original square (see figure).

The midpoint of the square M = (0.5, 0.5) is a special point. It corresponds to the fuzziest set of all, since it satisfies M∪M^C = < i>M∩M^C = M = M^C. It is a point equidistant from the 4 vertices of the square, and is equivalent to the metaphor of the half-full and half-empty bottle, which is where paradoxes appear.

The cardinality of a set A, represented as |A|, is the sum of its fit values:

|A| = 0.8   |A^C| = 1. 2   |A∪A^C| = 1.4   |A∩A^C| = 0.6

If we consider that A is the probability that a random variable has a value in the rectangle formed by the vertices (0, 0) and A, it holds:

P(A∪A^C) = 1   and   P(A∩A^C) = 0

Subconjunctivity

The degree of inclusion or subconjunctivity (subsethood) s of a set A in another B is a fit defined as:

s(A, B) = grado(A⊂B) = |a∩B|/|A|

This expression is reminiscent of the formula for the (Bayesian) probability of an event B conditional on an event A:

P(B|A) = P(A∩B)/P(A)

If A=(0.3, 0.3) and B=(0.5, 0.5), then s(A, B) = (0.3 + 0.3)/0.6 = 1.

If A=(0.5, 0.5) and B=(0.3, 0.3), then s(A, B) = (0.3 + 0.3)/1 = 0.6.


Fuzzy Entropy

According to Kosko, entropy is a general concept; it is not necessarily linked to probability theory. Entropy measures the uncertainty of a system or message.

Kosko introduced the concept of "fuzzy entropy," which serves to measure the degree of fuzziness of a set.

The entropy of A −symbolized by E(A)−, is the degree to which < i>A∪A^C is a subset of A∩A^C, that is, the degree to which the whole is included in the part, the degree of violation of the law of no−contradiction with respect to the law of the excluded third party:

E(A) = s(A∪A^C, A∩A^{< i>C}) = |A∩A^C|/|A∪A^C|

In the case of the set A=(0.2, 0.6), we have: E(A)= 0.6/1.4 = 0.42

The entropy of the vertices (crisp subsets) is 0, which is the minimum entropy. If A=A^C (center point), then E(A)=1, which is the maximum entropy. As we approach the center, the entropy grows and at the center the maximum entropy is obtained.


MENTAL: The Generalization of the Fuzzy
Fuzzy Universalism and Linguistics

Fuzzy theory has a universal orientation, reflected in its principle "Everything is a matter of degree". There is talk of fuzzy sets, fuzzy logic, fuzzy numbers, fuzzy operations, fuzzy rules, fuzzy algorithms, etc. But such universality in practice is very difficult to achieve because of the lack of a common formal language.

MENTAL provides a linguistic model on a common conceptual framework and a universal theory−practice that allows to express all kinds of fuzzy expressions like: fuzzy matrices, fuzzy sequences, fuzzy vectors, fuzzy equations, fuzzy predicates, fuzzy algorithms, etc. And every discipline can be formalized in a fuzzy way: fuzzy arithmetic, fuzzy geometry, fuzzy algebra, etc. This universality of MENTAL provides an integrative framework for Soft Computing techniques.

MENTAL is a model of the mind that integrates the two modes of consciousness, those associated with the cerebral hemispheres. The Aristotelian vision (of the left hemisphere, HI) and the Buddhist vision (of the right hemisphere, HD), the integration between the precise and the diffuse, between the quantitative and the qualitative, between the discrete and the continuous, between the digital and the analogical, between the superficial and the deep, between the analytical and the synthetic, between theory and practice, between the particular and the universal, and so on. What Kosko calls "Aristotle vs. Buddha" is thus really the dichotomy or duality HI−HD.

The sharp or precise is a particular case of the diffuse. The diffuse is the universal. The sharp or precise is the particular.

Fuzzy theory attempts to humanize logic and mathematics by bringing the formal closer to the conceptual or mental. But complete humanization is achieved with MENTAL and its universal semantic primitives.


Factored expressions

In MENTAL, any expression of any type (set, sequence, function, rule, procedure, vector, matrix, truth value (V), quality, etc.), can be factored. There are 3 types of factored expressions:

1. Crisp or precise expressions. These are expressions in which no qualitative values are involved. They correspond to the analytical mode of consciousness. For example,
   
   
   (3 7 5)   {a b c}   0.3*⟨( f(x y) = (x+y x*y )⟩
   
   
2. Fuzzy expressions. They are qualitative values of linguistic variables. They correspond to the synthetic mode of consciousness. For example: tall, rich, fast, young, etc.
   
   
3. Mixed expressions. They are of the type f*x, where f is a factor between 0 and 1, and x is a qualitative value, e.g., 0.7*tall, 0.3*young, etc. They correspond to the union of the two modes of consciousness (HI and HD), where the qualitative and the quantitative, the precise and the diffuse are united. This is precisely what fuzzy logic has done and which justifies its being considered a model of the human mind.

Quantitative and qualitative magnitudes

Magnitudes are of the form r*x, where r is a real number and x is a unit of measurement or a qualitative value.

If the factor r is greater than 1 and x is a unit of measurement, we have a quantitative quantity. For example, in magnitudes such as weight, height, where there is a unit of measure.

17*Kg   23*m   80*(Km÷h)

If r is less than 1 and x is a qualitative value, we have a qualitative magnitude. It is interpreted as a fraction or proportion of the expression x. For example,

0.7*high   0.5*young   0.8*hot

The factor can affect any quality, including the qualities of truth (V) or falsity (F). In this case, we have a qualitative truth magnitude. For example.

0.7*V   0.3*F

Factorization in variables and linguistic values

A linguistic (or qualitative) variable x and one of its qualitative values v is represented by x/v (the particularization of x with the value v). For example: speed/high, height/medium, age/young, etc.


Factorization in sets

In the case of the membership relation of an element a to a set C, factorization can affect the element, the set, or both. There are 8 possibilities:


Fuzzy logic operations

• Fuzzy logic negation:
  ⟨( (f*x)' = (1−f)*x )⟩
⟨( (f*x)' = f*x' )⟩
  
  For example, (0.7*high)' = 0.3*high = 0.7*low), assuming that
  (high' = low) and (low' = high)
  
  
• Fuzzy logic sum:
  ⟨( (f1*x +/d f2*x) = max(f1 f2)*x )⟩
  
  
• Fuzzy logic product:
  ⟨( (f1*x +/d f2*x) = min(f1 f2)*x )⟩
  
  where d is the "fuzzy" predicate applied to the sum operator, and max and min the maximum and minimum functions, respectively.
  
  ⟨( max(f1 f2) = f1 ← f1b>>f2 →' f2) )⟩
⟨( min(f1 f2) = f1 ← f1<f2 →' f2) )⟩
  
  For example:
  
  (0.3*a +/d 0.8*a) // ev. 0.8*a
(0.3*a */d 0.8*a) // ev. 0.3*a
(0.3 +/d 0.8) // ev. 0.8
(0.3 */d 0.8) // ev. 0.3
  
  
• Fuzzy logic implication.
  In MENTAL it makes no sense to assign a truth value to the implication because it is a "condition → action" type expression. Therefore, there is only one interpretation: when the condition expression exists, the action is evaluated. [see Applications - Logic - The Problem of Implication].

Operations with fuzzy sets

• Complementary fuzzy set:
  
  ⟨( (C('/d) = C/⟨(r*x = (1−r)*x)⟩ )⟩
  
  
• Fuzzy union of sets:
  
  ⟨( (C1 ∪/d C2) = (C1 ∪/d C2)/g1 )⟩
  
  being ( g1 = ⟨( (r1*x r2*x)↓ = max(r1 r2)*x )⟩ )

( C1 = {0.3*a 0.6*b 0.2*c} )
( C2 = {0.5*a 0.7*b} )
(C1 ∪/d C2) // ev. (0.5*a 0.7*b 0.2*c)
  
  
• Fuzzy set intersection:
  
  ⟨( (C1 ∩/d C2) = (C1 ∩/d C2)/g2 )⟩
  
  being ( g2 = ⟨( (r1*x r2*x)↓ = min(r1 r2)*x )⟩ )

( C1 = {0.3*a 0.6*b 0.2*c} )
( C2 = {0.5*a 0.7*b] )
(C1 ∩/d C2) // ev. (0.3*a 0.6*b)

Other fuzzy operations

• Vector sum and fuzzy vector sum:
  
  ⟨( (s1 +/v s2) = ( [⌊s1↓⌋ + ⌊s2↓⌋] ) )⟩
⟨( (s1 +/vd s2) = ( [⌊s1↓⌋ +/d ⌊s2↓⌋] ) ) )⟩
  
  Examples:
  
  (g1 = (0.3*a 0.6*b))
(g2 = (0.5*a 0.7*b))
(g1 +/v g2) // ev. (0.8*a 1.3*b)
(g1 +/vd g2) // ev. (0.5*a 0.7*b)
  
  
• Matrix sum and fuzzy matrix sum (for 2-dimensional matrices):
  
  ⟨( (m1 +/m m2) = ( [⌊m1↓⌋ +/v ⌊m2↓⌋] ) )⟩
⟨( (m1 +/md m2) = ( ( [⌊m1↓⌋ +/vm ⌊m2↓⌋]) ) )⟩
  
  Examples:
  
  ( ((1 2) (3 4)) +/m ((5 6) (7 8)) ) // ev. ((6 8) (10 12))
( ((1 2) (3 4)) +/md ((5 6) (7 8)) ) // ev. ((5 6) (7 8))

Example of fuzzy control system: fan

We can integrate the 3 phases of a fuzzy control system (fuzzification, evaluation of the fuzzy rules and defuzzification) into a single expression: an input function (the temperature t) and output the angular velocity w of the fan:

// control function. Input: t (temperature). Output: w (fan speed)
//
// auxiliary function for the calculation of the factor associated to each type of temperature.
//
⟨( y(x x1 y1 y1 x2 y2) = y1 + (y2 - y1)*(x − x1)÷(x2 − x1 )⟩
//
// reference values
//
(t1=10 t2=20 t3=30) // temperatures
(w1=30 w2=60 w3=120) // fan speeds
//
// rules
//
( r1 =: (t/Cold → w/Minimum) ) // rule 1

( r2 =: (t/Weather → w/Average) ) // rule 2

( r3 =: (t/Hot → w/Maximum) ) // rule 3

//
// fuzzification: Obtaining the qualitative values as a function of the temperature t
//
( t≤t1 → (f1=1 f2=0 f3=0) )

( (t>t1 ∧ t≤t2) → (f1=y(t t1 1 t2 0) f2=(y(t t1 0 t2 1) f3=0))) )

( (t>t2 ∧ t≤t3) →
(f1=0 f2=(y(t t2 1 t3 0) f3=(y(t t2 0 t3 1))) )

( t>t3 → (f1=0 f2=0 f3=1) ) )
(f1>0 → t/Cold) (f2>0 → t/Warm) (f3>0 → t/Hot)
//
// evaluation of the rules
//
(r1 r2 r3)
//
// defuzzification
//
( w = (w1*f1 + w2*f2 + w3*f3)÷(f1+f2+f3) )
¡w)! )⟩

---

Addenda
Possibility theory

Zadeh "fuzzyfied" probability and turned it into fuzzy probability, a generalization of probability theory that he called "possibility theory." Both theories (fuzzy logic and fuzzy probability) deal with uncertainty. There are 3 types of uncertainty:

1. The randomness of events. For example, rolling a die and coming up with 6. This is the field of probability.
   
   
2. Imprecision, caused by vagueness or ambiguity. For example, fuzzy predicates high, low, fast, slow, old, young, etc. that induce fuzzy sets.
   
   
3. A randomness with imprecision. For example, "whether or not it rains tomorrow", in which randomness (whether or not it rains) and imprecision (there are degrees of rain: fine rain, thunderstorm, etc.) appear.

This last alternative is the most generic and is the one studied by possibility theory.


MIQ (Machine Intelligence Quotient)

Fuzzy systems emulate the human mind, which is why Zadeh proposed the concept of the "intelligence quotient" of a system or machine. MIQ is a registered trademark of Zadeh.


Fuzzy logic applications

Fuzzy logic has been enormously successful because of its conceptual simplicity, ease of application, and effectiveness. Although Zadeh's initial intention was to create a formalism for human reasoning using imprecise concepts, it came as something of a surprise when success came in the field of automatic control of systems and processes.

Fuzzy control (or fuzzy control) systems are more efficient and consume less energy than classical systems. Today, a large number of product patents are based on fuzzy logic.

The first industrial application was the control of a cement kiln in Denmark in 1980 by F.L. Smidth&Co.

In the U.S., fuzzy systems were initially ignored because they were associated with artificial intelligence, a field lacking credibility after the disappointment of the expectations raised, especially in the 1980s.

It was the Japanese who first became interested in fuzzy logic, applying it to industrial and consumer systems:

• The Hitachi company, in 1985, applied it to the control systems of the subway trains in the Japanese city of Sendai, on the subject of acceleration, deceleration (braking) and stopping. The system has 59 rules. All are triggered to some degree as a function of speed. The ride proved to be smoother than all previously known, both human and mechanical. Braking accuracy was 7 cm, while that performed by a driver usually exceeds 20 cm. Accelerations and decelerations proved to be much smoother. The number of gear changes was reduced to one third of those of human driving or those controlled by non-fuzzy systems. This also resulted in energy savings of 10 %. The control system was inaugurated in 1987. Given its success, Hitachi was soon contracted by Tokyo City Hall to install fuzzy control systems in the Tokyo subway as well.
  
  
• In 1987 it was applied for fuzzy control in an "inverted pendulum" experiment, a classic control problem in which a vehicle tries to maintain the balance of a mast by means of a vertical hinge, moving it back and forth.
  
  
• Other Japanese applications of fuzzy control included: cameras with autofocus and image stabilization; televisions that adjust volume and brightness according to ambient noise and light; intelligent washing machines that selected the best washing program according to the amount of laundry and degree of soiling; microwaves that adjust according to ambient humidity; air conditioning systems, which regulate temperature according to ambient temperature; showers that control temperature by combining hot and cold water; manufacturing robots; etc.

Other applications, in the USA and Europe, are: energy-efficient electric motors; docking (docking) automatic space; control of gates in hydroelectric plants, expert systems; auto−learning systems; automatic gear changes, ABS braking control and cruise control in cars; pattern recognition; handwriting recognition; voice-controlled robots; traffic control; thermal power plants; computer vision (object tracking by video camera); vehicle control system (airplanes, elevators, helicopters); etc.


The diffusion of fuzzy logic

Nowadays, fuzzy logic is a very wide and important field of research, both at the theoretical and practical level. There are many specialized publications. Two journals stand out: "Fuzzy Sets and Systems" and "IEEE Transactions on fuzzy Systems". International congresses are organized. There are several associations. In Spain there is the Spanish Association of Fuzzy Logic.

In the push for fuzzy logic the Japanese stand out. The Japanese government has participated in the creation of two research centers: LIFE (Laboratory for International fuzzy Engineering), created in 1985, and the FLSI (fuzzy Logic Systems Institute), created in 1990. Both centers organize conferences every two years.


Critiques of fuzzy logic

Zadeh's unorthodox ideas were initially greeted with some skepticism, but progressively gained acceptance until today, when success is considered complete and utter. But along the way there were many criticisms:

General criticisms:

• The concept of "partial truth" has been the subject of controversy. Some logicians hold that truth is something absolute, which have no degrees.
  
  
• Partial membership in a set or category is a subjective issue, since there is no way to objectively determine the values of partial membership. For example, for one person a height of 1.65 meters may be short and for another it may be normal (neither tall nor short).

Authors' criticisms:

• Quine once described alternatives to bivalent logic as "deviant" and called some literature on fuzzy logic "irresponsible."
  
  
• Dennis Lindley, a Bayesian statistician, stated, "Probability is the only sensible description of uncertainty and is adequate for all problems involving uncertainty. All other methods are inadequate."
  
  
• "Blur is probability in disguise. I can design a controller with probability that can do the same things that can be done with fuzzy logic" (Mylon Tribus, on learning of its application to the Sendai subway, Japan).
  
  
• "The danger of fuzzy logic is that it gives wings to that kind of imprecise thinking that has brought us so many problems. Fuzzy logic is the cocaine of science" (William Kahan).
  
  
• "Fuzzy logic is a kind of scientific permissiveness. It tends to end up in socially attractive slogans that are not accompanied by the discipline of hard scientific work and patient observation" (Rudolf Kalman).

Fuzzy logic and the Internet

Fuzzy logic is well suited to represent knowledge, especially on the Internet. At Zadeh's own initiative, the first meeting on fuzzy logic and the Internet (FLIT 2001) was held at the University of Berkeley in 2001. The papers are published on the BISC website. The main idea is to use Soft Computing techniques for knowledge representation, information categorization, fuzzy search, autonomous intelligent agents, data mining, etc.


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