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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Provides readers with the foundations of fuzzy mathematics as well as more advanced topics A Modern Introduction to Fuzzy Mathematics provides a concise presentation of fuzzy mathematics., moving from proofs of important results to more advanced topics, like fuzzy algebras, fuzzy graph theory, and fuzzy topologies. The authors take the reader through the development of the field of fuzzy mathematics, starting with the publication in 1965 of Lotfi Asker Zadeh's seminal paper, Fuzzy Sets. The book begins with the basics of fuzzy mathematics before moving on to more complex topics, including: * Fuzzy sets * Fuzzy numbers * Fuzzy relations * Possibility theory * Fuzzy abstract algebra * And more Perfect for advanced undergraduate students, graduate students, and researchers with an interest in the field of fuzzy mathematics, A Modern Introduction to Fuzzy Mathematics walks through both foundational concepts and cutting-edge, new mathematics in the field.
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Table of Contents
Cover
Preface
Reference
Acknowledgments
1 Introduction
1.1 What Is Vagueness?
1.2 Vagueness, Ambiguity, Uncertainty, etc.
1.3 Vagueness and Fuzzy Mathematics
Exercises
Notes
2 Fuzzy Sets and Their Operations
2.1 Algebras of Truth Values
2.2 Zadeh's Fuzzy Sets
2.3 ‐Cuts of Fuzzy Sets
2.4 Interval-valued and Type 2 Fuzzy Sets
2.5 Triangular Norms and Conorms
2.6 ‐fuzzy Sets
2.7 “Intuitionistic” Fuzzy Sets and Their Extensions
2.8 The Extension Principle
2.9* Boolean‐Valued Sets
2.10* Axiomatic Fuzzy Set Theory
Exercises
Notes
3 Fuzzy Numbers and Their Arithmetic
3.1 Fuzzy Numbers
3.2 Arithmetic of Fuzzy Numbers
3.3 Linguistic Variables
3.4 Fuzzy Equations5
3.5 Fuzzy Inequalities
3.6 Constructing Fuzzy Numbers
3.7 Applications of Fuzzy Numbers
Exercises
Notes
4 Fuzzy Relations
4.1 Crisp Relations
4.2 Fuzzy Relations
4.3 Cartesian Product, Projections, and Cylindrical Extension
4.4 New Fuzzy Relations from Old Ones
4.5 Fuzzy Binary Relations on a Set
4.6 Fuzzy Orders
4.7 Elements of Fuzzy Graph Theory
4.8 Fuzzy Category Theory
4.9 Fuzzy Vectors5
4.10 Applications
Exercises
Notes
5 Possibility Theory
5.1 Fuzzy Restrictions and Possibility Theory
5.2 Possibility and Necessity Measures
5.3 Possibility Theory
5.4 Possibility Theory and Probability Theory
5.5 An Unexpected Application of Possibility Theory
Exercises
Note
6 Fuzzy Statistics
6.1 Random Variables
6.2 Fuzzy Random Variables
6.3 Point Estimation
6.4 Fuzzy Point Estimation
6.5 Interval Estimation
6.6 Interval Estimation for Fuzzy Data
6.7 Hypothesis Testing
6.8 Fuzzy Hypothesis Testing
6.9 Statistical Regression
6.10 Fuzzy Regression
Exercises
Notes
7 Fuzzy Logics
7.1 Mathematical Logic
7.2 Many‐Valued Logics
7.3 On Fuzzy Logics
7.4 Hájek's Basic Many‐Valued Logic
7.5 Łukasiewicz Fuzzy Logic
7.6 Product Fuzzy Logic
7.7 Gödel Fuzzy Logic
7.8 First‐Order Fuzzy Logics
7.9 Fuzzy Quantifiers
7.10 Approximate Reasoning
7.11 Application: Fuzzy Expert Systems
7.12 A Logic of Vagueness
Exercises
Notes
8 Fuzzy Computation
8.1 Automata, Grammars, and Machines
8.2 Fuzzy Languages and Grammars
8.3 Fuzzy Automata
8.4 Fuzzy Turing Machines
8.5 Other Fuzzy Models of Computation
9 Fuzzy Abstract Algebra
9.1 Groups, Rings, and Fields
9.2 Fuzzy Groups
9.3 Abelian Fuzzy Subgroups
9.4 Fuzzy Rings and Fuzzy Fields
9.5 Fuzzy Vector Spaces
9.6 Fuzzy Normed Spaces
9.7 Fuzzy Lie Algebras
Exercises
Note
10 Fuzzy Topology
10.1 Metric and Topological Spaces1
10.2 Fuzzy Metric Spaces
10.3 Fuzzy Topological Spaces6
10.4 Fuzzy Product Spaces
10.5 Fuzzy Separation
10.6 Fuzzy Nets
10.7 Fuzzy Compactness
10.8 Fuzzy Connectedness
10.9 Smooth Fuzzy Topological Spaces
10.10 Fuzzy Banach and Fuzzy Hilbert Spaces
10.11 Fuzzy Topological Systems
Exercises
Notes
11 Fuzzy Geometry
11.1 Fuzzy Points and Fuzzy Distance
11.2 Fuzzy Lines and Their Properties
11.3 Fuzzy Circles
11.4 Regular Fuzzy Polygons
11.5 Applications in Theoretical Physics
Exercises
12 Fuzzy Calculus
12.1 Fuzzy Functions
12.2 Integrals of Fuzzy Functions
12.3 Derivatives of Fuzzy Functions
12.4 Fuzzy Limits of Sequences and Functions
Exercises
A Fuzzy Approximation
A.1 Weierstrass and Stone–Weierstrass Approximation Theorems
A.2 Weierstrass and Stone–Weierstrass Fuzzy Analogs
Note
B Chaos and Vagueness
B.1 Chaos Theory in a Nutshell
B.2 Fuzzy Chaos
B.3 Fuzzy Fractals
Notes
Works Cited
Subject Index
Author Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 A table representation of the type 2 fuzzy set of Example 2.4.1.
Chapter 5
Table 5.1 Intention to engage in political violence (IEPV) and approval of politi...
List of Illustrations
Chapter 1
Figure 1.1 A drawing depicting an electron.
Chapter 3
Figure 3.1 A triangular fuzzy number.
Figure 3.2 A trapezoidal fuzzy number.
Figure 3.3 A Gaussian fuzzy number.
Figure 3.4 A quadratic fuzzy number.
Figure 3.5 An exponential fuzzy number.
Figure 3.6 An
–
fuzzy number.
Figure 3.7 The fuzzy numbers of example 3.2.3.
Figure 3.8 Room temperature as a linguistic variable quantified by some ling...
Chapter 4
Figure 4.1 A sagittal diagram of the fuzzy relation in Example 4.2.1.
Figure 4.2 A directed graph that depicts a fuzzy relation.
Figure 4.3 A typical hypergraph.
Figure 4.4 Types of fuzzy graphs.
Chapter 5
Figure 5.1 A scatter plot.
Chapter 7
Figure 7.1 Membership function of readings of a digital voltmeter.
Chapter 8
Figure 8.1 A typical Turing machine.
Guide
Cover
Table of Contents
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A Modern Introduction to Fuzzy Mathematics
Apostolos Syropoulos
Theophanes Grammenos
New York 2020
This edition first published 2020
© 2020 John Wiley & Sons, Inc
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Library of Congress Cataloging‐in‐Publication Data
Names: Syropoulos, Apostolos, author.
Title: A modern introduction to fuzzy mathematics / Apostolos Syropoulos, Theophanes Grammenos.
Description: First edition. | New York : Wiley, 2020. | Includes bibliographical references and index.
Identifiers: LCCN 2020004300 (print) | LCCN 2020004301 (ebook) | ISBN 9781119445289 (cloth) | ISBN 9781119445302 (adobe pdf) | ISBN 9781119445296 (epub)
Subjects: LCSH: Fuzzy mathematics.
Classification: LCC QA248.5 .S97 2020 (print) | LCC QA248.5 (ebook) | DDC 511.3/13—dc23
LC record available at https://lccn.loc.gov/2020004300
LC ebook record available at https://lccn.loc.gov/2020004301
Cover Design: Wiley
Cover Image: Courtesy of Apostolos Syropoulos
To my son Demetrios‐Georgios, to Koula and Linda. Apostolos Syropoulos
To my wife Andromache for her endurance, tolerance, and care. Theophanes Grammenos
Preface
In mathematics, we investigate the properties of abstract objects (e.g. numbers, geometric shapes, spaces) and the possible relationships between these objects. One basic characteristic of mathematics is that all these properties and relationships are absolute. Thus, properties and relationships are either true or false. Nothing else is meaningful. In other words, in mathematics there is no room for vagueness, for randomness, and for extremely small quantities. By introducing one of these qualities into mathematics, one can create alternative mathematics [1]. But how do we introduce vagueness into mathematics? One very simple way to achieve this is to allow notions like “small,” “large,” and “few.” However, another way is to modify the most basic object of mathematics, that is, to modify sets. In this respect, fuzzy mathematics is a form of alternative mathematics since it is based on a generalization of set membership. Simply put, in fuzzy mathematics, an element may belong to a degree to a set, while in ordinary mathematics, it either belongs or does not belong to a set. This simple idea has been applied to most fields of mathematics and so we can talk about fuzzy mathematics.
Even today, many researchers and thinkers consider fuzzy mathematics as a tool that can be used instead of probability theory to reason about or to work with a specific system. This text is based on the idea that vagueness is a basic notion and thus tries to present fuzzy mathematics as a form of alternative mathematics and not as an alternative to probability theory. In addition, this text is an introduction to fuzzy mathematics. This simply means that we have tried to cover as many fields of fuzzy mathematics as possible. Thus, this text is a compendium but not a handbook of fuzzy mathematics.
Chapter 1 of this book explains what vagueness means from a philosophical point of view. Also, it demonstrates the connection between vagueness and fuzzy mathematics.
In Chapter 2, we introduce the notion of fuzzy set as well as the basic set operations. In addition, we introduce a number of variants or extensions of fuzzy sets. The chapter concludes with a section marked with a star. There are a few such sections in the book and these are optional readings as they deal with quite advanced ideas.
Fuzzy number are special kinds of fuzzy sets that, in a way, have been introduced to generalize the notion of a number. In Chapter 3, we present various forms of fuzzy numbers and the basic arithmetic operations between them. We introduce linguistic variables, that is, terms such as “small,” “heavy,” “tall.” Also, we present fuzzy equations (e.g. simple equations like , where known and unknown quantities are fuzzy numbers) and how one can solve them. The chapter concludes with applications of fuzzy numbers. In general, most chapters that follow have a final section that presents applications of the main material presented in the specific chapter.
Fuzzy relations are a very important subject that is presented in Chapter 4. We discuss fuzzy relations, the Cartesian product of fuzzy sets and related notions, and fuzzy orders. Since graphs can be described by relations, we also discuss fuzzy graphs. Also, since precategories are also described by graphs, we discuss fuzzy categories.
Chapter 5 is devoted to possibility theory, that is, generally speaking, the fuzzy “version” of a probability theory. Also, we compare probability and possibility theories in order to see their differences.
Chapter 6 discusses fuzzy statistics. In this chapter, we discuss fuzzy random variables (in a way as something that corresponds to vague randomness …) and all related notions such as fuzzy regression and fuzzy point estimation.
In Chapter 7, we discuss many‐valued and fuzzy logics. We do not just present truth values and the basic logical operations, but we present complete logical systems. In addition, we discuss approximate reasoning and try to see what is a logic of vagueness.
Although computability theory is a very basic part of logic, we discuss in a separate chapter fuzzy computation (Chapter 8). In particular, we discuss fuzzy automata, fuzzy Turing machines, and other fuzzy models of computation.
In Chapter 9, we give a taste of fuzzy abstract algebra theory. We present fuzzy groups, fuzzy rings, fuzzy vector spaces, fuzzy normed spaces, and fuzzy Lie algebras.
Chapter 10 introduces the basic notions and ideas of fuzzy metric spaces and fuzzy topology. In addition, we briefly discuss fuzzy Banach spaces and fuzzy Hilbert spaces.
Fuzzy geometry is introduced in Chapter 11. We discuss the notion of fuzzy points and the distance between them, fuzzy lines, fuzzy circles, and fuzzy polygons.
Chapter 12 introduces the reader to fuzzy calculus. In particular, we discuss fuzzy functions, integrals and derivatives of fuzzy functions, and fuzzy limits of sequences and function. Furthermore, fuzzy (ordinary and partial) differential equations are presented.
The book includes two appendices: the first briefly presents fuzzy approximation and the second gives a taste of fuzzy chaos and fuzzy fractals.
Each chapter starts with a section that describes the nonfuzzy concepts whose fuzzy counterparts are presented in the rest of the chapter. We felt this was necessary since we present many and quite diverse topics and we cannot expect everyone to be familiar with all these notions and ideas. Also, most chapters have some exercises at the end. Readers are invited to work on them if they want to deepen their understanding of the ideas presented in the corresponding chapter. However, the chapter on fuzzy computation does not include exercises since the subject is not mature enough.
Apostolos SyropoulosXanthi, Greece, November 2019
Theophanes GrammenosVolos, Greece, November 2019
Reference
1
Van Bendegen, J.P. (2005). Can there be an alternative mathematics, really? In:
Activity and Sign: Grounding Mathematics Education
(ed. M.H. Hoffmann, J. Lenhard, and F. Seeger), 349–359. Boston, MA: Springer US.
Acknowledgments
We would like to thank Kathleen Pagliaro who believed in this project and helped us in every possible way to realize it. Also, we thank Christina E. Linda, our project editor, for her help and assistance. In addition, we thank Andromache Spanou for carefully reading drafts of the text and suggesting improvements and corrections. Also, we thank Athanasios Margaris and Basil K. Papadopoulos for reading parts of the book and making comments and suggestions.
1Introduction
Vagueness is a fundamental property of this world. Vague objects are real objects and exist in the real world. Fuzzy mathematics is mathematics of vagueness. The core of fuzzy mathematics is the idea that objects have a property to some degree.
1.1 What Is Vagueness?
When we say that something is vague, we mean that its properties and capacities are not sharply determined. In different words, a vague concept is one that is characterized by fuzzy boundaries (i.e. there are cases where it is not clear if an object has or does not have a specific property or capacity). Jiri Benovsky [27] put forth an objection to this idea by claiming that everybody who thinks that there are ordinary objects must accept that they are vague, whereas everybody must accept the existence of sharp boundaries to ordinary objects. This does not lead to a contradiction since the two claims do not concern the same “everybody”.
The Sorites Paradox (σόφισμα τοῦ σωρείτη), which was introduced by Eubulides of Miletus (Eὐβουλίδης ὁ Mιλήσιος),1 is a typical example of an argument that demonstrates what fuzzy boundaries are. The term “σωρείτες” (sorites) derives from the Greek word σωρός (soros), which means “heap.” The paradox is about the number of grains of wheat that makes a heap. All agree that a single grain of wheat does not comprise a heap. The same applies for two grains of wheat as they do not comprise a heap, etc. However, there is a point where the number of grains becomes large enough to be called a heap, but there is no general agreement as to where this occurs. Although there is no precise definition of vagueness, still most people would agree that adjectives such as tall, old, short, and young, express vague concepts since, for instance, a person who is 6 years old is definitely young but can we say the same for a person who is 30 years old? Moreover, there are objects that one can classify as vague. For example, a cloud is vague since its boundaries are not sharp. Also, a dog is a vague object since it loses hair all the time and so it is difficult to say what belongs to it.
To a number of people, these arguments look like sophisms. Others consider vagueness as a linguistic phenomenon, that is, something that exists only in the realm of natural languages and gives us greater expressive power. And there are others that think that vagueness is a property of the world. In summary, there are three views regarding the nature of vagueness2 : the ontic view, the semantic view, and the epistemic view. According to the ontic view the world itself is vague and, consequently, language is vague so to describe the world. The semantic view asserts that vagueness exists only in our language and our thoughts. In a way, this view is similar to the mental constructions of intuitionism, that is, things that exist in our minds but not in the real world. On the other hand, the epistemic view asserts that vagueness exists because we do not know where the boundaries exist for a “vague” concept. So we wrongly assume they are vague. In this book, we assume that onticism about vagueness is the right view. In different words, we believe that there are vague objects and that vagueness is a property of the real world. It seems that semanticism is shared by many people, engineers in particular who use fuzzy mathematics, while if epistemicism is true, then there is simply no need for fuzzy mathematics, and this book is useless.
Let us consider countries and lakes. These geographical objects do not have sharply defined boundaries since natural phenomena (e.g. drought or heavy rainfalls) may alter the volume of water contained in a lake. Thus, one can think these are vague objects. Nevertheless, vagueness can emerge from other unexpected observations. In 1967, Benoît Mandelbrot [208] argued that the measured length of the coastline of Great Britain (or any island for that matter) depends on the scale of measurement. Thus, Great Britain is a vague object since its boundaries are not sharp. Nevertheless, one may argue that here there is no genuine geographical vagueness, instead this is just a problem of representation. A response to this argument was put forth by Michael Morreau [224]. Obviously, if the existence of vague objects is a matter of representation, then there are obviously no vague objects including animals. Consider Koula the dog. Koula has hair that she will lose tonight, so it is a questionable part of her. Because Koula has many such questionable parts (e.g. nails, whisker), she is a vague dog. Assume that Koula is not a vague dog. Instead, assume that there are many precise mammals that must be dogs because they differ from each other around the edges of the hair. Obviously, all these animals are dogs that differ slightly when compared to Koula. All of these candidates are dogs, and they have very small differences between them. If vagueness is a matter of representation, then, wherever I own a dog, I own at least a thousand dogs. Clearly, this is not the case.
Gareth Evans [119] presented an argument that proves that there are no vague objects. Evans used the modality operator to express indeterminacy. Thus, is read as it is indeterminate whether. The dual of is the operator and is read as it is determinate that. Evans started his argument with the following premise:
This means that it is true that it is indeterminate whether and are identical. Next, he transformed this expression to an application of some sort of λ‐abstraction:
Of course, it is a fact that it is not indeterminate whether is identical to :
Using this “trick” to derive formula 1.2, one gets
Finally, he used the identity of indiscernibles principle to derive from 1.2 and 1.4:
meaning that and are not identical. So we started by assuming that it is indeterminate whether and are identical and concluded that they are not identical. In different words, indeterminate identities become nonidentities, which makes no sense, therefore, the assumption makes no sense. The identity of indiscernibles principle (see [125] for a thorough discussion of this principle) states that if, for every property , object has if and only if object has , then is identical to . This principle was initially formulated by Wilhelm Gottfried Leibniz.
A first response to this argument is that the logic employed to deliver this proof is not really adequate. Francis Jeffry Pelletier [240] points out that when one says that an object is vague, then this means that there is a predicate that neither applies nor does not apply to it. Thus when you have a meaningful predicate , it makes no sense to make it indeterminate by just prefixing it with the operator. Although this logic is not appropriate for vagueness, still this does not refute Evan's argument.
Edward Jonathan Lowe [196] put forth an argument that is a response to Evans' “proof”:
Suppose (to keep matters simple) that in an ionization chamber a free electron is captured by a certain atom to form a negative ion which, a short time later, reverts to a neutral state by releasing an electron . As I understand it, according to currently accepted quantum‐mechanical principles there may simply be no objective fact of the matter as to whether or not is identical with .
The idea behind this example is that “identity statements represented by ‘’ are ‘ontically’ indeterminate in the quantum mechanical context” [126] (for a thorough discussion of the problem of identity in physics see [127]). Lowe's argument prompted a series of responses, nevertheless, we are not going to describe them here and the interested reader can read a summary of these responses in [83]. In a way, these responses culminated to a revised à la Evans proof based on Lowe's initial argument:
(i) At
,
(
has been emitted).
(ii) So at
,
(
has been emitted)
.
(iii) But at
,
(
has been emitted).
(iv) So at
,
(
has been emitted)
.
(v) Therefore,
.
It is possible to provide a reinterpretation of quantum mechanics that does not use probabilities but possibilities instead [277]. This reinterpretation assumes that vagueness is a fundamental property of the world. Although the ideas involved are very simple, still they require a good background in quantum mechanics and in ideas that we are going to present in this book.
1.2 Vagueness, Ambiguity, Uncertainty, etc.
We explained that a vague concept is one that is characterized by fuzzy boundaries, still this is not a precise definition as the Sorites Paradox has demonstrated. Of course, one could say that it is an oxymoron to expect a precise definition which by its very nature is not precise. Not so surprisingly, at least to these authors, is the definition that is provided by Otávio Bueno and Mark Colyvan [51]:
Definition 1.2.1 A predicate is vague just in case it can be employed to generate a sorites argument.
To be fair, even the authors admit that this is not a new idea, but they were the first to systematically defend this idea. This definition of vague predicates is very useful in order to distinguish vagueness from ambiguity and uncertainty.
The term ambiguity refers to something that has more than one possible meaning, which may cause confusion. Of course, we encounter ambiguity only in language, but if vagueness is omnipresent, then it is definitely present in language. And this is the reason why it is confused with ambiguity. Examples of ambiguous sentences include Sarah gave a bath to her dog wearing a pink T‐shirt and Mary ate the biscuits on the couch. The first sentence is ambiguous because it is not clear who was wearing the pink T‐shirt: Sarah or the dog? The second sentence is ambiguous because it is not clear if the biscuits were on the couch or if Mary brought them and ate them on the couch. Of course, one can come up with many more examples of ambiguous sentences, but none of them can be employed to generate a sorites paradox.
Imprecision is a notion that is closer to vagueness than ambiguity. Typically, we employ this term when talking about things whose boundaries are not precise. Of course, not precise does not mean fuzzy. It would be sufficient to say that we encounter imprecision when our tools are not precise enough to perform a given task (e.g. measuring the length of a tiny object).
Generality is yet another word that is close to vagueness. General terms are words like “chair.” There are many and different kinds of chairs, nevertheless, all the chairs that one may imagine are still chairs. Thus, in generality we focus on a few common characteristics and use them to make up classes of objects.
It is not widely accepted that chance plays an important role in nature. There are people who believe that our universe is completely deterministic, but there are others who assume that the universe is nondeterministic. In such a universe, randomness has a central role to play. In a way, randomness is a guarantee that we have no way to say what will happen next. Naive probability theory [i.e. not the one formulated by Andrei Nikolaevich Kolmogorov [179] (Анрей Никоаевич Комогоров)] is an attempt to make predictions about future events. Naturally, these predictions depend on the assumptions we make. But then the real problem is to what extent these assumptions are meaningful. Certainly, we will not discuss what meaningful actually means…
One more term that is remotely related to vagueness is uncertainty. This notion is related to a situation where the consequences, the extent of circumstances, the conditions, or the events are unpredictable. In quantum mechanics, the uncertainty principle is, roughly, about our inability to accurately measure at the same time the location and the momentum of a subatomic particle (e.g. an electron). Figure 1.1 shows how an electron looks like at any given moment. Strictly speaking, an electron is a wave and not a “solid” particle, but for the sake of the argument, we can assume that it is a sphere that vibrates extremely fast. So an electron would be a vague object if it would be in all these positions at the same time, but the uncertainty springs from the fact that it moves so fast that we cannot spot its place. In reality, an electron as a wave is in all places at the same time. Moreover, one can say that an electron is actually a cloud, so an electron is actually a vague object. Interestingly, there is a deep connection between uncertainty and vagueness through possibility theory.
Figure 1.1 A drawing depicting an electron.
When Lotfi Aliasker Zadeh, the founder of fuzzy mathematics, introduced fuzzy sets [305], he justified his work by using examples of vague concepts like “the class of beautiful women,” and “the class of tall men.” Later on, people working on fuzzy mathematics were divided into two groups: those who believe that fuzzy mathematics are mathematics of vagueness, and those who believe that fuzziness and vagueness are two different things. Members of the second group argue that there is some sort of misunderstanding between the two communities:
One of the reasons for the misunderstanding between fuzzy sets and the philosophy of vagueness may lie in the fact that Zadeh was trained in engineering mathematics, not in the area of philosophy.
Dubois (2012) [108]
We find this argument completely silly as there is no engineering mathematics, but just mathematics. Also, one cannot compare (presumably applied) mathematics with philosophy as one cannot compare apples with flowers. On the other hand, we are in favor of an idea that is closer to what the first group is advocating.
1.3 Vagueness and Fuzzy Mathematics
Conferences and workshops are usually very interesting events. Typically, one has the chance to meet people and discuss new ideas. Some years ago, the first author was invited to talk about fuzzy computing in a workshop. After the talk, he had a chat with someone who really liked the idea of vagueness in computing. She noted that even our hardware is vague since we assume that it operates within a specific range, but in fact this is just an approximation that makes our life easier. Of course, hardware is not vague for this reason, but it gives an idea of why vagueness actually matters.
If we accept the ontic view of vagueness, then we must explore its use in science. This simply means to recognize that objects are vague and therefore their properties are not exact. For example, one of the first “applications” of vagueness was the definition of fuzzy algorithms and fuzzy conceptual computing devices (see [275] for details). In addition, vagueness should be used in order to provide alternative (and possibly better?) interpretations of quantum phenomena and all sciences that are affected by them (e.g. chemistry and biology at the molecular level). But even if we accept the semantic view, then vagueness should be relevant to law.
In our own opinion, vagueness is a fundamental property of this world. On the other hand, all sciences use the language of mathematics to express laws, make predictions, etc. Thus, if one wants to introduce vagueness in an “exact” science, there is a need for a new language. The question, of course, is: What kind of language should this be? Many people agree that sets can be used to describe the universe of all mathematical objects. Others believe that categories can play this role. And there are some who suggest that types should play this role. By extending the properties of the most fundamental building block, one can create new mathematics. Zadeh [305] extended the notion of set membership and so he defined his fuzzy sets. More specifically, an element belongs to a fuzzy set to some degree that is usually a number between 0 and 1. The next question is how this extension introduces vagueness into mathematics?
Consider a cloud, which is a vague object. At a given moment, we can consider that the cloud is a strange solid object in three‐dimensional space. Obviously, there are points that are definitely inside the cloud and points that are definitely outside the cloud. However, there are points for which we cannot say with certainty whether they are inside or outside the cloud. For these points, one can employ linguistic modifiers, that is, words such as “more”, or “less”, to express their membership degree or one can use a number that clearly expresses to what extent a point belongs to the cloud. In summary, fuzzy sets are a quantitative way to describe borderline cases. At this point, let us say that one can approach borderline cases by using two nonvague solids: one that is contained in the vague object and the other that contains the vague object. These two nonvague objects are called the lower and the upper approximation, and they form a rough set. These sets have been introduced by Zdzisław Pawlak [237].
Saunders Mac Lane, who cofounded category theory with Samuel Eilenberg, noted that “problems, generalizations, abstraction and just plain curiosity are some of the forces driving the development of the Mathematical network” [201, pp. 438–439]. Then, he goes on as follows [201, pp. 439–440]:
Not all outside influences are really fruitful. For example, one engineer came up with the notion of a fuzzy set—a set where a statement of membership may be neither true nor false but lies somewhere in between, say between 0 and 1. It was hoped that this ingenious notion would lead to all sorts of fruitful applications, to fuzzy automata, fuzzy decision theory and elsewhere. However, as yet most of the intended applications turn out to be just extensive exercises, not actually applicable; there has been a spate of such exercises. After all, if all Mathematics can be built up from sets, then a whole lot of variant (or, should we say, deviant) Mathematics can be built by fuzzifying these sets.
Of course, the fact that there are books like this one means that Mac Lane's prediction was not verified by reality. And yes, there are fuzzy automata and fuzzy decision theory is an established interdisciplinary topic. However, Mac Lane was not alone, as many researchers and scholars have expressed negative views about fuzzy sets. One obviously should learn to listen to the opposite view and try to understand the relevant arguments. But there are cases where this is simply impossible. A few years ago, the first author submitted a paper to a prestigious journal devoted to the international advancement of the theory and application of fuzzy sets. The paper described a model of computation built on the idea that vagueness is a fundamental property of our world. To his surprise, the paper was rejected mainly because three (!) reviewers insisted that it is a “fact that there is an equivalence between fuzzy set theory and probability theory.” The author was sure that the editor did not like him for some reason, so he asked the reviewers to reject the paper and they found the most stupid argument to do so. Of course, no one can be sure about anything, but the fact remains that a prestigious journal on fuzzy sets confuses fuzzy set theory with probability theory. And this is not the only confusion as we have already seen.
Contra to these nonsensical ideas, ontic vagueness and fuzzy mathematics are very serious and interesting ideas that took the scientific community by storm and now most mathematical structures have been fuzzyfied. In certain cases, some have gone too far. For example, since vagueness and probabilities are about different things, it makes sense to talk about fuzzy probabilities, however, it does not make sense to talk about fuzzy rough sets or rough fuzzy sets. These are completely nonsensical ideas as we are talking about two different mathematical methods to describe vagueness. Similarly, a car can have either a diesel engine or a gasoline engine or a compressed natural gas engine, but not a “hybrid” engine that burns either gasoline or petrol. So our motto could be expressed as generalization is useful, but too much generalization may lead to nonsense.
Exercises
1.1
Find an argument similar to the Sorites Paradox.
1.2
Name five vague properties and five vague objects.
1.3
Find two ambiguous sentences.
1.4
Name one more idea that led to the development of a new branch of mathematics.
1.5
Explain why vagueness and probability theory are different and unrelated things.
1.6
Multisets form another generalization of sets where elements may occur more than one time. Does it make sense to talk about fuzzy multisets?
Notes
1
2Fuzzy Sets and Their Operations
Based on the assumption that all mathematics can be built up from sets, one has to first define fuzzy sets and their variations. However, one needs to be familiar with certain notions and ideas in order to fully understand the notions that will be introduced. Thus, we start by exploring truth values and their algebras and continue with ‐norms and ‐conorms.
2.1 Algebras of Truth Values
Aristotle ('Aριστοτἑλης) was the first thinker who made a systematic study of logic in his Organon (Ὄργανον), a collection of six works on logic. However, the transformation of logic into something that looks like an algebra was started by the German philosopher and mathematician Gottfried Wilhelm Leibniz, who is primarily known as one of the two people who devised the differential calculus. Later on, George Boole in his The Mathematical Analysis of Logic [35] introduced in a way the truth values (true) and (false) and algebraic operations between them. This work was important because it showed that truth values can be represented by numbers and the various logical operations as “ordinary” number operations. To be accurate, Boole represented true with “” and false with “,” where is any variable. Having identified truth values with numbers, we are able to define conjunction, disjunction, and negation as arithmetic operations. In this way, one builds algebraic structures that describe truth values and the operations between them. However, numbers are ordered and this is used to define the logical operations. In general, these structures are studied in lattice theory1 and in the rest of this section, we are going to present basic ideas and results.
2.1.1 Posets
Definition 2.1.1 A partially ordered set or just poset is a set equipped with a binary relation2) (i.e. is a subset of ) that for all has the following properties:
Reflexivity
;
Transitivity
if
and
, then
;
Antisymmetry
if
and
, then
.
One can say that each element of a poset is a logical proposition and the operator “” is the “” operator. Thus, when , this actually means that entails .
Example 2.1.1 The set is a poset when has the usual meaning. Also, the set of all subsets of some set and the operation form a poset. Thus, if and only if .
We have seen how the truth values can be expressed as numbers and how the entails operator is represented. Let us continue with the and operator.
Definition 2.1.2 Assume that is a poset, , and . Then, is a greatest lower bound (glb or inf or meet) for if and only if:
when
, then
, that is,
is a
lower bound
of
, and
when
is any other lower bound for
, then
.
Typically, one writes . The expression can also be written as .
Proposition 2.1.1 Assume that is a poset and . Then, can have at most one glb.
Proof: Assume that and are two glbs of . Since both are glbs, it holds that and and by antisymmetry .
The least upper bound corresponds to or.
Definition 2.1.3 Assume that is a poset, , and . Then, is a least upper bound (lub or sup or join) for if and only if:
when
, then
, that is,
is an
upper bound
of
, and
when
is any other upper bound for
, then
.
Typically, one writes . The expression can also be written as . It is easy to prove the following proposition:
Proposition 2.1.2 Assume that is a poset and . Then, can have at most one lub.
Example 2.1.2 Consider the set . This set is trivially a poset. The glb of this set is , while its lub is .
2.1.2 Lattices
Definition 2.1.4 A poset is a lattice if and only if every finite subset of has both a glb and a lub.
In what follows, the symbol will denote the top (greatest) element and the bottom (least) element of a poset. The operators and have the following properties:
Commutativity
Associativity
Unit laws
Idempotence
Absorption
Definition 2.1.5 A lattice is distributive if and only if for all it holds that
Example 2.1.3 Consider a set and the set that consists of all subsets of . The latter is called the powerset of and it is denoted by . Given , then and . It is not difficult to show that the powerset is a lattice.
Definition 2.1.6 A poset with all lubs and all glbs is called a complete lattice.
Definition 2.1.7 A Heyting algebra is a lattice with and that has to each pair of elements and an exponential element . This element is usually written as and has the following property:
That is, is a lub for all those elements with .
2.1.3 Frames
Definition 2.1.8 A poset is a frame if and only if
every subset has a lub;
every finite subset has a glb; and
the operator
distributes over
:
Note that the structure just defined is called by some authors locale or complete Heyting algebra by some authors.
Example 2.1.4 The set , where , is a frame (why?).
2.2 Zadeh's Fuzzy Sets
In 1965, Zadeh published a paper entitled “Fuzzy Sets” [305] where he introduced his fuzzy sets. His motivation for introducing them is best explained in the introduction of his paper:
More often than not, the classes of objects encountered in the real physical world do not have precisely defined criteria of membership. For example, the class of animals clearly includes dogs, horses, birds, etc., as its members, and clearly excludes such objects as rocks, fluids, plants, etc. However, such objects as starfish, bacteria, etc., have an ambiguous status with respect to the class of animals.
From this statement, one can say that he actually supported the semantic view of vagueness. In classical set theory, one can define a set using one of the following methods:
List method.
A set is defined by naming all its members. This method can be used only for finite sets. For example, any set
, whose members are the elements
, where
is small enough, is usually written as
Rule method.
A set is defined by specifying a property that is satisfied by all its members. A common notation expressing this method is
where the symbol
denotes the phrase “such that,” and
designates a proposition of the form “
has the property
.”
Characteristic function.
Assume that
is a set, which is called a
universe
, and
. Then, the
characteristic function
of
is defined as follows:
For example, the characteristic function of the set of positive even numbers less or equal than 10 can be defined as follows:
Zadeh had opted to introduce his fuzzy sets by employing an extension of the characteristic function:
Definition 2.2.1 Let be a universe (i.e. an arbitrary set). A fuzzy subset of , is characterized by a function , which is called the membership function. For every , the value is called a degree to which elementbelongs to the fuzzy subset.
We say that a fuzzy subset is characterized by a function or not and that it is just a function because Zadeh used this term. However, we know that , the powerset of , is isomorphic to , the set of all functions from to , and similarly, , the set of all fuzzy subsets of , is isomorphic to the set of all fuzzy characteristic functions, . Therefore, there is no need to distinguish between “is” and “characterized.” If there is at least one element of a fuzzy subset such that , then is called normal. Otherwise, it is called subnormal. The height of a fuzzy subset is the maximum membership degree, that is,
provided that is finite, or
when is not finite. For reasons of simplicity, the term “fuzzy set” is preferred over the term “fuzzy subset.”
When the universe is a set with few elements, it is customary to write down a fuzzy set as follows:
where , is the degree to which belongs to , and means that belongs to with a degree that is equal to . Obviously, the symbol “+” does not denote addition, but it is some sort of metasymbol. Alternatively, one can write down a fuzzy set as follows:
In case is not finite (e.g. it is an interval of real numbers), a fuzzy set can be written down as
Note that the symbols and do not denote summation or integration. In the first case, ranges over a set of discrete values, while in the second case, it ranges over a continuum.
Example 2.2.1 Consider the following five squares:
Assume that the set forms a universe. Then, we can form a fuzzy subset of black squares of this set. Let us call this set . Then, the membership values of this set are , , , , and .
In Example 2.2.1, we assigned the membership degrees by consulting a gray color code table, but, in general, there is no method by which one can assign membership degrees to elements of a fuzzy subset. Instead, one should employ some rule of thumb to compute the membership degrees. Since the membership degree can be any number that belongs to , it follows that numbers like , , , and so on, can be used as membership degrees. However, these numbers do not have a finite decimal representation, and some people are not very comfortable with this idea. Instead, it is quite common to assume that membership degrees should be elements of the set , where is the set of rational numbers. On the other hand, irrational numbers are used very frequently in physics in order to describe the natural world.
The empty fuzzy subset of some universe is a set such that , for all . A fuzzy subset of such that for all is called a crisp or sharp set (i.e. an ordinary set).
Definition 2.2.2 Assume that and are two fuzzy sets of . Then,
their
union
is
their
intersection
is
the
complement
3
of
is the fuzzy set
their algebraic product is
is a
subset
of
, denoted by
, if and only if
the
scalar
cardinality
4
of
is
the fuzzy
powerset
of
(i.e. the set of all ordinary fuzzy subsets of
) is denoted by
.
Example 2.2.2 Suppose that is the set of all pupils of some class. Then, we can construct the fuzzy subset of tall pupils and the fuzzy subset of obese pupils. Let us call these sets and , respectively. Then, is the fuzzy set of pupils that are either tall or obese, while is the fuzzy subset of pupils that are tall and obese. In fact, there is a relation between set operations and logical connectives (see Section 7.1).
The operations between fuzzy sets have a number of properties that are similar to the properties of crisp sets. For example, it is trivial to prove that union and intersection are commutative operations:
Also, it is easy to show that the De Morgan's laws, named after Augustus De Morgan,
are valid. Indeed, we want to prove that for all . We note that the membership degree of is
which is the membership degree of . Similarly, we can prove the validity of the second rule. However, the law of the excluded middle and the law of contradiction
are not valid. Clearly, if , then but . Similarly, if , then , but . These laws are associated with two fundamental laws of logic. Since sets and their operations are models of logic, it follows that these laws are not valid in a fuzzy way of thinking. However, this is not true unless you assume that contradictions are meaningful.
An important property of fuzzy sets is convexity:
Definition 2.2.3 A fuzzy set is convex if and only if
for all and all . If
then is strongly convex.
Proposition 2.2.1 If and are convex fuzzy sets, so is their intersection.
The proof follows from the convexity of and and the “fact” that
where .
Over the years, it became clear that the scalar cardinality of fuzzy sets is problematic. Ronald Robert Yager [303
