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.

APOSTOLOS SYROPOULOS, PHD, is an independent scholar based in Xanthi, Greece. His research interests include fuzzy mathematics, the philosophy of vagueness, computability theory, category theory, and digital typography. He has authored or co-authored more than 10 books and more than 60 papers and articles. He has served in the program committee of numerous scientific conferences and has reviewed papers for many journals.

THEOPHANES GRAMMENOS, PHD, is Assistant Professor of Applied Mathematics in the Department of Civil Engineering, University of Thessaly, Greece. He received his PhD from University of Athens, Greece. Dr. Grammenos is a member of the editorial board for Applied Mathematics and the International Journal of Applied Mathematical Research.

# 1

Introduction

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 (s?f?sµa t?? s??e?t?), which was introduced by Eubulides of Miletus (E?ß????d?? ? M???s???),1 is a typical example of an argument that demonstrates what fuzzy boundaries are. The term "s??e?te?" (sorites) derives from the Greek word s???? (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:

(1.1) 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 ?-abstraction:

(1.2) Of course, it is a fact that it is not indeterminate whether is identical to :

(1.3) Using this "trick" to derive formula 1.2, one gets

(1.4) 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...