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## Über dieses Buch

Since their inception, fuzzy sets and fuzzy logic became popular. The reason is that the very idea of fuzzy sets and fuzzy logic attacks an old tradition in science, namely bivalent (black-or-white, all-or-none) judg­ ment and reasoning and the thus resulting approach to formation of scientific theories and models of reality. The idea of fuzzy logic, briefly speaking, is just the opposite of this tradition: instead of full truth and falsity, our judgment and reasoning also involve intermediate truth values. Application of this idea to various fields has become known under the term fuzzy approach (or graded truth approach). Both prac­ tice (many successful engineering applications) and theory (interesting nontrivial contributions and broad interest of mathematicians, logicians, and engineers) have proven the usefulness of fuzzy approach. One of the most successful areas of fuzzy methods is the application of fuzzy relational modeling. Fuzzy relations represent formal means for modeling of rather nontrivial phenomena (reasoning, decision, control, knowledge extraction, systems analysis and design, etc. ) in the pres­ ence of a particular kind of indeterminacy called vagueness. Models and methods based on fuzzy relations are often described by logical formulas (or by natural language statements that can be translated into logical formulas). Therefore, in order to approach these models and methods in an appropriate formal way, it is desirable to have a general theory of fuzzy relational systems with basic connections to (formal) language which enables us to describe relationships in these systems.

## Inhaltsverzeichnis

### Chapter 1. Preliminaries

Abstract
The aim of this chapter is to give an overview of some basic notions and results (with almost no proofs).
Radim Bělohlávek

### Chapter 2. Fuzzy Approach, Graded Truth, and Structures of Truth Values

Abstract
Paradoxes of vague terms. Human description of the world provides us with remarkable situations of a special kind. Consider the following form of the ancient sorites paradox (paradox of heap). “A heap consisting of just one grain of sand is small. If you add one grain to a small heap, it remains small. Therefore, each heap is small.” Note that various forms of sorites paradoxes (e.g. the phalakros paradox, i.e. a paradox of a bald man that will be mentioned shortly) were known in antiquity and their formulation is attributed to Eubulides of Miletus (fourth century B.C.).
Radim Bělohlávek

### Chapter 3. Fuzzy Structures

Abstract
This chapter is devoted to fuzzy structures — systems of fuzzy relations and functions on a set. We introduce and investigate basic structural notions connected with fuzzy structures, and basic relationships between fuzzy structures and a language of fuzzy predicate logic. As a matter of fact, what is developed here belongs traditionally to so-called model theory, in our case to model theory for fuzzy logic. However, this chapter is not to be understood to be a treatise on model theory for fuzzy logic. Rather, we present only notions and results that are more or less directly connected to fuzzy relational modeling. Notions and results of this chapter will be used in the subsequent chapters where several special fuzzy structures will be investigated.
Radim Bělohlávek

### Chapter 4. Binary Fuzzy Relations

Abstract
This chapter is devoted to binary fuzzy relations. For quite a variety of problems, binary relations turn out to be a simple but very powerful tool. Recall that a binary L-relation between (nonempty) sets X and Y is any mapping R: X × YL (L is the support set of the complete residuated lattice L; for xX and yY, R(x,y) ∈ L is the truth degree to which x and y are in the relation R). Thus, L X ×Y is the set of all binary L-relations between X and Y.
Radim Bělohlávek

### Chapter 5. Object-Attribute Fuzzy Relations and Fuzzy Concept Lattices

Abstract
Elementary knowledge: relation between objects and attributes. When exploring an unknown domain of interest, the primarily observable data are in the form of a collection of relevant objects (minerals, animals, cities, documents, etc.), attributes that may apply to the objects (to be hard, to be warm-blooded, to be a capital city, to have many inhabitants, etc.), 1 and a relationship between objects and attributes specifying what objects have what attributes (some minerals are hard, some not; a city is either a capital or not; etc.). Most of empirical attributes are vague—a given attribute applies to a given object to a certain (truth) degree (mineral is hard to a certain degree).
Radim Bělohlávek

### Chapter 6. Composition and Decomposition of Fuzzy Relations

Abstract
We are going to introduce is compositions of binary fuzzy relations. The basic situation is this: Given an L-relation R between X and Y and an L-relation S between Y and Z, it might be desirable to obtain from R and S a binary L-relation R*S between X and Z. R*S will be called the composition of R and S.
Radim Bělohlávek

### Chapter 7. Miscellanea

Abstract
Closure operators (and closure systems and related structures) play a significant role in both pure and applied mathematics. Once we adopt and use fuzzy approach to modeling, ordinary closure operators are no longer an appropriate tool and need to be replaced by a more general one. A general theory of closure operators and related structures from the point of view of fuzzy approach is the subject of this section.
Radim Bělohlávek

### Backmatter

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