Libor Běhounek

Fuzzy class theory

The paper introduces a simple, yet powerful axiomatization of Zadehs notion of fuzzy set, based on formal fuzzy logic. The presented formalism is strong enough to serve as foundations of a large part of fuzzy mathematics. Its essence is elementary fuzzy set theory, cast as two-sorted first-order theory over fuzzy logic, which is generalized to simple type theory. We show a reduction of the elementary fuzzy set theory to fuzzy propositional calculus and a general method of fuzzification of classical mathematical theories within this formalism. In this paper we restrict ourselves to set relations and operations that are definable without any structure on the universe of objects presupposed; however, we also demonstrate how to add structure to the universe of discourse within our framework.

From fuzzy logic to fuzzy mathematics: A methodological manifesto

The paper states the problem of fragmentation of contemporary fuzzy mathematics and the need of a unified methodology and formalism. We formulate several guidelines based on Hajeks methodology in fuzzy logic, which enable us to follow closely the constructions and methods of classical mathematics recast in a fuzzy setting. As a particular solution we propose a three-layer architecture of fuzzy mathematics, with the layers of formal fuzzy logic, a foundational theory, and individual mathematical disciplines developed within its framework. The ground level of logic being sufficiently advanced, we focus on the foundational level; the theory we propose for the foundations of fuzzy mathematics can be characterized as Henkin-style higher-order fuzzy logic. Finally, we give some hints on the further development of individual mathematical disciplines in the proposed framework, and proclaim it a research programme in formal fuzzy mathematics.

Relational compositions in Fuzzy Class Theory

We present a method for mass proofs of theorems of certain forms in a formal theory of fuzzy relations and classes. The method is based on formal identification of fuzzy classes and inner truth values with certain fuzzy relations, which allows transferring basic properties of sup-T and inf-R compositions to a family of more than 30 composition-related operations, including sup-T and inf-R images, pre-images, Cartesian products, domains, ranges, resizes, inclusion, height, plinth, etc. Besides yielding a large number of theorems on fuzzy relations as simple corollaries of a few basic principles, the method provides a systematization of the family of relational notions and generates a simple equational calculus for proving elementary identities between them, thus trivializing a large part of the theory of fuzzy relations.

Relations in Fuzzy Class Theory

This paper studies fuzzy relations in the graded framework of Fuzzy Class Theory (FCT). This includes (i) rephrasing existing work on graded properties of binary fuzzy relations in the framework of Fuzzy Class Theory and (ii) generalizing existing crisp results on fuzzy relations to the graded framework. Our particular aim is to demonstrate that Fuzzy Class Theory is a powerful and easy-to-use instrument for handling fuzzified properties of fuzzy relations. This paper does not rephrase the whole theory of (fuzzy) relations; instead, it provides an illustrative introduction showing some representative results, with a strong emphasis on fuzzy preorders and fuzzy equivalence relations.

On the difference between traditional and deductive fuzzy logic

In three case studies on notions of fuzzy logic and fuzzy set theory (Dubois--Prades gradual elements, the entropy of a fuzzy set, and aggregation operators), the paper exemplifies methodological differences between traditional and deductive fuzzy logic. While traditional fuzzy logic admits various interpretations of membership degrees, deductive fuzzy logic always interprets them as degrees of truth preserved under inference. The latter fact imposes several constraints on systems of deductive fuzzy logic, which need not be followed by mainstream fuzzy logic. That makes deductive fuzzy logic a specific area of research that can be characterized both methodologically (by constraints on meaningful definitions) and formally (as a specific class of logical systems). An analysis of the relationship between deductive and traditional fuzzy logic is offered.

Features of Mathematical Theories in Formal Fuzzy Logic

A genuine fuzzy approach to fuzzy mathematics consists in constructing axiomatic theories over suitable systems of formal fuzzy logic. The features of formal fuzzy logics (esp. the invalidity of the law of contraction) entail certain differences in form between theories axiomatized in fuzzy logic and usual theories known from classical mathematics. This paper summarizes the most important differences and presents guidelines for constructing new theories, defining new notions, and proving new theorems in formal fuzzy mathematics.

Towards Fuzzy Partial Set Theory

We sketch a simple theory of fuzzy partial sets, i.e., fuzzy sets that can have undefined membership degrees. The theory is developed in the semantic framework of a first-order extension of the recently proposed fuzzy partial propositional logic. We introduce a selection of basic notions of fuzzy partial set theory, discuss their variants, and present a few initial results on the properties of fuzzy partial class operations and relations.

Topology in Fuzzy Class Theory: Basic Notions

In the formal and fully graded setting of Fuzzy Class Theory (or higher-order fuzzy logic) we make an initial investigation into basic notions of fuzzy topology. In particular we study graded notions of fuzzy topology regarded as a fuzzy system of open or closed fuzzy sets and as a fuzzy system of fuzzy neighborhoods. We show their basic graded properties and mutual relationships provable in Fuzzy Class Theory and give some links to the traditional notions of fuzzy topology.

Graded properties of unary and binary fuzzy connectives

The paper studies basic graded properties of unary and binary fuzzy connectives, i.e., unary and binary operations on the set of truth degrees of a background fuzzy logic extending the logic MTL of left-continuous t-norms. The properties studied in this paper are graded generalizations of monotony, Lipschitz continuity, null and unit elements, idempotence, commutativity, and associativity. The paper elaborates the initial study presented in previous papers and focuses mainly on parameterization of graded properties by conjunction-multiplicities of subformulae in the defining formulae, preservation of graded properties under compositions and slight variations of fuzzy connectives, the values of graded properties for basic connectives of the ground logic, and the dependence of the values on the ground logic. The results are proved in the formal framework of higher-order fuzzy logic MTL, also known as Fuzzy Class Theory (FCT). General theorems provable in FCT are illustrated on several semantic examples.

Set Theory and Arithmetic in Fuzzy Logic

This chapter offers a review of Petr Hajek’s contributions to first-order axiomatic theories in fuzzy logic (in particular, ZF-style fuzzy set theories, arithmetic with a fuzzy truth predicate, and fuzzy set theory with unrestricted comprehension schema). Generalizations of Hajek’s results in these areas to MTL as the background logic are presented and discussed.