Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category Theoretic Approaches

@article{Gottwald2006UniversesOF,
  title={Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category Theoretic Approaches},
  author={S. Gottwald},
  journal={Studia Logica},
  year={2006},
  volume={84},
  pages={23-50}
}
  • S. Gottwald
  • Published 2006
  • Mathematics, Computer Science
  • Studia Logica
For classical sets one has with the cumulative hierarchy of sets, with axiomatizations like the system ZF, and with the category SET of all sets and mappings standard approaches toward global universes of all sets.We discuss here the corresponding situation for fuzzy set theory. Our emphasis will be on various approaches toward (more or less naively formed) universes of fuzzy sets as well as on axiomatizations, and on categories of fuzzy sets.What we give is a (critical) survey of quite a lot… Expand
Fuzzy Set Theory – 40 Years of Foundational Discussions
For classical sets one has the cumulative hierarchy of sets, and also the category SET of all sets and mappings as standard approaches toward the universe of all sets. Both of them discussed withinExpand
The Logic of Fuzzy Set Theory: A Historical Approach
TLDR
The chapter discusses aspects of the historical development of the relationship of fuzzy sets with formal logics suitable for a natural presentation of fuzzy set theory. Expand
L-Fuzzy Sets and Isomorphic Lattices: Are All the “New” Results Really New? †
We review several generalizations of the concept of fuzzy sets with two- or three-dimensional lattices of truth values and study their relationship. It turns out that, in the two-dimensional case,Expand
Variable-domain fuzzy sets - Part II: Apparatus
TLDR
The aim is to demonstrate the viability of variable-domain fuzzy set theory and highlight the similarity as well as differences between the fixed- and variable- domain treatments of fuzzy sets. Expand
Constructive Sets with Non-binary Inclusion
In this thesis we outline how to represent rough and fuzzy sets in intuitionistic type theory. The inherent meaning of such constructs are valuable when it comes to representing certainty in complexExpand
Fuzzy Association Rules Fuzzy Association Rules Table of Contents
The idea of empowering classical association rules by combining them with fuzzy set theory has already been around since several years. The original idea derives from attempts to deal withExpand
Fuzzy Categories
Since categories are graphs with additional “structure”, one should start from fuzzy graphs in order to define a theory of fuzzy categories. Thus is makes sense to introduce categories whoseExpand
Intervals and More: Aggregation Functions for Picture Fuzzy Sets
TLDR
The set of truth values for these fuzzy sets as well as aggregation functions for these truth values are discussed, paying special attention to t-norms and t-conorms. Expand
Mathematical Fuzzy Logics
  • S. Gottwald
  • Computer Science, Mathematics
  • Bulletin of Symbolic Logic
  • 2008
TLDR
The paper discusses the mathematical background for the interest in such systems of mathematical fuzzy logics, as well as the most important ones of them, and concentrates on the propositional cases. Expand
Implication structures, fuzzy subsets, and enriched categories
  • Dexue Zhang
  • Mathematics, Computer Science
  • Fuzzy Sets Syst.
  • 2010
TLDR
It is argued that the theory of enriched categories is a useful tool for fuzzy set theorists because some basic results in fuzzy set theory are applications of enriched category theory. Expand
...
1
2
3
4
...

References

SHOWING 1-10 OF 90 REFERENCES
Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based and Axiomatic Approaches
  • S. Gottwald
  • Mathematics, Computer Science
  • Stud Logica
  • 2006
TLDR
A (critical) survey of quite a lot of such approaches which have been offered in the last approximately 35 years of fuzzy set theory. Expand
Intuitionistic Fuzzy Logic and Intuitionistic Fuzzy Set Theory
TLDR
From a logical standpoint, each logic has its corresponding set theory in which each logical operation is translated into a basic operation for set theory; namely, the relation ⊆ and = on sets are translation of the logical operations → and ↔. Expand
Topoi and Categories of Fuzzy Sets
Motivation. In Chapters 2 and 3 we have introduced the notion of a topos and showed how classical many sorted logic can be interpreted in a topos. This fact has a very natural interpretation: byExpand
Topoi and categories of fuzzy sets
Abstract Let H be a completely distributive lattice and hence a Heyting algebra. Goguen's category of fuzzy sets Set(H) Eytan's logos Fuz(H) and the topos of sheaves on H, Sh(H), are interconnectedExpand
Fuzzy sets: A topos-logical point of view
Abstract Benabou deduction-categories are defined, with a set of additional assumptions that define categories with formal finite limits (resp. formal regular categories, formal logoi, formal topoi).Expand
Fuzzy logic and fuzzy set theory
TLDR
An intuitionistic logic and intuitionistic set theory based on Gentzen's intuitionistic predicate logic with additional axioms and inference rules are presented, which asserts that the truth value set is linearly ordered and dense cHa. Expand
A survey of fuzzy set and topos theory
Abstract This paper is a comparison and contrast of approaches to many-valued mathematics offered by Fuzzy Set Theory and topos theory. It gives a survey of the categorical foundations of Fuzzy SetExpand
Concept Representation in Natural and Artificial Languages: Axioms, Extensions and Applications for Fuzzy Sets
  • J. Goguen
  • Computer Science
  • Int. J. Man Mach. Stud.
  • 1974
TLDR
A system of axioms for a relatively simple form of fuzzy set theory is given, and used to consider the accuracy of representing concepts in various ways by fuzzy sets, and some implications for artificial intelligence are discussed. Expand
Foundations of fuzzy sets
Abstract This paper gives an overview of the origins of fuzzy set theory and the problems for the foundations of fuzzy sets arising from those origins and current practice. It then gives detailedExpand
Categories of Fuzzy Sets with Values in a Quantale or Projectale
Properties of the lattice L are reflected in the properties of the categories Set(L), Set(L)/(A, α), and the lattice u(A,α). The lattices u(A,α) best reflect the structures on the lattice if theExpand
...
1
2
3
4
5
...