Concept Equations

  title={Concept Equations},
  author={Radim Belohl{\'a}vek},
  journal={J. Log. Comput.},
Studied in the paper are systems of equations which naturally arise in the formalization of the Port-Royal theory of concepts. The unknown quantity is a relation between objects and attributes. We study the case where the relation is fuzzy with truth values in a complete residuated lattice, covering therefore the special cases of complete Boolean algebras, Heyting algebras, MV-algebras etc. We answer the question of solvability, structure of solutions, and show how solvability of non-solvable… 

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A flexible way to build concepts within fuzzy logic and set theory that is general enough to capture some important particular cases, with their own independent interpretations, but also to allow the two universes to be equipped each with its own truth structure.

Adjoint Triples and Residuated Aggregators

This paper presents the comparison of joint triples with other general operators considered in frameworks for fuzzy logic programming, formal concept analysis and fuzzy relation equations.

Non-dual fuzzy connections

Some kind of hardly describable ‘‘local preduality’’ still makes possible important parallel results and interesting new concepts besides antitone and isotone ones, that were classically reducible to the first, gain independency in fuzzy setting.

Multi‐adjoint Relation Equations: A Decision Support System for Fuzzy Logic

MARE is shown as a fundamental DSS in multi‐adjoint logic programming and two approximations of unsolvable equations will be obtained from a multiadjoint object‐oriented concept lattice theory.

On Boolean factor analysis with formal concept as factors

Several results are proved, e.g. that each matrix is conceptfactorizable, that concept-factorizability is the best way to factorize binary matrices, describe sets of good factors, mandatory factors, etc.

Optimal decompositions of matrices with entries from residuated lattices

It is shown that formal concepts of I, which play a central role in the Port-Royal approach to logic and which are the fixpoints of particular Galois connections associated to I, are optimal factors for decomposition of I in that they provide us with decompositions with the smallest number of factors.



Concept lattices and order in fuzzy logic

Logical Precision in Concept Lattices

It is argued that logical precision should be considered as a possible way to treat the problem of information granularity from the logical point of view.

Fuzzy Galois Connections

It is proved that fuzzy Galois connections are in one‐to‐one correspondence with binary fuzzy relations.

Reduction and a Simple Proof of Characterization of Fuzzy Concept Lattices

A simple proof of the characterization theorem for fuzzy concept lattices is obtained after a reduction of fuzzy Galois connections and fuzzy idea lattices to (crisp) Galoisconnections and concept lattice.

Formal Concept Analysis: Mathematical Foundations

From the Publisher: This is the first textbook on formal concept analysis. It gives a systematic presentation of the mathematical foundations and their relation to applications in computer science,

Similarity relations in concept lattices

A way to factorize (simplify) concept lattices by the similarity of concepts is shown and how to reduce the computation of the similarity relations is shown.

L-fuzzy sets

Metamathematics of Fuzzy Logic

  • P. Hájek
  • Philosophy, Computer Science
    Trends in Logic
  • 1998
This paper presents a meta-analysis of many-Valued Propositional Logic, focusing on the part of Lukasiewicz's Logic that deals with Complexity, Undecidability and Generalized Quantifiers and Modalities.

Lattices of Fixed Points of Fuzzy Galois Connections

It is shown that fixed points are naturally interpreted as concepts in the sense of traditional logic.

[Formal concept analysis].

In the present paper the authors show in detail howormal Concept Analysis can be applied to the study of results obtained in clinical practice.