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Connecting Systems of Mathematical Fuzzy Logic with Fuzzy Concept Lattices

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Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations (IPMU 2018)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 854))

Abstract

In this paper our aim is to explore a new look at formal systems of fuzzy logics using the framework of (fuzzy) formal concept analysis (FCA). Let L be an extension of MTL complete with respect to a given L-chain. We investigate two possible approaches. The first one is to consider fuzzy formal contexts arising from L where attributes are identified with L-formulas and objects with L-evaluations: every L-evaluation (object) satisfies a formula (attribute) to a given degree, and vice-versa. The corresponding fuzzy concept lattices are shown to be isomorphic to quotients of the Lindenbaum algebra of L. The second one, following an idea in a previous paper by two of the authors for the particular case of Gödel fuzzy logic, is to use a result by Ganter and Wille in order to interpret the (lattice reduct of the) Lindenbaum algebra of L-formulas as a (classical) concept lattice of a given context.

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Notes

  1. 1.

    MTL algebras are commutative integral bounded residuated lattices satisfying prelinearity [9].

  2. 2.

    We use \(\varphi ^2\) as a shorcut for \(\varphi \odot \varphi \).

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Acknowledgments

The authors are thankful to the anonymous reviewers for their helpful comments. Pietro Codara is supported by the INdAM-Marie Curie Cofund project LaVague (FP7-PEOPLE-2012-COFUND 600198). Francesc Esteva and Lluis Godo acknowledge partial support by the FEDER/MINECO project TIN2015-71799- C2-1-P.

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Authors and Affiliations

  1. Artificial Intelligence Research Institute (IIIA-CSIC), Campus de la UAB, 08193, Bellaterra, Barcelona, Spain

    Pietro Codara, Francesc Esteva & Lluis Godo

  2. Dipartimento di Informatica, Università degli Studi di Milano, Via Comelico 39, 20135, Milano, Italy

    Diego Valota

Authors
  1. Pietro Codara
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  2. Francesc Esteva
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  3. Lluis Godo
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  4. Diego Valota
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Corresponding author

Correspondence to Lluis Godo .

Editor information

Editors and Affiliations

  1. Universidad de Cádiz, Cádiz, Cadiz, Spain

    Jesús Medina

  2. Universidad de Málaga, Málaga, Málaga, Spain

    Manuel Ojeda-Aciego

  3. Universidad de Granada, Granada, Spain

    José Luis Verdegay

  4. Universidad de Granada, Granada, Spain

    David A. Pelta

  5. Universidad de Málaga, Málaga, Málaga, Spain

    Inma P. Cabrera

  6. LIP6, Université Pierre et Marie Curie, CNRS, Paris, France

    Bernadette Bouchon-Meunier

  7. Iona College, New Rochelle, New York, USA

    Ronald R. Yager

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Codara, P., Esteva, F., Godo, L., Valota, D. (2018). Connecting Systems of Mathematical Fuzzy Logic with Fuzzy Concept Lattices. In: Medina, J., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Foundations. IPMU 2018. Communications in Computer and Information Science, vol 854. Springer, Cham. https://doi.org/10.1007/978-3-319-91476-3_23

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  • DOI: https://doi.org/10.1007/978-3-319-91476-3_23

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  • Online ISBN: 978-3-319-91476-3

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