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Fuzzy Implications: Classification and a New Class

  • Chapter
Advances in Fuzzy Implication Functions

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 300))

Abstract

One of the most important and interesting topics in fuzzy mathematics is the study of fuzzy connectives and in particular fuzzy implications. Fuzzy implications are supposed to have at least some fundamental properties in common with the classical binary implication. Besides these fundamental properties there are many additional potential properties for fuzzy implications, among which eight are widely used in the literature. Fuzzy implications satisfying different subsets of these eight properties have been constructed and some interrelationships between these eight properties have been established. This paper aims to lay bare all the interrelationships between the eight additional properties. Where needed suitable counterexamples are provided. In our search for these counterexamples we discovered a new class of fuzzy implications that is completely determined by a fuzzy negation. For this new class we examine the conditions under which the eight properties are satisfied and we obtain the intersection with the class of strong and residual fuzzy implications.

Dedicated to the late Prof. Dr. Da Ruan.

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Author information

Authors and Affiliations

  1. Fuzziness and Uncertainty Modelling Research Unit Department of Applied Mathematics and Computer Science, Ghent University, Ghent, Belgium

    Yun Shi, Bart Van Gasse & Etienne E. Kerre

Authors
  1. Yun Shi
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  2. Bart Van Gasse
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  3. Etienne E. Kerre
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Corresponding author

Correspondence to Yun Shi .

Editor information

Editors and Affiliations

  1. , Institute of Mathematics, University of Silesia, ul. Bankowa 14, Katowice, 40-007, Poland

    Michał Baczyński

  2. School of Information Technology, Deakin University, Burwood Highway 221, Burwood, 3125, Victoria, Australia

    Gleb Beliakov

  3. Departamento de Automática y, Computación, Universidad Pública de Navarra, Campus Arrosadia Campus s/n, Pamplona, 31006, Spain

    Humberto Bustince Sola

  4. , Departamento de Ciencias de la, Universidad Rey Juan Carlos, c/ Tulipan s/n, Mostoles, 28933, Spain

    Ana Pradera

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Shi, Y., Van Gasse, B., Kerre, E.E. (2013). Fuzzy Implications: Classification and a New Class. In: Baczyński, M., Beliakov, G., Bustince Sola, H., Pradera, A. (eds) Advances in Fuzzy Implication Functions. Studies in Fuzziness and Soft Computing, vol 300. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35677-3_2

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  • DOI: https://doi.org/10.1007/978-3-642-35677-3_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-35676-6

  • Online ISBN: 978-3-642-35677-3

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