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Modus Tollens on Fuzzy Implication Functions Derived from Uninorms

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Fuzzy Logic and Information Fusion

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 339))

Abstract

The most used inference schemes in approximate reasoning are the so-called Modus Ponens for forward inferences, and Modus Tollens for backward inferences. In this way, finding new fuzzy implication functions satisfying these two properties has become an important topic for researchers. In the framework of fuzzy logic, they can be written as two inequalities involving fuzzy implication functions. In this paper, the property of Modus Tollens with respect to a continuous t-norm and a continuous fuzzy negation is studied for residual implication functions derived from uninorms, that is, for RU-implications. The corresponding inequality is solved in the cases of an RU-implication derived from a uninorm U in the class of \(\mathscr {U}_{\min }\), from an idempotent uninorm or from a representable uninorm.

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Acknowledgments

The authors want to dedicate this work to Professor Gaspar Mayor in his 70th birthday. We have had the pleasure of jointly work with him in many aspects of aggregation functions and fuzzy implications. He has been and he is, not only a very good teacher and colleague for us, but also a friend. This paper has been partially supported by the Spanish grant TIN2013-42795-P.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of the Balearic Islands, 07122, Palma de Mallorca, Spain

    M. Mas, J. Monreal, M. Monserrat, J. V. Riera & Joan Torrens Sastre

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  1. M. Mas
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  2. J. Monreal
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  3. M. Monserrat
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  4. J. V. Riera
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  5. Joan Torrens Sastre
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Correspondence to Joan Torrens Sastre .

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Editors and Affiliations

  1. University of Alcalá, Alcalá de Henares, Spain

    Tomasa Calvo Sánchez

  2. University of Balearic Islands, Palma, Spain

    Joan Torrens Sastre

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Mas, M., Monreal, J., Monserrat, M., Riera, J.V., Torrens Sastre, J. (2016). Modus Tollens on Fuzzy Implication Functions Derived from Uninorms. In: Calvo Sánchez, T., Torrens Sastre, J. (eds) Fuzzy Logic and Information Fusion. Studies in Fuzziness and Soft Computing, vol 339. Springer, Cham. https://doi.org/10.1007/978-3-319-30421-2_5

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  • DOI: https://doi.org/10.1007/978-3-319-30421-2_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-30419-9

  • Online ISBN: 978-3-319-30421-2

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