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Graduated Conjectures

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Towards the Future of Fuzzy Logic

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 325))

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Abstract

The study of the relationships between conjectures, hypotheses, refutations and speculations have been studied by Professor Enric Trillas and coworkers in the classical case to the point of having well clarified its main properties in rather general structures as the orthocomplemented lattices. In the framework of a possibilistic interpretation of fuzzy logic, these models have been studied from the point of view of a crisp reasoning. In this work these models defined by graduated consequences relations are studied under a fuzzy algebraic structure general enough that it can accommodate various common phenomena in natural language reasoning.

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Notes

  1. 1.

    It will be the pointwise equally between fuzzy sets, but it is not strictly necessary.

  2. 2.

    With \(T=\min \), the classical relations \(\preccurlyeq _{r}\) are reflexive and transitive relations.

  3. 3.

    If the negation is strong, if \(\mu \) is contradictory with \(\rho \) is equivalent to \(\rho \) is contradictory with \(\mu \) because \(0 < r < \mu \preccurlyeq \rho ' \le \rho '' \preccurlyeq \mu ' = \rho \preccurlyeq \mu '\).

  4. 4.

    If \(P\) is clear from the context we use \(\rho \) instead of \(\rho _P\) to reference the résumé of \(P.\)

  5. 5.

    Note that in a framework of a grade ordering relation, first condition is a strong one but necessary to avoid self-contradiction.

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Acknowledgments

I would like to thank Professor Enric Trillas for their support in carrying out this work and for an enriching collaboration during my research career. Also I acknowledge the support of the Spanish Ministry for Economy and Innovation and the European Regional Development Fund (ERDF/FEDER) under grant TIN2011-29827-C02-02.

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Authors and Affiliations

  1. Mechanical, Informatics and Aerospace Department, University of León, Leon, Spain

    Adolfo Rodríguez de Soto

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  1. Adolfo Rodríguez de Soto
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Correspondence to Adolfo Rodríguez de Soto .

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Editors and Affiliations

  1. Institut für Geschichte der Medizin, Naturwissenschaft und Technik, Friedrich-Schiller-Universität Jena, Berggasse, Jena, Germany

    Rudolf Seising

  2. Edificio de Investigación, 3ª Planta. C Gonzalo Gutiérrez Quirós S/N, European Centre for Soft Computing, Mieres, Asturias, Spain

    Enric Trillas

  3. Polish Academy of Sciences, Systems Research Institute, Warsaw, Poland

    Janusz Kacprzyk

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de Soto, A.R. (2015). Graduated Conjectures. In: Seising, R., Trillas, E., Kacprzyk, J. (eds) Towards the Future of Fuzzy Logic. Studies in Fuzziness and Soft Computing, vol 325. Springer, Cham. https://doi.org/10.1007/978-3-319-18750-1_16

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  • DOI: https://doi.org/10.1007/978-3-319-18750-1_16

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-18749-5

  • Online ISBN: 978-3-319-18750-1

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