Graduated Conjectures | SpringerLink
Skip to main content
Springer Nature Link
Log in

Graduated Conjectures

  • Chapter
  • First Online:
Towards the Future of Fuzzy Logic

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

  • 1100 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
JPY 3498
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
JPY 11439
Price includes VAT (Japan)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
JPY 14299
Price includes VAT (Japan)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
JPY 14299
Price includes VAT (Japan)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

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.

References

  1. Lewis, C.: What the tortoise said to Achilles. Mind 104(416), 691–693 (1995)

    Article  MathSciNet  Google Scholar 

  2. Itziar, G-H., Enric, T.: On an attempt to formalize guessing. In: Rudolf, S., González, V.S. (eds.) Soft Computing in Humanities and Social Sciences. Studies in Fuzziness and Soft Computing, vol 273, pp. 237–255, Springer, Berlin (2012)

    Google Scholar 

  3. John, M.: From here to human-level AI. Artif Intell 171(18), 1174–1182 (2007). Special Review Issue

    Article  Google Scholar 

  4. Marvin, M.: The Emotion Machine. Commonsense Thinking, Artificial Intelligence, and the Future of the Human Mind. Simon and Schuster (2007)

    Google Scholar 

  5. Nilsson, N.J.: Human-level artificial intelligence? Be serious!. AI Mag. 26(4), 68 (2005)

    Google Scholar 

  6. Popper, K.R.: Conjectures and Refutations: The Growth of Scientific Knowledge. Harper and Row, New York (1968)

    Google Scholar 

  7. Tarski, A.: Methodology of deductive sciences. In: Logic, S. (ed.) Metamathematics, vol. 24, pp. 60–109. Clarendon Press, Oxford (1956)

    Google Scholar 

  8. Trillas, E., Alsina, C.: Elkan’s theoretical argument, reconsidered. Int. J. Approx. Reason. 26(2), 145–152 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Enric, T., Alsina, C., Pradera, A.: On a class of fuzzy set theories. In: IEEE International Fuzzy Systems Conference, 2007. July 2007, pp. 1-5 (2007)

    Google Scholar 

  10. Trillas, E., Sanchez, D.: Conjectures in de morgan algebras. In: 2012 Annual Meeting of the North American, Fuzzy Information Processing Society (NAFIPS), pp. 1–6 (2012)

    Google Scholar 

  11. Trillas, E.: Non contradiction, excluded middle, and fuzzy sets. In: Fuzzy Logic and Applications, pp. 1–11, Springer (2009)

    Google Scholar 

  12. Trillas, E.: A model for “crisp reasoning” with fuzzy sets. Int. J. Intell. Syst. 27(10), 859–872 (2012)

    Article  Google Scholar 

  13. Trillas, E., Cubillo, S., Castiñeira, E.: On conjectures in orthocomplemented lattices. Artif. Intell. 117(2), 255–275 (2000)

    Article  MATH  Google Scholar 

  14. Trillas, E., García-Honrado, I., Pradera, A.: Consequences and conjectures in preordered sets. Inf. Sci. 180(19), 3573–3588 (2010)

    Article  MATH  Google Scholar 

  15. Valverde, L.: On the structure of F-indistinguishability operators. Fuzzy Sets Syst. 17(3), 313–328 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ying, M., Wang, H.: Lattice-theoretic models of conjectures, hypotheses and consequences. Artif. Intell. 139(2), 253–267 (2002)

    Article  MathSciNet  Google Scholar 

  17. Zadeh, L.A.: From computing with numbers to computing with words from manipulation of measurements to manipulation of perceptions. Int. J. Appl. Math. Comput. Sci. 12(3), 307–324 (2002)

    MATH  MathSciNet  Google Scholar 

  18. Zadeh, L.A.: Is there a need for fuzzy logic? Inf. Sci. 178(13), 2751–2779 (2008)

    Article  MATH  MathSciNet  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

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

    Adolfo Rodríguez de Soto

Authors
  1. Adolfo Rodríguez de Soto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adolfo Rodríguez de Soto .

Editor information

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

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

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

Download citation

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

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-18749-5

  • Online ISBN: 978-3-319-18750-1

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation