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Enric Trillas: A Passion for Fuzzy Sets pp 109–123Cite as

On Conjectures in t-Norm Based Fuzzy Logics

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Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 322))

Abstract

This paper is a humble homage to Enric Trillas. Following his foundational contributions on models of ordinary reasoning in an algebraic setting, we study here elements of these models, like conjectures and hypothesis, in the logical framework of continuous t-norm based fuzzy logics. We consider notions of consistency, conjecture and hypothesis arising from two natural families of consequence operators definable in these logics, namely the ones corresponding to the so-called truth-preserving and degree-preserving consequence relations. We pay special attention to the particular cases of three prominent fuzzy logics: Gödel, Product and Łukasiewicz logics

A previous version of this paper appeared in Actas del XVII Congreso Español sobre Tecnologías y Lógica Fuzzy (ESTYLF 2014), F. Bobillo et al. (eds.), pp. 435–440.

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Notes

  1. 1.

    Here by a theory we just mean a set of formulas, so not necessarily closed under consequence.

  2. 2.

    We assume readers to be familiar with the notions of t-norm, the three basic continuous t-norms, i.e. minimum, product and Łukaseewicz t-norm, and the notion of ordinal sum. We also assume familiarity with the decomposition of continuous t-norm as ordinal sums of isomorphic copies of the three basic continuous t-norms.

  3. 3.

    Recall that in a SBL-chain, both \(\lnot \lnot 0 = 0\) and \(\lnot \lnot x = 1\) if \(x > 0\). Moreover \(\lnot \lnot \) defines a morphism from the algebra \(([0, 1],\star ,\Rightarrow _\star ,0,1)\) into itself.

  4. 4.

    Here we use the local deduction theorem that is valid for all t-norm based logics, namely \(\varGamma \cup \{\varphi \} \models \psi \) iff there exists an \(n \in \mathbb {N}\) such that \( \varGamma \models \varphi \& \mathop {\ldots }\limits ^{n} \& \varphi \rightarrow \psi \) (see e.g. [7, 8]).

References

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

    Article  MATH  Google Scholar 

  2. García-Honrado, I., Trillas, E.: On an attempt to formalize guessing. In: Seising, R., Sanz, V. (eds.) Soft Computing in Humanities and Social Sciences, vol. 273, pp. 237–255. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

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

    Article  MATH  Google Scholar 

  4. Watanabe, S.: Knowing and guessing. A Quantitative Study of Inference and Information. Wiley, New York (1969)

    MATH  Google Scholar 

  5. Qiu, D.: A note on Trillas’ CHC models. Artif. Intell. 171, 239–254 (2007)

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. Cintula, P., Hájek, P., Noguera, C.: Handbook of Mathematical Fuzzy Logic (in 2 volumes). of Studies in Logic, Mathematical Logic and Foundations, vols. 37–38 College Publications, London (2011)

    Google Scholar 

  8. Hájek, P.: Metamathematics of Fuzzy Logic, Trends in Logic, vol. 4. Kluwer, Dordrecht (1998)

    Book  Google Scholar 

  9. Bou, F., Esteva, F., Font, J.M., Gil, A., Godo, L., Torrens, A., Verdú, V.: Logics preserving degrees of truth from varieties of residuated lattices. J. Log. Comput. 19(6), 1031–1069 (2009)

    Article  MATH  Google Scholar 

  10. Esteva, F., Godo, L., Hájek, P., Navara, M.: Residuated fuzzy logics with an involutive negation. Arch. Math. Log. 39(2), 103–124 (2000)

    Article  MATH  Google Scholar 

  11. Esteva, F., Godo, L., Montagna, F.: Equational characterization of the subvarieties of BL generated by t-norm algebras. Stud. Log. 76, 161–200 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cignoli, R., Esteva, F., Godo, L., Torrens, A.: Basic fuzzy logic is the logic of continuous t-norms and their residua. Soft Comput. 4(2), 106–112 (2000)

    Google Scholar 

  13. Ertola, R., Esteva, F., Flaminio, T., Godo, L., Noguera, C.: Paraconsistency properties in degree-preserving fuzzy logics. Soft Comput. J. 19(3), 531–546 (2015)

    Google Scholar 

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Acknowledgments

This work has been partially supported by the Spanish projects TIN2012-39348-C02-01 (Esteva and Godo) and TIN2011-29827-C02-01 (García-Honrado).

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Authors and Affiliations

  1. Artificial Intelligence Research Institute (IIIA-CSIC), Campus UAB s/n, 08193, Bellaterra, Spain

    Francesc Esteva & Lluís Godo

  2. Department of Statistics, Operational Research and Mathematical Education, University of Oviedo, Oviedo, Spain

    Itziar García-Honrado

Authors
  1. Francesc Esteva
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  2. Itziar García-Honrado
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  3. Lluís Godo
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Correspondence to Francesc Esteva .

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Editors and Affiliations

  1. European Centre for Soft Computing, Asturias, Spain

    Luis Magdalena

  2. Depto. de Ciencias de la Computación e Inteligencia Artificial, University of Granada, Granada, Spain

    Jose Luis Verdegay

  3. Spanish Council for Scientific Research (CSIC), Artificial Intelligence Research Institute (IIIA-CSIC), Catalonia, Spain

    Francesc Esteva

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Esteva, F., García-Honrado, I., Godo, L. (2015). On Conjectures in t-Norm Based Fuzzy Logics. In: Magdalena, L., Verdegay, J., Esteva, F. (eds) Enric Trillas: A Passion for Fuzzy Sets. Studies in Fuzziness and Soft Computing, vol 322. Springer, Cham. https://doi.org/10.1007/978-3-319-16235-5_9

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  • DOI: https://doi.org/10.1007/978-3-319-16235-5_9

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