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Petr Hájek on Mathematical Fuzzy Logic

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 6))

Abstract

Since Petr Hájek, the scientist we are going to celebrate, is the main contributor to Mathematical Fuzzy Logic, we will first spend a few words about this subject

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Notes

  1. 1.

    Deleting even the prelinearity condition had given the monoidal logic of  Höhle (1994, 1995). This logic is characterized by the class of all residuated lattices, but seems to be too general as a logic for t-norms.

References

  • Aglianò, P., Ferreirim, I. M. A., & Montagna, F. (2007). Basic hoops: An algebraic study of continuous T-norms. Studia Logica, 87(1), 73–98.

    Article  MathSciNet  MATH  Google Scholar 

  • Baader, F., Calvanese, D., McGuinness, D. L., Nardi, D., & Patel-Schneider, P. F. (Eds.). (2003). The description logic handbook: Theory, implementation, and applications. New York: Cambridge University Press.

    Google Scholar 

  • Baaz, M., Hájek, P., & Krajíček, J. (1998). Embedding logics into product logic. Studia Logica, 61(1), 35–47.

    Article  MathSciNet  MATH  Google Scholar 

  • Baaz, M., Hájek, P., Montagna, F., & Veith, H. (2002). Complexity of T-tautologies. Annals of Pure and Applied Logic, 113(1–3), 3–11.

    MathSciNet  MATH  Google Scholar 

  • Bobillo, F., Delgado, M., Gómez-Romero, J., & Straccia, U. (2009). Fuzzy description logics under Gödel semantics. International Journal of Approximate Reasoning, 50(3), 494–514.

    Article  MathSciNet  MATH  Google Scholar 

  • Borgwardt, S., Distel, F., & Peñaloza, R. (2012) How fuzzy is my fuzzy description logic? In: Proceedings of the 6th International Joint Conference on Automated Reasoning (IJCAR 12), Lecture Notes in Artificial Intelligence, (Vol. 7364, pp. 82–96).

    Google Scholar 

  • Borgwardt, S., & Peñaloza, R. (2011). Fuzzy ontologies over lattices with t-norms. In: Proceedings of the 24th International Workshop on Description Logics (DL 2011) (pp. 70–80).

    Google Scholar 

  • Bou, F., Esteva, F., Godo, L., & Rodríguez, R. O. (2011). On the minimum many-valued modal logic over a finite residuated lattice. Journal of Logic and Computation, 21(5), 739–790.

    Article  MathSciNet  MATH  Google Scholar 

  • Caicedo, X., & Rodríguez, R. O. (2010). Standard Gödel modal logics. Studia Logica, 94(2), 189–214.

    Article  MathSciNet  MATH  Google Scholar 

  • Caicedo, X., & Rodríguez, R. O. (2012). Bi-modal Gödel logic over [0, 1]-valued Kripke frames. Journal of Logic and Computation (in press).

    Google Scholar 

  • Cerami, M. (2012). Fuzzy description logics under a mathematical fuzzy logic point of view. Ph. D. thesis, University of Barcelona, October 16, 2012.

    Google Scholar 

  • Cerami, M., & Straccia, U. (2013). On the (un)decidability of fuzzy description logics under Łukasiewicz t-norm. Information Sciences, 227, 1–21.

    Article  MathSciNet  MATH  Google Scholar 

  • Cerami, M., Esteva, F., & Bou, F. (2010). Decidability of a description logic over infinite-valued product logic. In Proceedings of KR2010 (pp. 203–213).

    Google Scholar 

  • Chang, C. C. (1959). A new proof of the completeness of the Łukasiewicz axioms. Transactions of the American Mathematical Society, 93(1), 74–80.

    MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  • Cintula, P., Hájek, P., & Horčík, R. (2007). Formal systems of fuzzy logic and their fragments. Annals of Pure and Applied Logic, 150(1–3), 40–65.

    Google Scholar 

  • Cintula, P., & Hájek, P. (2009). Complexity issues in axiomatic extensions of Łukasiewicz logic. Journal of Logic and Computation, 19(2), 245–260.

    Article  MathSciNet  MATH  Google Scholar 

  • Cintula, P., Hájek, P., & Noguera, C. (Eds.). (2011). Handbook of mathematical fuzzy logic (in 2 volumes). In Studies in logic. Mathematical logic and foundations, (Vols. 37, 38). London: College Publications.

    Google Scholar 

  • Cintula, P., Haniková, Z., & Švejdar, V. (Eds.). (2009). Witnessed years–essays in honour of Petr Hájek. In Tributes, (vol 10). London: College Publications.

    Google Scholar 

  • Esteva, F., Godo, L., Hájek, P., & Navara, M. (2000). Residuated fuzzy logics with an involutive negation. Archive for Mathematical Logic, 39(2), 103–124.

    Article  MathSciNet  MATH  Google Scholar 

  • Esteva, F., & Godo, L. (2001). Monoidal t-norm based logic: Towards a logic for left-continuous t-norms. Fuzzy Sets and Systems, 124(3), 271–288.

    Article  MathSciNet  MATH  Google Scholar 

  • Esteva, F., Godo, L., Hájek, P., & Montagna, F. (2003). Hoops and fuzzy logic. Journal of Logic and Computation, 13(4), 532–555.

    Article  Google Scholar 

  • Esteva, F., & Godo, L. (2013). A logical approach to fuzzy truth hedges. Information Sciences, 232, 366–385.

    Google Scholar 

  • Fitting, M. C. (1992a). Many-valued modal logics. Fundamenta Informaticae, 15, 235–254.

    MathSciNet  Google Scholar 

  • Fitting, M. C. (1992b). Many-valued modal logics II. Fundamenta Informaticae, 17, 55–73.

    MathSciNet  MATH  Google Scholar 

  • Flaminio, T., & Godo, L. (2007). A logic for reasoning about probability of fuzzy events. Fuzzy Sets and Systems, 158, 625–638.

    Article  MathSciNet  MATH  Google Scholar 

  • Flaminio, T., Godo, L., & Marchioni, E. (2011). On the logical formalization of possibilistic counterparts of states over n-valued Łukasiewicz events. Journal of Logic and Computations, 21(3), 429–446.

    Article  MathSciNet  MATH  Google Scholar 

  • Flaminio, T., & Montagna, F. (2011). Models for many-valued probabilistic reasoning. Journal of Logic and Computation, 21(3), 447–464.

    Article  MathSciNet  MATH  Google Scholar 

  • Flaminio, T., Godo, L., & Marchioni, E. (2013). Logics for belief functions on MV-algebras. International Journal of Approximate Reasoning, 54(4), 491–512.

    Article  MathSciNet  MATH  Google Scholar 

  • Flondor, P., Georgescu, G., & Iorgulescu, A. (2001). Pseudo t-norms and pseudo-BL-algebras. Soft Computing, 5(5), 355–371.

    Article  MATH  Google Scholar 

  • Font, J. M., & Hájek, P. (2002). On Łukasiewicz’s four-valued modal logic. Studia Logica, 70(2), 157–182.

    Article  MathSciNet  MATH  Google Scholar 

  • García-Cerdaña, A., Armengol, E., & Esteva, F. (2010). Fuzzy description logics and t-norm based fuzzy logics. International Journal of Approximate Reasoning, 51(6), 632–655.

    Article  MathSciNet  MATH  Google Scholar 

  • García-Cerdaña, A., Cerami, M., & Esteva, F. (2010). From classical description logic to n-grade fuzzy description logics. In Proceedings of the FUZZ-IEEE2010 (pp. 187–192).

    Google Scholar 

  • Georgescu, G., & Iorgulescu, A. (2001). Pseudo-MV algebras. Multiple-Valued Logic, 6(1–2), 95–135.

    MathSciNet  MATH  Google Scholar 

  • Godo, L., Hájek, P., & Esteva, F. (2003). A fuzzy modal logic for belief functions. Fundamenta Informaticae, 57(2–4), 127–146.

    MathSciNet  MATH  Google Scholar 

  • Gottwald, S. (2001). A treatise on many-valued logics. In Studies in logic and computation (Vol. 9). Baldock: Research Studies Press.

    Google Scholar 

  • Hájek, P. (1998). Metamathematics of fuzzy logic. In Trends in logic (Vol. 4). Dordrecht: Kluwer.

    Google Scholar 

  • Hájek, P. (2006). What does mathematical fuzzy logic offer to description logic? In E. Sanchez (Ed.), Fuzzy logic and the semantic web, capturing intelligence (Chap. 5, pp. 91–100). Elsevier.

    Google Scholar 

  • Hájek, P. (2010). Towards metamathematics of weak arithmetic over fuzzy logic. Logic Journal of the Interest Group of Pure and Applied Logic, 19(3), 467–475.

    Google Scholar 

  • Hájek, P., Harmancová, D., & Verbrugge, R. (1995). A qualitative fuzzy possibilistic logic. International Journal of Approximate Reasoning, 12(1), 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, P., Godo, L., & Esteva, F. (1996). A complete many-valued logic with product conjunction. Archive for Mathematical Logic, 35(3), 191–208.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, P. (1998). Basic fuzzy logic and BL-algebras. Soft Computing, 2(3), 124–128.

    Article  Google Scholar 

  • Hájek, P., & Harmancová, D. (2000). A hedge for Gödel fuzzy logic. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 8(4), 495–498.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, P., Paris, J., & Shepherdson, J. C. (2000). Rational Pavelka logic is a conservative extension of Łukasiewicz logic. Journal of Symbolic Logic, 65(2), 669–682.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, P. (2001). On very true. Fuzzy Sets and Systems, 124(3), 329–333.

    Google Scholar 

  • Hájek, P., & Shepherdson, J. C. (2001). A note on the notion of truth in fuzzy logic. Annals of Pure and Applied Logic, 109(1–2), 65–69.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, P., & Tulipani, S. (2001). Complexity of fuzzy probability logics. Fundamenta Informaticae, 45, 207–213.

    MathSciNet  MATH  Google Scholar 

  • Hájek, P. (2002a). Observations on the monoidal t-norm logic. Fuzzy Sets and Systems, 132(1), 107–112.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, P. (2002b). Some hedges for continuous T-norms logics. Neural Network World, 12(2), 159–164.

    Google Scholar 

  • Hájek, P. (2003a). Embedding standard BL-algebras into non-commutative pseudo-BL-algebras. Tatra Mountains Mathematical Publications, 27(3), 125–130.

    MathSciNet  MATH  Google Scholar 

  • Hájek, P. (2003b). Fuzzy logics with noncommutative conjunctions. Journal of Logic and Computation, 13(4), 469–479.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, P. (2005a). Fleas and fuzzy logic. Journal of Multiple-Valued Logic and Soft Computing, 11(1–2), 137–152.

    MathSciNet  MATH  Google Scholar 

  • Hájek, P. (2005b). Making fuzzy description logic more general. Fuzzy Sets and Systems, 154(1), 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, P. (2005c). A non-arithmetical Gödel logic. Logic Journal of the Interest Group of Pure and Applied Logic, 13(5), 435–441.

    MATH  Google Scholar 

  • Hájek, P. (2005d). On arithmetic in Cantor-Łukasiewicz fuzzy set theory. Archive for Mathematical Logic, 44(6), 763–782.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, P. (2006a). Computational complexity of t-norm based propositional fuzzy logics with rational truth constants. Fuzzy Sets and Systems, 157(5), 677–682.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, P., & Cintula, P. (2006b). On theories and models in fuzzy predicate logics. Journal of Symbolic Logic, 71(3), 863–880.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, P. (2007a). Complexity of fuzzy probability logics II. Fuzzy Sets and Systems, 158, 2605–2611.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, P. (2007b). Mathematical fuzzy logic and natural numbers. Fundamenta Informaticae, 81(1–3), 155–163.

    Google Scholar 

  • Hájek, P., & Montagna, F. (2008). A note on the first-order logic of complete BL-chains. Mathematical Logic Quarterly, 54(4), 435–446.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, P. (2009). Arithmetical complexity of fuzzy predicate logics-a survey II. Annals of Pure and Applied Logic, 161(2), 212–219.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, P. (2010). On fuzzy modal logics \(S5({{C}})\). Fuzzy Sets and Systems, 161, 2389–2396.

    Article  MathSciNet  MATH  Google Scholar 

  • Hájek, P., Godo, L., & Esteva, F. (1995). Fuzzy logic and probability. In Proceedings of the 11th Annual Conference on Uncertainty in Artificial Intelligence UAI’95 (pp. 237–244). Montreal.

    Google Scholar 

  • Hájek, P., & Haniková, Z. (2003). A development of set theory in fuzzy logic. In M. C. Fitting & E. Orlowska (Eds.), Beyond two: Theory and applications of multiple-valued logic, Studies in fuzziness and soft computing (Vol. 114, pp. 273–285). Heidelberg: Physica.

    Google Scholar 

  • Hájek, P., & Harmancová, D. (1995). Medical fuzzy expert systems and reasoning about beliefs. In P. Barahona, M. Stefanelli, & J. Wyatt (Eds.), Artificial intelligence in medicine (pp. 403–404). Berlin: Springer.

    Chapter  Google Scholar 

  • Hájek, P., & Harmancová, D. (1996). A many-valued modal logic. In Proceedings of IPMU’96 (pp. 1021–1024). Granada.

    Google Scholar 

  • Hájek, P., Harmancová, D., Esteva, F., Garcia, P., & Godo, L. (1994). On modal logics for qualitative possibility in a fuzzy setting. In Proceedings of the 94 Uncertainty in Artificial Intelligence Conference (UAI’94) (pp. 278–285). Morgan Kaufmann.

    Google Scholar 

  • Hähnle, R. (1994). Many-valued logic and mixed integer programming. Annals of Mathematics and Artificial Intelligence, 12(34), 231–264.

    Article  MathSciNet  MATH  Google Scholar 

  • Haniková, Z. (2002). A note on the complexity of propositional tautologies of individual t-algebras. Neural Network World, 12(5), 453–460. (Special issue on SOFSEM 2002).

    Google Scholar 

  • Hansoul, G., & Teheux, B. (2013). Extending Łukasiewicz logics with a modality: Algebraic approach to relational semantics. Studia Logica, 101(3), 505–545.

    Article  MathSciNet  MATH  Google Scholar 

  • Höhle, U. (1994). Monoidal logic. In R. Palm, R. Kruse, & J. Gebhardt (Eds.), Fuzzy systems in computer science (pp. 233–243). Wiesbaden: Vieweg.

    Chapter  Google Scholar 

  • Höhle, U. (1995). Commutative, residuated l-monoids. In U. Höhle & E. P. Klement (Eds.), Non-classical logics and their applications to fuzzy subsets (pp. 53–106). Dordrecht: Kluwer.

    Chapter  Google Scholar 

  • Łukasiewicz, T., & Straccia, U. (2008). Managing uncertainty and vagueness in Description Logics for the Semantic Web. Journal of Web Semantics, 6(4), 291–308.

    Google Scholar 

  • Metcalfe, G., & Olivetti, N. (2011). Towards a proof theory of Gödel modal logics. Logical Methods in Computer Science, 7(2:10), 1–27.

    Google Scholar 

  • Mundici, D. (1987). Satisfiability in many-valued sentential logic is NP-complete. Theoretical Computer Science, 52(1–2), 145–153.

    Article  MathSciNet  MATH  Google Scholar 

  • Pavelka, J. (1979). On fuzzy logic I, II, III. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 25, 45–52, 119–134, 447–464.

    Google Scholar 

  • Sánchez, D., & Tettamanzi, A. G. B. (2006). Fuzzy logic and the semantic web (Capturing intelligence) In E. Sanchez (Ed.), Fuzzy quantification in fuzzy description logics (pp. 135–159). Amsterdam: Elsevier.

    Google Scholar 

  • Skolem, T. (1957). Bemerkungen zum Komprehensionsaxiom. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 3, 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  • Stoilos, G., Straccia, U., Stamou, G., & Pan, J. Z. (2006). General concept inclusions in fuzzy description logics. In Proceedings of the 17th European Conference on Artificial Intelligence (ECAI-06) (pp. 457–461). IOS Press.

    Google Scholar 

  • Straccia, U. (1998). A fuzzy description logic. In Proceedings of AAAI-98, 15th National Conference on Artificial Intelligence (pp. 594–599). MIT Press.

    Google Scholar 

  • Straccia, U. (2001). Reasoning within fuzzy description logics. Journal of Artificial Intelligence Research, 14, 137–166.

    MathSciNet  MATH  Google Scholar 

  • Straccia, U., & Bobillo, F. (2007). Mixed integer programming, general concept inclusions and fuzzy description logics. Mathware and Soft Computing, 14(3), 247–259.

    MathSciNet  MATH  Google Scholar 

  • Tresp, C. B., & Molitor, R. (1998). A description logic for vague knowledge. Technical report RWTH-LTCS report 98–01. Aachen University of Technology.

    Google Scholar 

  • Vychodil, V. (2006). Truth-depressing hedges and BL-logic. Fuzzy Sets and Systems, 157(15), 2074–2090.

    Google Scholar 

  • White, R. B. (1979). The consistency of the axiom of comprehension in the infinite-valued predicate logic of Łukasiewicz. Journal of Philosophical Logic, 8(1), 509–534.

    Article  MathSciNet  MATH  Google Scholar 

  • Yen, J. (1991). Generalizing term subsumption languages to fuzzy logic. In Proceedings of the 12th IJCAI (pp. 472–477). Sidney, Australia.

    Google Scholar 

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Esteva, F., Godo, L., Gottwald, S., Montagna, F. (2015). Introduction. In: Montagna, F. (eds) Petr Hájek on Mathematical Fuzzy Logic. Outstanding Contributions to Logic, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-06233-4_1

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