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
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
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.
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.
Baaz, M., Hájek, P., & Krajíček, J. (1998). Embedding logics into product logic. Studia Logica, 61(1), 35–47.
Baaz, M., Hájek, P., Montagna, F., & Veith, H. (2002). Complexity of T-tautologies. Annals of Pure and Applied Logic, 113(1–3), 3–11.
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.
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).
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).
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.
Caicedo, X., & Rodríguez, R. O. (2010). Standard Gödel modal logics. Studia Logica, 94(2), 189–214.
Caicedo, X., & Rodríguez, R. O. (2012). Bi-modal Gödel logic over [0, 1]-valued Kripke frames. Journal of Logic and Computation (in press).
Cerami, M. (2012). Fuzzy description logics under a mathematical fuzzy logic point of view. Ph. D. thesis, University of Barcelona, October 16, 2012.
Cerami, M., & Straccia, U. (2013). On the (un)decidability of fuzzy description logics under Łukasiewicz t-norm. Information Sciences, 227, 1–21.
Cerami, M., Esteva, F., & Bou, F. (2010). Decidability of a description logic over infinite-valued product logic. In Proceedings of KR2010 (pp. 203–213).
Chang, C. C. (1959). A new proof of the completeness of the Łukasiewicz axioms. Transactions of the American Mathematical Society, 93(1), 74–80.
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.
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.
Cintula, P., & Hájek, P. (2009). Complexity issues in axiomatic extensions of Łukasiewicz logic. Journal of Logic and Computation, 19(2), 245–260.
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.
Cintula, P., Haniková, Z., & Švejdar, V. (Eds.). (2009). Witnessed years–essays in honour of Petr Hájek. In Tributes, (vol 10). London: College Publications.
Esteva, F., Godo, L., Hájek, P., & Navara, M. (2000). Residuated fuzzy logics with an involutive negation. Archive for Mathematical Logic, 39(2), 103–124.
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.
Esteva, F., Godo, L., Hájek, P., & Montagna, F. (2003). Hoops and fuzzy logic. Journal of Logic and Computation, 13(4), 532–555.
Esteva, F., & Godo, L. (2013). A logical approach to fuzzy truth hedges. Information Sciences, 232, 366–385.
Fitting, M. C. (1992a). Many-valued modal logics. Fundamenta Informaticae, 15, 235–254.
Fitting, M. C. (1992b). Many-valued modal logics II. Fundamenta Informaticae, 17, 55–73.
Flaminio, T., & Godo, L. (2007). A logic for reasoning about probability of fuzzy events. Fuzzy Sets and Systems, 158, 625–638.
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.
Flaminio, T., & Montagna, F. (2011). Models for many-valued probabilistic reasoning. Journal of Logic and Computation, 21(3), 447–464.
Flaminio, T., Godo, L., & Marchioni, E. (2013). Logics for belief functions on MV-algebras. International Journal of Approximate Reasoning, 54(4), 491–512.
Flondor, P., Georgescu, G., & Iorgulescu, A. (2001). Pseudo t-norms and pseudo-BL-algebras. Soft Computing, 5(5), 355–371.
Font, J. M., & Hájek, P. (2002). On Łukasiewicz’s four-valued modal logic. Studia Logica, 70(2), 157–182.
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.
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).
Georgescu, G., & Iorgulescu, A. (2001). Pseudo-MV algebras. Multiple-Valued Logic, 6(1–2), 95–135.
Godo, L., Hájek, P., & Esteva, F. (2003). A fuzzy modal logic for belief functions. Fundamenta Informaticae, 57(2–4), 127–146.
Gottwald, S. (2001). A treatise on many-valued logics. In Studies in logic and computation (Vol. 9). Baldock: Research Studies Press.
Hájek, P. (1998). Metamathematics of fuzzy logic. In Trends in logic (Vol. 4). Dordrecht: Kluwer.
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.
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.
Hájek, P., Harmancová, D., & Verbrugge, R. (1995). A qualitative fuzzy possibilistic logic. International Journal of Approximate Reasoning, 12(1), 1–19.
Hájek, P., Godo, L., & Esteva, F. (1996). A complete many-valued logic with product conjunction. Archive for Mathematical Logic, 35(3), 191–208.
Hájek, P. (1998). Basic fuzzy logic and BL-algebras. Soft Computing, 2(3), 124–128.
Hájek, P., & Harmancová, D. (2000). A hedge for Gödel fuzzy logic. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 8(4), 495–498.
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.
Hájek, P. (2001). On very true. Fuzzy Sets and Systems, 124(3), 329–333.
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.
Hájek, P., & Tulipani, S. (2001). Complexity of fuzzy probability logics. Fundamenta Informaticae, 45, 207–213.
Hájek, P. (2002a). Observations on the monoidal t-norm logic. Fuzzy Sets and Systems, 132(1), 107–112.
Hájek, P. (2002b). Some hedges for continuous T-norms logics. Neural Network World, 12(2), 159–164.
Hájek, P. (2003a). Embedding standard BL-algebras into non-commutative pseudo-BL-algebras. Tatra Mountains Mathematical Publications, 27(3), 125–130.
Hájek, P. (2003b). Fuzzy logics with noncommutative conjunctions. Journal of Logic and Computation, 13(4), 469–479.
Hájek, P. (2005a). Fleas and fuzzy logic. Journal of Multiple-Valued Logic and Soft Computing, 11(1–2), 137–152.
Hájek, P. (2005b). Making fuzzy description logic more general. Fuzzy Sets and Systems, 154(1), 1–15.
Hájek, P. (2005c). A non-arithmetical Gödel logic. Logic Journal of the Interest Group of Pure and Applied Logic, 13(5), 435–441.
Hájek, P. (2005d). On arithmetic in Cantor-Łukasiewicz fuzzy set theory. Archive for Mathematical Logic, 44(6), 763–782.
Hájek, P. (2006a). Computational complexity of t-norm based propositional fuzzy logics with rational truth constants. Fuzzy Sets and Systems, 157(5), 677–682.
Hájek, P., & Cintula, P. (2006b). On theories and models in fuzzy predicate logics. Journal of Symbolic Logic, 71(3), 863–880.
Hájek, P. (2007a). Complexity of fuzzy probability logics II. Fuzzy Sets and Systems, 158, 2605–2611.
Hájek, P. (2007b). Mathematical fuzzy logic and natural numbers. Fundamenta Informaticae, 81(1–3), 155–163.
Hájek, P., & Montagna, F. (2008). A note on the first-order logic of complete BL-chains. Mathematical Logic Quarterly, 54(4), 435–446.
Hájek, P. (2009). Arithmetical complexity of fuzzy predicate logics-a survey II. Annals of Pure and Applied Logic, 161(2), 212–219.
Hájek, P. (2010). On fuzzy modal logics \(S5({{C}})\). Fuzzy Sets and Systems, 161, 2389–2396.
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.
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.
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.
Hájek, P., & Harmancová, D. (1996). A many-valued modal logic. In Proceedings of IPMU’96 (pp. 1021–1024). Granada.
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.
Hähnle, R. (1994). Many-valued logic and mixed integer programming. Annals of Mathematics and Artificial Intelligence, 12(34), 231–264.
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).
Hansoul, G., & Teheux, B. (2013). Extending Łukasiewicz logics with a modality: Algebraic approach to relational semantics. Studia Logica, 101(3), 505–545.
Höhle, U. (1994). Monoidal logic. In R. Palm, R. Kruse, & J. Gebhardt (Eds.), Fuzzy systems in computer science (pp. 233–243). Wiesbaden: Vieweg.
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.
Łukasiewicz, T., & Straccia, U. (2008). Managing uncertainty and vagueness in Description Logics for the Semantic Web. Journal of Web Semantics, 6(4), 291–308.
Metcalfe, G., & Olivetti, N. (2011). Towards a proof theory of Gödel modal logics. Logical Methods in Computer Science, 7(2:10), 1–27.
Mundici, D. (1987). Satisfiability in many-valued sentential logic is NP-complete. Theoretical Computer Science, 52(1–2), 145–153.
Pavelka, J. (1979). On fuzzy logic I, II, III. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 25, 45–52, 119–134, 447–464.
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.
Skolem, T. (1957). Bemerkungen zum Komprehensionsaxiom. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 3, 1–17.
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.
Straccia, U. (1998). A fuzzy description logic. In Proceedings of AAAI-98, 15th National Conference on Artificial Intelligence (pp. 594–599). MIT Press.
Straccia, U. (2001). Reasoning within fuzzy description logics. Journal of Artificial Intelligence Research, 14, 137–166.
Straccia, U., & Bobillo, F. (2007). Mixed integer programming, general concept inclusions and fuzzy description logics. Mathware and Soft Computing, 14(3), 247–259.
Tresp, C. B., & Molitor, R. (1998). A description logic for vague knowledge. Technical report RWTH-LTCS report 98–01. Aachen University of Technology.
Vychodil, V. (2006). Truth-depressing hedges and BL-logic. Fuzzy Sets and Systems, 157(15), 2074–2090.
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.
Yen, J. (1991). Generalizing term subsumption languages to fuzzy logic. In Proceedings of the 12th IJCAI (pp. 472–477). Sidney, Australia.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-06233-4_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06232-7
Online ISBN: 978-3-319-06233-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)