Abstract
In this paper we present completeness results of several fuzzy logics trying to capture different notions of necessity (in the sense of Possibility theory) for Gödel logic formulas. In a first attempt, based on different characterizations of necessity measures on fuzzy sets, a group of logics, with Kripke style semantics, are built over a restricted language, indeed a two level language composed of non-modal and modal formulas, the latter moreover not allowing for nested applications of the modal operator N. Besides, a full fuzzy modal logic for graded necessity over Gödel logic is also introduced together with an algebraic semantics, the class of NG-algebras.
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References
Aguzzoli, S., Gerla, B., Marra, V.: De Finetti’s no-Dutch-book criterion for Gödel logic. Studia Logica 90, 25–41 (2008)
Alsinet, T.: Logic Programming with Fuzzy Unification and Imprecise Constants: Possibilistic Semantics and Automated Deduction. Monografies de l’Institut d’Investigació en Intel·ligència Artificial, CSIC, Barcelona (2003)
Alsinet, T., Godo, L., Sandri, S.: On the Semantics and Automated Deduction for PLFC: a logic of possibilistic uncertainty and fuzziness. In: Proc. of 15th Conference on Uncertainty in Artificial Intelligence Conference UAI 1999, Stokholm, Sweden, pp. 3–12 (1999)
Alsinet, T., Godo, L., Sandri, S.: Two formalisms of extended possibilistic logic programming with context-dependent fuzzy unification: a comparative description. Electr. Notes Theor. Comput. Sci. 66(5) (2002)
Blok, W.J., Pigozzi, D.: Algebraizable logics. Memoirs of the American Mathematical Society A.M.S. 396 (1989)
Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Graduate texts in mathematics, vol. 78. Springer, Heidelberg (1981)
Dubois, D., Prade, H.: Possibility theory. Plenum Press, New York (1988)
Dubois, D., Prade, H.: Resolution principles in possibilistic logic. International Journal of Approximate Reasoning 4(1), 1–21 (1990)
Dubois, D., Lang, J., Prade, H.: Possibilistic logic. In: Gabbay, et al. (eds.) Handbook of Logic in Artificial Intelligence and Logic Programming. Nonmonotonic Reasoning and Uncertain Reasoning, vol. 3, pp. 439–513. Oxford University Press, Oxford (1994)
Esteva, F., Godo, L., Hájek, P., Navara, M.: Residuated fuzzy logics with an involutive negation. Archive for Mathematical Logic 39(2), 103–124 (2000)
Esteva, F., Gispert, J., Godo, L., Noguera, C.: Adding truth-constants to logics of continuous t-norms: Axiomatization and completeness results. Fuzzy Sets and Systems 158, 597–618 (2007)
Flaminio, T., Godo, L.: A logic for reasoning about the probability of fuzzy events. Fuzzy Sets and Systems 158, 625–638 (2007)
Flaminio, T., Godo, L., Marchioni, E.: On the Logical Formalization of Possibilistic Counterparts of States over n-Valued Łukasiewicz Events. Journal of Logic and Computation (in press), doi:10.1093/logcom/exp012
Flaminio, T., Montagna, F.: MV-algebras with internal states and probabilistic fuzzy logics. International Journal of Approximate Reasoning 50, 138–152 (2009)
Flaminio, T., Montagna, F.: An algebraic approach to states on MV-algebras. In: Proc. of EUSFLAT 2007, Ostrava, Czech Republic, vol. II, pp. 201–206 (2007)
Godo, L., Vila, L.: Possibilistic temporal reasoning based on fuzzy temporal constraints. In: Proceedings of IJCAI 1995, pp. 1916–1922. Morgan Kaufmann, San Francisco (1995)
Hájek, P.: Metamathematics of Fuzzy Logic. Trends in Logic, vol. 4. Kluwer Academic Publishers, Dordrecht (1998)
Hájek, P.: Complexity of fuzzy probability logics II. Fuzzy Sets and Systems 158, 2605–2611 (2007)
Hájek, P., Harmancová, D., Esteva, F., Garcia, P., Godo, L.: On Modal Logics for Qualitative Possibility in a Fuzzy Setting. In: Proc. of the 94 Uncertainty in Artificial Intelligence Conference (UAI 1994), pp. 278–285. Morgan Kaufmann, San Francisco (1994)
Halpern, J.Y.: Reasoning about uncertainty. MIT Press, Cambridge (2003)
Hodges, W.: Model Theory. In: Encyclopedia of Mathematics and its Applications, vol. 42. Cambridge University Press, Cambridge (1993)
Mundici, D.: Averaging the truth-value in Łukasiewicz logic. Studia Logica 55(1), 113–127 (1995)
Shafer, G.: A mathematical theory of evidence. Princeton University Press, Princeton (1976)
Sugeno, M.: Theory of Fuzzy Integrals and its Applications. PhD thesis, Tokyo Institute of Technology, Tokio, Japan (1974)
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Dellunde, P., Godo, L., Marchioni, E. (2009). Exploring Extensions of Possibilistic Logic over Gödel Logic. In: Sossai, C., Chemello, G. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2009. Lecture Notes in Computer Science(), vol 5590. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02906-6_79
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DOI: https://doi.org/10.1007/978-3-642-02906-6_79
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