Abstract
In this paper we provide a simplified, possibilistic semantics for the logics K45(G), i.e. a many-valued counterpart of the classical modal logic K45 over the [0, 1]-valued Gödel fuzzy logic \(\mathbf{G}\). More precisely, we characterize K45(G) as the set of valid formulae of the class of possibilistic Gödel frames \(\langle W, \pi \rangle \), where W is a non-empty set of worlds and \(\pi : W \mathop {\rightarrow }[0,1]\) is a possibility distribution on W. We provide decidability results as well. Moreover, we show that all the results also apply to the extension of K45(G) with the axiom (D), provided that we restrict ourselves to normalised Gödel Kripke frames, i.e. frames \(\langle W, \pi \rangle \) where \(\pi \) satisfies the normalisation condition \(\sup _{w \in W} \pi (w) = 1\).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bou, F., F. Esteva, L. Godo, and R. Rodriguez, On the minimum many-valued modal logic over a finite residuated lattice, Journal of Logic and Computation 21(5):739–790, 2011.
Bou, F., F. Esteva, and L. Godo, On possibilistic modal logics defined over MTL-chains, in F. Montagna, (ed.), Petr Hájek on Mathematical Fuzzy Logic, vol. 6 of Outstanding Contributions to Logic, Springer, 2015, pp. 225–244.
Bou, F., F. Esteva, L. Godo, and R. Rodriguez, Possibilistic semantics for a modal KD45 extension of Gödel fuzzy logic, in J.P. Carvalho, M.-J. Lesot, U. Kaymak, S. Vieira, B. Bouchon-Meunier, and R.R. Yager, (eds.), Information Processing and Management of Uncertainty in Knowledge-Based Systems. Proceedings of IPMU 2016 (Part II), vol. 611 of Communications in Computer and Information Science, Springer, 2016, pp. 123–135.
Busaniche, M., P. Cordero, and R. Rodriguez, Pseudo-monadic BL algebras, Soft Computing 23:2199–2212, 2019.
Caicedo, X., and R. Rodriguez, Standard Gödel modal logics, Studia Logica, 94(2):189–214, 2010.
Caicedo, X., and R. Rodriguez, Bi-modal Gödel modal logics, Journal of Logic and Computation 25-1:37–55, 2015.
Caicedo, X., G. Metcalfe, R. Rodriguez, and J. Rogger, A Finite Model Property for Gödel Modal Logics, in L. Libkin, U. Kohlenbach, and R. de Queiroz, (eds.), Logic, Language, Information, and Computation. Proceedings of WoLLIC 2013, vol. 8071 of Lecture Notes in Computer Science, Springer, 2013, pp. 226–237.
Caicedo, X., G. Metcalfe, R. Rodriguez, and O. Tuyt, The one-variable fragment of Corsi logic, in R. Iemhoff, M. Moortgat, and R. de Queiroz, (eds.), Logic, Language, Information, and Computation. Proceedings of WoLLIC 2019, vol. 11541 of Lecture Notes in Computer Science, Springer, 2019, pp. 70–83.
Dellunde, P., L. Godo, and E. Marchioni, Extending possibilistic logic over Gödel logic, International Journal of Approximate Reasoning 52(1): 63–75, 2011.
Dubois, D., J. Lang, and H. Prade, Possibilistic logic, in: D.M. Gabbay, C.J. Hogger, and J.A. Robinson, (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, Vol. 3: Nonmonotonic Reasoning and Uncertain Reasoning, Oxford UP, 1994, pp. 439–513.
Dubois, D., and H. Prade, Possibilistic logic: a retrospective and prospective view, Fuzzy Sets and Systems 144:3–23, 2004.
Dubois, D., and H. Prade, Inconsistency management from the standpoint of possibilistic logic, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 23(Suppl. 1):15–30, 2015.
Dubois, D., H. Prade, and S. Schockaert, Generalized possibilistic logic: foundations and applications to qualitative reasoning about uncertainty, Artificial Intelligence 252:139–174, 2017.
Fitting, M., Many-valued modal logics, Fundamenta Informaticae 15:325–254, 1991.
Fitting, M., Many-valued modal logics II, Fundamenta Informaticae 17:55–73, 1992.
Flaminio, T., L. Godo, and E. Marchioni, On the logical formalization of possibilistic counterparts of states over \(n\)-valued Łukasiewicz events, Journal of Logic and Computation 21(3):429–446, 2011.
Hájek, P., Metamathematics of Fuzzy Logic, vol. 4 of Trends in Logic, Kluwer Academic Publishers, 1998.
Hájek, P., D. Harmancová, F. Esteva, P. Garcia, and L. Godo, On modal logic for qualitative possibilisty in a fuzzy setting, in R.L. De Mantaras, and D. Poole, (eds.), Proceedings of the 94 Uncertainty in Artificial Intelligence Conference (UAI’94), Morgan Kaufmann Publishers, 1994, pp. 278–285.
Pietruszczak, A., Simplified Kripke style semantics for modal logics K45, KB4 and KD45, Bulletin of the Section of Logic 38(3/4):163–171, 2009.
Acknowledgements
Rodriguez acknowledges partial support of Argentinean projects: PICT-2019-2019-00882, UBA-CyT-20020190100021BA and PIP 112-2015-0100412 CO. Tuyt is supported by the Swiss National Science Foundation (SNF) grant 200021_184693. Esteva and Godo acknowledge partial support by the Spanish project PID2019-111544GB-C21 funded by MCIN/AEI/10.13039/501100011033. Finally, the authors acknowledge partial support by EU project 101007627 “MOSAIC”.
Funding
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Presented by Yde Venema
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Rodriguez, R.O., Tuyt, O.F., Esteva, F. et al. Simplified Kripke Semantics for K45-Like Gödel Modal Logics and Its Axiomatic Extensions. Stud Logica 110, 1081–1114 (2022). https://doi.org/10.1007/s11225-022-09987-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-022-09987-0