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Axiomatization of Crisp Gödel Modal Logic

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Abstract

In this paper we consider the modal logic with both \(\Box \) and \(\Diamond \) arising from Kripke models with a crisp accessibility and whose propositions are valued over the standard Gödel algebra \([0,1]_G\). We provide an axiomatic system extending the one from Caicedo and Rodriguez (J Logic Comput 25(1):37–55, 2015) for models with a valued accessibility with Dunn axiom from positive modal logics, and show it is strongly complete with respect to the intended semantics. The axiomatizations of the most usual frame restrictions are given too. We also prove that in the studied logic it is not possible to get \(\Diamond \) as an abbreviation of \(\Box \), nor vice-versa, showing that indeed the axiomatic system we present does not coincide with any of the mono-modal fragments previously axiomatized in the literature.

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Acknowledgements

The authors are thankful to the anonymous reviewers for their useful comments, that have helped to improve the layout of the paper. This project has received funding from the following sources: (1) the European Union’s Horizon 2020 Research and Innovation program under the Marie Sklodowska-Curie Grant Agreement No. 689176 (SYSMICS project); (2) the Grant No. CZ.02.2.69/0.0/0.0/17_050/0008361 of the Operational programme Research, Development, Education of the Ministry of Education, Youth and Sport of the Czech Republic, co-financed by the European Union; (3) the Spanish MINECO project RASO (TIN2015-71799-C2-1-P) and (4) the Argentinean project PIP CONICET 11220150100412CO and UBA-CyT 20020150100002BA.

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Authors and Affiliations

  1. UBA, FCEyN, Departamento de Computación, UBA-CONICET, Inst. de Invest. en Cs. de la Computación, Buenos Aires, Argentina

    Ricardo Oscar Rodriguez

  2. Department of Theoretical Computer Science, Institute of Computer Science of the Czech Academy of Sciences, Pod Vodárenskou věží 271/2, 182 07, Praha, Czech Republic

    Amanda Vidal

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  1. Ricardo Oscar Rodriguez
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  2. Amanda Vidal
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Correspondence to Amanda Vidal.

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Rodriguez, R.O., Vidal, A. Axiomatization of Crisp Gödel Modal Logic. Stud Logica 109, 367–395 (2021). https://doi.org/10.1007/s11225-020-09910-5

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