Abstract
Recent works about ecumenical systems, where connectives from classical and intuitionistic logics can co-exist in peace, warmed the discussion on proof systems for combining logics. This discussion has been extended to alethic modalities using Simpson’s meta-logical characterization: necessity is independent of the viewer, while possibility can be either intuitionistic or classical. In this work, we propose a pure, label free calculus for ecumenical modalities, \(\mathsf {nEK}\), where exactly one logical operator figures in introduction rules and every basic object of the calculus can be read as a formula in the language of the ecumenical modal logic \(\mathsf {EK}\). We prove that \(\mathsf {nEK}\) is sound and complete w.r.t. the ecumenical birelational semantics and discuss fragments and extensions.
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Acknowledgements
This work was partially financed by CNPq, CAPES and by the UK’s EPSRC through research grant EP/S013008/1. We would like to thank the anonymous reviewers for their suggestions and comments.
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Marin, S., Pereira, L.C., Pimentel, E., Sales, E. (2021). A Pure View of Ecumenical Modalities. In: Silva, A., Wassermann, R., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2021. Lecture Notes in Computer Science(), vol 13038. Springer, Cham. https://doi.org/10.1007/978-3-030-88853-4_24
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