Abstract
Building on early work by Girard (1987) and using closely related techniques from the proof theory of many-valued logics, we propose a sequent calculus capturing a hierarchy of notions of satisfaction based on the Strong Kleene matrices introduced by Barrio et al. (Journal of Philosophical Logic 49:93–120, 2020) and others. The calculus allows one to establish and generalize in a very natural manner several recent results, such as the coincidence of some of these notions with their classical counterparts, and the possibility of expressing some notions of satisfaction for higher-level inferences using notions of satisfaction for inferences of lower level. We also show that at each level all notions of satisfaction considered are pairwise distinct and we address some remarks on the possible significance of this (huge) number of notions of consequence.
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Acknowledgements
The authors wish to thank the referees for detailed and useful comments on previous versions of the paper. Luca Tranchini acknowledges the Deutsche Forschungsgemeinschaft for funding his research as part of the project “Falsity and Refutations. Understanding the negative side of logic” (DFG: TR1112/4-1). This research was also supported by the Ministry of Science, Innovation and Universities of the Government of Spain with the project ``Logic and Substructurality (FFI2017-84805-P).
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Cobreros, P., La Rosa, E. & Tranchini, L. Higher-level Inferences in the Strong-Kleene Setting: A Proof-theoretic Approach. J Philos Logic 51, 1417–1452 (2022). https://doi.org/10.1007/s10992-021-09639-z
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DOI: https://doi.org/10.1007/s10992-021-09639-z