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Cut elimination for entailment relations

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Abstract

Entailment relations, introduced by Scott in the early 1970s, provide an abstract generalisation of Gentzen’s multi-conclusion logical inference. Originally applied to the study of multi-valued logics, this notion has then found plenty of applications, ranging from computer science to abstract algebra. In particular, an entailment relation can be regarded as a constructive presentation of a distributive lattice and in this guise it has proven to be a useful tool for the constructive reformulation of several classical theorems in commutative algebra. In this paper, motivated by these concrete applications, we state and prove a cut-elimination result for inductively generated entailment relations. We analyse some of its consequences and describe the existing connections with analogous results in the literature.

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Acknowledgements

The research that has led to this paper was carried out within the project “Categorical localisation: methods and foundations” (CATLOC) funded by the University of Verona within the programme “Ricerca di Base 2015”; the related financial support is gratefully acknowledged. This work was further supported by the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA - INdAM). The final version of this paper was prepared within the project “A New Dawn of Intuitionism: Mathematical and Philosophical Advances” (ID 60842) funded by the John Templeton Foundation, as well as within the project “Dipartimenti di Eccellenza 2018–2022” of the Italian Ministry of Education, Universities and Research (MIUR). The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

We wish to express our gratitude to Peter Schuster for his encouragement and insights, and thank Henri Lombardi and Stefan Neuwirth for interesting discussions. Last but not least, we thank the anonymous referee, whose expertly remarks and valuable suggestions helped to improve our manuscript, and the anonymous referee of a forerunner of this paper for pointing out to us the connection with hyperresolution calculi.

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  1. Dipartimento di Informatica, Università di Verona, Strada le Grazie 15, 37134, Verona, Italy

    Davide Rinaldi & Daniel Wessel

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Rinaldi, D., Wessel, D. Cut elimination for entailment relations. Arch. Math. Logic 58, 605–625 (2019). https://doi.org/10.1007/s00153-018-0653-0

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