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Proof Theory for Positive Logic with Weak Negation

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Abstract

Proof-theoretic methods are developed for subsystems of Johansson’s logic obtained by extending the positive fragment of intuitionistic logic with weak negations. These methods are exploited to establish properties of the logical systems. In particular, cut-free complete sequent calculi are introduced and used to provide a proof of the fact that the systems satisfy the Craig interpolation property. Alternative versions of the calculi are later obtained by means of an appropriate loop-checking history mechanism. Termination of the new calculi is proved, and used to conclude that the considered logical systems are PSPACE-complete.

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Acknowledgements

We would like to thank Dick de Jongh for several discussions on the subject, and for his valuable remarks. We would also like to thank the referees for their careful reading, which has helped us to improve the paper. The first author has been supported by the grant GA17-04630S of the Czech Science Foundation. The research of the second author was supported by the Swiss National Science Foundation Grant 200021_165850. Moreover, this project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 689176.

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Authors and Affiliations

  1. The Czech Academy of Sciences, Institute of Computer Science, Prague, Czech Republic

    Marta Bílková

  2. Mathematical Institute, University of Bern, Bern, Switzerland

    Almudena Colacito

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  1. Marta Bílková
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  2. Almudena Colacito
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Correspondence to Almudena Colacito.

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Bílková, M., Colacito, A. Proof Theory for Positive Logic with Weak Negation. Stud Logica 108, 649–686 (2020). https://doi.org/10.1007/s11225-019-09869-y

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