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Higher-Order Categorical Substructural Logic: Expanding the Horizon of Tripos Theory

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Relational and Algebraic Methods in Computer Science (RAMiCS 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12062))

Abstract

Higher-order intuitionistic logic categorically corresponds to toposes or triposes; here we address what are toposes or triposes for higher-order substructural logics. Full Lambek calculus gives a framework to uniformly represent different logical systems as extensions of it. Here we define higher-order Full Lambek calculus, which boils down to higher-order intuitionistic logic when equipped with all the structural rules, and give categorical semantics for (any extension of) it in terms of triposes or higher-order Lawvere hyperdoctrines, which were originally conceived for intuitionistic logic, and yet are flexible enough to be adapted for substructural logics. Relativising the completeness result thus obtained to different axioms, we can obtain tripos-theoretical completeness theorems for a broad variety of higher-order logics. The framework thus developed, moreover, allows us to obtain tripos-theoretical Girard and Kolmogorov translation theorems for higher-order logics.

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Acknowledgements

The author would like to thank the reviewers of the paper for their numerous, substantial comments and suggestions for improvement. The author hereby acknowledges that this work was supported by JST PRESTO (grant code: JPMJPR17G9), JSPS Kakenhi (grant code: 17K14231), and the JSPS Core-to-Core Program “Mathematical Logic and its Applications”.

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Authors and Affiliations

  1. Research School of Computer Science, The Australian National University, Canberra, Australia

    Yoshihiro Maruyama

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  1. Yoshihiro Maruyama
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Correspondence to Yoshihiro Maruyama .

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Editors and Affiliations

  1. Laboratoire d’informatique (LIX), École Polytechnique, Palaiseau, France

    Uli Fahrenberg

  2. Faculty of Mathematics, Chapman University, Orange, CA, USA

    Peter Jipsen

  3. Department of Computer Science, Brock University, St. Catharines, ON, Canada

    Michael Winter

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Maruyama, Y. (2020). Higher-Order Categorical Substructural Logic: Expanding the Horizon of Tripos Theory. In: Fahrenberg, U., Jipsen, P., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2020. Lecture Notes in Computer Science(), vol 12062. Springer, Cham. https://doi.org/10.1007/978-3-030-43520-2_12

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  • DOI: https://doi.org/10.1007/978-3-030-43520-2_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-43519-6

  • Online ISBN: 978-3-030-43520-2

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