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Kripke Completeness of Bi-intuitionistic Multilattice Logic and its Connexive Variant

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Abstract

In this paper, bi-intuitionistic multilattice logic, which is a combination of multilattice logic and the bi-intuitionistic logic also known as Heyting–Brouwer logic, is introduced as a Gentzen-type sequent calculus. A Kripke semantics is developed for this logic, and the completeness theorem with respect to this semantics is proved via theorems for embedding this logic into bi-intuitionistic logic. The logic proposed is an extension of first-degree entailment logic and can be regarded as a bi-intuitionistic variant of the original classical multilattice logic determined by the algebraic structure of multilattices. Similar completeness and embedding results are also shown for another logic called bi-intuitionistic connexive multilattice logic, obtained by replacing the connectives of intuitionistic implication and co-implication with their connexive variants.

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Authors and Affiliations

  1. Department of Information and Electronic Engineering Faculty of Science and Engineering, Teikyo University, Toyosatodai 1-1, Utsunomiya, Tochigi,  320-8551, Japan

    Norihiro Kamide

  2. Department of Philosophy, Kryvyi Rih State Pedagogical University, prosp. Gagarina 54, Kryvyi Rih, 50086, Ukraine

    Yaroslav Shramko

  3. Department of Philosophy II, Ruhr-University Bochum, 44801, Bochum, Germany

    Heinrich Wansing

Authors
  1. Norihiro Kamide
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  2. Yaroslav Shramko
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  3. Heinrich Wansing
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Correspondence to Norihiro Kamide.

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Kamide, N., Shramko, Y. & Wansing, H. Kripke Completeness of Bi-intuitionistic Multilattice Logic and its Connexive Variant. Stud Logica 105, 1193–1219 (2017). https://doi.org/10.1007/s11225-017-9752-x

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