Studia Logica

, Volume 105, Issue 6, pp 1193–1219

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

• Norihiro Kamide
• Yaroslav Shramko
• Heinrich Wansing
Article

## 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.

## Keywords

First-degree entailment logic Multilattices Bi-intuitionistic logic Connexive logic

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## Authors and Affiliations

• Norihiro Kamide
• 1
• Yaroslav Shramko
• 2
• Heinrich Wansing
• 3
1. 1.Department of Information and Electronic Engineering Faculty of Science and EngineeringTeikyo UniversityUtsunomiyaJapan
2. 2.Department of PhilosophyKryvyi Rih State Pedagogical UniversityKryvyi RihUkraine
3. 3.Department of Philosophy IIRuhr-University BochumBochumGermany