# Two proofs of the algebraic completeness theorem for multilattice logic

Two proofs of the algebraic completeness theorem for multilattice logic AbstractShramko [(2016). Truth, falsehood, information and beyond: The American plan generalized. In K. Bimbo (Ed.), J. Michael Dunn on information based logics, outstanding contributions to logic (pp. 191–212). Dordrecht: Springer] formulated multilattice logic and the algebraic completeness theorem for it. However, the proof has not been presented. In this paper, we consider Kamide and Shramko's multilattice logic [Kamide & Shramko (2017a). Embedding from multilattice logic into classical logic and vice versa. Journal of Logic and Computation, 27(5), 1549–1575] which is an extension of Shramko's original multilattice logic by several implications and coimplications. Using the technique of algebraic embedding, we show that Kamide and Shramko's sequent calculus for multilattice logic is sound and complete with respect to multilattices. Moreover, we introduce yet another algebraic semantics for this logic based on the notion of a De Morgan multilattice. Using Lindenbaum-Tarski algebras, we show that is sound and complete with respect to De Morgan multilattices. Besides, we modify Kamide and Shramko's notion of modal multilattice [Kamide & Shramko (2017b). Modal multilattice logic. Logica Universalis, 11(3), 317–343], i.e. we present the concept of De Morgan modal multilattice. We prove that Kamide and Shramko's modal multilattice logic (Kamide & Shramko, 2017b) is adequate with respect to De Morgan modal multilattices. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied Non-Classical Logics Taylor & Francis

# Two proofs of the algebraic completeness theorem for multilattice logic

, Volume 29 (4): 24 – Oct 2, 2019

## Two proofs of the algebraic completeness theorem for multilattice logic

Abstract

AbstractShramko [(2016). Truth, falsehood, information and beyond: The American plan generalized. In K. Bimbo (Ed.), J. Michael Dunn on information based logics, outstanding contributions to logic (pp. 191–212). Dordrecht: Springer] formulated multilattice logic and the algebraic completeness theorem for it. However, the proof has not been presented. In this paper, we consider Kamide and Shramko's multilattice logic [Kamide & Shramko (2017a). Embedding from multilattice logic...  /lp/taylor-francis/two-proofs-of-the-algebraic-completeness-theorem-for-multilattice-YRVoCzfifL
Publisher
Taylor & Francis
ISSN
1958-5780
eISSN
1166-3081
DOI
10.1080/11663081.2019.1647654
Publisher site
See Article on Publisher Site

### Abstract

AbstractShramko [(2016). Truth, falsehood, information and beyond: The American plan generalized. In K. Bimbo (Ed.), J. Michael Dunn on information based logics, outstanding contributions to logic (pp. 191–212). Dordrecht: Springer] formulated multilattice logic and the algebraic completeness theorem for it. However, the proof has not been presented. In this paper, we consider Kamide and Shramko's multilattice logic [Kamide & Shramko (2017a). Embedding from multilattice logic into classical logic and vice versa. Journal of Logic and Computation, 27(5), 1549–1575] which is an extension of Shramko's original multilattice logic by several implications and coimplications. Using the technique of algebraic embedding, we show that Kamide and Shramko's sequent calculus for multilattice logic is sound and complete with respect to multilattices. Moreover, we introduce yet another algebraic semantics for this logic based on the notion of a De Morgan multilattice. Using Lindenbaum-Tarski algebras, we show that is sound and complete with respect to De Morgan multilattices. Besides, we modify Kamide and Shramko's notion of modal multilattice [Kamide & Shramko (2017b). Modal multilattice logic. Logica Universalis, 11(3), 317–343], i.e. we present the concept of De Morgan modal multilattice. We prove that Kamide and Shramko's modal multilattice logic (Kamide & Shramko, 2017b) is adequate with respect to De Morgan modal multilattices.

### Journal

Journal of Applied Non-Classical LogicsTaylor & Francis

Published: Oct 2, 2019

Keywords: Multilattice logic; algebraic semantics; algebraic embedding; Lindenbaum-Tarski algebra; sequent calculus; modal multilattice logic

Access the full text.