Two proofs of the algebraic completeness theorem for multilattice logic

Shramko [(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 MLn [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 MLn 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 MLn 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.