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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.

This paper reformulates and decides a certain conjecture in Dunn's ‘Relevant Predication 1: The Formal Theory’ (Journal of Philosophical Logic16, 347–381, 1987). This conjecture of Dunn's relates his object-language characterisation of a property's being relevant in a variable x to certain grammatical characterisations of relevance, analogous to some given by Helman, in ‘Relevant Implication and Relevant Functions’ (to appear in Entailment: The Logic of Relevance and Necessity, vol. 2, by Alan Ross Anderson, Nuel Belnap, and J. Michael Dunn et al.) In the course of the investigation this paper also investigates Kit Fine's semantics for quantified relevance logics, which appears in his appropriately titled ‘Semantics for Quantified Relevance Logics’

Book Information Algebraic Methods in Philosophical Logic. By J. Michael Dunn and Gary Hardegree. Clarendon Press. Oxford. 2001. Pp. xv + 470. £60.50.