This article presents some theorems for syntactic and semantic embeddings of a Gentzen-type sequent calculus MLn for multilattice logic into a Gentzen-type sequent calculus LK for classical logic and vice versa. These embedding theorems are used to prove cut-elimination, decidability and completeness theorems for MLn⁠, as well as a modified Craig interpolation theorem. Some of these results are then extended to the first-order system FMLn with implications and co-implications.

We introduce an abstract algebraic structure – a lattice defined on a generalized truth value space of constructive logic. For background one can refer to the idea of ‘under‐determined’ and ‘over‐determined’ valuations (Dunn), a ‘useful four‐valued logic’ (Belnap), and the notion of a bilattice (Ginsberg). We consider within one general framework the notions of constructive truth and constructive falsity, as well as the notions of non‐constructive truth and non‐constructive falsity. All possible combinations of the basic truth values give rise to an interesting ‘16‐valued logic’. It appears that these 16 truth values constitute what we call a trilattice – a natural mathematical structure with three partial orderings that represent respectively an increase in information, truth and constructivity. The presentation of the paper is essentially conceptual: the stress is laid on introducing new concepts and structures as well as on their general interpretation.