The Trilattice of Constructive Truth Values

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.