Journal of Logic and Computation, Volume 11, Issue 6, pp. 761-788: Abstract.
The Trilattice of Constructive Truth Values
Yaroslav Shramko1, J. Michael Dunn2, and Tatsutoshi Takenaka3
1Department of Philosophy, State Pedagogical University, 50086 Kryvyi Rih, Ukraine. E-mail: firstname.lastname@example.org
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.
Keywords: Under-, determined valuation, over-, determined valuation, truth value space, bilattice, trilattice, constructive truth values, many-, valued logic