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The trilattice of constructive truth values. (English) Zbl 0996.03014
Summary: 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 ‘usful 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.

MSC:
03B50 Many-valued logic
03G10 Logical aspects of lattices and related structures
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