Bilattices, Intuitionism and Truth-knowledge Duality: Concepts and Foundations
We propose a family of intuitionistic bilattices with full truth-knowledge duality (D-bilattices) for a logic programming. The first family of perfect D-bilattices is composed by Boolean algebras with even number of atoms: the simplest of them, based on intuitionistic truth-functionally complete extension of Belnap’s 4-valued bilattice, can be used in paraconsistent programming, that is, for knowledge bases with incomplete and inconsistent information. The other two families are useful for a probability theory where the uncertainty in the knowledge about a piece of information is in the form of belief types: as an interval (lower and upper boundary) probability or as a confidence level. Such programs can be parameterized by different kinds of probabilistic conjunctive/disjunctive strategies for their rules, based on intuitionistic implication. Such a frame-work offers a clear semantics for the satisfaction relation, and allows the extension of logic languages with intuitionistic implications also in the body of rules. From a theoretical point of view we introduce also a duality in higher-order bilatices, as in Temporal Probabilistic Logic, constructed as functional spaces over ordinary dual bilattices. Then we show the full truth-knowledge duality for a fixpoint semantics of logic programs based on dual bilattices. Finally we develop also an autoreferential version of Stone’s Representation Theorem for the dual bilattices.