Based on Fiting's Φ operator a unified framework for three-valued semantics of logic programming is presented. The truth space used in the framework is the class of partial interpretations. Underlying the truth space is two partial orderings, knowledge ordering and truth ordering. It turns out that the truth space with the truth ordering is a complete lattice and the truth space with knowledge ordering is a semi-complete lattice. Φ is proved to be continuous over the complete lattice and monotonic over the semi-complete lattice. With the use of Φ operator two well-known three-valued semantics for logic programming, Fitting's three-valued semantics and well-founded semantics, are characterized within the framework in a simple and elegant way. We show that Fitting's semantics is the least stable three-valued model with respect to the knowledge ordering and well-founded semantics is the least stable three-valued model with respect to the truth ordering.
Yang, Feng, "A Unified Framework For Three-Valued Semantical Treatments of Logic Programming" (1991). Electrical Engineering and Computer Science Technical Reports. 118.