Logic Programs under Three-Valued Lukasiewicz Semantics

If logic programs are interpreted over a three-valued logic, then often Kleene’s strong three-valued logic with complete equivalence and Fitting’s associated immediate consequence operator is used. However, in such a logic the least fixed point of the Fitting operator is not necessarily a model for the program under consideration. Moreover, the model intersection property does not hold. In this paper, we consider the three-valued Lukasiewicz semantics and show that fixed points of the Fitting operator are also models for the program under consideration and that the model intersection property holds. Moreover, we review a slightly different immediate consequence operator first introduced by Stenning and van Lambalgen and relate it to the Fitting operator under Lukasiewicz semantics. Some examples are discussed to support the claim that Lukasiewicz semantics and the Stenning and van Lambalgen operator is better suited to model common sense and human reasoning.