Three-valued constructive logic and logic programs

We propose three valued constructive logic with strong negation 3N by describing both proof and model theory. We relate the proposed three valued system to J. Lukasiewicz's (1920) three valued logic. We demonstrate that 3N is very useful to formalize a semantics for logic programs with negation as failure within the framework of constructive logic. We introduce the concept of N completion to provide an intuitive semantics in which negation as failure can be interpreted as strong negation. The N completion semantics is a natural alternative to K. Kunen's three valued completion semantics (1987; 1989).