Certain extensions of Nelson's constructive logic N with strong negation have recently become important in arti.cial intelligence and nonmonotonic reasoning, since they yield a logical foundation for answer set programming (ASP). In this paper we look at some extensions of Nelson's .rst-order logic as a basis for de.ning nonmonotonic inference relations that underlie the answer set programming semantics. The extensions we consider are those based on 2-element, here-and-there Kripke frames. In particular, we prove completeness for .rst-order here-and-there logics, and their (...) minimal strong negation extensions, for both constant and varying domains. We choose the constant domain version, which we denote by QNc5, as a basis for de.ning a .rst-order nonmonotonic extension called equilibrium logic. We establish several metatheoretic properties of QNc5, including Skolem forms and Herbrand theorems and Interpolation, and show that the .rst-oder version of equilibrium logic can be used as a foundation for answer set inference. (shrink)

We give an account of stable reasoning, a recent and novel approach to problem solving from a formal, logical point of view. We describe the underlying logic of stable reasoning and illustrate how it is used to model different domains and solve practical reasoning problems. We discuss some of the main differences with respect to reasoning in classical logic and we examine an ongoing research programme for the rational reconstruction of human knowledge that may be considered a successor to the (...) logical empiricists’ programme of the mid-twentieth century. (shrink)