This paper relates inference in extended logic programming with nonclassical, nonmonotonic logics. We define a nonmonotonic logic, called equilibrium logic, based on the least constructive extension, N2, of the intermediate logic of "here-and-there". We show that on logic programs equilibrium logic coincides with the inference operation associated with the stable model and answer set semantics of Gelfond and Lifschitz. We thereby obtain a very simple characterisation of answer set semantics as a form of minimal model reasoning in N2, while equilibrium logic itself provides a natural generalisation of this semantics to arbitrary theories. We discuss briefly some consequences and applications of this result. 1 Introduction By contrast with the minimal model style of reasoning characteristic of several approaches to the semantics of logic programs, the stable model semantics of Gelfond and Lifschitz [8] was, from the outset, much closer in spirit to the styles of reasoning found in othe...

Heinrich Wansing Dresden University of Technology The knowability paradox is an instance of a remarkable reasoning pattern (actually, a pair of such patterns), in the course of which an occurrence of the possibility operator, the diamond, disappears. In the present paper, it is pointed out how the unwanted disappearance of the diamond may be escaped. The emphasis is not laid on a discussion of the contentious premise of the knowability paradox, namely that all truths are possibly known, but on how from this assumption the conclusion is derived that all truths are, in fact, known. Nevertheless, the solution o#ered is in the spirit of the constructivist attitude usually maintained by defenders of the anti-realist premise. In order to avoid the paradoxical reasoning, a paraconsistent constructive relevant modal epistemic logic with strong negation is defined semantically. The system is axiomatized and shown to be complete.

Several constructive description logics, 12) in which classical negation was replaced by strong negation as a component to treat negative atomic information have been proposed as intuitionistic variants of description logics. For conceptual representation, strong negation alone and in a combination with classical negation seems to be useful and necessary due to their respective predicate denial (e.g., not happy) and predicate term negation (e.g., unhappy) properties. In this paper, we propose an alternative description logic ALC n ∼ with classical negation and strong negation. We adhere in particular to the notions of contraries, contradictories, and subcontraries (as discussed in 6)), generated from conceivable statement types using predicate denial and predicate term negation. To capture these notions, our formalization includes a semantics that suitably interprets various combinations of classical negation and strong negation. We show that our semantics preserves contradictoriness and contrariness for ALC n ∼-concepts, but the semantics of constructive description logic CALC 2 ∼ with Heyting negation and strong negation cannot preserve the property for CALC 2 ∼-concepts.

This paper considers Kripke completeness of Nelson's constructive predicate logic N and its several variants. N is an extension of intuitionistic predicate logic Int by an contructive negation operator called strong negation. The variants of N in consideration are by the axiom of constant domain 8x(A(x)B) ! 8xA(x)B, the axiom (A ! B)(B ! A), omitting the axiom A ! (A ! B) and the axiom ::(AA); the last one we would like to call the axiom of potential omniscience and can be interpreted that we can always verify or falsify a statement, with proper additional information.

ABSTRACT We present an investigation of duality in the traditional logical manner. We extend Nelson's symmetrization of intuitionistic logic, constructible falsity, to a self-dual logic-- constructible duality. We develop a self-dual model by considering an interval of worlds in an intuitionistic Kripke model. The duality arises through how we judge truth and falsity. Truth is judged forward in the Kripke model, as in intuitionistic logic, while falsity is judged backwards. We develop a self-dual algebra such that every point in the algebra is representable by some formula in the logic. This algebra arises as an instantiation of a Heyting algebra into several categorical constructions. In particular, we show that this algebra is an instantiation of the Chu construction applied to a Heyting algebra, the second Dialectica construction applied to a Heyting algebra, and as an algebra for the study of recursion and corecursion. Thus the algebra provides a common base for these constructions, and suggests itself as an important part of any constructive logical treatment of duality. Implicit programming is suggested as a new paradigm for computing with constructible duality as its formal system. We show that all the operators that have computable least fixed points are definable explicitly and all operators with computable optimal fixed points are definable implicitly within constructible duality. Implicit programming adds a novel definitional mechanism that allows functions to be defined implicitly. This new programming feature is especially useful for programming with co-recursively defined data-types such as circular lists. iii DEDICATION To my cousin Jordan Lackey (1963-1995) whose courage with AIDS was an inspiration. iv