Extended Stepping Theories of Active Logic: Declarative Semantics

Active Logic is a conceptual system with reasoning formalism that allows for correlation of their results with specific points in time and that has tolerance to inconsistencies. Currently, tolerance to inconsistencies (paraconsistency) in Active Logic systems is theoretically justified in the works of the authors of this paper and is attributed to the so-called formalisms of stepping theories, which integrate the principles of Active Logic and Logic Programming. More specifically, the argumentation semantics of so-called formalisms of stepping theories with two kinds of negation has been proved to be paraconsistent. This formalism has more expressive power than the other formalisms of stepping theories and to a greater extent satisfies the principles of Logic Programming. This case study proposes the declarative semantics for formalisms of stepping theories and represents its equivalency with respect to the argumentation semantics of this type of formalism. This, in turn, means that the proposed declarative semantics is also paraconsistent, and logical inconsistencies existing in these theories do not result in their destruction.