Negation as Inconsistency and Double Negation Law Problem

Abstract

In this paper, we deal with the database proposed by Gabbay, et al., which realizes negation as inconsistency, containing negation as failure, to focus our attention on its double negation problem. The database consists of a clausal form program as a positive declaration, and of a set as a negative declaration. A formula as a goal is expected to be transformed and added to a clausal form program and the negative declaration set. The negative declaration comes up with the idea that the inconsistency of the negative declaration with the program is expected to infer the negation of the goal. In the database, if the negative declaration is defined to be a set of formulas finitely failing in SLDNF resolutions, the negation as inconsistency is equivalent to the negation as failure. However, the double negation law does not necesserily hold. We give a class of databases in which the double negation law may hold. The database in the class consists of a general logic program as a positive declaration, and of a set, as a negative declaration, whose member is not in a stable model of the general logic program.