Four-Valued Semantics for Default Logic

Abstract

Reiter’s default logic suffers the triviality, that is, a single contradiction in the premise of a default theory leads to the only trivial extension which everything follows from. In this paper, we propose a default logic based on four-valued semantics, which endows default logic with the ability of handling inconsistency without leading to triviality. We define four-valued models for default theory such that the default logic has the ability of nonmonotonic paraconsistent reasoning. By transforming default rules in propositional language ${\mathcal L}$ into language $\overline{\mathcal{L}}^{+}$, a one-to-one relation between the four-valued models in ${\mathcal L}$ and the extensions in $\overline{\mathcal{L}}^{+}$ is proved, whereby the proof theory of Reiter’s default logic is remained.

This work was partially supported by NSFC (grant numbers 60373002 and 60496322) and by NKBRPC (2004CB318000).