Paraconsistent reasoning and preferential entailments by signed quantified Boolean formulae

Abstract

We introduce a uniform approach of representing a variety of paraconsistent nonmonotonic formalisms by quantified Boolean formulae (QBFs) in the context of multiple-valued logics. We show that this framework provides a useful platform for capturing, in a simple and natural way, a wide range of methods for preferential reasoning. The outcome is a subtle approach to represent the underlying formalisms, which induces a straightforward way to compute the corresponding entailments: By incorporating off-the-shelf QBF solvers it is possible to simulate within our framework various kinds of preferential formalisms, among which are Priest's logic LPm of reasoning with minimal inconsistency, Batens' adaptive logic ACLuNs2, Besnard and Schaub's inference relation &vbar;= n , a variety of formula-preferential systems, some bilattice-based preferential relations (e.g., &vbar;= I 1 and &vbar;= I 2 ), and consequence relations for reasoning with graded uncertainty, such as the four-valued logic &vbar;= 4 c .