Four-valued Logics for Paraconsistent Reasoning

A drawback of many logics for reasoning on Semantic Web ontologies is the Principle of Explosion in case of inconsistent data. Considering a four-valued semantics, inconsistencies can be tolerated such that paraconsistency is obtained. We extend Belnap’s four-valued logic to a four-valued first-order logic with implication and equality, and we develop a method for translating formulas to a representation in classical first-order logic. This procedure allows paraconsistent reasoning in classical first-order logic. We transfer our results to a setting of Description Logics by defining four-valued versions of ALC and SHIQ. This lays the foundations for implementing paraconsistent reasoning for KRR systems that work with classical logics. We exemplify a realisation with respect to the ontology management infrastructure KAON2. The crucial idea is enabling KAON2 to merge ontologies and to perform reasoning tasks on the resulting ontology even if this ontology is inconsistent.