On the Application of the Disjunctive Syllogism in Paraconsistent Logics Based on Four States of Information

We identify three classes of four-state paraconsistent logics according to their different approaches towards the disjunctive syllogism, and investigate three representatives of these approaches: Quasiclassical logic, which always accepts this principle, Belnap’s logic, that rejects the disjunctive syllogism altogether, and a logic of inconsistency minimization that restricts its application to consistent fragments only. These logics are defined in a syntactic and a semantic style, which are linked by a simple transformation. It is shown that the three formalisms accommodate knowledge minimization, and that the most liberal formalism towards the disjunctive syllogism is also the strongest among the three, while the most cautious logic is the weakest one.