Adding the disjunctive syllogism to relevant logics including TW plus the contraction and reductio rules (Q2904134)

The admissibility of disjunctive syllogism is one of major problems concerning relevant logics listed by Anderson. In this paper the authors investigate the role of disjunctive syllogism semantically, that is, they show how to define a Routley-Meyer-type ternary relational semantics for relevant logics including TW\(^\prime\) where the logic TW is obtained by removing the axioms of contraction and reductio from ticket entailment logic T, and TW\(^\prime\) by adding to TW the rules of contraction and reductio as well as disjunctive syllogism. On the base of the semantics for TW established by Routley, Meyer et al., semantical postulates corresponding to the three additional rules are specified, with which TW\(^\prime\) is proved to be sound and complele. Several standard relevant logics, such as T, E, R etc., are shown to be characterizable in the semantical framework, so that it is useful also for the study of their extensions admitting disjunctive syllogism.