Adding the Inference Rule Disjunctive Syllogism to Relevant Logics

In order to avoid trivial consequences, the derivability relation of relevant logics (RL) does not support the inference rule disjunctive syllogism (DS). In this paper, I will show that the inference rule DS can be added to this derivability relation in a way that doesn’t lead to trivial or irrelevant consequences. To do so, I will rely on the insights gained from the study of inconsistency–adaptive logics, the branch of adaptive logics that was devised to explicate reasoning based on inconsistent premise sets. Moreover, I will also point to the relations with some alternative proposals from the literature, and I will show that the inconsistency–adaptive framework has some major advantages over these.