RM and its Nice Properties

Dunn–McCall logic RM is by far the best understood and the most well-behaved logic in the family of logics developed by the school of Anderson and Belnap. However, it is not considered to be a relevant logic by the relevant logicians, since it fails to have the variable-sharing property. Instead, RM is usually characterized as being “semi-relevant,” without explaining what this notion means. In this paper we suggest a plausible definition of semi-relevance, and show that according to it, RM is a strongly maximal semi-relevant logic having a conjunction, a disjunction, and an implication. We also review and prove the most important nice properties of RM, especially strong completeness results about it (the full proofs of which are difficult to find in the literature).