An Avron rule for fragments of R-mingle

Axiomatic bases of admissible rules are obtained for fragments of the substructural logic R-mingle. In particular, it is shown that a ‘modus-ponens-like’ rule introduced by Arnon Avron forms a basis for the admissible rules of its implication and implication–fusion fragments, while a basis for the admissible rules of the full multiplicative fragment requires an additional countably infinite set of rules. Indeed, this latter case provides an example of a three-valued logic with a finitely axiomatizable consequence relation that has no finite basis for its admissible rules.