Multiset-Multiset Frames

This paper presents the notion of multiset-multiset frame (mm-frame for short), a frame equipped with a relation between (finite) multisets over the set of points which satisfies the condition called compositionality. This notion is an extension of Restall and Standefer’s multiset frame, a frame that relates a multiset to a single point. While multiset frames serve as frames for the positive fragments of relevant logics RW and R, mm-frames are for the full RW and R with negation. We show this by presenting a way of constructing an mm-frame from any GS-frame, a frame with two dual ternary relations in which the Routley star is definable.