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Schiller Joe Scroggs in [9] established remarkable facts concerning “normal” extensions of the modal sentential calculus S5, the most notable of these facts being that all such proper extensions have finite characteristic matrices. The major import of the present paper is that like facts hold for the relevant sentential calculus R-Mingle (RM). Robert K. Meyer in [6] has obtained an important completeness result for RM, which will play a central role in our results. However, in §2 we shall obtain a new proof of Meyer's result as a by-product of the algebraic logic that we develop in §1. Also in §2 we shall obtain the promised results for extensions of RM. In §3 we shall provide a strong completeness theorem for RM by generalizing the semantics of Meyer.

By γ, we mean the rule, “From ├ A and ├ Ā V B, infer ├ B”.1 This rule has played an important and a controversial role in a set of relevant logics free of certain well-known paradoxes of implication, like AĀ-→B and A-→(B-→B). Among these logics we count the pioneering systems of strenge Implikation presented by Ackermann in [1],2 as well as the Anderson-Belnap systems E of entail-ment and R of relevant implication.3

I The basic formal ideas of this interpretation were in a certain sense anticipated by E. W. Beth with his "reduced logic" in [3]. We are grateful to Professor Bas C. van Fraassen for bringing this to our attention. The idea has also been employed by P. T. Geach in his "Quantification theory and the problem of identifying objects of reference" in Acta Philosophica Fennica, 1963, and by Henry S. Leonard in his "Essences, attributes, and predicates" in the Proceedings of the American Philisophical Association, 1963-64. Further, H. Leblanc has studied the interpretation in a pair of forthcoming papers, "A simplified strong completeness proof for QC=," and "A simplified account of validity and implication for quantification logic."