In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.
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.
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