Entailment, Mingle and Binary Accessibility

Saul Kripke’s work on the semantics of modal logics is well known, unlike his work on Anderson and Belnap’s system \(\textbf{E}\) of Entailment (a modal relevance logic), which included his proof of the decidability of its implicational fragment \(\textbf{E}_\rightarrow \) , and also a counterexample to the conjecture of Belnap that \(\textbf{E}_\rightarrow \) is the intersection of the implicational fragments of the relevance logic \(\textbf{R}\) and the modal logic \(\textbf{S4}\) . This led to Storrs McCall’s suggesting that the “mingle” axiom \(A\rightarrow (A\rightarrow A)\) might be added to \(\textbf{E}_\rightarrow \) , and that this system might be used instead of \(\textbf{S4}_\rightarrow \) . We give a counterexample to this conjecture. We also disprove the conjecture Anderson and Belnap attribute to McCall, in which “restricted mingle” \(\overrightarrow{A}\rightarrow (\overrightarrow{A}\rightarrow \overrightarrow{A})\) is added to \(\textbf{E}_\rightarrow \) . We use a modal extension of the lattice \(M_0\) and follow ideas of Meyer. In the 1970s, Dunn showed that \(\textbf{RM}\) ( \(\textbf{R}\) with the mingle axiom) can be modeled using a binary accessibility relation, like the accessibility relation in Kripke’s semantics for normal modal logics and his semantics for intuitionistic logic. Dunn also gave a variant interpretation for \(\textbf{EM}\) ( \(\textbf{E}\) with the mingle axiom). We explore some other variations of these binary relational semantics.