R-Calculi for Post \(\textbf{L}_4\) -Valued DL

A 1/4-multisequent \(\Gamma \) is \(\textbf{M}^\texttt{t}_{1/4}/\textbf{N}^\texttt{t}_{1/4}\) -valid, denoted by \(\models ^{\texttt{t}}_{1/4}/\models ^{\ne \texttt{t}}_{1/4}\Gamma \) , if for any interpretation I, there is a statement \(A\in \Gamma \) such that \(I(A)=\texttt{t}/I(A)\ne \texttt{t}\) .