R-Calculus for Binary-Valued DL

Let \(*\in \{\texttt{t},\texttt{f}\}.\) A 1/2-sequent \(\textbf{X}\) is \(\textbf{M}^*_{1/2}/\textbf{N}^*_{1/2}\) -valid, denoted by \(\models ^*_{1/2}/\models ^{\ne *}_{1/2}\textbf{X},\) if for any interpretation I,  there is a statement \(X\in \textbf{X}\) such that \(I(X)=*/I(X)\ne *.\)