R-Calculus for  \(\mathbf{B^2_2}\) -Valued DL

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