\textbf{$\mathcal{T}$-satisfiablility.} A \emph{formula $\varphi$ is satisfiable in a theory $\mathcal{T}$}, or $\mathcal{T}$-satisfiable, if and only if there is a
model $\mathcal{M}$ within  $\mathcal{T}$ that satisfies $\varphi$ (i.e., $\mathcal{M} \models A$ for every $A \in \mathcal{A}$
and $\mathcal{M} \models\varphi$).

\textbf{$\mathcal{T}$-valid.} A \emph{formula $\varphi$ is valid in a theory $\mathcal{T}$}, or $\mathcal{T}$-valid, if and only if all
models within  $\mathcal{T}$ satisfy $\varphi$.