Theory-valid Formula (Satisfiability Modulo Theory)

Definition

Theory-valid Formula (Satisfiability Modulo Theory)

Let $\mathcal{T}$ be a theory over $\Sigma$. A $\Sigma$-formula $\phi$ is $\mathcal{T}$-valid, or valid modulo $\mathcal{T}$, if it is valid in every [[Knowledge/Theory Interpretation (Satisfiability Modulo Theory)|$\mathcal{T}$-interpretation]]:

$$\forall I:I\models \mathcal{T}⟹\phi \text{ is valid in }I.$$