Valid Formula in an Interpretation (Satisfiability Modulo Theory)

Definition

Valid Formula in an Interpretation (Satisfiability Modulo Theory)

Let $I$ be a $\Sigma$-interpretation and let $\phi$ be a $\Sigma$-formula with $\mathrm{FV}(\phi )=X$. The formula $\phi$ is valid in $I$ if every [[Knowledge/X-variant Interpretation (Satisfiability Modulo Theory)|$X$-variant]] $I^{\prime }$ of $I$ makes it true:

$$\forall I^{\prime }{\equiv }_{X}I:I^{\prime }(\phi )=1.$$

Thus, validity in $I$ fixes the carriers and meanings of all signature symbols while universally varying the values of the free variables of $\phi$.