Theories, models and semantic consequence

An $L$-theory $T$ is a set of $L$-sentences, without any requirement that it be deductively closed. A structure $\mathcal{M}$ is a model of $T$ if it satisfies every sentence of $T$. By coincidence this does not depend on the chosen assignment. Write $T\models \varphi$ if for every set $L$-structure $\mathcal{M}$ which is a model of $T$ and every assignment $s$, one has $\mathcal{M},s\models \varphi$. Thus if $T$ has no models the consequence condition is vacuous.

Validity means truth in every structure at every assignment; semantic equivalence means equality of truth under all such choices. The universal closure of $\varphi$ binds its finitely many free variables in increasing index order (and is $\varphi$ if none occur). Repeated use of the universal truth clause shows that a structure satisfies this closure exactly when $\varphi$ holds at every assignment. Quantification over all set structures is a first-order class description, not a set of all structures.

Conventions and prerequisites: Coincidence for term values and satisfaction.