Think of it this way.
There are no elements of the empty set $\emptyset$. Moreover, for all sets $A$, $\emptyset \subseteq A$.
But what does subset actually mean? $A \subseteq B$ means that any element of $A$ must be an element of $B$. That is, $\forall x \in A, x \in B$.
It follows that for all sets $A$, $\forall x \in \emptyset, x \in A$. Every single element of $\emptyset$, every one of them, every none of them, is an element of every other set.
Assuming that $A \wedge \neg A$ is false in your logic (there are logics that admit controlled contradiction), then there are no models satisfy $S_1$. It follows that all models that satisfy $S_1$, every one of them, every none of them, also satisfy $S_2$.