In proposition 1.1.8 of "Model Theory: An Introduction" David Marker proves that:
If $\cal M$ is an $\cal L$-substructure of $\cal N$, $\bar a \in M$, and $\phi(\bar v)$ is a quantifier-free formula, then $\cal M \models \phi(\bar a)\iff N\models \phi(\bar a)$
In the setting I'm working in, we have a finite language $\cal L$ and we are working with finite structures $\cal M,N$ with $\cal M$ being a structure $\cal N$.
Can we get anything stronger than proposition 1.1.8 just from the fact that both the language and the structures are finite? Are there any results like this that allow us to conclude any relationship between the formulas that are true in $\cal N$ and the formulas that are true in $\cal M$ or vice versa in this context? What if we assume extra conditions on the language/structures, can we get such results and what would the conditions be?
I know this question is a little too vague, so I'm adding the "reference-request" tag.