In Baldwin's "Model Theory and the Philosophy of Mathematical Practice" I read this phrase.
The interaction between the syntactically definable properties of theories and the Boolean algebra of definable sets in their models is crucial.
I was wondering what is meant by this "interaction", to me it is not evident that there should be such relation.
I also don't know what "interaction" exactly means, but if we have a first-order theory $T$, we may define equivalence $\sim$ on the set of all formulas (in given language) as: $$(\varphi \sim \psi) \Leftrightarrow (T \models (\varphi \leftrightarrow \psi)),$$ i. e. $\varphi \sim \psi$ holds iff $\varphi \leftrightarrow \psi$ is satisfied in every model of $T$.
This equivalence is a congruence with respect to operations $\wedge$, $\vee$ and $\neg$, so we may define these operations on the set $B_T$ of classes of equivalence $\sim$, and this gives the structure of Boolean algebra on $B_T$.
So if we have this, we may explore relationship between $B_T$ and Boolean algebras of definable sets in models of $T$. Maybe some of them are isomorphic.