Let's suppose we have propositional variables $x_1 ... x_n$. A valuation is an assignment $v$ s.t. $v(x_i)$ is an element of $\{false, true\}$ for $1 \leq i \leq n$. So, there are $2^n$ possible valuations. The number of sets of valuations is then $2^{2^n}$. If we associate to each formula $\varphi$ the set of valuations that satisfy it, we see that there are many sets of valuations that do not correspond to any 3CNF formula. This is because there are only polynomially many possible 3CNF clauses and any formula is a subset of such clauses.
Is there any theory about the structure of the sets of valuations representable by 3CNF formulas?