Descriptive complexity of 3SAT
The language consists of ternary relations $R_1, \dots, R_4$, which represent the clauses as follows:


$(x,y,z)\in R_1$ iff there is a clause $(x\lor y\lor z)$;
$(x,y,z)\in R_2$ iff there is a clause $(x\lor y\lor \neg z)$;
$(x,y,z)\in R_3$ iff there is a clause $(x\lor \neg y\lor \neg z)$;
$(x,y,z)\in R_4$ iff there is a clause $(\neg x\lor \neg y\lor \neg z)$.


($\lor$ commutes so you don't need to consider other patterns of negation.)

The formula then says "There exists a set&nbsp;$T$ of variables such that, if the variables in&nbsp;$T$ are set to true and the ones not in&nbsp;$T$ are set to false, the formula is satisfied."  That is,

$$\begin{align*}
\exists T\,\forall x\forall y\forall z\, \big[&amp;\big(R_1(x,y,z)\rightarrow (T(x)\lor T(y)\lor T(z))\big)\\
&amp;\quad \land \big(R_2(x,y,z)\rightarrow (T(x)\lor T(y)\lor \neg T(z))\big)\\
&amp;\quad \land \dots\big]\,.
\end{align*}$$