How to understand the notion of boolean query via Immerman's definition
Boolean queries are a special case of general queries. You can take $\tau$ to be some convenient vocabulary and choose a pair of $\tau$-structures to represent "true" and "false".

FOr a  concrete example, take $\tau$ to be the vocabulary with a single nullary relation symbol&nbsp;$T$. Now, associate "true" with the $\tau$-structure $\mathfrak{T}$ that has $|\mathfrak{T}|=\emptyset$ and $T^\mathfrak{T}=\{\langle\,\rangle\}$ (i.e., the nullary relation that contains the empty tuple) and "false" with the structure $\mathfrak{F}$ that has $|\mathfrak{F}|=\emptyset$ and $T^\mathfrak{F}=\emptyset$ (the nullary relation that doesn't contain the empty tuple).

Empty universes are a bit awkward, since they mean that $(\forall x\,\varphi) \rightarrow \exists x\,\varphi$ isn't a tautology. If you don't like that and/or you don't like nullary relations, make the relation&nbsp;$T$ unary and use a one-element universe.