Consider a map $f:\mathbb{R}\to\mathbb{R}$.

We can form multivalued maps $\mathcal{Q}(f) : 2^\mathbb{R}\to 2^\mathbb{R}$ from $f$ if we add quantifiers $\forall, \exists$ and boolian functions \begin{equation}sq(x, y) = \begin{cases}
    true & \text{if $x = y^2$} \\
    false & \text{otherwise}
  \end{cases}\end{equation}

\begin{equation}in(x) = \begin{cases}
    true & \text{if $x\in[1, 3]$} \\
    false & \text{otherwise}
  \end{cases}\end{equation}

For example $\mathcal{Q}(f):2^\mathbb{R}\to2^\mathbb{R}$, $$\mathcal{Q}(f)(S) = \{a\in\{f(x): x\in S \text{ and }\exists y\in S , sq(x, y)=true\}:in(a)=true\}.$$ Such functions, in the definition of which only one $ f $ is used, can be called linear quantification of $f$ with using boolean functions $sq, in$.

Question. Is there any reference to such linear Quantifierisation and how to describe them if we are given a map $f$ and a family of boolean functions?