What is the best way of specifying an element of a set of predicates on a variable $x$, which is a member of a set $X$?
By convention, I understand that for any sets X and Y, the set of all functions $f: X \rightarrow Y$ is denoted $Y^X$.
My preference would be to write $ p\in \Psi( x )$ specifying that $p$ is a predicate taking $x$ as a predicate variable, and $\Psi$ as the set of all predicates relating to $x \in X$. Is there anything wrong with this?
I would have thought this to be more intuitive than something like this; $ p\in \Psi$ where $\Psi( y ) = \{{\text{true}, \text{false}}\}^X$.