How would you define a predicate $P$ that asserts $P(x,y)$ which can also be reflexive?
The way I did it was:
$\quad$ $I(\exists$ $P.C)=\{x,y \quad | \quad \top \sqsubseteq \exists$ $P.Self$ $\land$ $P(x, y) \}$
I'm trying to state that for some $x$ and $y$, the model can assert $P(x,y)$ but also that $P$ is reflexive.
Is this even valid notation?
As far as notation goes, you can just use $P(x,y)$ ... just because $x$ and $y$ are different variables does not mean that $P$ cannot be reflexive: you can always pick the same object for both $x$ and $y$ if needed.