"I suggest making it explici..."

I suggest making it explicit that $P$ is a distribution over a (possibly infinite) set of variables (or propositions naming symbols, or whatever your preferred formalization is), and that $P(x)$ is shorthand for $P(X=x)$ when $X$ is unambiguous. This is one of those things that I had to figure out myself, which had confused me historically in my youth, and led me to think that all the $P$ notation was probably informal argument rather than formal math.