Consider the following statements, seen in this order in a paper ("An equilibrium characterization of the term structure" by Vasicek (1977)):
(1) "Let $P(t,s)$ denote the price."
(2) (after some assumption is made), "...this implies the price $P(t,s)$ is a function of $r(t)$, $P(t,s) = P(t,s,r(t))$."
Before statement (2), what should I be thinking of $P$ as, a function or some number? I.e, by the use of the author's notation $P(t,s)$, should I perhaps think of $P$ as $P: \mathbb{R} \times \mathbb{R} \to [0,\infty)$ (say)? But then, when he actually states it is a function of $r(t)$, what should I think of $P$ as? Either $P: (\text{range of $r$}) \to [0,\infty)$ or $P: \mathbb{R} \times \mathbb{R} \times (\text{range of } r) \to [0,\infty)$?
Right now I'm going with the latter, and so I read statement (2) as, "we've already defined $P$ to be a function of $(t,s)$, but by some assumption we've made $P$ is really also a function of $r(t)$." Is this correct?