I realized, in response to a student question this term, that the textbook isn't quite as explicit as I'd like it to be about how to construct an interpretation from an open branch.
Somewhere in Chapter 5 it should say something along the lines of: To construct a model that satisfies the root, look at a completed open branch. For any atomic sentence $\Phi$ in that branch, have your interpretation include $\Phi=1$. For any negated atomic sentence $\neg \Phi$ in the branch, have your interpretation include $\Phi=0$.
I realized, in response to a student question this term, that the textbook isn't quite as explicit as I'd like it to be about how to construct an interpretation from an open branch.
Somewhere in Chapter 5 it should say something along the lines of: To construct a model that satisfies the root, look at a completed open branch. For any atomic sentence $\Phi$ in that branch, have your interpretation include $\Phi=1$. For any negated atomic sentence $\neg \Phi$ in the branch, have your interpretation include $\Phi=0$.