Example of Kripke's model.

Question

Let's consider Kripke's model:

Why $b \Vdash (p \implies q)$?

Answer 1

See Kripke semantics for intuitionistic logic :

$b$ forces $(p → q)$ if, for every $b′ ≥ b$, if $b′$ forces $p$ then $b′$ forces $q$.

The only $b' \ge b$ is $b$ itself, and in $b$ both $p$ and $q$ hold.