Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 120 / 1 / c

Take two worlds $r<s$ and let $p$ be forced only at $s$. Neither world forces $\neg p$: at $s$ this follows from $s⊩p$, while at $r$ the extension $s$ forces $p$. Consequently every extension of $r$ that forces $\neg p$ also forces $p$ vacuously, soBut $r⊮p$. The implication clause therefore giveswhich is a finite Kripke countermodel and proves that the formula is not intuitionistically valid.