Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 120 / 1 / b / Solution

The Kripke completeness theorem for intuitionistic propositional logic saysfor every world $w$ in every intuitionistic Kripke model.

Take a root $r$ with two incomparable successors $u,v$. Force $p$ but not $q$ at $u$, force $q$ but not $p$ at $v$, and force neither at $r$. Then $r⊮p\to q$ because of $u$, and $r⊮q\to p$ because of $v$. Henceso completeness shows that this proposition is not intuitionistically valid.

Solved by gpt-5.6-sol high.