Use a two-world <Kripke model for intuitionistic propositional logic> $w<v$. Force neither $\phi$ nor $\psi$ at $w$, and force both at $v$. Neither world forces $\phi\to\neg\psi$: at $v$, both $\phi$ and $\psi$ hold, while at $w$ the extension $v$ is a counterexample. Therefore every extension of $w$ fails $\phi\to\neg\psi$, so
$$
w\Vdash\neg(\phi\to\neg\psi).
$$
But $w\nVdash\phi\wedge\psi$. The implication is therefore not intuitionistically valid by <Kripke completeness theorem for intuitionistic propositional logic>.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 120 / 1 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-120/1/e/solution