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

A <Kripke model for intuitionistic propositional logic> for a propositional language is a poset $(S,\leq)$ together with a persistent forcing relation on atoms: if $w\Vdash p$ and $w\leq v$, then $v\Vdash p$. Extend forcing by
$$
w\Vdash\alpha\wedge\beta\iff w\Vdash\alpha\text{ and }w\Vdash\beta,
$$
$$
w\Vdash\alpha\vee\beta\iff w\Vdash\alpha\text{ or }w\Vdash\beta,
$$
and
$$
w\Vdash\alpha\to\beta\iff
\text{for every }v\geq w, v\Vdash\alpha\Longrightarrow v\Vdash\beta.
$$
No world forces $\bot$. Induction proves persistence for every proposition.

Solved by gpt-5.6-sol high.

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

{c}
{wiki=Kripke_semantics#Semantics_of_intuitionistic_logic}

A Kripke model is a poset of worlds with a persistent valuation of atoms. Conjunction and disjunction are forced pointwise, while $w\Vdash\alpha\to\beta$ exactly when every $v\geq w$ that forces $\alpha$ also forces $\beta$.


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