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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-120/1/a/solution