Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 120 / 3 / b

By part (a), such a term would provein intuitionistic propositional logic. Consider the two-world Kripke model for intuitionistic propositional logic $r<s$. Let $\sigma$ hold only at $s$ and let $\tau$ hold nowhere. At both worlds $\sigma \to \tau$ fails, so $(\sigma \to \tau )\to \sigma$ holds at $r$ vacuously, while $\sigma$ does not hold at $r$. The displayed formula therefore fails at $r$. By the Kripke completeness theorem for intuitionistic propositional logic, it is not derivable, so no simply typed lambda term inhabits that type.