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

No. Although $\neg (\varphi \to \neg \psi )$ is classically equivalent to $\varphi \wedge \psi$, the reverse implication $\neg (\varphi \to \neg \psi )\to \varphi \wedge \psi$ is not intuitionistically valid, as part e shows. More generally, the standard normal-form separation theorem for the implicational fragment with falsity says that no formula built uniformly from $\varphi ,\psi ,\to ,\perp$ has both the pairing introduction rule and the two projection elimination rules of conjunction. Hence conjunction is not definable from implication and falsity in Intuitionistic propositional logic.

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