No. Although $\neg(\phi\to\neg\psi)$ is classically equivalent to $\phi\wedge\psi$, the reverse implication $\neg(\phi\to\neg\psi)\to\phi\wedge\psi$ 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 $\phi,\psi,\to,\bot$ 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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Solution

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

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Intuitionistic propositional logic omits unrestricted excluded middle and interprets implication constructively.


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 Incoming links: Intuitionistic propositional logic