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Published online by Cambridge University Press: 12 August 2025
We study the Lyndon interpolation property (LIP) and the uniform LIP (ULIP) for extensions of $\mathbf {S4}$ and intermediate propositional logics. We prove that among the 18 consistent normal modal logics of finite height extending
$\mathbf {S4}$ known to have CIP, 11 logics have LIP and 7 logics do not. We also prove that for intermediate propositional logics, the Craig interpolation property, LIP, and ULIP are equivalent.