Uniform Lyndon Interpolation for Basic Non-normal Modal and Conditional Logics
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In this paper, a proof-theoretic method to prove uniform Lyndon interpolation for non-normal modal and conditional logics is introduced and applied to show that the logics 𝖤, 𝖬, 𝖤𝖭, 𝖬𝖭, 𝖬𝖢, 𝖪, and their conditional versions, 𝖢𝖤, 𝖢𝖬, 𝖢𝖤𝖭, 𝖢𝖬𝖭, 𝖢𝖬𝖢, 𝖢𝖪, in addition to 𝖢𝖪𝖨𝖣 have that property. In particular, it implies that these logics have uniform interpolation. Although for some of them the latter is known, the fact that they have uniform Lyndon interpolation is new. Also, the proof-theoretic proofs of these facts are new, as well as the constructive way to explicitly compute the interpolants that they provide. On the negative side, it is shown that the logics 𝖢𝖪𝖢𝖤𝖬 and 𝖢𝖪𝖢𝖤𝖬𝖨𝖣 enjoy uniform interpolation but not uniform Lyndon interpolation. Moreover, it is proved that the non-normal modal logics 𝖤𝖢 and 𝖤𝖢𝖭 and their conditional versions, 𝖢𝖤𝖢 and 𝖢𝖤𝖢𝖭, do not have Craig interpolation, and whence no uniform (Lyndon) interpolation.
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