Uniform lyndon interpolation for basic non-normal modal and conditional logics

In this paper, a proof-theoretic method to prove uniform Lyndon interpolation (ULIP) for non-normal modal and conditional logics is introduced and applied to show that the logics, $\textsf{E}$, $\textsf{M}$, $\textsf{EN}$, $\textsf{MN}$, $\textsf{MC}$, $\textsf{K}$, and their conditional versions, $\textsf{CE}$, $\textsf{CM}$, $\textsf{CEN}$, $\textsf{CMN}$, $\textsf{CMC}$, $\textsf{CK}$, in addition to $\textsf{CKID}$ have that property. In particular, it implies that these logics have uniform interpolation (UIP). Although for some of them the latter is known, the fact that they have uniform LIP 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 $\textsf{CKCEM}$ and $\textsf{CKCEMID}$ enjoy UIP but not uniform LIP. Moreover, it is proved that the non-normal modal logics, $\textsf{EC}$ and $\textsf{ECN}$, and their conditional versions, $\textsf{CEC}$ and $\textsf{CECN}$, do not have Craig interpolation, and whence no uniform (Lyndon) interpolation.