论文标题
基本非正常模态和条件逻辑的均匀林登插值
Uniform Lyndon Interpolation for Basic Non-normal Modal and Conditional Logics
论文作者
论文摘要
在本文中，引入并应用了一种证明非正常模态和条件逻辑的统一的林登插值的证明理论方法，以表明逻辑$ \ Mathsf {e} $，$ \ Mathsf {m} {m} $ \ MATHSF {K} $及其条件版本，$ \ Mathsf {Ce} $，$ \ Mathsf {Cm} $，$ \ Mathsf {Cen} $，$ \ Mathsf {cmn} $ $ \ mathsf {ckid} $具有该属性。特别是，这意味着这些逻辑具有统一的插值。尽管对于其中一些人来说，后者是已知的，但它们具有统一的林登插值的事实是新的。同样，这些事实的证明理论证明是新的，也是明确计算它们提供的插值的建设性方法。负面的一面，据表明，逻辑$ \ mathsf {ckcem} $和$ \ mathsf {ckcemid} $享受统一的插值，但不符合统一的林登插值。此外，事实证明，非正态模态逻辑$ \ mathsf {ec} $和$ \ mathsf {ecn} $及其条件版本，$ \ mathsf {cec} $ and $ \ mathsf {cecn} $，没有craig interpolation，no craig interpolation and no supercration（lysiration）（lynifity）（lynor）（lynonon）。
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 $\mathsf{E}$, $\mathsf{M}$, $\mathsf{EN}$, $\mathsf{MN}$, $\mathsf{MC}$, $\mathsf{K}$, and their conditional versions, $\mathsf{CE}$, $\mathsf{CM}$, $\mathsf{CEN}$, $\mathsf{CMN}$, $\mathsf{CMC}$, $\mathsf{CK}$, in addition to $\mathsf{CKID}$ 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 $\mathsf{CKCEM}$ and $\mathsf{CKCEMID}$ enjoy uniform interpolation but not uniform Lyndon interpolation. Moreover, it is proved that the non-normal modal logics $\mathsf{EC}$ and $\mathsf{ECN}$ and their conditional versions, $\mathsf{CEC}$ and $\mathsf{CECN}$, do not have Craig interpolation, and whence no uniform (Lyndon) interpolation.
