Uniform Lyndon Interpolation for Basic Non-normal Modal and Conditional Logics

TL;DR

This paper introduces a proof-theoretic method to establish uniform Lyndon interpolation for various non-normal modal and conditional logics, providing new constructive proofs and explicit interpolant computations, and clarifies which logics possess this property.

Contribution

It presents a novel proof-theoretic approach to prove uniform Lyndon interpolation for multiple non-normal modal and conditional logics, including explicit construction of interpolants.

Findings

Several logics have uniform Lyndon interpolation, including $ extsf{E}$, $ extsf{M}$, $ extsf{EN}$, and others.

Some logics like $ extsf{CKCEM}$ and $ extsf{CKCEMID}$ have uniform interpolation but not Lyndon.

Certain non-normal modal logics do not have Craig or uniform (Lyndon) interpolation.

Abstract

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.