🤖 AI Summary
This study resolves the long-standing open question of whether six specific normal extensions of the modal logic S4 possess the Craig interpolation property. Building on Smoryński’s approach, the authors introduce Fine’s frame formulas for splitting clusters into interpolation analysis for the first time, integrating modal semantics, model-theoretic techniques, and classical interpolation proof strategies to develop a novel methodological framework. This work provides a complete solution to Maksimova’s classification problem concerning interpolation in S4 extensions, establishing that all six previously unresolved cases indeed enjoy the Craig interpolation property. The results not only settle a decades-old conjecture but also open new avenues for research in interpolation within modal logic.
📝 Abstract
We complete Maksimova's classification of the normal extensions of S4 with interpolation. In particular, we prove Craig interpolation for the six extensions of S4 for which Craig interpolation was still open. The proof strategy builds upon the ideas of Smoryński, but employs a novel approach using Fine's frame formulas for splitting clusters.