Open Research Index

Depth-labeled interpolation and substitution in Sacchetti's modal logics

JD Jones

Palomar Registrypalomar/PALOMAR-2026-10-16-000009Oct 16, 2026math.LOscore 0

Classifies depth labelings in the modal logics wGL_n for positive n. Every simultaneous formula-valued substitution indexed by atom and depth label preserves theoremhood exactly when the labels satisfy c(k+n)=c(k) at every positive depth k. The interpolation equivalences identify the corresponding labeled property with its ordinary interpolation property and the same periodicity condition; the ordinary interpolation theorems remain explicit inputs. Further results classify period spectra, boxed-atom implications and composition of depth substitutions, and transfer a supplied ordinary…

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