A Brief Analysis of Interpolation in Universal Algebraic Logic

This paper offers a brief historical and conceptual overview of interpolation properties within the framework of Universal Algebraic Logic (UAL). Although the main theme is Craig’s interpolation theorem our focus revolves around the generalizations and refinements that the algebraic treatment of logic offers in the context of the Budapest School of Logic. We examine various formulations of the interpolation property including the standard and modelwise variants, and revisit their connections with definability, amalgamation, and the Robinson property. In particular, we highlight how recent developments in UAL clarify the structural nuances of interpolation across different logical systems including conditionally algebraizable ones. The discussion also addresses the methodological significance of interpolation in the logical reconstruction of scientific theories, emphasizing the contrast between global and model-relative notions of truth. Parts of this exposition draw on an unpublished chapter of the author’s doctoral dissertation.