Harmonious logic: Craig’s interpolation theorem and its descendants

Though deceptively simple and plausible on the face of it, Craig’s interpolation theorem (published 50 years ago) has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. Craig’s theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of subsequent generalizations and applications, especially of many-sorted interpolation theorems. Attention is also paid tomethodological considerations, since the Craig theorem and its generalizations were initially obtained by proof-theoretic arguments while most of the applications are model-theoretic in nature. The article concludes with the role of the interpolation property in the quest for “reasonable” logics extending first-order logic within the framework of abstract model theory.