Interpolation Properties, Beth Definability Properties and Amalgamation Properties for Substructural Logics

This article develops a comprehensive study of various types of interpolation properties and Beth definability properties (BDPs) for substructural logics, and their algebraic characterizations through amalgamation properties (APs) and epimorphisms surjectivity. In general, substructural logics are algebraizable but lack many of the basic logical properties that modal and superintuitionistic logics enjoy [Gabbay and Maksimova (2005, Oxford Logic Guides, Vol. 46)]. In this case, careful examination is necessary to see how these logical and algebraic properties are related. To describe these relations exactly, many variants of interpolation properties and BDPs, and also corresponding algebraic properties, are introduced. Because of their generality, the results reported here hold not only for substructural logics, but can also be extended to a more general setting such as abstract algebraic logic [Andreka, Nemeti and Sain (Handbook of Philosophical Logic, Vol. 2, 2nd edn, pp. 133-247) and Czelakowski and Pigozzi (1999, Vol. 203 of Lecture Notes in Pure and Applied Mathematics, pp. 187-265)].