Institution-independent Ultraproducts

We generalise the ultraproducts method from conventional model theory to an institution-independent (i.e. independent of the details of the actual logic formalised as an institution) framework based on a novel very general treatment of the semantics of some important concepts in logic, such as quantification, logical connectives, and ground atomic sentences. Unlike previous abstract model theoretic approaches to ultraproducts based on category theory, our work makes essential use of concepts central to institution theory, such as signature morphisms and model reducts. The institution-independent fundamental theorem on ultraproducts is presented in a modular manner, different combinations of its various parts giving different results in different logics or institutions. We present applications to institution-independent compactness, axiomatizability, and higher order sentences, and illustrate our concepts and results with examples from four different algebraic specification logics. In the introduction we also discuss the relevance of our institution-independent approach to the model theory of algebraic specification and computing science, but also to classical and abstract model theory.