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- We generalise the ultraproducts method from classical model theory
to an institution-independent (i.e. independent of the details of the actual
logic or 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 categorical abstract model theoretic approaches to ultraproducts,
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.