A Topological Approach to Universal Logic: Model-Theoretical Abstract Logics

In this paper we present a model-theoretic approach to a general theory of logics. We define a model-theoretical abstract logic as a structure consisting of a set of expressions, a class of interpretations and a satisfaction relation between interpretations (models) and expressions. The main idea is to use the observation that there exist in some sense pre-topological structures on the set of theories and on the class of interpretations. For example, if the logic has conjunction, then these structures turn out to be topological spaces. Properties of a given abstract logic now are reflected in topological properties of these spaces. One of the aim of this research is to investigate relationships between abstract logics. We introduce the concept of a logic-homomorphism between abstract logics by means of topological terms. This leads in a natural way to the notion of a logic-isomorphism, a mapping that preserves all structural properties of a logic. We study in detail variations of logic-homomorphisms and their properties. One of the main results is that logic-homomorphisms with a special property satisfy a condition which has the same form as the satisfaction axiom of institutions. This fact can serve in future work to investigate possible connections between (classes of) model-theoretical abstract logics and a resulting institution. At the end of the paper we sketch out this idea. Furthermore, we outline two examples of model-theoretical abstract logics and the respective logic-homomorphisms. However, a systematic study of relevant logics as abstract logics, together with their logic-homomorphisms and further relationships, remains to be done in future work.