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

Abstract

In this paper we develop a topological approach to a theory of model-theoretical abstract logics. A model-theoretical abstract logic is given by a set of expressions (formulas), a class of interpretations (models) and a satisfaction relation between interpretations and expressions. Notions such as theory, consequence, etc. are derived in a natural way. We define topologies on the space of (prime) theories and on the space of models. Then structural properties of a logic are mirrored in the respective topological spaces and can be studied now by topological means. We introduce the notion of logic-homomorphism, a map between model-theoretical abstract logics that preserve structural (topological) properties. We study in detail conditions under which logic-homomorphisms determine continuous or/and open functions between the respective topological spaces. We define a logic-isomorphism as a bijective logic-homomorphism. Such a map forces a homeomorphism on the corresponding topological spaces.

Moreover, we show that certain maps between logics lead to a condition which has the same form as the satisfaction axiom of institutions. This promising result may serve in future research to establish a connection between our approach and the well-known category-theoretical concept of institution.

This research was supported by CNPq grants 150309/2003-1 and 350092/2006-0.