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

Steffen Lewitzka ()
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Steffen Lewitzka: Federal University of Bahia — UFBA, Instituto de Matemática Departamento de Ciência da Computação Laboratório de Sistemas Distribuídos — LaSiD

A chapter in Logica Universalis, 2007, pp 35-61 from Springer

Abstract: 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.

Keywords: Abstract logic; universal logic; general logic theory; topology; institutions (search for similar items in EconPapers)
Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-7643-8354-1_3

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DOI: 10.1007/978-3-7643-8354-1_3

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