An ontology is a theory in a logic, and a database is a model of a theory in a logic, but which logic? And what is a logic? And while we are asking such abstract questions, what precisely does it mean to integrate databases? Or to integrate ontologies? What if the logics involved are different? (Languages for expressing ontologies include Owl, Ontologic, Flora, KIF, and RDF.) And what about schemas and the languages in which they are expressed? Or their underlying data models (which include relational, object oriented, spreadsheet, formatted file, ...)?
This talk sketches how such questions can be answered using institutions, an axiomatization of the notion of logical system, based on Tarski's idea of taking the notion of satisfaction as central. We sketch institutions and their theory, avoiding category theory as much as possible, but if you are brave, you might want to look at the following paper, since institution morphisms are what translate between logics, while theory morphisms are what translate ontologies and schemas:
The slide webpages for a prior lecture, Ontologies, Ontology Languages, and Data Integration by Metadata Integration, especially the page Federating the Kingdoms of Ontology, can provide some intuition for this lecture; see also the Data Integration page, and my Notes for the SEEK Project page.