Theory — Introduction

To support the development and use of ontologies, both, top-level ontologies such as BFO, GFO, DOLCE, and UFO, and upper ontologies such as schema.org, and SUMO have been developed for more than two decades. Top-level ontologies assist modelers in the development of domain ontologies such as the GeneOntology. Upper level ontologies serve as frameworks that modelers can use directly or extend as needed. Both types of ontologies contain tens to thousands of concepts, properties and axioms to model reality. Top-level ontologies provide more abstract concepts such as action, achievement, activity, amount of matter, agent, dimension, entity, event, goal, instrument, location, matter, occurrent, part, path, perdurant, process, quality, quantity, result, role, space, region, time, theme, and so on. Upper level ontologies provide more concrete concepts such as animal, article, artwork, body part, body of water, building, city, competition, contract, country, corporation, device, document, food, game, hospital, marriage, nuclide, person, plant, product, room, school, surgery, team, work, and so on. Driven in particular by the developments in the area of the Semantic Web, there has been a great deal of fundamental research into the methodology of ontology development. The works of Noy and McGuiness [NoMG2000], Arp et al. [ArSm2015], and Allemang et al. [AlHe2020] are representative for that. This was accompanied by the development of comprehensive axiom systems as presented by Masolo et al. in [MaBo2003] by Guizzardi et al. in [GuBe2021], and by Selway et al. [SeSt2017]. We consider such comprehensive axiom systems to be Special Ontology Theories (SOT). In physics and chemistry there are essentially only a few building blocks such as atoms, neutrons, protons and electrons, from which complex molecules can be assembled. The interaction of these building blocks is governed by the laws of nature. By analogy, the question arises as to whether we can find a similarly small set of building blocks to create ontologies and what rules govern the interaction of their building blocks. Therefore, we focus on the design of a General Ontology Theory (GOT) that identifies a minimal set of ontology building blocks and a set of definitions and axioms to describe the laws of their interaction. The goal is to capture the fundamental structures and principles that hold across top-level, upper level, and domain ontologies, potentially addressing overarching functionalities such as abstraction, aggregation, classification, inheritance, subsumption, instantiation, and identity. Although there is already a lot of preliminary work in this field of research, there is no mathematically formalized axiom system that describes the laws between the basic building blocks of ontologies. Here we want to close this gap.

Our research approach is based on the paper of Nunamaker et al. [NuCh1991] and that of Hevner et al. [HeMa2004] which discusses ‘Design Science in Information Systems Research’. The central research question is: What might be a minimal set of ontology building blocks that would form the basis for expressive conceptual and meta-level modeling? This leads to further questions: How can we use mathematical set theory to express the principle of ordering relations of  classes and relationship types, and what are the resulting regularities? Finally, what is the relationship between extension and intension of classes and relationship types?

The rest of this paragraph is organized as follows. In the Theory - Related Work, we discuss related work in the areas of top-level ontologies and their axiomatic systems. In the Theory - Foundations, we discuss naming conventions, non-strictly ordered sets, and the ordering relations for defining structures of classes, relationship types and data properties. We also define rules for instantiating data and object properties. The results is our minimal ontology blueprint. The Theory - Axioms derives subsumption axioms based on the intension and extension of universals. In the Theory - Results, we compare our methodology with other minimal top ontologies. In the Theory - Results we summarize the results.