Theory - Results

As a fundamental artifact in the sense of [HeMa2004], with O4Top we have developed a minimal top ontology. This is the basis for an axiom system that formally describes the mathematical relationships between the elements of O4Top. It also contains definitions for the intension and extension of classes and laws for inheritance and instantiation. The Class Subsumption Axiom CSAx and the Relationship Type Subsumption Axiom RTSAx were derived from this as new laws. The RT Subsumption Theorem RTSTh was then used to prove that strict inheritance relationships apply between classes involved in sub-relationships. This also made it possible to define a Class Identity Axiom CIdAX, which establishes a connection between the intensions and extensions of classes and the relationship types involved in them. Compared to other larger top-level ontologies such as BFO, DOLCE, GFO and UFO, the top-level ontology O4Top presented in the Axioms section is minimal. We believe that O4Top is general and robust enough to model not only other top-level ontologies but also domain ontologies. Therefore, we also assume that the axiom systems developed elsewhere can still be used. In our opinion, the axioms discussed in the previous sections could be added under other ontological axiom systems. The utility of our method for the streamlining of conceptual modeling has been discussed in the author‘s paper [Bens2023a] (The goal of design-science is utility: [HeMa2004], page 80). Our method also supports the detection and elimination of inconsistencies in conceptual modeling with the help of build-and-evaluate loops (The goal of behavioral-science is truth: [HeMa2004], page 78, 80).

Guarino in [Guar1998] discusses a “Minimal Ontology of Universals”. We would map his universals to the modeling elements of our O4Top ontology blueprint as follows: ^Class and ◊ObjectProperty (relationship type) are our top universals. Our understanding of “Property” is similar in that we consider the “Attribution” to represent Data Properties (DP) and the other properties are Object Properties (OP). We think that “Type”, “Category”, and “Role” can be represented by ^Class. Our top universal ◊ObjectProperty corresponds to "Relation".

Guarino suggests having two separate ontologies for particulars and universals, keeping lexical items out of the domain. In our approach, particulars and universals are part of the same ontology. They just reside in the two different layers PL and SL. The YAGO ontology needs quadruples to represent so-called reification graphs ([SuKa2007], page 10 ff.). This has the advantage in contrast to RDFS and OWL, that n-ary relations can be modeled easily. However, when using triple stores, there is a mapping overhead due to the additional reification requirements. On the other hand, we see the Yago approach as a confirmation that a general ontology theory like ours can get by with a small set of model elements and axioms.

In this article, we have proposed a General Ontology Theory (GOT). The foundation is the O4Top ontology blueprint which is based on a minimal set of ontological concepts. Using the transitive relationship types subClassOf, subOpOf and subDpOf and applying the principal ideal filter, we derived the hierarchy structures for classes, object and data properties. After elaborating the difference between elements of classes and particulars of classes, and the relationship between extension and intension we were able to define the particulars-elements correspondence axiom PECAx. Then, applying axioms from Formal Concept Analysis, we obtained the extended class subsumption axiom CSAx. This allows a hierarchy of formal concepts to be derived only from the set of objects and the sets of attributes of those objects. That is, a modeler can start with a table of objects and their attributes as input and get the underlying hierarchy of formal concepts as result. Applying the relationship type subsumption theorem RTSTh it becomes possible to infer hierarchy relationships of formal concepts solely from object property instantiations, and to check the structural correctness of ontologies with respect to the ordering relations of object properties.