Subtyping, description logic, and owl
Same Order Pattern Across Different Fields
Many systems use the same mathematical backbone: a specialization partial order.
Type Theory
In type systems, subtyping is written $S <: T$ and means:
- a value of type $S$ can be used where $T$ is expected
- $S$ is more specific than $T$
So subtyping is an order by specificity.
Description Logic (DL)
Description logic is a formal language for classes (concepts) and relations (roles).
Its core reasoning task is subsumption:
\[C \sqsubseteq D\]meaning every instance of $C$ is also an instance of $D$.
OWL
OWL is the W3C web ontology language built on description-logic foundations.
It provides:
- class hierarchies
- property constraints
- logical inference over subsumption and equivalence
So OWL ontologies also organize concepts via specialization order.
Category-Theoretic Connection
Each of these can be viewed as a poset category:
- objects are types/classes/concepts
- there is a morphism $A \to B$ exactly when $A \sqsubseteq B$
This gives a uniform way to reason about hierarchy, abstraction, and compatibility.
See: