A negative property assertion as defined in the upcoming OWL 2 states that a given individual i is never connected to a given individual j by a given property expression P. In other words, asserting that i is connected to j by P results in an inconsistent ontology. In this sense this assertion can be considered as a constraint that should not be violated. In contrast, considering an ontology where it cannot be inferred that i is connected to j by P does not necessarily mean that there cannot be such a connection - in fact, it is merely not modeled.

[1] Patel-Schneider, P.F., Hayes, P., Horrocks, I.: OWL Web Ontology Language Semantics and Abstract Syntax, W3C Recommendation 10 February 2004.

[2] Motik, B., Patel-Schneider, P.F., Parsia, B.: OWL 2 Structural Specification and Functional-Style Syntax. W3C Candidate Recommendation 11 June 2009, 2009.

SubClassOf(ObjectOneOf(i1), ObjectComplementOf(ObjectSomeValuesFrom(prop, ObjectOneOf(i2))))

Let C and D be concepts. Then C and D are disjoint if, and only if, C is subsumed by the complement of D, i.e., '( SubClassOf( C ObjectComplementOf(D) ).

The equivalence is correct because of the duality of disjointness, equivalence, and unsatisfiability: C is subsumed by D if, and only if, ObjectIntersectionOf( C ObjectComplementOf(D) ) is unsatisfiable, and the intersection of C and D is unsatisfiable if, and only if, C' and D are disjoint.

One also reminds that the extension of the concept ObjectSomeValuesFrom(prop C) is the set of individuals i which are connected to an individual j that is in the extension of the concept C, by the property prop.

Let NegativePropertyAssertion(p a b) be a negative property assertion axiom, i.e., the individual a is not related to b by the property p. Then the extension of ObjectSomeValuesFrom( p ObjectOneOf(b) ) which contain all individuals that are connected to b by p must not contain a. This is true, if, and only if ObjectOneOf(a) is disjoint to ObjectSomeValuesFrom( p ObjectOneOf(b) )

Individual i2

ObjectProperty prop

Let Adam and Eve be two persons and like a property (A likes B). Furthermore we know that Adam does not like Eve but we have no dislike relationship. Moreover, our language (such as OWL 1) does not have any NPA axiom constructor. The sample ontology is interpreted with respect to the open-world semantics, i. e. , one can not infer the dislike merely from the lack of a property assertion axiom ObjectPropertyAssertion(like Adam Eve). Then this fact can be expressed with the following axiom (we will also use the OWL 2 Abstract Syntax here):

SubClassOf (Adam ComplementOf(ObjectSomeValuesFrom( likes ObjectOneOf(Eve))))