25 
 
unlike many languages it does not have a unique names assumption. Unless specifically stated otherwise, 
the reasoner can infer that two individuals with different names are actually the same individual. It should 
be noted however, that if Peggy and Margaret were explicitly stated to be two diﬀerent individuals then 
the above statements would lead the reasoner to infer that there was an inconsistency in the ontology. We 
will discuss names more in chapter 7.  
In section 4.16 we will discuss cardinality restrictions on properties. E.g., that the hasWheel property of 
the Bicycle class has a minimum of 2 (allowing for training wheels) whereas hasWheel for the 
Unicycle class is defined to be exactly 1. A functional property is equivalent to a property with a 
cardinality restriction that says it has a maximum of 1 value. The term functional is from mathematics 
where a function is defined as a relation where each member of the domain has at most one value. For 
example, the greaterThan relation is not functional since for any number X many (in fact an infinite 
number) can be greaterThan X but the plusOne relation is functional since for any number X 
plusOne always results in one unique value.  
4.8.2 Inverse Functional Properties 
If a property is inverse functional then it means that the inverse property is functional. For a given 
individual, there can be at most one individual related to that individual via the property. Following our 
example from section 4.8.1 the inverse of hasBirthMother would be isBirthMotherOf. The 
isBirthMotherOf property would not be functional since a woman can be the birth mother of several 
children. However, it would be inverse functional since each person has exactly one mother.  
4.8.3 Transitive Properties 
 
If a property P is transitive, and P relates individual a to 
individual b, and also individual b to individual c, then we can 
infer that individual a is related to individual c via property P. 
For example, Figure 4.13 shows an example of the transitive 
property hasAncestor. If the individual Diya has an ancestor 
that is Fatima, and Fatima has an ancestor that is Arjun, then 
we can infer that Diya has an ancestor that is Arjun – this is 
indicated by the curved line in Figure 4.13. 
An example of the transitive property in mathematics is the > 
relation. If  x > y and y > z then x > z.  
Note that if a property is transitive it cannot be functional. Also, 
if a property is transitive then its inverse property must also be 
transitive. E.g., the inverse of  > is < and < is also transitive. We 
will see an example of this in chapter 6. 
 
 
4.8.4 Symmetric and Asymmetric Properties 
If a property P is symmetric, and the property relates individual a to individual b then individual b is also 
related to individual a via property P. The hasSibling property or hasSpouse are examples of 
Arjun 
Diya 
Fatima 
hasAncestor 
hasAncestor 
hasAncestor 
Figure 4.13 Transitive  
Properties