How to infer isBrotherOf property between two individuals

Use Property Chains and Rolification

Antoine Zimmermann's answer is a very good start to this problem, and touches on the major point that you need to solve this sort of task:

From x to each of x's brothers, there is a path of the form hasFather o hasFather^-1.

Now, that's actually not true of just brothers, though. That's true for all siblings and for x itself. This means you'll have the following definition of hasSibling:

hasSibling ≡ hasFather o hasFather^-1

(Actually, that's really just hasPaternalSibling; a more refined definition would use hasParent.) Now, using this, you could ask for brothers, which are simply siblings who are men, of x with a query like:

(hasSibling value x) and Man

However, it would be nice to define a proper hasBrother property. You could do this with a property chain and hasSibling if you had a special property that linked each Man to himself, and only linked males to themselves:

hasBrother ≡ hasSibling o specialProperty

In fact, such a property is what we get from a technique called rolification, which has been described more in a question, OWL 2 rolification, and its answer. The idea is that for the class Man, we create a new property r_Man and add the equivalence:

Man ≡ r_Man some Self

which says that each Man is related to himself by the r_Man property, and that only instances of Man are so connected. This is exactly the property that we need as specialProperty above. Thus, we end up with the following definitions of Man, hasSibling, and hasBrother:

Now we can ask for the brothers of x with a query like:

hasBrother^-1 value x

For instance, we can ask for Greg's siblings, and get Peter, Greg (himself), and Bobby.

Sample Ontology

Here's that ontology in Turtle:

@prefix :      <http://www.example.org/brady#> .
@prefix rdfs:  <http://www.w3.org/2000/01/rdf-schema#> .
@prefix brady: <http://www.example.org/brady#> .
@prefix owl:   <http://www.w3.org/2002/07/owl#> .
@prefix xsd:   <http://www.w3.org/2001/XMLSchema#> .
@prefix rdf:   <http://www.w3.org/1999/02/22-rdf-syntax-ns#> .

brady:hasSibling  a             owl:ObjectProperty ;
        owl:propertyChainAxiom  ( brady:hasFather [ owl:inverseOf
                          brady:hasFather ] ) .

brady:Carol  a  owl:NamedIndividual , brady:Woman .

brady:hasBrother  a             owl:ObjectProperty ;
        owl:propertyChainAxiom  ( brady:hasSibling brady:r_Man ) .

<http://www.example.org/brady>
        a       owl:Ontology .

brady:Woman  a               owl:Class ;
        rdfs:subClassOf      brady:Person ;
        owl:equivalentClass  [ a               owl:Restriction ;
                               owl:hasSelf     true ;
                               owl:onProperty  brady:r_Woman
                             ] .

brady:hasFather  a  owl:ObjectProperty .

brady:Person  a  owl:Class .

brady:Man  a                 owl:Class ;
        rdfs:subClassOf      brady:Person ;
        owl:equivalentClass  [ a               owl:Restriction ;
                               owl:hasSelf     true ;
                               owl:onProperty  brady:r_Man
                             ] .

brady:r_Woman  a  owl:ObjectProperty .

brady:r_Man  a  owl:ObjectProperty .

brady:Marcia  a          owl:NamedIndividual , brady:Woman ;
        brady:hasFather  brady:Mike .

brady:Peter  a           owl:NamedIndividual , brady:Man ;
        brady:hasFather  brady:Mike .

brady:Jan  a             owl:NamedIndividual , brady:Woman ;
        brady:hasFather  brady:Mike .

brady:Cindy  a           owl:NamedIndividual , brady:Woman ;
        brady:hasFather  brady:Mike .

brady:Bobby  a           owl:NamedIndividual , brady:Man ;
        brady:hasFather  brady:Mike .

brady:Greg  a            owl:NamedIndividual , brady:Man ;
        brady:hasFather  brady:Mike .

brady:Mike  a   owl:NamedIndividual , brady:Man .

Assuming that everyone is their own brother, and that sisters are brothers too:

:hasBrother  owl:propertyChainAxiom  (:hasFather [ owl:inverseOf :hasFather ]) .

It is hardly possible to define :hasBrother more precisely, excluding female brothers and self-brotherhood. But you can infer all the brothers of Lisa as follows:

:Female  a  owl:Class .
:Male  a  owl:Class;
  owl:disjointWith  :Female .
:Lisa  a  :Female .
:Bart  a  :Male .
:Homer a  :Male .
:hasFather  a  owl:ObjectProperty;
  rdfs:range  :Male .
:hasBrother a  owl:ObjectProperty;
  rdfs:range  :Male .
:hasSiblingOrSelf  owl:propertyChainAxiom  ( :hasFather [ :hasFather ] ) .
:LisaBrother  owl:equivalentClass  [
    a  owl:Restriction;
    owl:onProperty  [ owl:inverseOf  :hasBrother ];
    owl:hasValue  :Lisa
  ], [
    a  owl:Class;
    owl:intersectionOf  (
      [ a owl:Restriction; owl:onProperty :hasSiblingOrSelf; owl:hasValue :Lisa ]
      :Male
    )
  ] .