Exceptions in Ontologies: A Theoretical Model for Deducing Properties from Topological Axioms

Introduction

Some entities belong more or less to a class. In particular, some individual entities are attached to classes whereas they do not check all the properties of the class. To illustrate this phenomenon, let us consider the ontological network below (see Figure 1). This network corresponds to the seven following declarative statements:

Figure 1.

The element [paul] does not satisfy all the properties of the class [human-being]

• 1.

  A human being has 46 chromosomes

• 2.

  Peter is a human being

• 3.

  Paul is a human being

• 4.

  Paul has 45 chromosomes

• 5.

  Paul lives in Paris

• 6.

  Paul has a bike

• 7.

  One thing can not have at the same time 46 chromosomes and 45 chromosomes

Because [Paul] is a [Human-being], he inherits all the typical properties of [Human-being], in particular [To-have-46-chromosomes]. A paradox is introduced by the statement (7) because “A human-being has 46 chromosomes” is a general fact but not a universal fact. The statement (1) means “In general, human beings have 46 chromosomes but there are some exceptions to this rule”.

A similar phenomenon can be observed with distributive classes. Some subclasses are attached more or less to a general class because some of theirs elements may not check all the properties of this general class. To illustrate this phenomenon, let us consider the ontological network below (see Figure 2).

Figure 2.

The individual entity [paul’s motorcycle] does not satisfy all the properties of the class [motorcycle]. The subclass [motorcycle-with-3-wheels] does not satisfy all the properties of the class [motorcycle]

This network corresponds to the ten following declarative statements:

• 8.

  A thing which has an engine is a vehicle

• 9.

  A thing which has two wheels is a vehicle

• 10.

  A thing which has three wheels is a vehicle

• 11.

  A motorcycle has two wheels

• 12.

  A motorcycle has an engine

• 13.

  There are motorcycles with three wheels

• 14.

  A thing can not simultaneously have two wheels and three wheels

• 15.

  Paul's motorcycle is a motorcycle

• 16.

  Paul's motorcycle has three wheels

• 17.

  Paul's motorcycle is red

Because “Paul's motorcycle is a motorcycle”, as introduced in the statement (15), it inherits all the typical properties of [Motorcycle] class, in particular, [Thing-which-has-2-wheels]. A paradox is introduced by the statement (16) because “A motorcycle has two wheels” is a general fact but not a universal one. The statement (11) “A motorcycle has two wheels” means that “In general, a motorcycle has two wheels but there are some exceptions to this law”.

In Artificial Intelligence, a solution for this kind of problem is default reasoning: an individual A belonging to a concept F inherits concepts subsuming F except contrary indications. The technique of default reasoning led for example Reiter (Reiter 1981) to propose Default Logic (see Section 2).