Sifting between orders of Logic
Ontologies are used to define the nature of entities within a specific domain of knowledge. A method to understand how best to model entities within a domain is to create 3 levels of distinction. The first is to examine the nature of the entity independent of all other external influence. If the entity were to be viewed as an isolated island, devoid of any relationships with any other entity, what would its characteristics be? Next, if the entity exists in an ecosystem with several other entities, what are those relationships, and what is the role of the entity in each of those relationships? Finally, what is the collective behavior of entities in specific circumstances, where each entity is constrained to defined criteria? These 3 questions, respectively, form the basis of 1st, 2nd, and 3rd order ontologies. To summarize:
1st order ontology: Only defines the entities that are important within a domain of knowledge in complete isolation from other entities, without specifying the relationship of those isolated entities with other entities, or the behavior of those isolated entities in specific situations. Attributes of the entity are also a part of the 1st order ontology. It is often difficult to distinguish whether a concept is an attribute of an entity, or entirely different entity. A good rule-of-thumb is to try phrase the entity with its characteristics, and see if the inversing the phrase also makes equal sense. For instance, stating that a "Car has a Color" makes sense, but stating that a "Color has a Car" does not, validating that Color is an attribute of a Car, and should be contained within the 1st order model of Cars.
Fig.1: 1st and 2nd Order Ontologies
2nd order ontology: Defines the relationships between isolated entities within a domain of knowledge. These relationships are usually defined in conjunction with the role the entities would assume in order to fulfill the relationship between each other. The relationship should not define constraints, like how many of each entity is required for the relationship, or the circumstances under which the relationship is created or negated. To return to our example with Cars, a Car uses wheels, and Wheels are attached to a Car. Both entities can exist independently of each other, although neither is useful by themselves. Usefulness is achieved through a role, such as being a vehicle, when the Car and the Wheels, along with all the other entities such as the Engine, have to work together to achieve motion. Constraints such as the number of wheels that a Car must possess is 3rd order logic. This distinction is important as constraints are not always true. Some cars have 3 wheels, such as the Reliant Robin, or even 6 wheels, such as the Tyrell P34. When a car is being repaired or manufactured, or it has been parked in a dubious neighborhood for too long, it can have 2, 3, or even no wheels.
Fig.2: 1st and 2nd Order Ontologies of Cars and Drivers
3rd order ontology: Defines the behavior of entities within the domain of knowledge, where each entity is constrained to defined states and criteria. Also called axiomatic definitions, or popularly referred to in computing circles as "Business Rules". To expand on the 1st and 2nd order definitions of a Car and Driver: For a Car to be driven by a Driver in Germany, the Driver must be older than 17 years old, and possess a B License, and the Car must be certified as being roadworthy by TUV Rheinland. In another country, or in some cases, in a different State or Province of the same country, the rules for certifying a Car or the legal driving age can be completely different. Designing a flexible systems that can be reused in a variety of situations necessitates the separation of 1st, 2nd, and 3rd order logic. The structures of the 1st and 2nd order logic within a system should be devoid of constraints, ensuring their repeatability.
Fig.3: 3rd order ontology reflecting constraints and rules for Cars driven in Germany
Approaching the 3 orders
A "top-down" approach to create a structured ontology starts by creating the 1st order ontology by defining the significant entities in isolation from all other entities within a domain of knowledge. Alternatively, it is also possible to use a "bottom-up" approach to first identify the manner of constraints that exist for entities within a domain. In both approaches, it is important to continuously validate the model against a large collection of facts. The optimal approach is to model an ontology from "both ends", disposing of abstractions in 1st and 2nd order logic whenever a large body of facts negates them.
In creating a 1st order ontology, it is difficult to view an entity in complete isolation from its environment. For instance, is a phone (a device used to communicate) truly a phone without a network connection, without which it is reduced to an non‐functional object (a device that cannot communicate)? Or is pond a pond without the variety of life it contains, or a workplace a workplace without the people that occupy it? The exercise to model entities in isolation is necessary for the purposes of establishing a computational model, and it is a reductionist exercise. The inter‐connected nature of entities can re‐established to some extent with 2nd and 3rd order logic, but it is important to recognize that computational models by their very nature are reductionist, and are at best shallow reflections on the complexity of actuality and facts.
Example of the 3 orders of logic
To illustrate the distillation of the different orders of logic, let us consider glossaries, where the domain of knowledge is words, and a well‐written glossary will systematically provide definitions of words. The glossary in this case is a 1st order ontology as it only provides a systematic definition of words, and it leaves the human reader to make inferences about similar words or how the word should be used with other words. The relationship of a word with other words, such as the synonyms or antonyms of a word, is systematically covered in a thesaurus, and example of a 2nd order ontology. An intuitive reader could figure out the meaning of a word through its synonyms and antonyms, but unless the thesaurus explicitly provides a definition of the word, it is only a 2nd order ontology. A thesaurus that also provides a definition of the word combines 1st and 2nd order ontologies. Finally, grammar guides provide structured rules for the appropriate usage of words, which is an example of a 3rd order ontology.
Fig.4: An Ontology of Lexicons
The level of specification required from an ontology depends upon its intended usage. A formal system must be specified to the level of a 3rd order ontology, particularly in the areas where creating room for interpretation would lead to different results. For instance, the customs department of a country can define a 1st order ontology by creating a list of things upon which a duty can be levied (e.g. alcohol, jewelry, electronics, etc.). Next, it could further define a generic charging structure where all items in a specific category would be levied the same charge (e.g. some alcoholic beverages can be liable for a 10% duty). Finally, it could define specific rules for charging specific items under specific circumstances (e.g. only alcoholic beverages that are above 40% volume and contained in unopened bottles greater than 500 ml will be applicable for a 10% duty). If the ontology of dutiable items were to neglect specifying things on the level of a 3rd order ontology, individual customs officers would be able to interpret the 2nd order ontology subjectively, unless the 2nd order ontology is specified without any ambiguity (e.g. all alcoholic beverages will be liable for a 10% duty).
