Type structures, taxonomies and ontologies

Copyright 2017 Graham Berrisford. One of about 300 papers at http://avancier.website. Last updated 03/03/2017 18:35


Preface. 1

Sets and types. 1

Taxonomies as classification structures. 2

Ontologies as richer type structures. 4

Conclusions and remarks. 4



Describers observe and envisage realities - past, present or future.

Practical idealism


<create and use>              <idealise>

Describers   <observe and envisage>   Realities


A reality is anything we can idealise/conceptualise into some kind of description.

If there is a reality out there that we can’t describe, it can play no role in science.


Taxonomy and ontology start with conceptualising the properties of similar things as types.



<create and use>       <idealise>

Ontologists    <observe and envisage>   Things


Taxonomists and ontologists create structures that relate types in what are called taxonomies and ontologies.

What is the difference? There does not seem to be a universally accepted distinction.

So this paper attempts to draw one; but first, we have to understand what a type is.

Sets and types

All animal intelligence involves abstracting types from realities.

Learning depends on matching newly observed things to past ones.


Human language, description and logic are based on types.

“In describing a situation, one is not merely registering a [perception], one is classifying it in some way, and this means going beyond what is immediately given.” A J Ayer.

A feudal society is organised according to a defined structure of types (e.g. King, Bishops, Lords etc.)

The 11th century Domesday Book described England using a data structure that could be recognised by a computer.

In short, there were type structures before set theory


Static (enumerable) sets

In basic set theory, a set is its members; it cannot gain or lose members.

E.g. The set of single digit integers, or even numbers, never changes.

There are two ways to define a set.

·         Enumeration: Define set members by listing them or defining a process to generate them.

·         Typification: Define one set member by forming an "intensional definition".


Dynamic (typified) sets

People in natural and computing sciences speak of sets that can gain and lose members.

The set members cannot be defined by enumeration (unless you stop the universe).

Therefore, a set member must be defined by a type.

A type implies the existence of a dynamic (typified) set.



“Planet” a large physical body orbiting the sun

“Human Resource” a person playing a role in a business

Things that act as instances

Mercury, Venus, Mars etc.

Joel Smart, Joe Smith, Joan Soap etc.


A type name alone does not define a type.

A type must be accompanied by a definition of its properties.

A taxonomy goes some way to define the properties of a type.

Taxonomies as classification structures

A taxonomy constrains the vocabulary used in a "bounded context" or "domain of knowledge".


Simple taxonomies -single inheritance

The hierarchy can be built from the top-down, from the bottom-up, or both.


The “base” type at the top of the hierarchy (e.g. resource) is definable by narrative, examples or axiom.

The base type is divided into subtypes (e.g. human workers and machines).

Subtypes are further sub-divisible into subtypes (e.g. human workers may be subtyped into employees and contractors).


To describe a thing using a type name is to presume it inherits the properties of all types above it.

The additional properties of a specific type may be defined more or less formally.


A simple taxonomy has one base type and one inheritance route to that from any subtype.

Taxonomies are used to help you find information.

If you find one term/type you can see all more generic types – which define that term.

You can find all next-more specialised types – and narrow your search.


Complex taxonomies - multiple inheritance

One thing may exhibit/embody the properties of types in different taxonomies.

e.g. Joan Soap is an instance of “Human Resource” and “Woman” and “Soap family member”.

Also, one type may specialise more than one supertype.

e.g. Female Employee is a kind of both “Human Resource” and “Woman”.


Two-dimensional inheritance tables

Types drawn from two taxonomies can be mapped to each other in a tabular form.

It can be of interest to define a compound type at the intersection between two types.

E.g. The table below maps types in a Material taxonomy to types in a Product taxonomy.

 “Wooden Chair” is a type that inherits from types in both taxonomies.








Wooden Chair



The periodic table of chemical elements can be viewed as a mapping between two taxonomies.

It was initially used to predict the existence of an element where a cell in the table contained no known element.


This two-dimensional taxonomy of theories is interesting.

Theories of the universe


Discrete (quantum leaps)



Most system modelling


Social systems thinking?

Quantum Mechanics


Note that continuous systems are usually modelled as discrete event-driven systems.

E.g. Forrester’s “System Dynamics” models continuous systems this way.

And precisely what system thinkers mean by “continuous” or “probabilistic” is not always clear.


Coordinating disparate taxonomies

Is it possible to superimpose an “upper level taxonomy” on different domains of knowledge?

Many have tried to build such a generic taxonomy.

Different attempts divide a base "Thing" between (for example):

·         Tangible Thing or Intangible Thing

·         Type (descriptive thing) or Instance (described thing)

·         Structure/Continuant Thing or Behavioral/Occurrent Thing

·         Space-Bound Thing or Time-Bound Thing.

Ontologies as richer type structures

People use taxonomies to classify entities and events in the natural or business world.

However, imposing a hierarchical classification on natural and business things is an artificial exercise.

Difficulties include not only multiple inheritance, but also that real-world things can change type over time

That is why ontologies can include association relationships as well as or instead of classification relationships


Data models are not (as some say) based on basic mathematical set theory.

The entities (named at the ends of relationships) are types.

An instance of an entity type belongs to a “dynamic” set, rather than a static one as in basic set theory

It belongs also to as many dynamic sets as that type has N-1 association and generalisation relationships.

Data modellers used generalisation relationships before “object orientation” came along.

But in models of persistent data, associations are more common than (relatively fragile) generalisations.


There is a further complication, especially in modelling entities and events in the natural world.

There are monothetic types, which exactly define and constrain the members of a set.

And polythetic types, whose properties may be neither necessary nor sufficient to define a member.

This is where fuzzy logic and artificial intelligence enter the picture.

Conclusions and remarks

We order and organise our many views of the world various ways.

We use types, taxonomies, ontologies, class diagrams, data models etc.

All of these are descriptions imposed on realities - they are models or theories.

The degree to which a reality matches a theory can and should be determined by testing and measurement of the reality.


Set theory is relevant to ontology in so far as a set member can be defined by a type.

Types exist as descriptions, as theories to which realities may or may not conform.

Anything that exists in the natural or business world can be regarded as an instance of one or more types (a member of one or more sets).

But only in so far as it can be measured or judged as matching a set's type definition, near enough to satisfy interested parties.


In the natural world, types are polythetic and near-enough/fuzzy matching of things to types is acceptable.

Maths, computing and hard sciences are special cases in which perfect matching of realities to descriptions is expected.

But these can be seen as peculiar exceptions to the general case.




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