This web page summarizes the top levels of the KR Ontology,
which is defined in the book
by John F. Sowa. Figure 1 shows
of the top-level categories discussed in Chapter 2 of that book.
These categories have been derived from a synthesis of various sources,
but the two major influences are the semiotics of Charles Sanders Peirce
and the categories of existence of Alfred North Whitehead.
Figure 1: Hierarchy of top-level categories
Any category in Figure 1 can be abbreviated by the initials
of the primitive categories above it:
Independent, Relative, or Mediating; Physical or Abstract;
Continuant or Occurrent. Actuality, for example, may be abbreviated as
IP for Independent Physical, and Purpose as MAO for Mediating Abstract
Occurrent. The twelve categories displayed in the center of the lattice
and the primitives from which they are generated can also be arranged
in the matrix of Figure 2.
Figure 2: Matrix of the twelve central categories
The lattice and the matrix are different ways of displaying the
combinatorial structure of the categories. The two kinds of diagrams
highlight different aspects:
- The lattice can display intermediate categories that are formed
from the options taken two at a time, such as Actuality (IP),
Proposition (RA), or Nexus (MP).
- But a diagram of the full lattice may become too cluttered if
all options are displayed, and some possible combinations, such as
AbstractOccurrent (AO) or RelativeContinuant (RC), were omitted to
simplify the diagram.
- When all possible combinations are meaningful,
the matrix is often the simplest way to display them.
But if some combinations are ruled out because of other
constraints, some boxes in the matrix may be empty.
All the categories defined by such lattices and matrices can be
represented as monadic predicates defined by conjunctions of simpler
monadic predicates. For example, the category Participation (RPO)
corresponds to a predicate participation(x), which
is defined by the following conjunction:
- participation(x) º relative(x)
Ù physical(x) Ù occurrent(x).
Categories defined by conjunctions of primitives are useful
for generating the structural backbone of the type hierarchy.
But the more specialized categories in an ontology
may require more complex logical expressions.
For further discussion of the problems and issues of defining
a large ontology of concepts with detailed semantic representations,
see the article "Concepts in the Lexicon".
The categories in Figure 1 are listed below.
Nine primitive categories have associated axioms:
Independent, Relative, Mediating, Physical, Abstract, Continuant, and
Occurrent. Each subtype is defined as the infimum
(greatest common subtype, represented by the symbol
of two supertypes, whose axioms it inherits. For example, the type
Form is defined as IndependentÇAbstract;
it therefore inherits the axioms of Independent and Abstract,
and it is abbreviated IA to indicate its two supertypes.
See the glossary for definitions of the
techniques and metalevel conventions used to define these categories.
See the tutorial for a review
of the definitions and notations for sets, functions,
relations, graphs, lattices, and logic.
The primitive categories of any theory are undefinable in terms of
anything more primitive. The axioms associated with the categories
are not closed-form definitions, but constraints on how instances
of those categories are related to instances of other categories,
many of which are not primitives. The only two categories in this
list whose axioms are completely formalized are T and ^.
The other axioms cannot be stated formally until a great deal more
has been fully formalized. The axioms for Physical, for example,
use the categories Place and Time and the predicates loc and pTim.
A complete formalization of those axioms would depend on a fully
developed Grand Unified Theory of physics -- a task that
the physicists are far from completing.
- T ().
- The universal type, which has no
T is a primitive that satisfies the following axioms:
- There exists something: ($x)T(x).
- Everything is an instance of T:
- Every type is a subtype of T:
All other types are defined by adding differentiae to T to show
how they are distinguished from T and from one another.
The type Entity is a pronounceable synonym for T.
- ^ (IRMPACO).
- The absurd type, which inherits all differentiae. Formally,
^ is a primitive that satisfies the following axioms:
- Nothing is an instance of ^:
- Every type is a supertype of ^:
Since ^ is the inconsistent conjunction of all differentiae,
it is not possible for any existing entity to be an instance of ^.
Two types s and t are said to be incompatible
if their only common subtype sÇt is ^.
For example, DogÇCat = ^ because it is not possible for
anything to be both a dog and a cat at the same time.
- Abstract (A).
- Pure information as distinguished from any particular encoding
of the information in a physical medium.
Formally, Abstract is a primitive that satisfies the following axioms:
- No abstraction has a location in space:
- No abstraction occurs at a point in time:
As an example, the information you are now reading is encoded
on a physical object in front of your eyes, but it is also encoded
on paper, magnetic spots, and electrical currents at several other
locations. Each physical encoding is said to represent
the same abstract information.
(IRMPACO) = ^.
- A pronounceable synonym for ^. It cannot
be the type of anything that exists.
(IP) = IndependentÇPhysical.
- A physical entity (P) whose existence is independent (I)
of any other entity. As instances, the category Actuality includes
both objects and processes.
The term is taken from Whitehead, who used it as a synonym
for actual entity, which he considered the equivalent
of Aristotle's ousia and Descartes's res vera.
- Continuant (C).
- An entity whose identity continues to be recognizable over some
extended interval of time. Formally, Continuant is a primitive that
satisfies the following axioms:
- A continuant x has only spatial parts and no temporal parts.
At any time t when x exists, all of x exists
at the same time t. New parts of a continuant x
may be acquired and old parts may be lost, as when a snake sheds its skin.
Parts that have been lost may cease to exist, but everything that remains
a part of x continues to exist at the same time as x.
- The identity conditions
for a continuant are independent of time.
If c is a subtype of Continuant, then the identity predicate
Idc(x,y) for identifying two instances
x and y of type c does not depend on time.
A physical continuant is an object, and an abstract continuant is
a schema that may be used to characterize some object.
(RAC) = PropositionÇContinuant.
- A proposition (RA) about a continuant (C). A description is
a proposition that states how some schema characterizes some
aspect or configuration of a continuant.
() = T.
- A pronounceable synonym for T.
Entity can be used as the default type for anything of any category.
(IA) = AbstractÇIndependent.
- Abstract information (A) independent (I) of any encoding or
embodiment. Forms can be said to exist in the same sense as
mathematical objects such as sets and relations, but instances
of forms cannot exist at a particular place and time without
some physical encoding or embodiment. Whitehead called them
"eternal objects" because they are independent of space and time.
(RAO) = PropositionÇOccurrent.
- A proposition (RA) about an occurrent (O).
A history is a proposition (RA) that relates some script (IAO)
to the stages of some occurrent (O).
A computer program, for example, is a script (IAO); a computer executing
the program is a process (IPO); and the abstract information
(A) encoded in a trace of the instructions executed is a history (RAO).
Like any proposition, a history need not be true, and it need not be
predicated of the past: a myth is a history of an imaginary past;
a prediction is a history of an expected future; and a scenario
is a history of some hypothetical occurrent.
- Independent (I).
- An entity characterized by some inherent Firstness,
independent of any relationships it may have to other entities.
Formally, Independent is a primitive for which the
has-test of Section 2.4 need not apply. If x is
an independent entity, it is not necessary that there exists
an entity y such that x has y or y
(MA) = AbstractÇMediating.
- Abstraction (A) considered as mediating (M) other entities.
Examples of intentions include the hopes, fears, wishes, and purposes
that mediate some agent's actions.
(RPC) = PrehensionÇContinuant.
- A prehension (RP) considered as a continuant (C) during some time
interval. The prehending entity is an object (IPC) in a stable
relationship to some prehended entity during that interval.
An example of a juncture is the relationship between two adjacent stones
in an arch. The arch itself is a nexus that both mediates and consists
of the multiple junctures.
- Mediating (M).
- An entity characterized by some Thirdness that brings other entities
into a relationship. An independent entity need not have any relationship
to anything else, a relative entity must have some relationship to
something else, and a mediating entity creates a relationship between
two other entities. An example of a mediating entity is a marriage,
which creates a relationship between a husband and a wife.
According to Peirce, the defining aspect of Thirdness is "the conception
of mediation, whereby a first and a second are brought into relation."
That property could be expressed in second-order logic:
(($R,S:Relation)(R(m,x) Ù S(m,y)))
This formula says that for any mediating entity m and
any other entities x and y,
if there exist relations R and S that
relate m to x and m to y,
then it is necessarily true that
there exists some relation T that relates x to y.
For example, if m is a marriage,
R relates m to a husband x,
S relates m to a wife y, then
T relates the husband to the wife (or the wife to the husband).
Instead of a second-order formula, an equivalent first-order axiom
could be stated in terms of the primitive has relation, which is
discussed in Section 2.4 of the book Knowledge Representation:
Ù has(m,y)) É
o(has(x,y) Ú has(y,x)).
This formula says that for any mediating entity m
and any other entities x and y, if m
has x and m has y, then it is necessary
that x has y or y has x.
In effect, the has relation in this formula is a generalization
of the relations R, S, and T in the second-order formula.
For example, if m is a marriage that has a husband x
and a wife y, then the husband has the wife or the wife has
the husband (or both).
(MP) = PhysicalÇMediating.
- A physical entity (P) mediating (M) two or more other entities.
Each nexus is a bundle of prehensions, which may be the junctures
of an object or the participants of a process.
Examples include an arch that consists of junctures of stones or
an action that consists of what one participant called an agent
is doing to another participant called a patient.
(IPC) = ActualityÇContinuant.
- Actuality (IP) considered as a continuant (C), which retains
its identity over some interval of time. Although no physical entity
is ever permanent, an object can be recognized by identity conditions
that remain stable during its lifetime. The type Object includes
ordinary physical objects as well as the instantiations of classes
in object-oriented programming languages.
- Occurrent (O).
- An entity that does not have a stable identity during any interval
of time. Formally, Occurrent is a primitive that satisfies
the following axioms:
- The temporal parts of an occurrent, which are called stages,
exist at different times.
- The spatial parts of an occurrent, which are called
participants, may exist at the same time, but an occurrent
may have different participants at different stages.
- There are no identity conditions that can be used to identify
two occurrents that are observed in nonoverlapping space-time regions.
A person's lifetime, for example, is an occurrent. Different stages
of a life cannot be reliably identified unless some continuant, such as
the person's fingerprints or DNA, is recognized by suitable identity
conditions at each stage. Even then, the identification depends on an
inference that presupposes the uniqueness of the identity conditions.
(RPO) = PrehensionÇOccurrent.
- A prehension (RP) considered as an occurrent (O) during the interval
of interest. The prehending entity is a process (IPO), and the
prehended entity is called a participant.
- Physical (P).
- An entity that has a location in space-time.
Formally, Physical is a primitive that satisfies the following axiom:
- Anything physical is located in some place:
- Anything physical occurs at some point in time:
More detailed axioms that relate physical entities to space, time,
matter, and energy would involve a great deal physical theory,
which is beyond the scope of the KR book.
(IPO) = ActualityÇOccurrent.
- Actuality (IP) considered as an occurrent (O) during the interval of interest.
Depending on the time scale and level of detail, the same actual entity may
be viewed as a stable object or a dynamic process. Even an entity as stable
as a diamond could be considered a process when viewed over a long time period
or at the atomic level of vibrating particles. For further discussion, see
the web page on processes.
- Prehension (RP).
- A physical entity (P) relative (R) to some entity or entities.
The has-test is used to check whether an entity x
prehends an entity y. If so, the prehension may be expressed
- Proposition (RA).
- An abstraction (A) that relates (R) some entity or entities.
In logic, the assertion of a proposition is a claim that the abstraction
corresponds to some aspect or configuration of the entity or entities
involved. As an example, the statement cat(Yojo) expresses a proposition
that the form labeled Cat characterizes the entity named Yojo.
According to Peirce and Whitehead, more complex propositions are asserted
by constructing a compound predicate, such as a mathematical expression
or a diagram, and using it to characterize the prehensions that relate
(MAO) = IntentionÇOccurrent.
- Intention (MA) that has the form of an occurrent (O).
As an example, the words and notes of the song "Happy Birthday"
constitute a script (IAO); a description of how people at a party sang
the song is history (RAO); and the intention (MA) that explains
the situation (MPO) is a purpose (MAO).
The basic axioms for Purpose are inherited from its supertypes Mediating,
Abstract, and Occurrent. Lower-level axioms relate purposes to actions
- Time sequence.
If an agent x performs an act y whose purpose is
a situation z, the start of y occurs before the start
If an agent x performs an act y whose purpose is
a situation z
described by a proposition p, then it is possible that z
might not occur or that p might not be true of z.
- Success or failure.
If an agent x performs an act y whose purpose is
a situation z described by a proposition p, then x
is said to be successful if z occurs and p is true
of z; otherwise, x is said to have failed.
For further discussion, see the web page
(MAC) = IntentionÇContinuant.
- Intention (MA) that has the form of a continuant (C).
Unlike a simple description (Secondness), a reason explains an entity
in terms of an intention (Thirdness).
For a birthday party, a description might list the presents,
but a reason would explain why the presents are relevant to the party.
- Relative (R).
- An entity in a relationship to some other entity.
Formally, Relative is a primitive for which the has-test
- ("x:Relative)o($y)(has(x,y) Ú has(y,x)).
For any relative x,
there must exist some y such that x has y or y has x.
(IAC) = FormÇContinuant.
- A form (IA) that has the structure of a continuant (C).
A schema is an abstract form (IA) whose structure
does not specify time or timelike relationships. Examples include
geometric forms, the syntactic structures of sentences in some
language, or the encodings of pictures in a multimedia system.
(IAO) = FormÇOccurrent.
- A form (IA) that has the structure of an occurrent (O).
A script is an abstract form (IA) that represents
time sequences. Examples include computer programs, a recipe for baking
a cake, a sheet of music to be played on a piano, or a differential
equation that governs the evolution of a physical process.
A movie can be described by several different kinds of scripts:
the first is a specification of the actions and dialog to be
acted out by humans; but the sequence of frames in a reel
of film is also a script that determines a process carried out
by a projector that generates flickering images on a screen.
(MPO) = NexusÇOccurrent.
- A nexus (MP) considered as an occurrent (O).
A situation mediates the participants of some process,
whose stages may involve different participants at different times.
(MPC) = NexusÇContinuant.
- A nexus (MP) considered as a continuant (C).
A structure mediates multiple objects whose junctures constitute
The task of formalizing everything is like the construction of a medieval
cathedral: it takes centuries to complete, and when it is done,
someone else will have a plan for an even grander cathedral.
Whitehead's motto is the best guideline: "We must be systematic,
but we should keep our systems open." For further discussion of the
problems and issues, see Chapter 6 on "Knowledge Soup" in the book
Send comments to John F. Sowa.