Top-Level Categories

This web page summarizes the top levels of the KR Ontology, which is defined in the book Knowledge Representation by John F. Sowa. Figure 1 shows a lattice 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:

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:

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: T, ^, 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 IndependentAbstract; 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.

T ().
The universal type, which has no differentiae. Formally, T is a primitive that satisfies the following axioms:

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.

The absurd type, which inherits all differentiae. Formally, ^ is a primitive that satisfies the following axioms:

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 st is ^. For example, DogCat = ^ 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:

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.

Absurdity (IRMPACO) = ^.
A pronounceable synonym for ^. It cannot be the type of anything that exists.

Actuality (IP) = IndependentPhysical.
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 physical continuant is an object, and an abstract continuant is a schema that may be used to characterize some object.

Description (RAC) = PropositionContinuant.
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.

Entity () = T.
A pronounceable synonym for T. Entity can be used as the default type for anything of any category.

Form (IA) = AbstractIndependent.
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.

History (RAO) = PropositionOccurrent.
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 has x:

Intention (MA) = AbstractMediating.
Abstraction (A) considered as mediating (M) other entities. Examples of intentions include the hopes, fears, wishes, and purposes that mediate some agent's actions.

Juncture (RPC) = PrehensionContinuant.
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:

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:

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).

Nexus (MP) = PhysicalMediating.
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.

Object (IPC) = ActualityContinuant.
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:

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.

Participation (RPO) = PrehensionOccurrent.
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:

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.

Process (IPO) = ActualityOccurrent.
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 has(x,y).

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 multiple entities.

Purpose (MAO) = IntentionOccurrent.
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 and agents:

For further discussion, see the web page on agents.

Reason (MAC) = IntentionContinuant.
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 must apply:

For any relative x, there must exist some y such that x has y or y has x.

Schema (IAC) = FormContinuant.
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.

Script (IAO) = FormOccurrent.
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.

Situation (MPO) = NexusOccurrent.
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.

Structure (MPC) = NexusContinuant.
A nexus (MP) considered as a continuant (C). A structure mediates multiple objects whose junctures constitute the structure.
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.

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 Knowledge Representation.

Send comments to John F. Sowa.

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