THEORY OF CONCEPTUAL GRAPHS

CHAPTER 6 - DEFINITION

Basic Ideas

So far we have placed type and relation labels into hierarchies but we must go further than this by providing definition mechanisms for them. Each type inherits the definition of its immediate supertype modified by the addition of some extra information. The common subtype of two types inherits the definitions of both types. The entire type hierarchy can be seen as a hierarchy of definitions.

Definitions take the form of a graph which describes and is equivalent to a single concept of the type being defined. There are two views about this. One view, the Aristotelian view, is that the definition represents all and only those characteristics of a type that must be present in an individual and the other view, the Wittgensteinian view, is that the definition represents that to which there should be a close “family resemblance”. There is probably even more to this and so we will extend and modify the semantics.

Aristotelian Semantics

For Aristotle a definition consisted of two parts: the genus and the differentiae. For some type t being defined in terms of its immediate supertype s the type s is the genus of the definition and the characteristics or attributes that differentiate t from all other kinds of s are the differentiae. The differentiae describe the necessary and sufficient conditions that make an entity of type t. There is the implicit assumption that any entity of type t will always possess all and only those characteristics. Sowa provides a notation for definitions but even though the genus and differentiae are shown as a conceptual graph (maybe in linear form) it is not strictly graphical. The following is a possible but maybe simplistic type definition of the elephant in terms of a more general mammal:

Fig 6.1

Notice that there is a coreference link between the MAMMAL concept (the genus) and the header of the definition. Under the Aristotelian assumptions this definition categorically states that if some thing is an elephant it must have a trunk.

Wittgensteinian Semantics

Wittgenstein saw type definitions as descriptions to which entities would be similar. Entities of the same type would share family resemblances. This is a much looser form of definition than is Aristotelian definition and indeed any type might have more than one definition if it suits the purpose. Taking the definition in Fig. 6.1 as an example Aristotle would say that any animal without a trunk was not an elephant, including any elephant that had lost its trunk in some accident. Clearly, humans do not say that anything which is obviously an elephant even though its trunk is missing is not an elephant at all and so Aristotelian semantics are not realistic in a flexible system. Nevertheless there may be times when a human would choose to apply definitions rigorously and others when a more pragmatic approach is more useful.

Sowa does not give a full account of how different forms of definition should be represented. We would suggest that the differences are more in the ways that the definitions are used and not in how they are notated. Sowa provided schemata and prototypes which were descriptions of typical examples of entities and situations that entities might find themselves in, represented exactly as type definitions but with “schema” or “prototype” in the headers. Maybe he intended that clusters of schemata were to provide the Wittgensteinian semantics.

Combining And Extending Definition Semantics

Our proposal is that Sowa's type definition notation can be taken to be either Aristotelian or Wittgensteinian and that schemata are not adopted as definitional representations (they are reserved for a purpose to be described later in this chapter). We suggest that since Sowa's prototypes define a typical instance of a type that this be used as a definition for a type. In this way any elephant that had lost its trunk would not stop being an elephant. As a result we will have the following definition for an elephant which does not require that all parts of the description of a typical elephant are present, except that [ MAMMAL ] will be true:

Fig. 6.2

To summarise, the type definition as in Sowa's formalism provides an Aristotelian view of definition and the prototype provides a Wittgensteinian view.

Lambda Abstractions

It is possible to place a type definition in the type field of a concept, in which case there will be a coreference link between the main concept and the definition:

Fig. 6.3

These graphical forms are more easily processed by the graph operations to be described later. The problem with including a type definition in a concept is that the definition is local, that is it applies only to that concept. The example in fig 6.3 only states that there exists that particular case of a mammal with a trunk and certainly does not state that all elephants are mammals with trunks. In fact it does not mention elephants at all. Fig 6.3 shows that the concept is defined by a lambda abstraction. A lambda abstraction is a nameless function. The *x is the formal parameter.

Relation Definitions

Relations are defined in an analogous manner to types. Definitions can be shown by the “header and body” notation or by lambda abstractions within a relation node, in which case the formal parameters are the referents of concepts attached to the relation node. The following is an example:

Fig. 6.4

The lambda form of relation definitions is slightly different from that of type definitions because the relation definitions require that the relation nodes are shown attached to the concepts between which the relation is being defined otherwise the formal parameters cannot be linked to the correct concepts.

Semantic Units

The term “semantic units” refers to groups of relations that always occur together. In principle a conceptual graph can consist of whatever concepts and relations the writer wants to include but in reality many concepts must have a particular set of relations attached to them. For example, any action represented by a verb will have an agent, the thing that is performing the action. Therefore the presence of a concept for that action will imply the presence of an AGENT relation. Similarly any action that is done to something by something will have an AGENT (the entity performing the action) and a PATIENT (the entity that is affected by the action). In some cases an action may be done by something to something with something and such an action could not be done if any one of these three entities were absent. In each case we have what we call a “semantic unit”. In natural language any part of a semantic unit may be omitted but conceptually it is nevertheless present.

All English language sentences contain a verb and it is usually the case that sentences in other languages contain verbs but there are rare examples that do not. It would seem that the verb, or the action that it represents, is a fundamental part of human reasoning. Therefore it seems reasonable to assume that actions form the “core” of any conceptual graph and that for any sentence that does not include all the participants in the particular action the unmentioned ones can be assumed.

Examples might be:

John eats. (but must be eating something).

Peter is travelling. (but must be going from somewhere to somewhere).

Patricia is being stabbed. (but must be being stabbed by someone with something).

The following show the simple graphs that these sentences might produce along with the more complete graphs that they imply:

Fig. 6.5

In each case, whilst the left hand graph is correct and true the right hand graph is implied by the semantics of the action being described. The graphs on the right are instantiations of semantic units which are represented as schemata as in the next figure:

Fig. 6.6

We could be more general and specify schemata for whole classes of actions. For example, any intransitive verb has a subject, or agent, and any transitive verb also has a direct object, or patient. In addition some actions can only be carried out with an indirect object, or instrument. We could specify second order definitions for each of these kinds of verbs or actions and an example might be:

Fig. 6.7

This is given as a type definition, with the Aristotelian assumption, but could also be shown as a schema specifying the context in which all transitive actions exist.

Order Abstraction And Order Reduction

Figures 6.6 and 6.7 hint at a relationship between schemata of different orders. In this section we introduce the notion that such a relationship exists and provide two operations for linking schemata of different orders. The following diagram will illustrate both the general case and an example:

Fig. 6.8

In the diagram above ACT_I is an intransitive act (one which is not done to something), AGENT² is a 2^nd order AGENT relation and AGENT¹ is a 1^st order AGENT relation. The difference between these AGENT relations is that the 2^nd order one shows the relationship between a kind of act and the kind of entity that performs it whilst the 1^st order relation shows the relationship between a particular instance of the act and the particular individual that did it. These are not the same.

It should be clear from fig. 6.8 that in any one graph the type and relation labels must always be of the same order and the referents must be of one order less. However, it is quite permissible to link a referent in a 2^nd order concept to a type or relation field in a 1^st order concept.

Updated 21^st December, 2006