01 May 2006

Concept based ontology system described using set theory

If my system (based on concepts) is right, then the universe of discourse can be described something like:

1. There exist a finite number of properties, the set of which we will refer to as P. There is, likewise, a set of property values, which we shall call Pv. Each property in P can be given quantification and qualification via 1 or more members of Pv which are mapped the member of P. Together this makes a P-PV mapped, which we shall describe as Pr{P, Pv}. When discussed in abstract, it is possible to acknowledge that a particular {P, Pv} pairing will have a range for its property values, rather than a specific value. Then when the property is applied with specificity, a value within that range reveals itself.

2. There exist a finite number of concepts, each of which have some subset of Pr that it exhibits. A concept, as described here, is an agreed to mental idea of a trait common to everything in the universe of discourse said to exhibit the concept (a somewhat circular definition, but it is being worked on). The set of all concepts, we shall call C. The idea of a propertied concept, that is the concept considered with all of it's exhibitted properties, exists, and the set of all of these propertied concepts we will call Cp. The properties in question come from Pr (and may, therefor, have the Pv expressed as a range, when discussed in the abstract, or an actual value, when discussed with instance specificity). Each member of Cp contains one or more members from Pr, which we signify by writing Cp(Pr*).

3. Within the universe of discourse, there are a number of entities. Entities are "things" within the universe of discourse. By "things", it is meant to consider both mental and physical things, as well as processes and events. For each of these, there are both abstract types as well as specific instances. The complete set of entities we shall call E.

4. There is a power set of Cp, which we shall refer to as power(Cp).

5. Each entity can definited by one subset from power(Cp). It is highly likely that within the universe of discousrse that members of power(Cp) will not be applicable to any of the entities within that universe of discourse, due to the conflicting nature of the concepts involved.

6. There is a universal entity type which is the most general of all entity types, which has no defining concepts other than those required to establish the existence of an entity. We shall call this universal entity U. U will have several children entities, each of which are disjoint from each other and also disjoint from their common parent by having a different subset of power(Cp) as its defining list of concepts, properties, and property values.

7. Each entity within the universe of discourse evolves as a child entity of some more general (i.e. - closer to U in heritage) entity. Each child entity inherits all of power(Cp) that its parent, yet is disjoint from its parent, and all its siblings, by having a different subset from power(Cp) that defines this. Note by 2 above, Pr is definied as a pairing of P and Pv, and that Pv can be a range or can be a value. If the only difference between siblings, or between a child and a parent, is a different specific value for Pv, then this required disjointness is satisfied.

8. Starting with U and tracing the "heritage" (path from parent to child to grandchild, etc) will give a series of propertied concepts, all of which accumulate with each new generation in the heritage.

9. Depending on the nature of the universe of discourse, there will be other semantic links between entities that are not parents or children or siblings of each other, depending on the entities in question and their relationship to each other within the universe of discourse. These semantic links we will call relations, and the set of all possible relations we will call Re. Each Re can have any number of satisfying strings, a string consisting of two (or more) entities that are linked together by the relation. We shall write this as Re(E1, E2), where E1 and E2 are entities (types, instances, objects, events, or processes).

10. It is possible that a number of entities that have semantic links (relations) can be combined to make a statement within the universe of discourse. In order to ensure that related entities are combined in the correct manner, there should be a number of guidelines ensuring that any declarative statements made within the universe of discourse create analytic statements (statements that have a priori truth) that are internally correct and make good semantic sense. These will be called the Internal Rules, and all of the possible such rules make up a set, which we shall call RuI. Such a set of rules could be considered a grammar. Each rule will have a condition describing when it could be applied, and the condition can have a conditional value. Condition values can exist as a specific value, or can exist as an allowed for range (i.e. - the Condition Value CoV(x) for Condition C(X) must be within the range of y to z in order for the RuI(a) to be true). Each member of RuI will then have a number of strings containing a relation, and a condition-condition_value pairing. We shall call conditions Co, and condition values CoV. The RuI string will be written RuI(Re,(Co,CoV)).

11. It is possible to make statements within the universe of discourse that have a number of contextual affects on when it is proper to make such a statement. There are a set of rules describing such statements, and these are referred to as the external rules. The set of all such external rules we will refer to as RuE. Rather than being analytic statements, which have a priori truth, these are synthetic statements, which are only true within a certain context. It is therefore important to capture the context state, and any values that the context state may have in order to describe when these rules may be applied. Note that as with conditions above, contexts values may exist as specific values, or defined as a satisfying range. A context will be writen as Ct, and a context value will be written as CtV. Therefore the total set of external rules will contain, for each RuE, a number of strings with the value RuE(Re,(Ct,CtV)).