The second principal axis of representation is the intensional – extensional axis. All representations have intensional and extensional aspects. The intension of a representation can be stated equivalently as that set of conditions which must be satisfied by any object, within the given universe of discourse, for the representation to exemplify the object. If an object satisfies the intensional conditions, then it is an exemplar of the object.
In indirect representations, the intension of a concept represents the syntactic definition of that which it represents. In indirect representations, the epistemic meaning of a concept can be inferred or interpreted from the representation of the concept’s intension by an intelligent observer, but not by the object itself. For example, a dictionary definition cannot read itself. It does not understand the meaning of the words it contains. A computer does not, and cannot, understand the meaning of its data. Indirect intensions represented using information contain syntax but not semantics. Consequently, the representation of indirect intensions using information is incomplete.
In direct representations, Representation = Existence. Direct representations represent things in terms of how they relate to the things that compose them and in terms of how they relate to those things that affect them in their external environment. The intension is directly composed of the extensions of the set of objects and object relationships that the object being represented is composed of and related to. The existence of the set of objects and object relationships that represent the objects that compose the object being represented and the existence of the relationships between its component objects and those objects it is related to in its external environment directly represent the existence of the object being represented. In a direct representation, the extension IS the representation and existence of the intension. Symbolically we have: Existence(Extension) = Existence(Intension). This is equivalent to Existence = Representation. In a direct representation, the representation of an objects’ intension encapsulates the representation and thus the existence of the extensions of the objects and object relationships that the object being represented is composed of and related to. (Intensional relations exist between the objects that compose the object being represented and between the object being represented and those it is related to in its external environment). Direct representations represent existence as an nth order relational hierarchy of composition. Objects can be composed of objects that are composed of objects that are composed of objects, to any required degree. Any object can be composed of zero or more objects, each of which may be composed of zero or more objects. The same is true of relations. There can be first order relations between objects: O R O, second order relations between relations: O R(R) O, third order relations: O R(R(R)) O, etc to any required degree. At the most fundamental level of the representation of existence, both the objects and the relations are composed of the same thing and represented the same way, by differences (i.e., asymmetries) in incomplete, inconsistent nonexistence. We’ll have a lot more to say about this when we cover the detailed representation of existence later.
The extension enumerates all members of the set that satisfy the relational conditions specified by the intension.
In a direct representation, the extension represents, and is, the only instance of the object represented by the intension. The existence of the intension IS the existence of the extension. In a direct representation, the cardinality of the extensional set is always 1. The extension represents the existence of the intension. In a direct representation, the extension can be represented by one bit of information, irrespective of the complexity or level of composition of the intension. However, the bit has no meaning in and of itself. Its existence simply signals the existence of the extension, and thus the satisfaction of the intensional conditions represented by the extension. If the intensional conditions are not met, the extension does not exist. This explains why fundamental particles cannot be split and why energy is quantized. It is impossible to divide the fundamental quanta of existence. All quanta exist completely, or not at all. Their representation is always complete and consistent. It is impossible for a quanta to have a partial state of existence. It cannot be part in the universe and part outside it. Creation and destruction of quanta are instantaneous, indivisible events. If the representation of a quanta is made inconsistent or incomplete, it ceases to exist. For this reason, the representation of existence is complete and consistent. It is impossible for it to become incomplete or inconsistent, so there is no need to represent or enforce ontological consistency.
In an indirect representation, the extension enumerates all members of the set that satisfy the relational conditions specified in the intension and that represent exemplars or instances of the object or concept being represented. The members of the extension represent the instances of that which is represented by satisfaction of the intensional conditions, they are not the instances themselves.
A universal representation combines intensional and extensional representation. In a universal representation, the intension is direct, but its extension is indirect. Consequently the intension defines meaning in context from the direct first person perspective. The intension of a direct universal representation defines and understands its own meaning from the first person perspective in context. The indirect extension can be used in multiple contexts and be included as part of the representation of multiple intensions. In this case, the representation of the intension is represented directly by value, but the extension is represented indirectly by reference. This will be covered in much more detail when we derive the representation of thought.
Table 1 provides an overview of the three classes of representation, their relationships, and some examples of things represented by each class.
The representations in column 1 are direct. Direct representations represent the physical existence of everything in the universe. We will refer to representation at the level of physical existence as direct representation, level 1 representation, type 1 representation or existential representation.
The representations in column 2 are universal; thus, they are both direct and indirect. Universal representations are used to represent thought. In the rest of this book, universal representations will be called type 2 representations. We will also refer to representation at the level of thought as level 2 representation.
The representations in column 3 are indirect. Indirect representation is the type of representation used to represent information. Most human generated representations are of this type. This type of representation is required for communication. It is used for logic, mathematics, writing, speaking, drawing, and computation. Indirect, or information based representations will be called type 3 representations. Representation at the level of information will also be called level 3 representation.
1-Direct (Existence) | 2 –Universal (Thought) | 3 –Indirect (Information) | |
A -Intensional | A1 Intensional Representation Conservation Symmetry, | A2:Universal* Intensional Representation Meaning Neurons’ | A3:Indirect* Intensional Representation Logical Mathematical |
B- Universal | B1:Direct* Universal Representation Evolution of the universe. Representation | B2:Universal* Universal Representation Cogito | B3:Indirect* Universal Representation First |
C -Extensional | C1:Direct* Extensional Representation Existence | C2:Universal* Extensional Representation Existence Neurons’ | C3:Indirect* Extensional Representation Mathematical |
Table 1: Types of Representation
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