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Revision as of 17:38, 20 August 2011
, 17:38, 20 August 2011→8.2.1.1. Sign Relations: + text
=====8.2.1.1. Sign Relations=====
=====8.2.1.1. Sign Relations=====
Conceived in logical terms, a "sign relation" R is a certain kind of three place relationship that exists among the elements of three domains: the object domain O, the sign domain S, and the interpretant domain I. To qualify as a sign relation in this setting, a three place relation R is required to satisfy a few additional properties to be named later.
Expressed in terms of its set theoretic extension, a sign relation R is associated with a set of ordered triples <o, s, i> that forms a subset of the cartesian product OxSxI. The notation R = Set(R) c OxSxI can be used to single out this interpretation.
Expressed in terms of its computational intension, a sign relation R is associated with a predicate or program that values ordered triples <o, s, i> according to their fitness for the logical functions of the intended sign relationship. The notation R = Fun(R) : OxSxI > B, where B = {0, 1}, can be used to single out this interpretation.
<pre>
Ways of Knowing a Relation
Knowledge by acquaintance: extension or enumeration;
Knowledge by description: intension or comprehension;
Knowledge by regulation: intention or competence.
</pre>
When the extension of a concept is infinite, or for any reason inconvenient/ or inconvenient for pragmatic reasons to enumerate in detail, then knowledge of its objects must be achieved by means of description rather than acquaintance.
When the extension of a concept is infinite, or inconvenient for pragmatic reasons to enumerate in final detail, then a finite agent's comprehension of it is required to be knowledge by indirect description rather than knowledge by direct acquaintance (Russell). That is, the objects of the concept are known by means of the concept's intensions, the common properties of its objects as expressed in symbolic reminders. But the intensions of a concept can be still more numerous than the objects of its extension, and even when a finite selection of these intensions is enough to specify the extension uniquely, there can always be many different collections which do so, and many different ways of approaching the concept by proceeding through a sequence of features in the subset chosen.
Thus, knowledge by description is approximate knowledge, knowledge whose quality and character can depend on the current stage and overall manner of approach. This sort of knowledge is contingent on and biased by each agent's particular way of approaching the objective, or the objects of the concept in question.
Due to the inclusion of these secondary facets in the character of the knowledge cut out, not every aspect of it is invariant over changes in the means of approach. The artifacts of the resulting knowledge that are not indifferent to the path of approach are called intensional features of the method, procedure, or computation. For example, programs that effectively compute the same function, the same set of ordered pairs in extension, but do it with non identical profiles of efficiency are said to differ in their intensional properties. Here, it is not the intensions of the functions as objects which differ, but the intensions of the programs as objects which do. The intensions of a program are related to the intensions of a function in the complex way that information about functional domain elements is traded for information about functional range elements throughout the progress of a computation.
Sometimes our grasp of the objects coming under a concept is even more tentative and tenuous.
Sometimes the mind's reach toward an objective is still more tentative and tenuous, exceeding the grasp of any present concept or familiar description, but represented only in the hope that certain rules of procedure or regulative principles are bound to converge on it in time. Thus, the object of knowledge is the object of an intention, and so is the hopeful knowledge itself. This kind of epistemological stance or orientation toward knowledge can be called "intentional knowledge" or "knowledge by intention", but is really more like an "intention to know".
To the extent that the computational intentions of this project are successful, more and more of the theoretical concepts employed in the unformalized parts of the inquiry will be operationalized as computable functions, serving to accomplish the actions or recognize the objects intended by each concept. In practical terms, this means that the functional interpretation of relational concepts, including the notion of a sign relation that founds the whole enterprise, will become paramount to the approach I have chosen. However, ...
The letters "o", "s", and "i" are examples of identifier names (variables or constants) that are used in discussing sign relations. They denote elements of the relational domains that fill the object, the sign, and the interpretant roles of the sign relation in question.
When the object is a formal system then its elements are regarded as signs (words or phrases, terms or formulas, pixels or pictures).
When the object is a dynamic system then its elements are regarded as states (points, moments, positions, vectors, configurations).
In general, the only constraint placed on a sign relation is the following definition.
: (Peirce, NEM)
To complete this definition, it would be necessary to say what is meant by the notions of "determination" and "correspondence" that it invokes. This I defer to a later discussion. For now, I can limit discussion to the kinds of sign relations that are useful in systems theory and that occur in the computational representation of formal systems.
For the purposes of systems theory, and staying within the frame of computable representations, a number of additional restrictions and simplifying assumptions can be attached to the generic specifications of a sign system.
Because this discussion will stay within the framework of systems theory and limit its scope to the computational representation of formal systems, a number of restrictions and simplifications can be imposed on the general definition of a sign relation.
The roles of the sign relation are filled by systems or states of systems.
The object system o is a member of the object domain O.
In the cases of interest here, the object name (variable or identifier) "o" refers to a system or a state of system.
=====8.2.1.2. Types of Signs=====
=====8.2.1.2. Types of Signs=====
