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===6.32. Partiality : Selective Operations===
===6.32. Partiality : Selective Operations===
<pre>
One of the main subtasks of this project is to develop a computational framework for carrying out set theoretic operations on abstractly represented classes and for reasoning about their indicated results. This effort has the general aim of enabling one to articulate the structures of n place relations and the special aim of allowing one to reflect theoretically on the properties and projections of sign relations. A prototype system that makes a beginning in this direction has already been implemented, to which the current work contributes a major part of the design philosophy and technical documentation. This section presents the rudiments of set theoretic notation in a way that conforms to these goals, taking the development only so far as needed for immediate application to sign relations like A and B.
One of the most important design considerations that goes into building the requisite software system is how well it furthers certain lines of abstraction and generalization. One of these dimensions of abstraction or directions of generalization is discussed in this section, where I attempt to unify its many appearances under the theme of "partiality". This name is chosen to suggest the desired sense of abstract intention since the extensions of concepts that it favors and for which it leaves room are outgrowths of the limitation that finite signs and expressions can never provide more than partial information about the richness of individual detail that is always involved in any real object. All in all, this modicum of tolerance for uncertainty is the very play in the wheels of determinism that provides a significant chance for luck to play a part in the finer steps toward finishing every real objective.
If one needs a slogan to entitle this form of propagation, it is only that "Necessity is the mother of invention". In other words, it is precisely this lack of perfect information that yields the opportunity for novel forms of speciation to develop among finitely informed creatures (FIC's), and just this need of perfect information that drives the evolving forms of independent determination and spontaneous creation in any area, no matter how well the arena is circumscribed by the restrictions of signs.
In tracing the echoes of this theme, it is necessary to reflect on the circumstance that degenerate sign relations happen to be perfectly possible in practice, and it is desirable to provide a critical method that can address the facts of their flaws in theoretically insightful terms. Relative to particular environments of interpretation, nothing proscribes the occurrence of sign relations that are defective in any of their various facets, namely: (1) with signs that fail to denote or connote, (2) with interpretants that lack of being faithfully represented or reliably objectified, and (3) with objects that make no impression or remain ineffable in the preferred medium.
A cursory examination of the topic of "partiality", as just surveyed, reveals two strains fixing how this "quality of murky" in general reigns. This division depends on the disposition of n tuples as the individual elements that inhabit an n place relation.
1. If the integrity of elementary relations as n tuples is maintained, then the predicate of "partiality" characterizes only the state of information that one has, either about elementary relations or about entire relations, or both. Thus, this strain of partiality affects the determination of relations at two distinct levels of their formation:
a. At the level of elementary relations, it frees up the point to which n tuples are pinned down by signs or expressions of relations by modifying the name that indicates or the formula that specifies a relation.
b. At the level of entire relations, it relaxes the grip that axioms and constraints have on the character of a relation by modifying the strictness or generalizing the form of their application.
2. If "partial n tuples" are admitted, and not permitted to be confused with "<n tuples", then one arrives at the concept of an "n place relational complex".
Relational complex?
R = R(1) U ... U R(n)
Sign relational complex?
R = R(1) U R(2) U R(3)
It is possible to see two directions of remove that signs and concepts can take in departing from complete specifications of individual objects, and thus to see two dimensions of variation in the requisite varieties of partiality, each of which leads off into its own distinctive realm of abstraction.
1. In a direction of "generality", with "general" signs and concepts, one loses an amount of certainty as to exactly what object the sign or concept applies at any given moment, and thus this can be recognized as an extensional type of abstraction.
2. In a direction of "vagueness", with "vague" signs and concepts, one loses a degree of security as to exactly what property the sign or concept implies in the current context, and thus this can be classified as an intensional mode of abstraction.
The first order of business is to draw some distinctions, and at the same time to note some continuities, between the varieties of partiality that remain to be sufficiently clarified and the more mundane brands of partiality that are already familiar enough for present purposes, but lack perhaps only the formality of being recognized under that heading.
The most familiar illustrations of information theoretic "partiality", "partial indication", or "signs bearing partial information about objects" occur every time one uses a general name, for example, the name of a genus, class, or set. Almost as commonly, the formula that expresses a logical proposition can be regarded as a partial specification of its logical models or satisfying interpretations. Just as the name of a genus or class can be taken as a "partially informed reference" or a "plural indefinite reference" (PIR) to one of its species or elements, so the name of an n place relation can be viewed as a PIR to one of its elementary relations or n tuples, and the formula or expression of a proposition can be understood as a PIR to one its models or satisfying interpretations. For brevity, this variety of referential indetermination can be called the "generic partiality" of signs as information bearers.
Note. In this discussion I will not systematically distinguish between the logical entity typically called a "proposition" or "statement" and the syntactic entity usually called an "expression", "formula", or "sentence". Instead, I work on the assumption that both types of entity are always involved in everything one proposes and also on the hope that context will determine which aspect of proposing is most apt. For precision, the abstract category of propositions proper will have to be reconstituted as logical equivalence classes of syntactically diverse expressions. For the present, I will use the phrase "propositional expression" whenever it is necessary to call particular attention to the syntactic entity. Likewise, I will not always separate "higher order propositions" (HOP's), that is, propositions about propositions, from their corresponding formulations in the guise of "higher order propositional expressions" (HOPE's).
Even though "partial information" is the usual case of information (as rendered by signs about objects) I will continue to use this phrase, for all its informative redundancy, to emphasize the issues of partial definition, specification, and determination that arise under the pervasive theme of "partiality".
In talking about properties and classes of relations, one would like to allude to "all relations" as the implicit domain of discussion, setting each particular topic against this optimally generous and neutral background. But even before discussion is restricted to a computational framework the notion of "all" (of almost anything) proves to be problematic in its very conception, not always amenable to assuming a consistent concept. So the connotation of "all relations" — really just a passing phrase that pops up in casual and careless discussions — must be relegated to the status of an informal concept, one that takes on definite meaning only when related to a context of constructive examples and formal models.
Thus, in talking "sensibly" about properties and classes of relations, one is always invoking, explicitly or implicitly, a preconceived domain of discussion or an established universe of discourse X, and in relation to this X, one is always talking, expressly or otherwise, about a selected subset S c X that exhibits the property in question or a binary valued selector function f : X > B that picks out the class in question.
When the subject matter of discussion is bounded by a universal set X, out of which all objects referred to must come, then every PIR to an object can be identified with the name or formula (sign or expression) of a subset S c X, or with that of its selector function S# : X > B. Conceptually, one imagines generating all the objects in X and then selecting out the ones that satisfy some test for membership in S.
In a realistic computational framework, however, when the domain of interest is given generatively in a genuine sense of the word, that is, defined solely in terms of the primitive elements and operations that are needed to generate it, and when the resource limitations in actual effect make it impractical to enumerate all the possibilities in advance of selecting the adumbrated subset, then the implementation of PIR's becomes a genuine computational problem.
Considered in its application to n place relations, the generic brand of partial specification constitutes a rather limited type of partiality, in that every element conceived as falling under the specified relation, no matter how indistinctly indicated, is still envisioned to maintain its full arity and to remain every bit a complete, though unknown, n tuple. Still, there is a simple way to extend the concept of generic partiality in a significant fashion, achieving a form of PIR's to relations by making use of "higher order propositions" (HOP's).
Extending the concept of generic partiality, by iterating the principle on which it is based, leads to higher order propositions about elementary relations, or propositions about relations, as one way to achieve partial specifications of relations, or PIR's to relations.
This direction of generalization expands the scope of PIR's by means of an analogical extension, and can be charted in the following manner. If the sign or expression (name or formula) of an n place relation can be interpreted as a proposition about n tuples and thus as a PIR to an elementary relation, then a higher order proposition about n tuples is a proposition about n place relations that can be used to formulate a PIR to an n place relation.
In order to formalize these ideas, it is helpful to have notational devices for switching back and forth among different ways of exemplifying what is abstractly the same contents of information, in particular, for translating among sets, their logical expressions, and their functional indications.
If S c X is a set contained in a universal set or domain X, then "S#", read as "S sharp" or "S selective", denotes the "selector function" of S, defined as:
S# : X > B with S#(x) = 1 iff x C S.
Other names for the same concept, appearing under various notations, are the "indicator function" or the "characteristic function" of a set.
Conversely, if one has a binary valued function f : X > B, then "f#", read as "f numbd" or "f selection", denotes the "selected set" of f, defined as:
f# c X with f# = f 1(1) = {x C X : f(x) = 1}.
Other names for this subset are the "fiber", "pre image", "level set", or "antecedents" of 1 under the mapping f.
Obviously, the relation between these operations is such that:
S## = S and f## = f.
It will facilitate future discussions to explicitly go through the details of applying these selective operations to the case of n place relations. If R c X1x...xXn is an n place relation, then R# : X1x...xXn > B is the selector of R defined by:
R#(<x1, ... , xn>) = 1 iff <x1, ... , xn> C R.
</pre>
===6.33. Sign Relational Complexes===
===6.33. Sign Relational Complexes===