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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.

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 a slogan is need to charge 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 (FICs), 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.

If a slogan is needed to charge 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 (FICs), 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.

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.

In speaking of 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.

In speaking of 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 <math>X,\!</math> and in relation to this <math>X\!</math> one is always talking, expressly or otherwise, about a selected subset <math>A \subset X\!</math> that exhibits the property in question and a binary-valued selector function <math>f_A : X \to \mathbb{B}\!</math> that picks out the class in question.

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.

When the subject matter of discussion is bounded by a universal set <math>X,\!</math> 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 <math>A \subseteq X\!</math> or else with that of its selector function <math>f_A : X \to \mathbb{B}.\!</math> Conceptually, one imagines generating all the objects in <math>X\!</math> and then selecting the ones that satisfy a definitive test for membership in <math>A.\!</math>

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 PIRs becomes a genuine computational problem.

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 PIRs becomes a genuine computational problem.