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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 one needs a slogan 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 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.

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.

The most familiar illustrations of information-theoretic partiality, partial indication, or &ldquo;signs bearing partial information about objects&rdquo; occur every time one uses a general name, for example, the name of a class, genus, 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 class or genus can be taken as a ''partially informed reference'' or a ''plural indefinite reference'' (PIR) to one of its elements or species, so the name of an <math>n\!</math>-place relation can be viewed as a PIR to one of its elementary relations or <math>n\!</math>-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.

The most familiar illustrations of information-theoretic partiality, partial indication, or &ldquo;signs bearing partial information about objects&rdquo; occur every time one uses a general name, for example, the name of a class, genus, 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 class or genus can be taken as a ''partially informed reference'' or a ''plural indefinite reference'' (PIR) to one of its elements or species, so the name of an <math>n\!</math>-place relation can be viewed as a PIR to one of its elementary relations or <math>n\!</math>-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 a ''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'', that is, propositions about propositions, from their corresponding formulations in the guise of ''higher order propositional expressions''.

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" (HOPs), that is, propositions about propositions, from their corresponding formulations in the guise of "higher order propositional expressions" (HOPEs).

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".

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, determination, and specification 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.

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'' &mdash; really just a passing phrase that pops up in casual and careless discussions &mdash; 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.

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.