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The pending example of a POSR is, of course, the system composed of a pair of sign relations <math>\{ L(\text{A}), L(\text{B}) \},\!</math> where the nouns and pronouns in each sign relation refer to the hypostatic agents <math>\text{A}\!</math> and <math>\text{B}\!</math> that are known solely as embodiments of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math>  But this example, as reduced as it is, already involves an order of complexity that needs to be approached in more discrete stages than the ones enumerated in the current account.  Therefore, it helps to take a step back from the full variety of sign relations and to consider related classes of POSRs that are typically simpler in principle.

The pending example of a POSR is, of course, the system composed of a pair of sign relations <math>\{ L(\text{A}), L(\text{B}) \},\!</math> where the nouns and pronouns in each sign relation refer to the hypostatic agents <math>\text{A}\!</math> and <math>\text{B}\!</math> that are known solely as embodiments of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math>  But this example, as reduced as it is, already involves an order of complexity that needs to be approached in more discrete stages than the ones enumerated in the current account.  Therefore, it helps to take a step back from the full variety of sign relations and to consider related classes of POSRs that are typically simpler in principle.

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1. The first class of POSRs I want to consider is diverse in form and content and has many names, but the feature that seems to unite all its instances is a "self commenting" or "self documenting" character.  Typically, this means a "partially self documenting" (PSD) character.  As species of formal structures, PSD data structures are rife throughout computer science, and PSD developmental sequences turn up repeatedly in mathematics, logic, and proof theory.  For the sake of euphony and ease of reference I collect this class of PSD POSRs under the name of "auto graphs" (AGs).

 

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<p>The first class of POSRs I want to consider is diverse in form and content and has many names, but the feature that seems to unite all its instances is a ''self-commenting'' or ''self-documenting'' character.  Typically, this means a ''partially self-documenting'' (PSD) character.  As species of formal structures, PSD data structures are rife throughout computer science, and PSD developmental sequences turn up repeatedly in mathematics, logic, and proof theory.  For the sake of euphony and ease of reference I collect this class of PSD POSRs under the name of ''auto-graphs'' (AGs).</p>

 

The archetype of all auto graphs is perhaps the familiar model of the natural numbers N as a sequence of sets, each of whose successive sets collects all and only the previous sets of the sequence:

: <p><math>\{\}, \quad \{\{\}\}, \quad \{\{\}, \{\{\}\}\}, \quad \{\{\}, \{\{\}\}, \{\{\}, \{\{\}\}\}\}, \quad \ldots\!</math></p>

{},  {{}}, {{}, {{}}},  {{}, {{}}, {{}, {{}}}},  ...

<p>This is the purest example of a PSD developmental sequence, where each member of the sequence documents the prior history of the development. This AG is akin to many kinds of PSD data structures that are found to be of constant use in computing. As a natural precursor to many kinds of ''intelligent data structures'', it forms the inveterate backbone of a primitive capacity for intelligence. That is, this sequence has the sort of developing structure that can support the initial growth of learning in many species of creature constructions with adaptive constitutions, while it remains supple enough to supply an articulate skeleton for the evolving process of reflective inquiry. But this takes time to see.</p>

For future reference, I dub this "model of natural numbers" as "MON".  The very familiarity of this MON means that one reflexively proceeds from reading the signs of its set notation to thinking of its sets as mathematical objects, with little awareness of the sign relation that mediates the process, or even much reflection after the fact that is independent of the reflections recorded.  Thus, even though this MON documents a process of reflective develoment, it need inspire no extra reflection on the acts of understanding needed to follow its directions.

For future reference, I dub this "model of natural numbers" as "MON".  The very familiarity of this MON means that one reflexively proceeds from reading the signs of its set notation to thinking of its sets as mathematical objects, with little awareness of the sign relation that mediates the process, or even much reflection after the fact that is independent of the reflections recorded.  Thus, even though this MON documents a process of reflective develoment, it need inspire no extra reflection on the acts of understanding needed to follow its directions.