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===6.19. Examples of Self Reference===
===6.19. Examples of Self Reference===
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For ease of repeated reference, I introduce the following terminology. With respect to the empirical dimension, a "good" POSR is described as an "exculpable self reference" (ESR) while a "bad" POSR is described as an "indictable self reference" (ISR). With respect to the intuitive dimension, a "good" POSR is depicted as an "explicative self reference" (ESR) while a "bad" POSR is depicted as an "implicative self reference" (ISR). Here, underscored acronyms are used to mark the provisionally settled, hypothetically tentative, or "status quo" status of these casually intuitive categories.
These categories of POSR's can be discussed in greater detail as follows:
1. There is an "empirical distinction" that appears to impose itself on the varieties of self reference, separating the forms that lead to trouble in thought and communication from the forms that do not. And there is a pragmatic reason for being interested in this distinction, the motive being to avoid the corresponding types of trouble in reflective thinking. Whether this apparent distinction can hold up under close examination is a good question to consider at a later point. But the real trouble to be faced at the moment is that an empirical distinction is a post hoc mark, a difference that makes itself obvious only after the possibly unpleasant facts to be addressed are already present in experience. Consequently, its certain recognition comes too late to avert the adverse portions of those circumstances that its very recognition is desired to avoid.
According to the form of this empirical distinction, a POSR can be classified either as an "exculpable self reference" (ESR) or as an "indictable self reference" (ISR). The distinction and the categories to either side of it are intended to sort out the POSR's that are safe and effective to use in thought and communication from the POSR's that can be hazardous to the health of inquiry.
More explicitly, the distinction between ESR's and ISR's is intended to capture the differences that exist between the following cases:
a. ESR's are POSR's that cause no apparent problems in thought or communication, often appearing as practiaclly useful in many contexts and even as logically necessary in some contexts.
b. ISR's are POSR's that lead to various sorts of trouble in the attempt to reason with them or to reason about them, that is, to use them consistently or even to decide for or against their use.
I refer to this as an "empirical distinction", in spite of the fact that the domain of experience in question is decidedly a formal one, because it rests on the kinds of concrete experiences and grows through the kinds of unforseen developments that are ever the hallmark of experimental knowledge.
There is a pragmatic motive involved in this effort to classify the forms of self reference, namely, to avoid certain types of trouble that seem to arise in reasoning by means of self referent forms. Accordingly, there is an obvious difference in the uses of self referent forms that is of focal interest here, but it presents itself as an empirical distinction, that is, an after the fact feature or post hoc mark. Namely, there are forms of self reference that prove themselves useful in practice, being conducive to both thought and communciation, and then there are forms that always seem to lead to trouble. The difference is evident enough after the impact of their effects has begun to set in, but it is not always easy to recognize these facts in advance of risking the very circumstances of confusion that one desires a classification to avoid.
In summary, one has the following problem. There is found an empirical distinction between different kinds of self reference, one that becomes evident and is easy to judge after the onset of their effects has begun to set in, between the kinds of self reference that lead to trouble and the kinds that do not. But what kinds of intuitive features, properties that one could recognize before the fact, would serve to distinguish the immanent and imminent empirical categories before one has gone through the trouble of suffering their effects?
Thus, one has the problem of translating between a given collection of empirical categories and a suitable collection of intuitive categories, the latter being of a kind that can be judged before the facts of experience have become inevitable, hoping thereby to correlate the two dimensions in such a way that the categories of intuition about POSR's can foretell the categories of experience with POSR's.
2. In a tentative approach to the subject of self reference, I notice a principled distinction between two varieties of self reference, that I call "constitutional", "implicative", or "intrinsic self reference" (ISR) and "extra constitutional", "explicative", or "extrinsic self reference" (ESR), respectively.
a. ISR
b. ESR
In the rest of this section I put aside the question of defining a thing, symbol, or concept in terms of itself, which promises to be an exercise in futility, and consider only the possibility of explaining, explicating, or elaborating a thing, symbol, or concept in terms of itself. In this connection I attach special importance to a particular style of exposition, one that reformulates one's initial idea of an object in terms of the active implications or the effective consequences that its presence in a situation or its recognition and use in an application constitutes for the practical agent concerned. This style of "pragmatic reconstruction" can serve a useful purpose in clarifying the information one possesses about the object, sign, or idea of concern. Properly understood, it is marks the effective reformulation of ideas in ways that are akin to the more reductive sorts of "operational definition", but overall is both more comprehensive and more pointedly related to the pragmatic agent, or the actual interpreter of the symbols and concepts in question.
The pending example of a POSR is, of course, the system composed of a pair of sign relations {A, B}, where the nouns and pronouns in each sign relation refer to the hypostatic agents A and B that are known solely as embodiments of the sign relations A and B. 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 POSR's that are typically simpler in principle.
1. The first class of POSR's 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 POSR's under the name of "auto graphs" (AG's).
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:
{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}, ...
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.
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.
In order to render this MON instructive for the development of a RIF, something intended to be a deliberately "self conscious" construction, it is important to remedy the excessive lucidity of this MON's reflections, the confusing mix of opacity and transparency that comes in proportion to one's very familiarity with an object and that is compounded by one's very fluency in a language. To do this, it is incumbent on a proper analysis of the situation to slow the MON down, to interrupt one's own comprehension of its developing intent, and to articulate the details of the sign process that mediates it much more carefully than is customary.
These goals can be achieved by singling out the formal language that is used by this MON to denote its set theoretic objects. This involves separating the object domain O = OMON from the sign domain S = SMON, paying closer attention to the naive level of set notation that is actually used by this MON, and treating its primitive set theoretic expressions as a formal language all its own.
Thus, I need to discuss a variety of formal languages on the following alphabet:
X = XMON = { " " , "," , "{" , "}" }.
Because references to an alphabet of punctation marks can be difficult to process in the ordinary style of text, it helps to have alternative ways of naming these symbols.
First, I use raised angle brackets (<...>), or "supercilia", as alternate forms of quotation marks.
X = XMON = { < > , <,> , <{> , <}> }.
Second, I use a collection of conventional names to refer to the symbols.
X = XMON = { blank, comma, enbow, exbow }.
Although it is possible to present this MON in a way that dispenses with blanks and commas, the more expansive language laid out here turns out to have capacities that are useful beyond this immediate context.
2. Reflection principles in propositional calculus. Many statements about the order are also statements in the order. Many statements in the order are already statements about the order.
3. Next, I consider a class of POSR's that turns up in group theory./ The next class of POSR's I want to discuss is one that arises in group theory.
Although it is seldom recognized, a similar form of self reference appears in the study of "group representations", and more generally, in the study of homomorphic representations of any mathematical structure. In particular, this type of ESR arises from the "regular representation" of a group in terms of its action on itself, that is, in the collection of effects that each element has on the all the individual elements of the group.
There are several ways to side step the issue of self reference in this situation. Typically, they are used in combination to avoid the problematic features of a self referential procedure and thus to effectively rationalize the representation.
As a preliminary study, it is useful to take up the slightly simpler brand of self reference occuring in the topic of regular representations and to use it to make a first reconnaissance of the larger terrain.
As a first foray into the area I use the topic of group representations to illustrate the theme of "extra constitutional self reference". To provide the discussion with concrete material I examine a couple of small groups, picking examples that incidentally serve a double purpose and figure more substantially in a later stage of this project.
Each way of rationalizing the apparent self reference begins by examining more carefully one of the features of the ostensibly circular formulation:
xi = {<x1, x1*xi>, ..., <xn, xn*xi>}.
a. One approach examines the apparent equality of the expressions.
b. Another approach examines the nature of the objects that are invoked.
Fragments
In previous work I developed a version of propositional calculus based on C.S. Peirce's "existential graphs" (PEG) and implemented this calculus in computational form as a "sentential calculus interpreter" (SCI). Taking this calculus as a point of departure, I devised a theory of "differential extensions" for propositional domains that can be used, figuratively speaking, to put universes of discourse "in motion", in other words, to provide qualitative descriptions of processes taking place in logical spaces. See (Awbrey, 1989 & 1994) for an account of this calculus, documentation of its computer program, and a detailed treatment of differential extensions.
In previous work (Awbrey, 1989) I described a system of notation for propositional calculus based on C.S. Peirce's "existential graphs" (PEG), documented a computer implementation of this formalism, and showed how to provide this calculus with a "differential extension" (DEX) that can be used to describe changing universes of discourse. In subsequent work (Awbrey, 1994) the resulting system of "differential logic" was applied to give qualitative descriptions of change in discrete dynamical systems. This section draws on that earlier work, summarizing the conceptions that are needed to give logical representations of sign relations and recording a few changes of a minor nature in the typographical conventions used.
Abstractly, a domain of propositions is known by the axioms it satisfies. Concretely, one thinks of a proposition as applying to the objects it is true of.
Logically, a domain of properties or propositions is known by the axioms it is subject to. Concretely, a property or proposition is known by the things or situations it is true of. Typically, the signs of properties and propositions are called "terms" and "sentences", respectively.
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===6.20. Three Views of Systems===
===6.20. Three Views of Systems===
