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Because POSRs span the spectrum from the moderately straightforward to the deliberately misleading, there is a need for ways to tell them apart, at least, before pursuing their consequences too far.  Of course, if one cannot rest without having all computable functions at one's command, then no program can tell all the good and bad programs apart.  But if one can be satisfied with a somewhat more modest domain, then there is hope for a way, an experimental, fallible, and incremental way, but a way nonetheless, that eventually leads one to know the good and ultimately keeps one away from the bad.

Because POSRs span the spectrum from the moderately straightforward to the deliberately misleading, there is a need for ways to tell them apart, at least, before pursuing their consequences too far.  Of course, if one cannot rest without having all computable functions at one's command, then no program can tell all the good and bad programs apart.  But if one can be satisfied with a somewhat more modest domain, then there is hope for a way, an experimental, fallible, and incremental way, but a way nonetheless, that eventually leads one to know the good and ultimately keeps one away from the bad.

When it comes to their propriety, POSRs are found on empirical grounds to fall into two varieties:  the "exculpable" and the "indictable" kinds.  Thus, it is reasonable to attempt an empirical distinction, proposing to let experience mark each POSR as an "excusable self-reference" (ESR) or an "improper self reference" (ISR), as the case may be.  But empirical grounds can be a hard basis to fall back on, since a recourse to actual experience with POSRs can risk an agent's participation in pretended sign relations and promissory representations that amount in the end to nothing more than forms of interpretive futility.  Therefore, one seeks an arrangement of methods in general or an ordering of options in these special cases that makes the empirical trial a court of last resort and that avoids resorting to the actual experience of interpretation as a routine matter of course.

When it comes to their propriety, POSRs are found on empirical grounds to fall into two varieties:  the "exculpable" and the "indictable" kinds.  Thus, it is reasonable to attempt an empirical distinction, proposing to let experience mark each POSR as an "excusable self-reference" (ESR) or an "improper self-reference" (ISR), as the case may be.  But empirical grounds can be a hard basis to fall back on, since a recourse to actual experience with POSRs can risk an agent's participation in pretended sign relations and promissory representations that amount in the end to nothing more than forms of interpretive futility.  Therefore, one seeks an arrangement of methods in general or an ordering of options in these special cases that makes the empirical trial a court of last resort and that avoids resorting to the actual experience of interpretation as a routine matter of course.

First, I recognize an "empirical distinction" that seems to exist between the less problematic and the more problematic varieties of self reference, allowing POSRs to be sorted according to the consequential features that they have in actual experience.  There are the "good" sorts, those cleared up to the limits of accumulated experience as innocuous usages and even as probable utilities, and then there are the "bad" sorts, those marked by hard experience as definitely problematic.

First, I recognize an "empirical distinction" that seems to exist between the less problematic and the more problematic varieties of self-reference, allowing POSRs to be sorted according to the consequential features that they have in actual experience.  There are the "good" sorts, those cleared up to the limits of accumulated experience as innocuous usages and even as probable utilities, and then there are the "bad" sorts, those marked by hard experience as definitely problematic.

Next, I search for an "intuitive distinction" that can be supposed to exist between the good and the bad sorts of POSRs, invoking a formal character or computable predicate of a POSR whose prior inspection can provide interpreters with a definitive indication or a decisive piece of information as to whether a POSR is good or bad, without forcing them to undergo the consequences of its actual use.

Next, I search for an "intuitive distinction" that can be supposed to exist between the good and the bad sorts of POSRs, invoking a formal character or computable predicate of a POSR whose prior inspection can provide interpreters with a definitive indication or a decisive piece of information as to whether a POSR is good or bad, without forcing them to undergo the consequences of its actual use.

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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.

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 POSRs can be discussed in greater detail as follows:

These categories of POSRs 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.

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 POSRs that are safe and effective to use in thought and communication from the POSRs that can be hazardous to the health of inquiry.

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 POSRs that are safe and effective to use in thought and communication from the POSRs that can be hazardous to the health of inquiry.

More explicitly, the distinction between ESRs and ISRs is intended to capture the differences that exist between the following cases:

More explicitly, the distinction between ESRs and ISRs is intended to capture the differences that exist between the following cases:

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.

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.

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?

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 POSRs can foretell the categories of experience with POSRs.

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 POSRs can foretell the categories of experience with POSRs.

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.

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

a. ISR

3. Next, I consider a class of POSRs that turns up in group theory./  The next class of POSRs I want to discuss is one that arises in group theory.

3. Next, I consider a class of POSRs that turns up in group theory./  The next class of POSRs 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.

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.

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 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.

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:

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>}.

xi  =  {<x1, x1*xi>, ..., <xn, xn*xi>}.