Changes

==6. Reflective Interpretive Frameworks==

==6. Reflective Interpretive Frameworks==

We continue the discussion of formalization in terms of concrete examples and detail the construction of a ''reflective interpretive framework'' (RIF).  This is a special type of sign-theoretic setting, illustrated in the present case by building on the sign relations <math>L(A)\!</math> and <math>L(B),\!</math> but intended more generally to form a fully-developed environment of objective and interpretive resources, in the likes of which an &ldquo;inquiry into inquiry&rdquo; can reasonably be expected to find its home.  We begin by presenting an outline of the developments ahead, working through the motivation, construction, and application of a RIF that is broad enough to mediate the dialogue of the interpreters <math>A\!</math> and <math>B.\!</math>  The first fifteen Sections (&sect;&sect;&nbsp;1&ndash;15) deal with a selection of preliminary topics and techniques that are involved in approaching the construction of a RIF.  The topics of these sections are described in greater detail below.

We continue the discussion of formalization in terms of concrete examples and detail the construction of a ''reflective interpretive framework'' (RIF).  This is a special type of sign-theoretic setting, illustrated in the present case by building on the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> but intended more generally to form a fully-developed environment of objective and interpretive resources, in the likes of which an &ldquo;inquiry into inquiry&rdquo; can reasonably be expected to find its home.  We begin by presenting an outline of the developments ahead, working through the motivation, construction, and application of a RIF that is broad enough to mediate the dialogue of the interpreters <math>A\!</math> and <math>B.\!</math>  The first fifteen Sections (&sect;&sect;&nbsp;1&ndash;15) deal with a selection of preliminary topics and techniques that are involved in approaching the construction of a RIF.  The topics of these sections are described in greater detail below.

The first section (&sect;&nbsp;1) takes up the phenomenology of reflection.  The next three sections (&sect;&sect;&nbsp;2&ndash;4) are allotted to surveying the site of the planned construction, presenting it from three different points of view.  An introductory discussion (&sect;&nbsp;2) presents the main ideas that lead up to the genesis of a RIF.  These ideas are treated at first acquaintance in an informal manner, located within a broader cultural context, and put in relation to the ways that intelligent agents can come to develop characteristic belief systems and communal perspectives on the world.  The next section (&sect;&nbsp;3) points out a specialized mechanism that serves to make inobvious types of observation of a reflective character.  The last section (&sect;&nbsp;4) takes steps to formalize the concepts of a ''point of view'' (POV) and a ''point of development'' (POD).  These ideas characterize the outlooks, perspectives, world views, and other systems of belief, knowledge, or opinion that are employed by agents of inquiry, with especial regard to the ways that these outlooks develop over time.

The first section (&sect;&nbsp;1) takes up the phenomenology of reflection.  The next three sections (&sect;&sect;&nbsp;2&ndash;4) are allotted to surveying the site of the planned construction, presenting it from three different points of view.  An introductory discussion (&sect;&nbsp;2) presents the main ideas that lead up to the genesis of a RIF.  These ideas are treated at first acquaintance in an informal manner, located within a broader cultural context, and put in relation to the ways that intelligent agents can come to develop characteristic belief systems and communal perspectives on the world.  The next section (&sect;&nbsp;3) points out a specialized mechanism that serves to make inobvious types of observation of a reflective character.  The last section (&sect;&nbsp;4) takes steps to formalize the concepts of a ''point of view'' (POV) and a ''point of development'' (POD).  These ideas characterize the outlooks, perspectives, world views, and other systems of belief, knowledge, or opinion that are employed by agents of inquiry, with especial regard to the ways that these outlooks develop over time.

&sect; 9. &nbsp; This section introduces ''higher order'' sign relations, which are used to formalize the process of reflection on interpretation.  The discussion is approaching a point where multiple levels of signs are becoming necessary, mainly for referring to previous levels of signs as the objects of an extended sign relation, and thereby enabling a process of reflection on interpretive conduct.  To begin dealing with this issue, I take advantage of a second look at <math>A\!</math> and <math>B\!</math> to introduce the use of ''raised angle brackets'' <math>({}^{\langle}~{}^{\rangle}),</math> also called ''supercilia'' or ''arches'', as quotation marks.  Ordinary quotation marks <math>({}^{\backprime\backprime}~{}^{\prime\prime})</math> have the disadvantage, for formal purposes, of being used informally for many different tasks.  To get around this obstacle, I use the ''arch'' operator to formalize one specific function of quotation marks in a computational context, namely, to create distinctive names for syntactic expressions, or what amounts to the same thing, to signify the generation of their gödel numbers.

&sect; 9. &nbsp; This section introduces ''higher order'' sign relations, which are used to formalize the process of reflection on interpretation.  The discussion is approaching a point where multiple levels of signs are becoming necessary, mainly for referring to previous levels of signs as the objects of an extended sign relation, and thereby enabling a process of reflection on interpretive conduct.  To begin dealing with this issue, I take advantage of a second look at <math>A\!</math> and <math>B\!</math> to introduce the use of ''raised angle brackets'' <math>({}^{\langle}~{}^{\rangle}),</math> also called ''supercilia'' or ''arches'', as quotation marks.  Ordinary quotation marks <math>({}^{\backprime\backprime}~{}^{\prime\prime})</math> have the disadvantage, for formal purposes, of being used informally for many different tasks.  To get around this obstacle, I use the ''arch'' operator to formalize one specific function of quotation marks in a computational context, namely, to create distinctive names for syntactic expressions, or what amounts to the same thing, to signify the generation of their gödel numbers.

&sect; 10. &nbsp; Returning to the sign relations <math>L(A)\!</math> and <math>L(B),\!</math> various kinds of higher order signs are exemplified by considering a series of higher order sign relations based on these two examples.

&sect; 10. &nbsp; Returning to the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> various kinds of higher order signs are exemplified by considering a series of higher order sign relations based on these two examples.

&sect; 11. &nbsp; In this section the tools that come with the theory of higher order sign relations are applied to an illustrative exercise, roughing out the shape of a complex form of interpreter.

&sect; 11. &nbsp; In this section the tools that come with the theory of higher order sign relations are applied to an illustrative exercise, roughing out the shape of a complex form of interpreter.

The next four sections (&sect;&sect;&nbsp;22&ndash;25) give examples of ERs and IRs, indicate the importance of forming a computational bridge between them, and discuss the conceptual and technical obstacles that will have to be faced in doing so.

The next four sections (&sect;&sect;&nbsp;22&ndash;25) give examples of ERs and IRs, indicate the importance of forming a computational bridge between them, and discuss the conceptual and technical obstacles that will have to be faced in doing so.

&sect; 22. &nbsp; For ease of reference, this section collects previous materials that are relevant to discussing the ERs of the sign relations <math>L(A)\!</math> and <math>L(B),\!</math> and explicitly details their dyadic projections.

&sect; 22. &nbsp; For ease of reference, this section collects previous materials that are relevant to discussing the ERs of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> and explicitly details their dyadic projections.

&sect; 23. &nbsp; This section discusses a number of general issues that are associated with the IRs of sign relations.  Because of the great degree of freedom there is in selecting among the potentially relevant properties of any real object, especially when the context of relevance to the selection is not known in advance, there are many different ways, perhaps an indefinite multitude of ways, to represent the sign relations <math>L(A)\!</math> and <math>L(B)\!</math> in terms of salient properties of their elementary constituents.  In this connection, the next two sections explore a representative sample of these possibilities, and illustrate several different styles of approach that can be used in their presentation.

&sect; 23. &nbsp; This section discusses a number of general issues that are associated with the IRs of sign relations.  Because of the great degree of freedom there is in selecting among the potentially relevant properties of any real object, especially when the context of relevance to the selection is not known in advance, there are many different ways, perhaps an indefinite multitude of ways, to represent the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> in terms of salient properties of their elementary constituents.  In this connection, the next two sections explore a representative sample of these possibilities, and illustrate several different styles of approach that can be used in their presentation.

&sect; 24. &nbsp; A transitional case between ERs and IRs of sign relations is found in the concept of a ''literal intensional representation'' (LIR).

&sect; 24. &nbsp; A transitional case between ERs and IRs of sign relations is found in the concept of a ''literal intensional representation'' (LIR).

&sect; 25. &nbsp; A fully fledged IR is one that accomplishes some measure of analytic work, bringing to the point of salient notice a selected array of implicit and otherwise hidden features of its object.  This section presents a variety of these ''analytic intensional representations'' (AIRs) for the sign relations <math>L(A)\!</math> and <math>L(B).\!</math>

&sect; 25. &nbsp; A fully fledged IR is one that accomplishes some measure of analytic work, bringing to the point of salient notice a selected array of implicit and otherwise hidden features of its object.  This section presents a variety of these ''analytic intensional representations'' (AIRs) for the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math>

'''Note for future reference.'''  The problem so naturally encountered here, due to the embarrassment of riches that presents itself in choosing a suitable IR, and tracing its origin to the wealth of properties that any real object typically has, is a precursor to one of the deepest issues in the pragmatic theory of inquiry:  ''the problem of abductive reasoning''.  This topic will be discussed at several later stages of this investigation, where it typically involves the problem of choosing, among the manifold aspects of an objective phenomenon or a problematic objective, only the features that are:  (1) relevant to explaining a present fact, or (2) pertinent to achieving a current purpose.

'''Note for future reference.'''  The problem so naturally encountered here, due to the embarrassment of riches that presents itself in choosing a suitable IR, and tracing its origin to the wealth of properties that any real object typically has, is a precursor to one of the deepest issues in the pragmatic theory of inquiry:  ''the problem of abductive reasoning''.  This topic will be discussed at several later stages of this investigation, where it typically involves the problem of choosing, among the manifold aspects of an objective phenomenon or a problematic objective, only the features that are:  (1) relevant to explaining a present fact, or (2) pertinent to achieving a current purpose.

&sect; 34. &nbsp; First, I consider a number of set-theoretic operations that can be utilized in discussing these ''identification'', ''reducibility'', or ''reconstruction'' questions.  Once a level of general discussion has been surveyed enough to make a start, these tools can be specialized and applied to concrete examples in the realm of sign relations and also applied in the neighborhood of closely associated triadic relations.

&sect; 34. &nbsp; First, I consider a number of set-theoretic operations that can be utilized in discussing these ''identification'', ''reducibility'', or ''reconstruction'' questions.  Once a level of general discussion has been surveyed enough to make a start, these tools can be specialized and applied to concrete examples in the realm of sign relations and also applied in the neighborhood of closely associated triadic relations.

&sect; 35. &nbsp; This section considers the positive case of reducibility, presenting examples of triadic relations that can be reconstructed from their dyadic projections.  In fact, it happens that the sign relations <math>L(A)\!</math> and <math>L(B)\!</math> fall into this category of dyadically reducible triadic relations.

&sect; 35. &nbsp; This section considers the positive case of reducibility, presenting examples of triadic relations that can be reconstructed from their dyadic projections.  In fact, it happens that the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> fall into this category of dyadically reducible triadic relations.

&sect; 36. &nbsp; This section considers the negative case of reducibility, presenting examples of ''irreducibly triadic relations'', or triadic relations that cannot be reconstructed from their lower dimensional projections or ''faces''.

&sect; 36. &nbsp; This section considers the negative case of reducibility, presenting examples of ''irreducibly triadic relations'', or triadic relations that cannot be reconstructed from their lower dimensional projections or ''faces''.

As I understand them, variables are a class of beneficially ambiguous or usefully equivocal signs.  In effect, variables are just signs, but signs possessed of a more adaptive constitution or affected by a more flexible interpretation than signs of the usual, more constant variety.  These forms of employment turn variables into a class of reusable signs, converting them into sustainable resources for meaning that can be used in a plurality of ways and deployed to articulate different choices at different times from among the available points of thematic variation.

As I understand them, variables are a class of beneficially ambiguous or usefully equivocal signs.  In effect, variables are just signs, but signs possessed of a more adaptive constitution or affected by a more flexible interpretation than signs of the usual, more constant variety.  These forms of employment turn variables into a class of reusable signs, converting them into sustainable resources for meaning that can be used in a plurality of ways and deployed to articulate different choices at different times from among the available points of thematic variation.

The next major task of this discussion, while continuing to take its bearings from examples as concrete as <math>L(A)\!</math> and <math>L(B),\!</math> is to develop systematic methods for divining the bearing of such isolated examples on issues of real concern.  This involves two stages:

The next major task of this discussion, while continuing to take its bearings from examples as concrete as <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> is to develop systematic methods for divining the bearing of such isolated examples on issues of real concern.  This involves two stages:

# One needs to detect the invariant features of the currently known examples, in other words, the dimensions along which their values are, knowingly or unknowingly, held to be constant.

# One needs to detect the invariant features of the currently known examples, in other words, the dimensions along which their values are, knowingly or unknowingly, held to be constant.

# One needs to try varying the features that are presently held to be constant by imagining new examples that are able to realize alternative features.

# One needs to try varying the features that are presently held to be constant by imagining new examples that are able to realize alternative features.

Viewed from a standpoint in the pragmatic theory of signs, computation is a process that trades a sign for a ''better'' sign of the same object.  Thus, a computation is an interpretive process whose passage from sign to interpretant sign ''improves'' the indication of the object in some way.  The dimensions along which signs can be compared are various, usually being described as measures of ''clarity'', ''distinctness'', or ''usability'' of the information conveyed, but all such measures are ''interpretive'' in character.  That is, the sense in which a computation improves its signs is relative to the purpose actualized in a given moment of interpretation.

Viewed from a standpoint in the pragmatic theory of signs, computation is a process that trades a sign for a ''better'' sign of the same object.  Thus, a computation is an interpretive process whose passage from sign to interpretant sign ''improves'' the indication of the object in some way.  The dimensions along which signs can be compared are various, usually being described as measures of ''clarity'', ''distinctness'', or ''usability'' of the information conveyed, but all such measures are ''interpretive'' in character.  That is, the sense in which a computation improves its signs is relative to the purpose actualized in a given moment of interpretation.

It is probably worth emphasizing this point.  There need be nothing intrinsic to a sign itself that makes it better or worse than another.  This is apparent from examples as simple as the sign relations <math>L(A)\!</math> and <math>L(B),\!</math> where nothing intrinsic to the grammatical categories of signs makes either the nouns or the pronouns essentially better than the others in every situation.  In general, a preference defined on signs need reflect nothing more than the purpose or caprice of a particular interpreter at a given moment of interpretation.  Of course, one is usually interested in cases where a measure of aptness, quality, or utility can be justified on more stable and substantial grounds.

It is probably worth emphasizing this point.  There need be nothing intrinsic to a sign itself that makes it better or worse than another.  This is apparent from examples as simple as the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> where nothing intrinsic to the grammatical categories of signs makes either the nouns or the pronouns essentially better than the others in every situation.  In general, a preference defined on signs need reflect nothing more than the purpose or caprice of a particular interpreter at a given moment of interpretation.  Of course, one is usually interested in cases where a measure of aptness, quality, or utility can be justified on more stable and substantial grounds.

Computation adds to the bare conception of a sign relation a notion of progress, which implies in turn:  (1) the dynamic notion of a temporal process taking place between signs, and (2) the evaluative notion of a utility measure rating each sign's relative virtue as a sign.

Computation adds to the bare conception of a sign relation a notion of progress, which implies in turn:  (1) the dynamic notion of a temporal process taking place between signs, and (2) the evaluative notion of a utility measure rating each sign's relative virtue as a sign.

===6.10. Higher Order Sign Relations : Examples===

===6.10. Higher Order Sign Relations : Examples===

In considering the higher order sign relations that stem from the examples <math>L(A)\!</math> and <math>L(B),\!</math> it appears that annexing the first level of HA signs is tantamount to adjoining or instituting an auxiliary interpretive framework, one that has the semantic equations shown in Table&nbsp;36.

In considering the higher order sign relations that stem from the examples <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> it appears that annexing the first level of HA signs is tantamount to adjoining or instituting an auxiliary interpretive framework, one that has the semantic equations shown in Table&nbsp;36.

<br>

<br>

<br>

<br>

The rest of this section discusses the relationship between higher order signs and a concept called the ''reflective extension'' of a sign relation.  Reflective extensions will be subjected to a more detailed study in a later part of this work.  For now, just to see how the process works, the sign relations <math>L(A)\!</math> and <math>L(B)\!</math> are taken as starting points to illustrate the more common forms of reflective development.

The rest of this section discusses the relationship between higher order signs and a concept called the ''reflective extension'' of a sign relation.  Reflective extensions will be subjected to a more detailed study in a later part of this work.  For now, just to see how the process works, the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> are taken as starting points to illustrate the more common forms of reflective development.

In the most typical scenario, higher order sign relations come into being as the reflective extensions of simpler, possibly unreflective sign relations.  Conversely, the incorporation of higher order signs within a sign relation leads to a larger sign relation that constitutes one of its reflective extensions.  In general, there are many different ways that a reflective extension can get started and many different structures that can result.

In the most typical scenario, higher order sign relations come into being as the reflective extensions of simpler, possibly unreflective sign relations.  Conversely, the incorporation of higher order signs within a sign relation leads to a larger sign relation that constitutes one of its reflective extensions.  In general, there are many different ways that a reflective extension can get started and many different structures that can result.

In the initial slice of semantics presented for the sign relations <math>L(A)\!</math> and <math>L(B),\!</math> the sign domain <math>S\!</math> is identical to the interpretant domain <math>I,\!</math> and this set is disjoint from the object domain <math>O.\!</math>  In order for this discussion to develop more interesting examples of sign relations these constraints will need to be generalized.  As a start in this direction, one can preserve the identification of the syntactic domain as <math>S = I\!</math> and contemplate ways of varying the pattern of intersection between <math>S\!</math> and <math>O.\!</math>

In the initial slice of semantics presented for the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> the sign domain <math>S\!</math> is identical to the interpretant domain <math>I,\!</math> and this set is disjoint from the object domain <math>O.\!</math>  In order for this discussion to develop more interesting examples of sign relations these constraints will need to be generalized.  As a start in this direction, one can preserve the identification of the syntactic domain as <math>S = I\!</math> and contemplate ways of varying the pattern of intersection between <math>S\!</math> and <math>O.\!</math>

One direction of generalization is motivated by the desire to give interpreters a measure of &ldquo;reflective capacity&rdquo;.  This is a property of sign relations that can be associated with the overlap of <math>O\!</math> and <math>S\!</math> and gauged by the extent to which <math>S\!</math> is contained in <math>O.\!</math>  In intuitive terms, interpreters are said to have a reflective capacity to the extent that they can refer to their own signs independently of their denotations.  An interpretive system with a sufficient amount of reflective capacity can support the maintenance and manipulation of textual objects like expressions and programs without necessarily having to evaluate the expressions or execute the programs.

One direction of generalization is motivated by the desire to give interpreters a measure of &ldquo;reflective capacity&rdquo;.  This is a property of sign relations that can be associated with the overlap of <math>O\!</math> and <math>S\!</math> and gauged by the extent to which <math>S\!</math> is contained in <math>O.\!</math>  In intuitive terms, interpreters are said to have a reflective capacity to the extent that they can refer to their own signs independently of their denotations.  An interpretive system with a sufficient amount of reflective capacity can support the maintenance and manipulation of textual objects like expressions and programs without necessarily having to evaluate the expressions or execute the programs.

In ordinary discourse HA signs are usually generated by means of linguistic devices for quoting pieces of text.  In computational frameworks these quoting mechanisms are implemented as functions that map syntactic arguments into numerical or syntactic values.  A quoting function, given a sign or expression as its single argument, needs to accomplish two things:  first, to defer the reference of that sign, in other words, to inhibit, delay, or prevent the evaluation of its argument expression, and then, to exhibit or produce another sign whose object is precisely that argument expression.

In ordinary discourse HA signs are usually generated by means of linguistic devices for quoting pieces of text.  In computational frameworks these quoting mechanisms are implemented as functions that map syntactic arguments into numerical or syntactic values.  A quoting function, given a sign or expression as its single argument, needs to accomplish two things:  first, to defer the reference of that sign, in other words, to inhibit, delay, or prevent the evaluation of its argument expression, and then, to exhibit or produce another sign whose object is precisely that argument expression.

The rest of this section considers the development of sign relations that have moderate capacities to reference their own signs as objects.  In each case, these extensions are assumed to begin with sign relations like <math>L(A)\!</math> and <math>L(B)\!</math> that have disjoint sets of objects and signs and thus have no reflective capacity at the outset.  The status of <math>L(A)\!</math> and <math>L(B)\!</math> as the reflective origins of the associated reflective developments is recalled by saying that <math>L(A)\!</math> and <math>L(B)\!</math> themselves are the ''zeroth order reflective extensions'' of <math>L(A)\!</math> and <math>L(B),\!</math> in symbols, <math>L(A) = \operatorname{Ref}^0 L(A)\!</math> and <math>L(B) = \operatorname{Ref}^0 L(B).\!</math>

The rest of this section considers the development of sign relations that have moderate capacities to reference their own signs as objects.  In each case, these extensions are assumed to begin with sign relations like <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> that have disjoint sets of objects and signs and thus have no reflective capacity at the outset.  The status of <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> as the reflective origins of the associated reflective developments is recalled by saying that <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> themselves are the ''zeroth order reflective extensions'' of <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> in symbols, <math>L(\text{A}) = \operatorname{Ref}^0 L(\text{A})\!</math> and <math>L(\text{B}) = \operatorname{Ref}^0 L(\text{B}).\!</math>

The next set of Tables illustrates a few the most common ways that sign relations can begin to develop reflective extensions.  For ease of reference, Tables&nbsp;40 and 41 repeat the contents of Tables&nbsp;1 and 2, respectively, merely replacing ordinary quotes with arch quotes.

The next set of Tables illustrates a few the most common ways that sign relations can begin to develop reflective extensions.  For ease of reference, Tables&nbsp;40 and 41 repeat the contents of Tables&nbsp;1 and 2, respectively, merely replacing ordinary quotes with arch quotes.

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

|+ style="height:30px" | <math>\text{Table 40.} ~~ \text{Reflective Origin} ~ \operatorname{Ref}^0 L(A)\!</math>

|+ style="height:30px" | <math>\text{Table 40.} ~~ \text{Reflective Origin} ~ \operatorname{Ref}^0 L(\text{A})\!</math>

|- style="height:40px; background:#f0f0ff"

|- style="height:40px; background:#f0f0ff"

| <math>\text{Object}\!</math>

| <math>\text{Object}\!</math>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Reflective Origin} ~ \operatorname{Ref}^0 L(B)\!</math>

|+ style="height:30px" | <math>\text{Table 41.} ~~ \text{Reflective Origin} ~ \operatorname{Ref}^0 L(\text{B})\!</math>

|- style="height:40px; background:#f0f0ff"

|- style="height:40px; background:#f0f0ff"

| <math>\text{Object}\!</math>

| <math>\text{Object}\!</math>

<br>

<br>

Tables&nbsp;42 and 43 show one way that the sign relations <math>L(A)\!</math> and <math>L(B)\!</math> can be extended in a reflective sense through the use of quotational devices, yielding the ''first order reflective extensions'', <math>\operatorname{Ref}^1 L(A)\!</math> and <math>\operatorname{Ref}^1 L(B).\!</math>  These extensions add one layer of HA signs and their objects to the sign relations <math>L(A)\!</math> and <math>L(B),\!</math> respectively.  The new triples specify that, for each <math>{}^{\langle} x {}^{\rangle}\!</math> in the set <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \},\!</math> the HA sign of the form <math>{}^{\langle\langle} x {}^{\rangle\rangle}\!</math> connotes itself while denoting <math>{}^{\langle} x {}^{\rangle}.\!</math>

Tables&nbsp;42 and 43 show one way that the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> can be extended in a reflective sense through the use of quotational devices, yielding the ''first order reflective extensions'', <math>\operatorname{Ref}^1 L(\text{A})\!</math> and <math>\operatorname{Ref}^1 L(\text{B}).\!</math>  These extensions add one layer of HA signs and their objects to the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> respectively.  The new triples specify that, for each <math>{}^{\langle} x {}^{\rangle}\!</math> in the set <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \},\!</math> the HA sign of the form <math>{}^{\langle\langle} x {}^{\rangle\rangle}\!</math> connotes itself while denoting <math>{}^{\langle} x {}^{\rangle}.\!</math>

Notice that the semantic equivalences of nouns and pronouns referring to each interpreter do not extend to semantic equivalences of their higher order signs, exactly as demanded by the literal character of quotations.  Also notice that the reflective extensions of the sign relations <math>L(A)\!</math> and <math>L(B)\!</math> coincide in their reflective parts, since exactly the same triples were added to each set.

Notice that the semantic equivalences of nouns and pronouns referring to each interpreter do not extend to semantic equivalences of their higher order signs, exactly as demanded by the literal character of quotations.  Also notice that the reflective extensions of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B})\!</math> coincide in their reflective parts, since exactly the same triples were added to each set.

<br>

<br>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Higher Ascent Sign Relation} ~ \operatorname{Ref}^1 L(A)\!</math>

|+ style="height:30px" | <math>\text{Table 42.} ~~ \text{Higher Ascent Sign Relation} ~ \operatorname{Ref}^1 L(\text{A})\!</math>

|- style="height:40px; background:#f0f0ff"

|- style="height:40px; background:#f0f0ff"

| <math>\text{Object}\!</math>

| <math>\text{Object}\!</math>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Higher Ascent Sign Relation} ~ \operatorname{Ref}^1 L(B)\!</math>

|+ style="height:30px" | <math>\text{Table 43.} ~~ \text{Higher Ascent Sign Relation} ~ \operatorname{Ref}^1 L(\text{B})\!</math>

|- style="height:40px; background:#f0f0ff"

|- style="height:40px; background:#f0f0ff"

| <math>\text{Object}\!</math>

| <math>\text{Object}\!</math>

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There are many ways to extend sign relations in an effort to develop their reflective capacities.  The implicit goal of a reflective project is to reach a condition of ''reflective closure'', a configuration satisfying the inclusion <math>S \subseteq O,\!</math> where every sign is an object.  It is important to note that not every process of reflective extension can achieve a reflective closure in a finite sign relation.  This can only happen if there are additional equivalence relations that keep the effective orders of signs within finite bounds.  As long as there are higher order signs that remain distinct from all lower order signs, the sign relation driven by a reflective process is forced to keep expanding.  In particular, the process that is ''freely'' suggested by the formation of <math>\operatorname{Ref}^1 L(A)\!</math> and <math>\operatorname{Ref}^1 L(B)\!</math> cannot reach closure if it continues as indicated, without further constraints.

There are many ways to extend sign relations in an effort to develop their reflective capacities.  The implicit goal of a reflective project is to reach a condition of ''reflective closure'', a configuration satisfying the inclusion <math>S \subseteq O,\!</math> where every sign is an object.  It is important to note that not every process of reflective extension can achieve a reflective closure in a finite sign relation.  This can only happen if there are additional equivalence relations that keep the effective orders of signs within finite bounds.  As long as there are higher order signs that remain distinct from all lower order signs, the sign relation driven by a reflective process is forced to keep expanding.  In particular, the process that is ''freely'' suggested by the formation of <math>\operatorname{Ref}^1 L(\text{A})\!</math> and <math>\operatorname{Ref}^1 L(\text{B})\!</math> cannot reach closure if it continues as indicated, without further constraints.

Tables&nbsp;44 and 45 present ''higher import extensions'' of <math>L(A)\!</math> and <math>L(B),\!</math> respectively.  These are just higher order sign relations that add selections of higher import signs and their objects to the underlying set of triples in <math>L(A)\!</math> and <math>L(B).\!</math>  One way to understand these extensions is as follows.  The interpreters <math>A\!</math> and <math>B\!</math> each use nouns and pronouns just as before, except that the nouns are given additional denotations that refer to the interpretive conduct of the interpreter named.  In this form of development, using a noun as a canonical form that refers indifferently to all the <math>(o, s, i)\!</math> triples of a sign relation is a pragmatic way that a sign relation can refer to itself and to other sign relations.

Tables&nbsp;44 and 45 present ''higher import extensions'' of <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> respectively.  These are just higher order sign relations that add selections of higher import signs and their objects to the underlying set of triples in <math>L(\text{A})\!</math> and <math>L(\text{B}).\!</math>  One way to understand these extensions is as follows.  The interpreters <math>A\!</math> and <math>B\!</math> each use nouns and pronouns just as before, except that the nouns are given additional denotations that refer to the interpretive conduct of the interpreter named.  In this form of development, using a noun as a canonical form that refers indifferently to all the <math>(o, s, i)\!</math> triples of a sign relation is a pragmatic way that a sign relation can refer to itself and to other sign relations.

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{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

|+ style="height:30px" | <math>\text{Table 44.} ~~ \text{Higher Import Sign Relation} ~ \operatorname{HI}^1 L(A)\!</math>

|+ style="height:30px" | <math>\text{Table 44.} ~~ \text{Higher Import Sign Relation} ~ \operatorname{HI}^1 L(\text{A})\!</math>

|- style="height:40px; background:#f0f0ff"

|- style="height:40px; background:#f0f0ff"

| <math>\text{Object}\!</math>

| <math>\text{Object}\!</math>

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:80%"

|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Higher Import Sign Relation} ~ \operatorname{HI}^1 L(B)\!</math>

|+ style="height:30px" | <math>\text{Table 45.} ~~ \text{Higher Import Sign Relation} ~ \operatorname{HI}^1 L(\text{B})\!</math>

|- style="height:40px; background:#f0f0ff"

|- style="height:40px; background:#f0f0ff"

| <math>\text{Object}\!</math>

| <math>\text{Object}\!</math>

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Several important facts about the class of higher order sign relations in general are illustrated by these examples.  First, the notations appearing in the object columns of <math>\operatorname{HI}^1 L(A)\!</math> and <math>\operatorname{HI}^1 L(B)\!</math> are not the terms that these newly extended interpreters are depicted as using to describe their objects, but the kinds of language that you and I, or other external observers, would typically make available to distinguish them.  The sign relations <math>L(A)\!</math> and <math>L(B),\!</math> as extended by the transactions of <math>\operatorname{HI}^1 L(A)\!</math> and <math>\operatorname{HI}^1 L(B),\!</math> respectively, are still restricted to their original syntactic domain <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \}.\!</math>  This means that there need be nothing especially articulate about a HI sign relation just because it qualifies as higher order.  Indeed, the sign relations <math>\operatorname{HI}^1 L(A)\!</math> and <math>\operatorname{HI}^1 L(B)\!</math> are not very discriminating in their descriptions of the sign relations <math>L(A)\!</math> and <math>L(B),\!</math> referring to many different things under the very same signs that you and I and others would explicitly distinguish, especially in marking the distinction between an interpretive agent and any one of its individual transactions.

Several important facts about the class of higher order sign relations in general are illustrated by these examples.  First, the notations appearing in the object columns of <math>\operatorname{HI}^1 L(\text{A})\!</math> and <math>\operatorname{HI}^1 L(\text{B})\!</math> are not the terms that these newly extended interpreters are depicted as using to describe their objects, but the kinds of language that you and I, or other external observers, would typically make available to distinguish them.  The sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> as extended by the transactions of <math>\operatorname{HI}^1 L(\text{A})\!</math> and <math>\operatorname{HI}^1 L(\text{B}),\!</math> respectively, are still restricted to their original syntactic domain <math>\{ {}^{\langle} \text{A} {}^{\rangle}, {}^{\langle} \text{B} {}^{\rangle}, {}^{\langle} \text{i} {}^{\rangle}, {}^{\langle} \text{u} {}^{\rangle} \}.\!</math>  This means that there need be nothing especially articulate about a HI sign relation just because it qualifies as higher order.  Indeed, the sign relations <math>\operatorname{HI}^1 L(\text{A})\!</math> and <math>\operatorname{HI}^1 L(\text{B})\!</math> are not very discriminating in their descriptions of the sign relations <math>L(\text{A})\!</math> and <math>L(\text{B}),\!</math> referring to many different things under the very same signs that you and I and others would explicitly distinguish, especially in marking the distinction between an interpretive agent and any one of its individual transactions.

In practice, it does an interpreter little good to have the higher import signs for referring to triples of objects, signs, and interpretants if it does not also have the higher ascent signs for referring to each triple's syntactic portions.  Consequently, the higher order sign relations that one is likely to observe in practice are typically a mixed bag, having both higher ascent and higher import sections.  Moreover, the ambiguity involved in having signs that refer equivocally to simple world elements and also to complex structures formed from these ingredients would most likely be resolved by drawing additional information from context and fashioning more distinctive signs.

In practice, it does an interpreter little good to have the higher import signs for referring to triples of objects, signs, and interpretants if it does not also have the higher ascent signs for referring to each triple's syntactic portions.  Consequently, the higher order sign relations that one is likely to observe in practice are typically a mixed bag, having both higher ascent and higher import sections.  Moreover, the ambiguity involved in having signs that refer equivocally to simple world elements and also to complex structures formed from these ingredients would most likely be resolved by drawing additional information from context and fashioning more distinctive signs.