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==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(\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.

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>\text{A}\!</math> and <math>\text{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; 8. &nbsp; The notion of computation that makes sense in this setting is one of a process that replaces an arbitrary sign with a better sign of the same object.  In other words, computation is an interpretive process that improves the indications of intentions.  To deal with computational processes it is necessary to extend the pragmatic theory of signs in a couple of new but coordinated directions.  To the basic conception of a sign relation is added a notion of progress, which implies a notion of process together with a notion of quality.

&sect; 8. &nbsp; The notion of computation that makes sense in this setting is one of a process that replaces an arbitrary sign with a better sign of the same object.  In other words, computation is an interpretive process that improves the indications of intentions.  To deal with computational processes it is necessary to extend the pragmatic theory of signs in a couple of new but coordinated directions.  To the basic conception of a sign relation is added a notion of progress, which implies a notion of process together with a notion of quality.

&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>\text{A}\!</math> and <math>\text{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(\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; 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.

A logical calculus cannot initiate reflection on a text, but it can help to support and maintain it.  The raw ability to perceive selected features of an ongoing text and the basic language of primitive terms, that allow one to mark the presence and note the passing of these features, have to be supplied from outside the calculus at the outset of its calculations.  In the present text, the means to support critical reflection on its own POV and others are implemented in the form of a propositional calculus.  Given the raw ability of a perceptive interpreter to form glosses on the text and to reflect on the contents of its current POV, a logical calculus can serve to augment the text and assist its critique by catalyzing the consideration of alternative POVs and facilitating reasoning about the wider implications of any particular POV.

A logical calculus cannot initiate reflection on a text, but it can help to support and maintain it.  The raw ability to perceive selected features of an ongoing text and the basic language of primitive terms, that allow one to mark the presence and note the passing of these features, have to be supplied from outside the calculus at the outset of its calculations.  In the present text, the means to support critical reflection on its own POV and others are implemented in the form of a propositional calculus.  Given the raw ability of a perceptive interpreter to form glosses on the text and to reflect on the contents of its current POV, a logical calculus can serve to augment the text and assist its critique by catalyzing the consideration of alternative POVs and facilitating reasoning about the wider implications of any particular POV.

The discussion so far has dwelt at length on a particular scene, returning periodically to the fragmentary but concrete situation of a dialogue between <math>A</math> and <math>B,</math> poring over the formal setting and teasing out the casual surroundings of a circumscribed pair of sign relations.  If the larger inquiry into inquiry is ever to lift itself off from these concrete and isolated grounds, then there is need for a way to extract the lessons of this exercise for reuse on other occasions.  If items of knowledge with enduring value are to be found in this arena, then they ought to be capable of application to broader areas of interest and to richer domains of inquiry, and this demands ways to test their tentative findings in analogous and alternative situations of a more significant stripe.  One way to do this is to identify properties and details of the selected examples that can be varied within the bounds of a common theme and treated as parameters whose momentary values convey the appearance of complete individuality to each particular case.

The discussion so far has dwelt at length on a particular scene, returning periodically to the fragmentary but concrete situation of a dialogue between <math>\text{A}</math> and <math>\text{B},</math> poring over the formal setting and teasing out the casual surroundings of a circumscribed pair of sign relations.  If the larger inquiry into inquiry is ever to lift itself off from these concrete and isolated grounds, then there is need for a way to extract the lessons of this exercise for reuse on other occasions.  If items of knowledge with enduring value are to be found in this arena, then they ought to be capable of application to broader areas of interest and to richer domains of inquiry, and this demands ways to test their tentative findings in analogous and alternative situations of a more significant stripe.  One way to do this is to identify properties and details of the selected examples that can be varied within the bounds of a common theme and treated as parameters whose momentary values convey the appearance of complete individuality to each particular case.

Typically, a movement from reduced examples to realistic exercises takes a definite but gradual progression of steps, moving forward through the paces of abstraction, generalization, transformation, and re application.  The prospects of success in these stages of development are associated with the introduction of certain formal devices.  Principal among these are the explicit recognition of sets of ''parameters'' and their expression in terms of lists of ''variables''.

Typically, a movement from reduced examples to realistic exercises takes a definite but gradual progression of steps, moving forward through the paces of abstraction, generalization, transformation, and re application.  The prospects of success in these stages of development are associated with the introduction of certain formal devices.  Principal among these are the explicit recognition of sets of ''parameters'' and their expression in terms of lists of ''variables''.

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.

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(\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.

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>\text{A}\!</math> and <math>\text{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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