12,080
edits
Changes
MyWikiBiz, Author Your Legacy — Sunday April 28, 2024
Jump to navigationJump to searchDirectory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1 (view source)
Revision as of 13:40, 19 April 2012
, 13:40, 19 April 2012Force PNG rendering of LaTeX
Compositions of ''faculties'' are indicated by concatenating their names, posed in the sense that the right-indicated faculty applies to the left-indicated faculty, in the following form:
Compositions of ''faculties'' are indicated by concatenating their names, posed in the sense that the right-indicated faculty applies to the left-indicated faculty, in the following form:
:: <math>f \cdot g</math>
:: <math>f \cdot g\!</math>
A notation of the form
A notation of the form
The coset notation
The coset notation
:: <math>F \cdot G</math>
:: <math>F \cdot G\!</math>
indicates a class of ''faculties'' of the form
indicates a class of ''faculties'' of the form
:: <math>f \cdot g</math>,
:: <math>f \cdot g\!</math>,
with <math>f\!</math> in <math>F\!</math> and <math>g\!</math> in <math>G\!</math>.
with <math>f\!</math> in <math>F\!</math> and <math>g\!</math> in <math>G\!</math>.
Notations like
Notations like
:: <math>\{ ? \} ~,~ \{ ? , ? \} ~,~ \{ ? , ? , ?\} ~,~ \ldots</math>
:: <math>\{ ? \} ~,~ \{ ? , ? \} ~,~ \{ ? , ? , ?\} ~,~ \ldots\!</math>
serve as proxies for unknown components and indicate tentative analyses of faculties in question.
serve as proxies for unknown components and indicate tentative analyses of faculties in question.
If the faculty of inquiry is a coherent power, then it has an active or instrumental face, a passive or objective face, and a substantial body of connections between them.
If the faculty of inquiry is a coherent power, then it has an active or instrumental face, a passive or objective face, and a substantial body of connections between them.
:: <math>y = \{ ? \}</math>
:: <math>y = \{ ? \}\!</math>
In giving the current inquiry a reflexive cast, as inquiry into inquiry, I have brought inquiry face to face with itself, inditing it to apply its action in pursuing a knowledge of its passion.
In giving the current inquiry a reflexive cast, as inquiry into inquiry, I have brought inquiry face to face with itself, inditing it to apply its action in pursuing a knowledge of its passion.
:: <math>y_0 = y \cdot y = \{ ? \} \{ ? \}</math>
:: <math>y_0 = y \cdot y = \{ ? \} \{ ? \}\!</math>
If this juxtaposition of characters is to have a meaningful issue, then the fullness of its instrumental and objective aspects must have recourse to easier actions and simpler objects.
If this juxtaposition of characters is to have a meaningful issue, then the fullness of its instrumental and objective aspects must have recourse to easier actions and simpler objects.
:: <math>y >\!\!= \{ ? , ? \}</math>
:: <math>y >\!\!= \{ ? , ? \}\!</math>
Looking for an edge on each face of inquiry, as a plausible option for beginning to apply one to the other, I find what seems a likely pair. I begin with an aspect of instrumental inquiry that is easy to do, namely ''discussion'', along with an aspect of objective inquiry that is unavoidable to discuss, namely ''formalization''.
Looking for an edge on each face of inquiry, as a plausible option for beginning to apply one to the other, I find what seems a likely pair. I begin with an aspect of instrumental inquiry that is easy to do, namely ''discussion'', along with an aspect of objective inquiry that is unavoidable to discuss, namely ''formalization''.
:: <math>y >\!\!= \{ \operatorname{discussion} , \operatorname{formalization} \}</math>
:: <math>y >\!\!= \{ \operatorname{discussion} , \operatorname{formalization} \}\!</math>
In accord with this plan, the body of this section is devoted to a discussion of formalization.
In accord with this plan, the body of this section is devoted to a discussion of formalization.
:: <math>y_0 = y \cdot y >\!\!= \{ d , f \} \{ d , f \} >\!\!= \{ f \} \{ d \}</math>
:: <math>y_0 = y \cdot y >\!\!= \{ d , f \} \{ d , f \} >\!\!= \{ f \} \{ d \}\!</math>
====1.3.2. Discussion of Discussion====
====1.3.2. Discussion of Discussion====
But first, I nearly skipped a step. Though it might present itself as an interruption, a topic so easy that I almost omitted it altogether deserves at least a passing notice.
But first, I nearly skipped a step. Though it might present itself as an interruption, a topic so easy that I almost omitted it altogether deserves at least a passing notice.
:: <math>y_0 = y \cdot y >\!\!= \{ d , f \} \{ d , f \} >\!\!= \{ d \} \{ d \}</math>
:: <math>y_0 = y \cdot y >\!\!= \{ d , f \} \{ d , f \} >\!\!= \{ d \} \{ d \}\!</math>
Discussion is easy in general because its termination criterion is relaxed to the point of becoming otiose. A discussion of things in general can be pursued as an end in itself, with no consideration of any purpose but persevering in its current form, and this accounts for the virtually perpetual continuation of many a familiar and perennial discussion.
Discussion is easy in general because its termination criterion is relaxed to the point of becoming otiose. A discussion of things in general can be pursued as an end in itself, with no consideration of any purpose but persevering in its current form, and this accounts for the virtually perpetual continuation of many a familiar and perennial discussion.
An immediate application of the above rules is presented here, in hopes of giving the reader a concrete illustration of their use in a ready example, but the issues raised can quickly diverge into yet another distracting digression, one not so easily brought under control as the discussion of discussion, but whose complexity probably approaches that of the entire task. Therefore, a partial adumbration of its character will have to suffice for the present.
An immediate application of the above rules is presented here, in hopes of giving the reader a concrete illustration of their use in a ready example, but the issues raised can quickly diverge into yet another distracting digression, one not so easily brought under control as the discussion of discussion, but whose complexity probably approaches that of the entire task. Therefore, a partial adumbration of its character will have to suffice for the present.
:: <math>y_0 = y \cdot y >\!\!= \{ d , f \} \{ d , f \} >\!\!= \{ f \} \{ f \}</math>
:: <math>y_0 = y \cdot y >\!\!= \{ d , f \} \{ d , f \} >\!\!= \{ f \} \{ f \}\!</math>
To illustrate the formal charge by taking the present matter to task, the word ''formalization'' is itself exemplary of the ''-ionized'' terms falling under the charge, and so it can be lionized as the nominal head of a prospectively formal discussion. The reader has a right to object at this point that I have not described what particular action I intend to convey under the heading of ''formalization'', by no means enough to begin applying it to any term, much less itself. However, anyone can recognize on syntactic grounds that the word is an instance of the formal rule, purely from the character of its terminal ''-ion'', and this can be done aside from all clues about the particular meaning that I intend it to have at the end of formalization.
To illustrate the formal charge by taking the present matter to task, the word ''formalization'' is itself exemplary of the ''-ionized'' terms falling under the charge, and so it can be lionized as the nominal head of a prospectively formal discussion. The reader has a right to object at this point that I have not described what particular action I intend to convey under the heading of ''formalization'', by no means enough to begin applying it to any term, much less itself. However, anyone can recognize on syntactic grounds that the word is an instance of the formal rule, purely from the character of its terminal ''-ion'', and this can be done aside from all clues about the particular meaning that I intend it to have at the end of formalization.
The previous section took the concept of ''formalization'' as an example of a topic that a writer might try to translate from informal to formal discussion, perhaps as a way of clarifying the general concept to an optimal degree, or perhaps as a way of communicating a particular concept of it to a reader. In either case the formalization process, that aims to translate a concept from informal to formal discussion, is itself mediated by a form of discussion: (1) that interpreters conduct as a part of their ongoing monologue with themselves, or (2) that a writer (speaker) conducts in real or imagined dialogue with a reader (hearer). In view of this, I see no harm in letting the concept of discussion be stretched to cover all attempted processes of formalization.
The previous section took the concept of ''formalization'' as an example of a topic that a writer might try to translate from informal to formal discussion, perhaps as a way of clarifying the general concept to an optimal degree, or perhaps as a way of communicating a particular concept of it to a reader. In either case the formalization process, that aims to translate a concept from informal to formal discussion, is itself mediated by a form of discussion: (1) that interpreters conduct as a part of their ongoing monologue with themselves, or (2) that a writer (speaker) conducts in real or imagined dialogue with a reader (hearer). In view of this, I see no harm in letting the concept of discussion be stretched to cover all attempted processes of formalization.
:: <math>F \subseteq D</math>
:: <math>F \subseteq D\!</math>
In this section, I step back from the example of ''formalization'' and consider the general task of clarifying and communicating concepts by means of a properly directed discussion. Let this kind of ''motivated'' or ''measured'' discussion be referred to as a ''meditation'', that is, "a discourse intended to express its author's reflections or to guide others in contemplation" (Webster's). The motive of a meditation is to mediate a certain object or intention, namely, the system of concepts intended for clarification or communication. The measure of a meditation is a system of values that permits its participants to tell how close they are to achieving its object. The letter "M" will be used to annotate this form of meditation.
In this section, I step back from the example of ''formalization'' and consider the general task of clarifying and communicating concepts by means of a properly directed discussion. Let this kind of ''motivated'' or ''measured'' discussion be referred to as a ''meditation'', that is, "a discourse intended to express its author's reflections or to guide others in contemplation" (Webster's). The motive of a meditation is to mediate a certain object or intention, namely, the system of concepts intended for clarification or communication. The measure of a meditation is a system of values that permits its participants to tell how close they are to achieving its object. The letter "M" will be used to annotate this form of meditation.
:: <math>F \subseteq M \subseteq D</math>
:: <math>F \subseteq M \subseteq D\!</math>
This brings the discussion around to considering the intentional objects of measured discussions and the qualifications of a writer so motivated. Just what is involved in achieving the object of a motivated discussion? Can these intentions be formalized?
This brings the discussion around to considering the intentional objects of measured discussions and the qualifications of a writer so motivated. Just what is involved in achieving the object of a motivated discussion? Can these intentions be formalized?
:: <math>y_0 = y \cdot y >\!\!= \{ d , f \} \{d , f \} >\!\!= \{ d \} \{ f \}</math>
:: <math>y_0 = y \cdot y >\!\!= \{ d , f \} \{d , f \} >\!\!= \{ d \} \{ f \}\!</math>
* The writer's task is not to create meaning from nothing, but to construct a relation from the typical meanings that are available in ordinary discourse to the particular meanings that are intended to be the effects of a particular discussion.
* The writer's task is not to create meaning from nothing, but to construct a relation from the typical meanings that are available in ordinary discourse to the particular meanings that are intended to be the effects of a particular discussion.
=====1.3.4.3. Semiotic Equivalence Relations=====
=====1.3.4.3. Semiotic Equivalence Relations=====
If one examines the sign relations <math>L_\text{A}</math> and <math>L_\text{B}</math> that are associated with the interpreters <math>\text{A}</math> and <math>\text{B}</math>, respectively, one observes that they have many contingent properties that are not possessed by sign relations in general. One nice property possessed by the sign relations <math>L_\text{A}</math> and <math>L_\text{B}</math> is that their connotative components <math>\text{A}_{SI}</math> and <math>\text{B}_{SI}</math> constitute a pair of [[equivalence relation]]s on their common syntactic domain <math>S = I</math>. It is convenient to refer to such structures as ''[[semiotic equivalence relation]]s'' (SERs) since they equate signs that mean the same thing to somebody. Each of the SERs, <math>\text{A}_{SI}, \text{B}_{SI} \subseteq S \times I = S \times S</math>, partitions the whole collection of signs into ''[[semiotic equivalence class]]es'' (SECs). This makes for a strong form of representation in that the structure of the participants' common object domain is reflected or reconstructed, part for part, in the structure of each of their ''[[semiotic partition]]s'' (SEPs) of the syntactic domain.
If one examines the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}\!</math> that are associated with the interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math>, respectively, one observes that they have many contingent properties that are not possessed by sign relations in general. One nice property possessed by the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}\!</math> is that their connotative components <math>\text{A}_{SI}\!</math> and <math>\text{B}_{SI}\!</math> constitute a pair of [[equivalence relation]]s on their common syntactic domain <math>S = I\!</math>. It is convenient to refer to such structures as ''[[semiotic equivalence relation]]s'' (SERs) since they equate signs that mean the same thing to somebody. Each of the SERs, <math>\text{A}_{SI}, \text{B}_{SI} \subseteq S \times I = S \times S\!</math>, partitions the whole collection of signs into ''[[semiotic equivalence class]]es'' (SECs). This makes for a strong form of representation in that the structure of the participants' common object domain is reflected or reconstructed, part for part, in the structure of each of their ''[[semiotic partition]]s'' (SEPs) of the syntactic domain.
The main trouble with this notion of semantics in the present situation is that the two semiotic partitions for <math>\text{A}</math> and <math>\text{B}</math> are not the same, indeed, they are orthogonal to each other. This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or invariant reality, independent of the individual interpreter's point of view.
The main trouble with this notion of semantics in the present situation is that the two semiotic partitions for <math>\text{A}\!</math> and <math>\text{B}\!</math> are not the same, indeed, they are orthogonal to each other. This makes it difficult to interpret either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of objective structure or invariant reality, independent of the individual interpreter's point of view.
Information about the different forms of semiotic equivalence induced by the interpreters <math>\text{A}</math> and <math>\text{B}</math> is summarized in Tables 3 and 4. The form of these Tables should suffice to explain what is meant by saying that the SEPs for <math>\text{A}</math> and <math>\text{B}</math> are orthogonal to each other.
Information about the different forms of semiotic equivalence induced by the interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math> is summarized in Tables 3 and 4. The form of these Tables should suffice to explain what is meant by saying that the SEPs for <math>\text{A}\!</math> and <math>\text{B}\!</math> are orthogonal to each other.
<br>
<br>
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:60%"
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Semiotic Partition of Interpreter A}</math>
|+ style="height:30px" | <math>\text{Table 3.} ~~ \text{Semiotic Partition of Interpreter A}\!</math>
|
|
{| align="center" border="0" cellpadding="4" cellspacing="0" style="text-align:center; width:100%"
{| align="center" border="0" cellpadding="4" cellspacing="0" style="text-align:center; width:100%"
| width="50%" | <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}</math>
| width="50%" | <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math>
| width="50%" | <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}</math>
| width="50%" | <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math>
|}
|}
|-
|-
|
|
{| align="center" border="0" cellpadding="4" cellspacing="0" style="text-align:center; width:100%"
{| align="center" border="0" cellpadding="4" cellspacing="0" style="text-align:center; width:100%"
| width="50%" | <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}</math>
| width="50%" | <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math>
| width="50%" | <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}</math>
| width="50%" | <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math>
|}
|}
|}
|}
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:60%"
{| align="center" border="1" cellpadding="12" cellspacing="0" style="text-align:center; width:60%"
|+ style="height:30px" | <math>\text{Table 4.} ~~ \text{Semiotic Partition of Interpreter B}</math>
|+ style="height:30px" | <math>\text{Table 4.} ~~ \text{Semiotic Partition of Interpreter B}\!</math>
|
|
{| align="center" border="0" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"
{| align="center" border="0" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"
| <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math>
|-
|-
| <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math>
|}
|}
|
|
{| align="center" border="0" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"
{| align="center" border="0" cellpadding="12" cellspacing="0" style="text-align:center; width:50%"
| <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math>
|-
|-
| <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math>
|}
|}
|}
|}
<br>
<br>
To discuss these types of situations further, I introduce the square bracket notation <math>[x]_E</math> for ''the equivalence class of the element <math>x</math> under the equivalence relation <math>E</math>''. A statement that the elements <math>x</math> and <math>y</math> are equivalent under <math>E</math> is called an ''equation'', and can be written in either one of two ways, as <math>[x]_E = [y]_E</math> or as <math>x =_E y</math>.
To discuss these types of situations further, I introduce the square bracket notation <math>[x]_E\!</math> for ''the equivalence class of the element <math>x\!</math> under the equivalence relation <math>E\!</math>''. A statement that the elements <math>x\!</math> and <math>y\!</math> are equivalent under <math>E\!</math> is called an ''equation'', and can be written in either one of two ways, as <math>[x]_E = [y]_E\!</math> or as <math>x =_E y\!</math>.
In the application to sign relations I extend this notation in the following ways. When <math>L</math> is a sign relation whose ''syntactic projection'' or connotative component <math>L_{SI}</math> is an equivalence relation on <math>S</math>, I write <math>[s]_L</math> for ''the equivalence class of <math>s</math> under <math>L_{SI}</math>''. A statement that the signs <math>x</math> and <math>y</math> are synonymous under a semiotic equivalence relation <math>L_{SI}</math> is called a ''semiotic equation'' (SEQ), and can be written in either of the forms: <math>[x]_L = [y]_L</math> or <math>x =_L y</math>.
In the application to sign relations I extend this notation in the following ways. When <math>L\!</math> is a sign relation whose ''syntactic projection'' or connotative component <math>L_{SI}\!</math> is an equivalence relation on <math>S\!</math>, I write <math>[s]_L\!</math> for ''the equivalence class of <math>s\!</math> under <math>L_{SI}\!</math>''. A statement that the signs <math>x\!</math> and <math>y\!</math> are synonymous under a semiotic equivalence relation <math>L_{SI}\!</math> is called a ''semiotic equation'' (SEQ), and can be written in either of the forms: <math>[x]_L = [y]_L\!</math> or <math>x =_L y\!</math>.
In many situations there is one further adaptation of the square bracket notation that can be useful. Namely, when there is known to exist a particular triple <math>(o, s, i) \in L</math>, it is permissible to use <math>[o]_L</math> to mean the same thing as <math>[s]_L</math>. These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions.
In many situations there is one further adaptation of the square bracket notation that can be useful. Namely, when there is known to exist a particular triple <math>(o, s, i) \in L\!</math>, it is permissible to use <math>[o]_L\!</math> to mean the same thing as <math>[s]_L\!</math>. These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible with the frequent use of similar devices for the denotations of signs and expressions.
In these terms, the SER for interpreter <math>\text{A}</math> yields the semiotic equations:
In these terms, the SER for interpreter <math>\text{A}\!</math> yields the semiotic equations:
{| cellpadding="10"
{| cellpadding="10"
| width="10%" |
| width="10%" |
| <math>[{}^{\backprime\backprime} \text{A} {}^{\prime\prime}]_\text{A}</math>
| <math>[{}^{\backprime\backprime} \text{A} {}^{\prime\prime}]_\text{A}\!</math>
| <math>=</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{i} {}^{\prime\prime}]_\text{A}</math>
| <math>[{}^{\backprime\backprime} \text{i} {}^{\prime\prime}]_\text{A}\!</math>
| width="20%" |
| width="20%" |
| <math>[{}^{\backprime\backprime} \text{B} {}^{\prime\prime}]_\text{A}</math>
| <math>[{}^{\backprime\backprime} \text{B} {}^{\prime\prime}]_\text{A}\!</math>
| <math>=</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{u} {}^{\prime\prime}]_\text{A}</math>
| <math>[{}^{\backprime\backprime} \text{u} {}^{\prime\prime}]_\text{A}\!</math>
|-
|-
| width="10%" | or
| width="10%" | or
| <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math>
| <math>=_\text{A}</math>
| <math>=_\text{A}\!</math>
| <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math>
| width="20%" |
| width="20%" |
| <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math>
| <math>=_\text{A}</math>
| <math>=_\text{A}\!</math>
| <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math>
|}
|}
and the semiotic partition: <math>\{ \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \} , \{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \}</math>.
and the semiotic partition: <math>\{ \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \} , \{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \}\!</math>.
In contrast, the SER for interpreter <math>\text{B}</math> yields the semiotic equations:
In contrast, the SER for interpreter <math>\text{B}\!</math> yields the semiotic equations:
{| cellpadding="10"
{| cellpadding="10"
| width="10%" |
| width="10%" |
| <math>[{}^{\backprime\backprime} \text{A} {}^{\prime\prime}]_\text{B}</math>
| <math>[{}^{\backprime\backprime} \text{A} {}^{\prime\prime}]_\text{B}\!</math>
| <math>=</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{u} {}^{\prime\prime}]_\text{B}</math>
| <math>[{}^{\backprime\backprime} \text{u} {}^{\prime\prime}]_\text{B}\!</math>
| width="20%" |
| width="20%" |
| <math>[{}^{\backprime\backprime} \text{B} {}^{\prime\prime}]_\text{B}</math>
| <math>[{}^{\backprime\backprime} \text{B} {}^{\prime\prime}]_\text{B}\!</math>
| <math>=</math>
| <math>=\!</math>
| <math>[{}^{\backprime\backprime} \text{i} {}^{\prime\prime}]_\text{B}</math>
| <math>[{}^{\backprime\backprime} \text{i} {}^{\prime\prime}]_\text{B}\!</math>
|-
|-
| width="10%" | or
| width="10%" | or
| <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math>
| <math>=_\text{B}</math>
| <math>=_\text{B}\!</math>
| <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math>
| width="20%" |
| width="20%" |
| <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math>
| <math>=_\text{B}</math>
| <math>=_\text{B}\!</math>
| <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}</math>
| <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math>
|}
|}
and the semiotic partition: <math>\{ \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} , \{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \} \}</math>.
and the semiotic partition: <math>\{ \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} , \{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \} \}\!</math>.
=====1.3.4.4. Graphical Representations=====
=====1.3.4.4. Graphical Representations=====
The dyadic components of sign relations can be given graph-theoretic representations, as ''digraphs'' (or ''directed graphs''), that provide concise pictures of their structural and potential dynamic properties.
The dyadic components of sign relations can be given graph-theoretic representations, as ''digraphs'' (or ''directed graphs''), that provide concise pictures of their structural and potential dynamic properties.
By way of terminology, a directed edge <math>(x, y)</math> is called an ''arc'' from point <math>x</math> to point <math>y</math>, and a self-loop <math>(x, x)</math> is called a ''sling'' at <math>x</math>.
By way of terminology, a directed edge <math>(x, y)\!</math> is called an ''arc'' from point <math>x\!</math> to point <math>y\!</math>, and a self-loop <math>(x, x)\!</math> is called a ''sling'' at <math>x\!</math>.
The denotative components <math>\operatorname{Den}(\text{A})</math> and <math>\operatorname{Den}(\text{B})</math> can be represented as digraphs on the six points of their common world set <math>W = O \cup S \cup I = \{ \text{A}, \text{B}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}</math>. The arcs are given as follows:
The denotative components <math>\operatorname{Den}(\text{A})\!</math> and <math>\operatorname{Den}(\text{B})\!</math> can be represented as digraphs on the six points of their common world set <math>W = O \cup S \cup I = \{ \text{A}, \text{B}, {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math>. The arcs are given as follows:
:: <math>\operatorname{Den}(\text{A})</math> has an arc from each point of <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}</math> to <math>\text{A}</math> and from each point of <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}</math> to <math>\text{B}</math>.
:: <math>\operatorname{Den}(\text{A})\!</math> has an arc from each point of <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}\!</math> to <math>\text{A}\!</math> and from each point of <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> to <math>\text{B}\!</math>.
:: <math>\operatorname{Den}(\text{B})</math> has an arc from each point of <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}</math> to <math>\text{A}</math> and from each point of <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}</math> to <math>\text{B}</math>.
:: <math>\operatorname{Den}(\text{B})\!</math> has an arc from each point of <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math> to <math>\text{A}\!</math> and from each point of <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}\!</math> to <math>\text{B}\!</math>.
<math>\operatorname{Den}(\text{A})</math> and <math>\operatorname{Den}(\text{B})</math> can be interpreted as ''transition digraphs'' that chart the succession of steps or the connection of states in a computational process. If the graphs are read this way, the denotational arcs summarize the ''upshots'' of the computations that are involved when the interpreters <math>\text{A}</math> and <math>\text{B}</math> evaluate the signs in <math>S</math> according to their own frames of reference.
<math>\operatorname{Den}(\text{A})\!</math> and <math>\operatorname{Den}(\text{B})\!</math> can be interpreted as ''transition digraphs'' that chart the succession of steps or the connection of states in a computational process. If the graphs are read this way, the denotational arcs summarize the ''upshots'' of the computations that are involved when the interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math> evaluate the signs in <math>S\!</math> according to their own frames of reference.
The connotative components <math>\operatorname{Con}(\text{A})</math> and <math>\operatorname{Con}(\text{B})</math> can be represented as digraphs on the four points of their common syntactic domain <math>S = I = \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}</math>. Since <math>\operatorname{Con}(\text{A})</math> and <math>\operatorname{Con}(\text{B})</math> are SERs, their digraphs conform to the pattern that is manifested by all digraphs of equivalence relations. In general, a digraph of an equivalence relation falls into connected components that correspond to the parts of the associated partition, with a complete digraph on the points of each part, and no other arcs. In the present case, the arcs are given as follows:
The connotative components <math>\operatorname{Con}(\text{A})\!</math> and <math>\operatorname{Con}(\text{B})\!</math> can be represented as digraphs on the four points of their common syntactic domain <math>S = I = \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math>. Since <math>\operatorname{Con}(\text{A})\!</math> and <math>\operatorname{Con}(\text{B})\!</math> are SERs, their digraphs conform to the pattern that is manifested by all digraphs of equivalence relations. In general, a digraph of an equivalence relation falls into connected components that correspond to the parts of the associated partition, with a complete digraph on the points of each part, and no other arcs. In the present case, the arcs are given as follows:
:: <math>\operatorname{Con}(\text{A})</math> has the structure of a SER on <math>S</math>, with a sling at each of the points in <math>S</math>, two-way arcs between the points of <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}</math>, and two-way arcs between the points of <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}</math>.
:: <math>\operatorname{Con}(\text{A})\!</math> has the structure of a SER on <math>S\!</math>, with a sling at each of the points in <math>S\!</math>, two-way arcs between the points of <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}\!</math>, and two-way arcs between the points of <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math>.
:: <math>\operatorname{Con}(\text{B})</math> has the structure of a SER on <math>S</math>, with a sling at each of the points in <math>S</math>, two-way arcs between the points of <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}</math>, and two-way arcs between the points of <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}</math>.
:: <math>\operatorname{Con}(\text{B})\!</math> has the structure of a SER on <math>S\!</math>, with a sling at each of the points in <math>S\!</math>, two-way arcs between the points of <math>\{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \}\!</math>, and two-way arcs between the points of <math>\{ {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime} \}\!</math>.
Taken as transition digraphs, <math>\operatorname{Con}(\text{A})</math> and <math>\operatorname{Con}(\text{B})</math> highlight the associations that are permitted between equivalent signs, as this equivalence is judged by the interpreters <math>\text{A}</math> and <math>\text{B}</math>, respectively.
Taken as transition digraphs, <math>\operatorname{Con}(\text{A})\!</math> and <math>\operatorname{Con}(\text{B})\!</math> highlight the associations that are permitted between equivalent signs, as this equivalence is judged by the interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math>, respectively.
The theme running through the last three subsections, that associates different interpreters and different aspects of interpretation with different sorts of relational structures on the same set of points, heralds a topic that will be developed extensively in the sequel.
The theme running through the last three subsections, that associates different interpreters and different aspects of interpretation with different sorts of relational structures on the same set of points, heralds a topic that will be developed extensively in the sequel.
=====1.3.4.5. Taking Stock=====
=====1.3.4.5. Taking Stock=====
So far, my discussion of the discussion between <math>\text{A}</math> and <math>\text{B}</math>, in the picture that it gives of sign relations and their connection to the imagined processes of interpretation and inquiry, can best be described as fragmentary. In the story of <math>\text{A}</math> and <math>\text{B}</math>, a sample of typical language use has been drawn from the context of informal discussion and partially formalized in the guise of two independent sign relations, but no unified conception of the commonly understood interpretive practices in such a situation has yet been drafted.
So far, my discussion of the discussion between <math>\text{A}\!</math> and <math>\text{B}\!</math>, in the picture that it gives of sign relations and their connection to the imagined processes of interpretation and inquiry, can best be described as fragmentary. In the story of <math>\text{A}\!</math> and <math>\text{B}\!</math>, a sample of typical language use has been drawn from the context of informal discussion and partially formalized in the guise of two independent sign relations, but no unified conception of the commonly understood interpretive practices in such a situation has yet been drafted.
It seems like a good idea to pause at this point and reflect on the state of understanding that has been reached. In order to motivate further developments it will be useful to inventory two types of shortfall in the present state of discussion, the first having to do with the defects of my present discussion in revealing the relevant attributes of even so simple an example as the one I used to begin, the second having to do with the defects that this species of example exhibits within the genus of sign relations it is intended to illustrate.
It seems like a good idea to pause at this point and reflect on the state of understanding that has been reached. In order to motivate further developments it will be useful to inventory two types of shortfall in the present state of discussion, the first having to do with the defects of my present discussion in revealing the relevant attributes of even so simple an example as the one I used to begin, the second having to do with the defects that this species of example exhibits within the genus of sign relations it is intended to illustrate.
As a general schema, I describe these respective limitations as the ''rhetorical'' and the ''objective'' defects that a discussion can have in addressing its intended object. The immediate concern is to remedy the insufficiencies of analysis that affect the treatment of the current case. The overarching task is to address the atypically simplistic features of this example as it falls within the class of sign relations that are relevant to actual inquiry.
As a general schema, I describe these respective limitations as the ''rhetorical'' and the ''objective'' defects that a discussion can have in addressing its intended object. The immediate concern is to remedy the insufficiencies of analysis that affect the treatment of the current case. The overarching task is to address the atypically simplistic features of this example as it falls within the class of sign relations that are relevant to actual inquiry.
The next few subsections will be concerned with the most problematic features of the <math>\text{A}</math> and <math>\text{B}</math> dialogue, especially with the sorts of difficulties that are clues to significant deficits in theory and technique, and that point out directions for future improvements.
The next few subsections will be concerned with the most problematic features of the <math>\text{A}\!</math> and <math>\text{B}\!</math> dialogue, especially with the sorts of difficulties that are clues to significant deficits in theory and technique, and that point out directions for future improvements.
=====1.3.4.6. The “Meta” Question=====
=====1.3.4.6. The “Meta” Question=====
There is one point of common contention that I finessed from play in my handling of the discussion between <math>\text{A}</math> and <math>\text{B}</math>, even though it lies in plain view on both their Tables. This is the troubling business, recalcitrant to analysis precisely because its operations race on so heedlessly ahead of thought and grind on so routinely beneath its notice, that concerns the placement of object languages within the frame of a meta-language.
There is one point of common contention that I finessed from play in my handling of the discussion between <math>\text{A}\!</math> and <math>\text{B}\!</math>, even though it lies in plain view on both their Tables. This is the troubling business, recalcitrant to analysis precisely because its operations race on so heedlessly ahead of thought and grind on so routinely beneath its notice, that concerns the placement of object languages within the frame of a meta-language.
Numerous bars to insight appear to interlock here. Each one is forged with a good aim in mind, if a bit single-minded in its coverage of the scene, and the whole gang is set to work innocently enough in the unavoidable circumstances of informal discussion. But a failure to absorb their amalgamated impact on the figurative representations and the analytic intentions of sign relations can lead to several types of false impression, both about the true characters of the tables presented here and about the proper utilities of their graphical equivalents to be implemented as data structures in the computer. The next few remarks are put forth in hopes of averting these brands of misreading.
Numerous bars to insight appear to interlock here. Each one is forged with a good aim in mind, if a bit single-minded in its coverage of the scene, and the whole gang is set to work innocently enough in the unavoidable circumstances of informal discussion. But a failure to absorb their amalgamated impact on the figurative representations and the analytic intentions of sign relations can lead to several types of false impression, both about the true characters of the tables presented here and about the proper utilities of their graphical equivalents to be implemented as data structures in the computer. The next few remarks are put forth in hopes of averting these brands of misreading.
The general character of this question can be expressed in the schematic terms that were used earlier to give a rough sketch of the modeling activity as a whole. How do the isolated SOIs of <math>\text{A}</math> and <math>\text{B}</math> relate to the interpretive framework that I am using to present them, and how does this IF operate, not only to objectify <math>\text{A}</math> and <math>\text{B}</math> as models of interpretation (MOIs), but simultaneously to embrace the present and the prospective SOIs of the current narrative, the implicit systems of interpretation that embody in turn the initial conditions and the final intentions of this whole discussion?
The general character of this question can be expressed in the schematic terms that were used earlier to give a rough sketch of the modeling activity as a whole. How do the isolated SOIs of <math>\text{A}\!</math> and <math>\text{B}\!</math> relate to the interpretive framework that I am using to present them, and how does this IF operate, not only to objectify <math>\text{A}\!</math> and <math>\text{B}\!</math> as models of interpretation (MOIs), but simultaneously to embrace the present and the prospective SOIs of the current narrative, the implicit systems of interpretation that embody in turn the initial conditions and the final intentions of this whole discussion?
One way to see how this issue arises in the discussion of <math>\text{A}</math> and <math>\text{B}</math> is to recognize that each table of a sign relation is a complex sign in itself, each of whose syntactic constituents plays the role of a simpler sign. In other words, there is nothing but text to be seen on the page. In comparison to what it represents, the table is like a sign relation that has undergone a step of ''semantic ascent''. It is as if the entire contents of the original sign relation have been transposed up a notch on the scale that registers levels of indirectness in reference, each item passing from a more objective to a more symbolic mode of presentation.
One way to see how this issue arises in the discussion of <math>\text{A}\!</math> and <math>\text{B}\!</math> is to recognize that each table of a sign relation is a complex sign in itself, each of whose syntactic constituents plays the role of a simpler sign. In other words, there is nothing but text to be seen on the page. In comparison to what it represents, the table is like a sign relation that has undergone a step of ''semantic ascent''. It is as if the entire contents of the original sign relation have been transposed up a notch on the scale that registers levels of indirectness in reference, each item passing from a more objective to a more symbolic mode of presentation.
Sign relations themselves, like any real objects of discussion, are either too abstract or too concrete to reside in the medium of communication, but can only find themselves represented there. The tables and graphs that are used to represent sign relations are themselves complex signs, involving a step of denotation to reach the sign relation intended. The intricacies of this step demand interpretive agents who are able, over and above executing all the rudimentary steps of denotation, to orchestrate the requisite kinds of concerted steps. This performance in turn requires a whole array of techniques to match the connotations of complex signs and to test their alternative styles of representation for semiotic equivalence. Analogous to the ways that matrices represent linear transformations and that multiplication tables represent group operations, a large part of the usefulness of these complex signs comes from the fact that they are not just conventional symbols for their objects but iconic representations of their structure.
Sign relations themselves, like any real objects of discussion, are either too abstract or too concrete to reside in the medium of communication, but can only find themselves represented there. The tables and graphs that are used to represent sign relations are themselves complex signs, involving a step of denotation to reach the sign relation intended. The intricacies of this step demand interpretive agents who are able, over and above executing all the rudimentary steps of denotation, to orchestrate the requisite kinds of concerted steps. This performance in turn requires a whole array of techniques to match the connotations of complex signs and to test their alternative styles of representation for semiotic equivalence. Analogous to the ways that matrices represent linear transformations and that multiplication tables represent group operations, a large part of the usefulness of these complex signs comes from the fact that they are not just conventional symbols for their objects but iconic representations of their structure.
What does appear in one of these Tables? It is not the objects that appear under the ''Object'' heading, but only the signs of these objects. It is not even the signs and interpretants themselves that appear under the ''Sign'' and ''Interpretant'' headings, but only the remoter signs of them that are formed by quotation. The unformalized sign relation in which these signs of objects, signs of signs, and signs of interpretants have their role as such is not the one Tabled, but another one that operates behind the scenes to bring its image and intent to the reader.
What does appear in one of these Tables? It is not the objects that appear under the ''Object'' heading, but only the signs of these objects. It is not even the signs and interpretants themselves that appear under the ''Sign'' and ''Interpretant'' headings, but only the remoter signs of them that are formed by quotation. The unformalized sign relation in which these signs of objects, signs of signs, and signs of interpretants have their role as such is not the one Tabled, but another one that operates behind the scenes to bring its image and intent to the reader.
To understand what the Table is meant to convey the reader has to participate in the informal and more accessory sign relation in order to follow its indications to the intended and more accessible sign relation. As logical or mathematical objects, the sign relations of <math>\text{A}</math> and <math>\text{B}</math> do not exist in the medium of their Tables but are represented there by dint of the relevant structural properties that they share with these Tables. As fictional characters, the interpretive agents <math>\text{A}</math> and <math>\text{B}</math> do not exist in a uniquely literal sense but serve as typical literary figures to convey the intended formal account, standing in for concrete experiences with language use the likes of which are familiar to writer and reader alike.
To understand what the Table is meant to convey the reader has to participate in the informal and more accessory sign relation in order to follow its indications to the intended and more accessible sign relation. As logical or mathematical objects, the sign relations of <math>\text{A}\!</math> and <math>\text{B}\!</math> do not exist in the medium of their Tables but are represented there by dint of the relevant structural properties that they share with these Tables. As fictional characters, the interpretive agents <math>\text{A}\!</math> and <math>\text{B}\!</math> do not exist in a uniquely literal sense but serve as typical literary figures to convey the intended formal account, standing in for concrete experiences with language use the likes of which are familiar to writer and reader alike.
The successful formalization of a focal sign relation cannot get around its reliance on prior forms of understanding, like the raw ability to follow indications whose components of competence are embodied in the vaster and largely unarticulated context of a peripheral sign relation. But the extent to which the analysis of a formal sign relation depends on a particular context or a particular interpreter is the extent to which an opportunity for understanding is undermined by a prior petition of the very principles to be explained. Thus, there is little satisfaction in special pleadings or ad hoc accounts of interpretive practice that cannot be transported across a multitude of contexts, media, and interpreters.
The successful formalization of a focal sign relation cannot get around its reliance on prior forms of understanding, like the raw ability to follow indications whose components of competence are embodied in the vaster and largely unarticulated context of a peripheral sign relation. But the extent to which the analysis of a formal sign relation depends on a particular context or a particular interpreter is the extent to which an opportunity for understanding is undermined by a prior petition of the very principles to be explained. Thus, there is little satisfaction in special pleadings or ad hoc accounts of interpretive practice that cannot be transported across a multitude of contexts, media, and interpreters.
One task is to eliminate several types of formal confound that currently affect this investigation. Even though there is an essential tension to be maintained down the lines between casual and formal discussion, the traffic across these realms needs to be monitored carefully. There are identifiable sources of confusion that devolve from the context of informal discussion and invade the arena of formal study, subverting its necessary powers of reflection and undermining its overall effectiveness.
One task is to eliminate several types of formal confound that currently affect this investigation. Even though there is an essential tension to be maintained down the lines between casual and formal discussion, the traffic across these realms needs to be monitored carefully. There are identifiable sources of confusion that devolve from the context of informal discussion and invade the arena of formal study, subverting its necessary powers of reflection and undermining its overall effectiveness.
One serious form of contamination can be traced to the accidental circumstance that <math>\text{A}</math> and <math>\text{B}</math> and I all use the same proper names for <math>\text{A}</math> and <math>\text{B}</math>. This renders it is impossible to tell, purely from the tokens that are being tendered, whether it is a formal or a casual transaction that forms the issue of the moment. It also means that a formalization of the writer's and the reader's accessory sign relations would have several portions that look identical to pieces of those Tables under formal review.
One serious form of contamination can be traced to the accidental circumstance that <math>\text{A}\!</math> and <math>\text{B}\!</math> and I all use the same proper names for <math>\text{A}\!</math> and <math>\text{B}\!</math>. This renders it is impossible to tell, purely from the tokens that are being tendered, whether it is a formal or a casual transaction that forms the issue of the moment. It also means that a formalization of the writer's and the reader's accessory sign relations would have several portions that look identical to pieces of those Tables under formal review.
=====1.3.4.8. The Conflict of Interpretations=====
=====1.3.4.8. The Conflict of Interpretations=====
One discrepancy that needs to be documented can be observed in the conflict of interpretations between <math>\text{A}</math> and <math>\text{B}</math>, as reflected in the lack of congruity between their semiotic partitions of the syntactic domain. This is a problematic but realistic feature of the present example. That is, it represents a type of problem with the interpretation of pronouns (indexical signs or bound variables) that actually arises in practice when attempting to formalize the semantics of natural, logical, and programming languages. On this account, the deficiency resides with the present analysis, and the burden remains to clarify exactly what is going on here.
One discrepancy that needs to be documented can be observed in the conflict of interpretations between <math>\text{A}\!</math> and <math>\text{B}\!</math>, as reflected in the lack of congruity between their semiotic partitions of the syntactic domain. This is a problematic but realistic feature of the present example. That is, it represents a type of problem with the interpretation of pronouns (indexical signs or bound variables) that actually arises in practice when attempting to formalize the semantics of natural, logical, and programming languages. On this account, the deficiency resides with the present analysis, and the burden remains to clarify exactly what is going on here.
Notice, however, that I have deliberately avoided dealing with indexical tokens in the usual ways, namely, by seeking to eliminate all semantic ambiguities from the initial formalization. Instead, I have preserved this aspect of interpretive discrepancy as one of the essential phenomena or inescapable facts in the realm of pragmatic semantics, tantamount to the irreducible nature of perspective diversity. I believe that the desired competence at this faculty of language will come, not from any strategy of substitution that constantly replenishes bound variables with their objective referents on every fixed occasion, but from a pattern of recognition that keeps indexical signs persistently attached to their interpreters of reference.
Notice, however, that I have deliberately avoided dealing with indexical tokens in the usual ways, namely, by seeking to eliminate all semantic ambiguities from the initial formalization. Instead, I have preserved this aspect of interpretive discrepancy as one of the essential phenomena or inescapable facts in the realm of pragmatic semantics, tantamount to the irreducible nature of perspective diversity. I believe that the desired competence at this faculty of language will come, not from any strategy of substitution that constantly replenishes bound variables with their objective referents on every fixed occasion, but from a pattern of recognition that keeps indexical signs persistently attached to their interpreters of reference.
=====1.3.4.11. Review and Prospect=====
=====1.3.4.11. Review and Prospect=====
What has been learned from the foregoing study of icons and indices? The import of this examination can be sized up in two stages, at first, by reflecting on the action of both the formal and the casual signs that were found to be operating in and around the discussion of <math>\text{A}</math> and <math>\text{B}</math>, and then, by taking up the lessons of this circumscribed arena as a paradigm for future investigation.
What has been learned from the foregoing study of icons and indices? The import of this examination can be sized up in two stages, at first, by reflecting on the action of both the formal and the casual signs that were found to be operating in and around the discussion of <math>\text{A}\!</math> and <math>\text{B}\!</math>, and then, by taking up the lessons of this circumscribed arena as a paradigm for future investigation.
In order to explain the operation of sign relations corresponding to iconic and indexical signs in the <math>\text{A}</math> and <math>\text{B}</math> example, it becomes necessary to refer to potential objects of thought that are located, if they exist at all, outside the realm of the initial object set, that is, lying beyond the objects of thought present at the outset of discussion that one initially recognizes as objects of formally identified signs. In particular, it is incumbent on a satisfying explanation to invoke the abstract properties of objects and the actual instances of objects, where these properties and instances are normally assumed to be new objects of thought that are distinct from the objects to which they refer.
In order to explain the operation of sign relations corresponding to iconic and indexical signs in the <math>\text{A}\!</math> and <math>\text{B}\!</math> example, it becomes necessary to refer to potential objects of thought that are located, if they exist at all, outside the realm of the initial object set, that is, lying beyond the objects of thought present at the outset of discussion that one initially recognizes as objects of formally identified signs. In particular, it is incumbent on a satisfying explanation to invoke the abstract properties of objects and the actual instances of objects, where these properties and instances are normally assumed to be new objects of thought that are distinct from the objects to which they refer.
In the pragmatic account of things, thoughts are just signs in the mind of their thinker, so every object of a thought is the object of a sign, though perhaps in a sign relation that has not been fully formalized. Considered on these grounds, the search for a satisfactory context in which to explain the actions and effects of signs turns into a recursive process that potentially calls on ever higher levels of properties and ever deeper levels of instances that are found to stem from whatever objects instigated the search.
In the pragmatic account of things, thoughts are just signs in the mind of their thinker, so every object of a thought is the object of a sign, though perhaps in a sign relation that has not been fully formalized. Considered on these grounds, the search for a satisfactory context in which to explain the actions and effects of signs turns into a recursive process that potentially calls on ever higher levels of properties and ever deeper levels of instances that are found to stem from whatever objects instigated the search.
In the immediate context the objective project and the three levels of objective description are presented in broad terms. In the process of surveying a variety of problems that serve to instigate efforts in this general direction, I explore the prospects of a particular ''organon'', or ''instrumental scheme for the analysis and synthesis of objects'', that is intended to address these issues, and I give an overview of its design. In interpreting the sense of the word ''objective'' as it is used in this application, it may help to regard this objective project in the light of a telescopic analogy, with an ''objective'' being "a lens or system of lenses that forms an image of an object" (Webster's).
In the immediate context the objective project and the three levels of objective description are presented in broad terms. In the process of surveying a variety of problems that serve to instigate efforts in this general direction, I explore the prospects of a particular ''organon'', or ''instrumental scheme for the analysis and synthesis of objects'', that is intended to address these issues, and I give an overview of its design. In interpreting the sense of the word ''objective'' as it is used in this application, it may help to regard this objective project in the light of a telescopic analogy, with an ''objective'' being "a lens or system of lenses that forms an image of an object" (Webster's).
In the next three subsections after this one the focus returns to the separate levels of object structure, starting with the highest level of specification and treating the supporting levels in order of increasing detail. At each stage, the developing tools are applied to the analysis of concrete problems that arise in trying to clarify the structure and function of sign relations. For the present task, elaborations of this perspective are kept within the bounds of what is essential to deal with the example of <math>\text{A}</math> and <math>\text{B}</math>.
In the next three subsections after this one the focus returns to the separate levels of object structure, starting with the highest level of specification and treating the supporting levels in order of increasing detail. At each stage, the developing tools are applied to the analysis of concrete problems that arise in trying to clarify the structure and function of sign relations. For the present task, elaborations of this perspective are kept within the bounds of what is essential to deal with the example of <math>\text{A}\!</math> and <math>\text{B}\!</math>.
My use of the word ''object'' derives from the stock of the Greek root ''pragma'', which captures all the senses needed to suggest the stability of concern and the dedication to purpose that are forever bound up in the constitution of objects and the institution of objectives. What it implies is that every object, objective, or objectivity is always somebody's object, objective, or objectivity.
My use of the word ''object'' derives from the stock of the Greek root ''pragma'', which captures all the senses needed to suggest the stability of concern and the dedication to purpose that are forever bound up in the constitution of objects and the institution of objectives. What it implies is that every object, objective, or objectivity is always somebody's object, objective, or objectivity.
Accordingly, one of the roles intended for this OF is to provide a set of standard formulations for describing the moment to moment uncertainty of interpretive systems. The formally definable concepts of the MOI (the objective case of a SOI) and the IM (the momentary state of a SOI) are intended to formalize the intuitive notions of a generic mental constitution and a specific mental disposition that usually serve in discussing states and directions of mind.
Accordingly, one of the roles intended for this OF is to provide a set of standard formulations for describing the moment to moment uncertainty of interpretive systems. The formally definable concepts of the MOI (the objective case of a SOI) and the IM (the momentary state of a SOI) are intended to formalize the intuitive notions of a generic mental constitution and a specific mental disposition that usually serve in discussing states and directions of mind.
The structures present at each objective level are formulated by means of converse pairs of ''staging relations'', prototypically symbolized by the signs <math>\lessdot</math> and <math>\gtrdot</math>. At the more generic levels of OFs and OGs the ''staging operations'' associated with the generators <math>\lessdot</math> and <math>\gtrdot</math> involve the application of dyadic relations analogous to class membership <math>\in\!</math> and its converse <math>\ni\!</math>, but the increasing amounts of parametric information that are needed to determine specific motives and detailed motifs give OMs the full power of triadic relations. Using the same pair of symbols to denote staging relations at all objective levels helps to prevent an excessive proliferation of symbols, but it means that the meaning of these symbols is always heavily dependent on context. In particular, even fundamental properties like the effective ''arity'' of the relations signified can vary from level to level.
The structures present at each objective level are formulated by means of converse pairs of ''staging relations'', prototypically symbolized by the signs <math>\lessdot\!</math> and <math>\gtrdot\!</math>. At the more generic levels of OFs and OGs the ''staging operations'' associated with the generators <math>\lessdot\!</math> and <math>\gtrdot\!</math> involve the application of dyadic relations analogous to class membership <math>\in\!</math> and its converse <math>\ni\!</math>, but the increasing amounts of parametric information that are needed to determine specific motives and detailed motifs give OMs the full power of triadic relations. Using the same pair of symbols to denote staging relations at all objective levels helps to prevent an excessive proliferation of symbols, but it means that the meaning of these symbols is always heavily dependent on context. In particular, even fundamental properties like the effective ''arity'' of the relations signified can vary from level to level.
The staging relations divide into two orientations, <math>\lessdot</math> versus <math>\gtrdot</math>, indicating opposing senses of direction with respect to the distinction between analytic and synthetic projects:
The staging relations divide into two orientations, <math>\lessdot\!</math> versus <math>\gtrdot\!</math>, indicating opposing senses of direction with respect to the distinction between analytic and synthetic projects:
: The ''standing relations'', indicated by <math>\lessdot</math>, are analogous to the ''element of'' or membership relation <math>\in\!</math>. Another interpretation of <math>\lessdot</math> is the ''instance of'' relation. At least with respect to the more generic levels of analysis, any distinction between these readings is immaterial to the formal interests and structural objectives of this discussion.
: The ''standing relations'', indicated by <math>\lessdot\!</math>, are analogous to the ''element of'' or membership relation <math>\in\!</math>. Another interpretation of <math>\lessdot\!</math> is the ''instance of'' relation. At least with respect to the more generic levels of analysis, any distinction between these readings is immaterial to the formal interests and structural objectives of this discussion.
: The ''propping relations'', indicated by <math>\gtrdot</math>, are analogous to the ''class of'' relation or converse of the membership relation. An alternate meaning for <math>\gtrdot</math> is the ''property of'' relation. Although it is possible to maintain a distinction here, this discussion is mainly interested in a level of formal structure to which this difference is irrelevant.
: The ''propping relations'', indicated by <math>\gtrdot\!</math>, are analogous to the ''class of'' relation or converse of the membership relation. An alternate meaning for <math>\gtrdot\!</math> is the ''property of'' relation. Although it is possible to maintain a distinction here, this discussion is mainly interested in a level of formal structure to which this difference is irrelevant.
Although it may be logically redundant, it is useful in practice to introduce efficient symbolic devices for both directions of relation, <math>\lessdot</math> and <math>\gtrdot</math>, and to maintain a formal calculus that treats analogous pairs of relations on an equal footing. Extra measures of convenience come into play when the relations are used as assignment operations to create titles, define terms, and establish offices of objects in the active contexts of given relations. Thus, I regard these dual relationships as symmetric primitives and use them as the ''generating relations'' of all three objective levels.
Although it may be logically redundant, it is useful in practice to introduce efficient symbolic devices for both directions of relation, <math>\lessdot\!</math> and <math>\gtrdot\!</math>, and to maintain a formal calculus that treats analogous pairs of relations on an equal footing. Extra measures of convenience come into play when the relations are used as assignment operations to create titles, define terms, and establish offices of objects in the active contexts of given relations. Thus, I regard these dual relationships as symmetric primitives and use them as the ''generating relations'' of all three objective levels.
Next, I present several different ways of formalizing objective genres and motives. The reason for employing multiple descriptions is to capture the various ways that these patterns of organization appear in practice.
Next, I present several different ways of formalizing objective genres and motives. The reason for employing multiple descriptions is to capture the various ways that these patterns of organization appear in practice.
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>G = \{ G_j \} = \{ G_j : j \in J \} ~\text{with}~ G_j \subseteq P_j \times Q_j ~ (\forall j \in J)</math>.
| <math>G = \{ G_j \} = \{ G_j : j \in J \} ~\text{with}~ G_j \subseteq P_j \times Q_j ~ (\forall j \in J)\!</math>.
|}
|}
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>X_j = P_j \cup Q_j</math>,
| <math>X_j = P_j \cup Q_j\!</math>,
| <math>P = \textstyle \bigcup_j P_j</math>,
| <math>P = \textstyle \bigcup_j P_j\!</math>,
| <math>Q = \textstyle \bigcup_j Q_j</math>.
| <math>Q = \textstyle \bigcup_j Q_j\!</math>.
|}
|}
Ultimately, all of these ''totalitarian'' strategies end the same way, at first, by envisioning a domain <math>X\!</math> that is big enough to encompass all the objects of thought that might demand entry into a given discussion, and then, by invoking one of the following conventions:
Ultimately, all of these ''totalitarian'' strategies end the same way, at first, by envisioning a domain <math>X\!</math> that is big enough to encompass all the objects of thought that might demand entry into a given discussion, and then, by invoking one of the following conventions:
: Rubric of Universal Inclusion: <math>X = \textstyle \bigcup_j (P_j \cup Q_j)</math>.
: Rubric of Universal Inclusion: <math>X = \textstyle \bigcup_j (P_j \cup Q_j)\!</math>.
: Rubric of Universal Equality: <math>X = P_j = Q_j\ (\forall j \in J)</math>.
: Rubric of Universal Equality: <math>X = P_j = Q_j\ (\forall j \in J)\!</math>.
Working under either of these assumptions, <math>G\!</math> can be provided with a simplified form of presentation:
Working under either of these assumptions, <math>G\!</math> can be provided with a simplified form of presentation:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>G = \{ G_j \} = \{ G_j : j \in J \} ~\text{with}~ G_j \subseteq X \times X ~ (\forall j \in J)</math>.
| <math>G = \{ G_j \} = \{ G_j : j \in J \} ~\text{with}~ G_j \subseteq X \times X ~ (\forall j \in J)\!</math>.
|}
|}
It is important to emphasize that the index set <math>J\!</math> and the particular attachments of indices to dyadic relations are part and parcel to <math>G\!</math>, befitting the concrete character intended for the concept of an objective genre, which is expected to realistically embody in the character of each <math>G_j\!</math> both ''a local habitation and a name''. For this reason, among others, the <math>G_j\!</math> can safely be referred to as ''individual dyadic relations''. Since the classical notion of an ''individual'' as a ''perfectly determinate entity'' has no application in finite information contexts, it is safe to recycle this term to distinguish the ''terminally informative particulars'' that a concrete index <math>j\!</math> adds to its thematic object <math>G_j\!</math>.
It is important to emphasize that the index set <math>J\!</math> and the particular attachments of indices to dyadic relations are part and parcel to <math>G\!</math>, befitting the concrete character intended for the concept of an objective genre, which is expected to realistically embody in the character of each <math>G_j\!</math> both ''a local habitation and a name''. For this reason, among others, the <math>G_j\!</math> can safely be referred to as ''individual dyadic relations''. Since the classical notion of an ''individual'' as a ''perfectly determinate entity'' has no application in finite information contexts, it is safe to recycle this term to distinguish the ''terminally informative particulars'' that a concrete index <math>j\!</math> adds to its thematic object <math>G_j\!</math>.
Depending on the prevailing direction of interest in the genre <math>G\!</math>, <math>\lessdot</math> or <math>\gtrdot</math>, the same symbol is used equivocally for all the relations <math>G_j\!</math>. The <math>G_j\!</math> can be regarded as formalizing the objective motives that make up the genre <math>G\!</math>, provided it is understood that the information corresponding to the parameter <math>j\!</math> constitutes an integral part of the ''motive'' or ''motif'' of <math>G_j\!</math>.
Depending on the prevailing direction of interest in the genre <math>G\!</math>, <math>\lessdot\!</math> or <math>\gtrdot\!</math>, the same symbol is used equivocally for all the relations <math>G_j\!</math>. The <math>G_j\!</math> can be regarded as formalizing the objective motives that make up the genre <math>G\!</math>, provided it is understood that the information corresponding to the parameter <math>j\!</math> constitutes an integral part of the ''motive'' or ''motif'' of <math>G_j\!</math>.
In this formulation, <math>G\!</math> constitutes ''ontological hierarchy'' of a plenary type, one that determines the complete array of objects and relationships that are conceivable and describable within a given discussion. Operating with reference to the global field of possibilities presented by <math>G\!</math>, each <math>G_j\!</math> corresponds to the specialized competence of a particular agent, selecting out the objects and links of the generic hierarchy that are known to, owing to, or owned by a given interpreter.
In this formulation, <math>G\!</math> constitutes ''ontological hierarchy'' of a plenary type, one that determines the complete array of objects and relationships that are conceivable and describable within a given discussion. Operating with reference to the global field of possibilities presented by <math>G\!</math>, each <math>G_j\!</math> corresponds to the specialized competence of a particular agent, selecting out the objects and links of the generic hierarchy that are known to, owing to, or owned by a given interpreter.
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>G = \{ (j, p, q) \} \subseteq J \times P \times Q</math>,
| <math>G = \{ (j, p, q) \} \subseteq J \times P \times Q\!</math>,
|}
|}
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>G = \{ (j, x, y) \} \subseteq J \times X \times X</math>.
| <math>G = \{ (j, x, y) \} \subseteq J \times X \times X\!</math>.
|}
|}
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>G\!\uparrow ~=~ \{ (j, q, p) \in J \times Q \times P : (j, p, q) \in G \}</math>,
| <math>G\!\uparrow ~=~ \{ (j, q, p) \in J \times Q \times P : (j, p, q) \in G \}\!</math>,
|}
|}
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>G\!\uparrow ~=~ \{ (j, y, x) \in J \times X \times X : (j, x, y) \in G \}</math>.
| <math>G\!\uparrow ~=~ \{ (j, y, x) \in J \times X \times X : (j, x, y) \in G \}\!</math>.
|}
|}
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>:\!\lessdot ~\subseteq~ J \times P \times Q</math>,
| <math>:\!\lessdot ~\subseteq~ J \times P \times Q\!</math>,
|-
|-
| <math>:\!\lessdot ~\subseteq~ J \times X \times X</math>.
| <math>:\!\lessdot ~\subseteq~ J \times X \times X\!</math>.
|}
|}
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>:\!\gtrdot ~\subseteq~ J \times Q \times P</math>,
| <math>:\!\gtrdot ~\subseteq~ J \times Q \times P\!</math>,
|-
|-
| <math>:\!\gtrdot ~\subseteq~ J \times X \times X</math>.
| <math>:\!\gtrdot ~\subseteq~ J \times X \times X\!</math>.
|}
|}
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>\textstyle \bigcup_J G = \textstyle \bigcup_j G_j = \{ (x, y) \in X \times X : (x, y) \in G_j ~ (\exists j \in J) \}</math>.
| <math>\textstyle \bigcup_J G = \textstyle \bigcup_j G_j = \{ (x, y) \in X \times X : (x, y) \in G_j ~ (\exists j \in J) \}\!</math>.
|}
|}
When the relevant genre is contemplated as a triadic relation, <math>G \subseteq J \times X \times X</math>, then one is dealing with the projection of <math>G\!</math> on the object dyad <math>X \times X</math>.
When the relevant genre is contemplated as a triadic relation, <math>G \subseteq J \times X \times X\!</math>, then one is dealing with the projection of <math>G\!</math> on the object dyad <math>X \times X\!</math>.
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>G_{XX} = \operatorname{proj}_{XX}(G) = \{ (x, y) \in X \times X : (j, x, y) \in G ~ (\exists j \in J) \}</math>.
| <math>G_{XX} = \operatorname{proj}_{XX}(G) = \{ (x, y) \in X \times X : (j, x, y) \in G ~ (\exists j \in J) \}\!</math>.
|}
|}
On these occasions, the assertion that <math>(x, y)\!</math> is in <math>\cup_J G = G_{XX}</math> can be indicated by any one of the following equivalent expressions:
On these occasions, the assertion that <math>(x, y)\!</math> is in <math>\cup_J G = G_{XX}\!</math> can be indicated by any one of the following equivalent expressions:
{| align="center" cellpadding="8" style="text-align:center; width:75%"
{| align="center" cellpadding="8" style="text-align:center; width:75%"
| <math>G : x \lessdot y</math>,
| <math>G : x \lessdot y\!</math>,
| <math>x \lessdot_G y</math>,
| <math>x \lessdot_G y\!</math>,
| <math>x \lessdot y : G</math>,
| <math>x \lessdot y : G\!</math>,
|-
|-
| <math>G : y \gtrdot x</math>,
| <math>G : y \gtrdot x\!</math>,
| <math>y \gtrdot_G x</math>,
| <math>y \gtrdot_G x\!</math>,
| <math>y \gtrdot x : G</math>.
| <math>y \gtrdot x : G\!</math>.
|}
|}
{| align="center" cellpadding="8" style="text-align:center; width:75%"
{| align="center" cellpadding="8" style="text-align:center; width:75%"
| <math>j : x \lessdot y</math>,
| <math>j : x \lessdot y\!</math>,
| <math>x \lessdot_j y</math>,
| <math>x \lessdot_j y\!</math>,
| <math>x \lessdot y : j</math>,
| <math>x \lessdot y : j\!</math>,
|-
|-
| <math>j : y \gtrdot x</math>,
| <math>j : y \gtrdot x\!</math>,
| <math>y \gtrdot_j x</math>,
| <math>y \gtrdot_j x\!</math>,
| <math>y \gtrdot x : j</math>.
| <math>y \gtrdot x : j\!</math>.
|}
|}
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:#f0f0ff; text-align:left; width:100%"
{| align="center" border="0" cellpadding="4" cellspacing="0" style="background:#f0f0ff; text-align:left; width:100%"
|-
|-
| width="50%" | <math>j : x \lessdot y</math>
| width="50%" | <math>j : x \lessdot y\!</math>
| width="50%" | <math>j : y \gtrdot x</math>
| width="50%" | <math>j : y \gtrdot x\!</math>
|-
|-
| <math>x \lessdot_j y</math>
| <math>x \lessdot_j y\!</math>
| <math>y \gtrdot_j x</math>
| <math>y \gtrdot_j x\!</math>
|-
|-
| <math>x \lessdot y : j</math>
| <math>x \lessdot y : j\!</math>
| <math>y \gtrdot x : j</math>
| <math>y \gtrdot x : j\!</math>
|}
|}
|-
|-
|
|
{| align="center" border="0" cellpadding="4" cellspacing="0" style="text-align:left; width:100%"
{| align="center" border="0" cellpadding="4" cellspacing="0" style="text-align:left; width:100%"
| width="50%" | <math>j ~\text{sets}~ x ~\text{in}~ y.</math>
| width="50%" | <math>j ~\text{sets}~ x ~\text{in}~ y.\!</math>
| width="50%" | <math>j ~\text{sets}~ y ~\text{on}~ x.</math>
| width="50%" | <math>j ~\text{sets}~ y ~\text{on}~ x.\!</math>
|-
|-
| <math>j ~\text{makes}~ x ~\text{an instance of}~ y.</math>
| <math>j ~\text{makes}~ x ~\text{an instance of}~ y.\!</math>
| <math>j ~\text{makes}~ y ~\text{a property of}~ x.</math>
| <math>j ~\text{makes}~ y ~\text{a property of}~ x.\!</math>
|-
|-
| <math>j ~\text{thinks}~ x ~\text{an instance of}~ y.</math>
| <math>j ~\text{thinks}~ x ~\text{an instance of}~ y.\!</math>
| <math>j ~\text{thinks}~ y ~\text{a property of}~ x.</math>
| <math>j ~\text{thinks}~ y ~\text{a property of}~ x.\!</math>
|-
|-
| <math>j ~\text{attests}~ x ~\text{an instance of}~ y.</math>
| <math>j ~\text{attests}~ x ~\text{an instance of}~ y.\!</math>
| <math>j ~\text{attests}~ y ~\text{a property of}~ x.</math>
| <math>j ~\text{attests}~ y ~\text{a property of}~ x.\!</math>
|-
|-
| <math>j ~\text{appoints}~ x ~\text{an instance of}~ y.</math>
| <math>j ~\text{appoints}~ x ~\text{an instance of}~ y.\!</math>
| <math>j ~\text{appoints}~ y ~\text{a property of}~ x.</math>
| <math>j ~\text{appoints}~ y ~\text{a property of}~ x.\!</math>
|-
|-
| <math>j ~\text{witnesses}~ x ~\text{an instance of}~ y.</math>
| <math>j ~\text{witnesses}~ x ~\text{an instance of}~ y.\!</math>
| <math>j ~\text{witnesses}~ y ~\text{a property of}~ x.</math>
| <math>j ~\text{witnesses}~ y ~\text{a property of}~ x.\!</math>
|-
|-
| <math>j ~\text{interprets}~ x ~\text{an instance of}~ y.</math>
| <math>j ~\text{interprets}~ x ~\text{an instance of}~ y.\!</math>
| <math>j ~\text{interprets}~ y ~\text{a property of}~ x.</math>
| <math>j ~\text{interprets}~ y ~\text{a property of}~ x.\!</math>
|-
|-
| <math>j ~\text{contributes}~ x ~\text{to}~ y.</math>
| <math>j ~\text{contributes}~ x ~\text{to}~ y.\!</math>
| <math>j ~\text{attributes}~ y ~\text{to}~ x.</math>
| <math>j ~\text{attributes}~ y ~\text{to}~ x.\!</math>
|-
|-
| <math>j ~\text{determines}~ x ~\text{an example of}~ y.</math>
| <math>j ~\text{determines}~ x ~\text{an example of}~ y.\!</math>
| <math>j ~\text{determines}~ y ~\text{a quality of}~ x.</math>
| <math>j ~\text{determines}~ y ~\text{a quality of}~ x.\!</math>
|-
|-
| <math>j ~\text{evaluates}~ x ~\text{an example of}~ y.</math>
| <math>j ~\text{evaluates}~ x ~\text{an example of}~ y.\!</math>
| <math>j ~\text{evaluates}~ y ~\text{a quality of}~ x.</math>
| <math>j ~\text{evaluates}~ y ~\text{a quality of}~ x.\!</math>
|-
|-
| <math>j ~\text{proposes}~ x ~\text{an example of}~ y.</math>
| <math>j ~\text{proposes}~ x ~\text{an example of}~ y.\!</math>
| <math>j ~\text{proposes}~ y ~\text{a quality of}~ x.</math>
| <math>j ~\text{proposes}~ y ~\text{a quality of}~ x.\!</math>
|-
|-
| <math>j ~\text{musters}~ x ~\text{under}~ y.</math>
| <math>j ~\text{musters}~ x ~\text{under}~ y.\!</math>
| <math>j ~\text{marshals}~ y ~\text{over}~ x.</math>
| <math>j ~\text{marshals}~ y ~\text{over}~ x.\!</math>
|-
|-
| <math>j ~\text{indites}~ x ~\text{among}~ y.</math>
| <math>j ~\text{indites}~ x ~\text{among}~ y.\!</math>
| <math>j ~\text{ascribes}~ y ~\text{about}~ x.</math>
| <math>j ~\text{ascribes}~ y ~\text{about}~ x.\!</math>
|-
|-
| <math>j ~\text{imputes}~ x ~\text{among}~ y.</math>
| <math>j ~\text{imputes}~ x ~\text{among}~ y.\!</math>
| <math>j ~\text{imputes}~ y ~\text{about}~ x.</math>
| <math>j ~\text{imputes}~ y ~\text{about}~ x.\!</math>
|-
|-
| <math>j ~\text{judges}~ x ~\text{beneath}~ y.</math>
| <math>j ~\text{judges}~ x ~\text{beneath}~ y.\!</math>
| <math>j ~\text{judges}~ y ~\text{beyond}~ x.</math>
| <math>j ~\text{judges}~ y ~\text{beyond}~ x.\!</math>
|-
|-
| <math>j ~\text{finds}~ x ~\text{preceding}~ y.</math>
| <math>j ~\text{finds}~ x ~\text{preceding}~ y.\!</math>
| <math>j ~\text{finds}~ y ~\text{succeeding}~ x.</math>
| <math>j ~\text{finds}~ y ~\text{succeeding}~ x.\!</math>
|-
|-
| <math>j ~\text{poses}~ x ~\text{before}~ y.</math>
| <math>j ~\text{poses}~ x ~\text{before}~ y.\!</math>
| <math>j ~\text{poses}~ y ~\text{after}~ x.</math>
| <math>j ~\text{poses}~ y ~\text{after}~ x.\!</math>
|-
|-
| <math>j ~\text{forms}~ x ~\text{below}~ y.</math>
| <math>j ~\text{forms}~ x ~\text{below}~ y.\!</math>
| <math>j ~\text{forms}~ y ~\text{above}~ x.</math>
| <math>j ~\text{forms}~ y ~\text{above}~ x.\!</math>
|}
|}
|}
|}
By way of anticipating the nature of the problem, consider the following examples to illustrate the contrast between logical and cognitive senses:
By way of anticipating the nature of the problem, consider the following examples to illustrate the contrast between logical and cognitive senses:
:* In a cognitive context, if <math>j\!</math> is a considered opinion that <math>S\!</math> is true, and <math>j\!</math> is a considered opinion that <math>T\!</math> is true, then it does not have to automatically follow that <math>j\!</math> is a considered opinion that the conjunction <math>S\ \operatorname{and}\ T</math> is true, since an extra measure of consideration might conceivably be involved in cognizing the conjunction of <math>S\!</math> and <math>T\!</math>.
:* In a cognitive context, if <math>j\!</math> is a considered opinion that <math>S\!</math> is true, and <math>j\!</math> is a considered opinion that <math>T\!</math> is true, then it does not have to automatically follow that <math>j\!</math> is a considered opinion that the conjunction <math>S\ \operatorname{and}\ T\!</math> is true, since an extra measure of consideration might conceivably be involved in cognizing the conjunction of <math>S\!</math> and <math>T\!</math>.
:* In a logical context, if <math>j\!</math> is a piece of evidence that <math>S\!</math> is true, and <math>j\!</math> is a piece of evidence that <math>T\!</math> is true, then it follows by these very facts alone that <math>j\!</math> is a piece of evidence that the conjunction <math>S\ \operatorname{and}\ T</math> is true. This is analogous to a situation where, if a person <math>j\!</math> draws a set of three lines, <math>AB,\!</math> <math>BC,\!</math> and <math>AC,\!</math> then <math>j\!</math> has drawn a triangle <math>ABC,\!</math> whether <math>j\!</math> recognizes the fact on reflection and further consideration or not.
:* In a logical context, if <math>j\!</math> is a piece of evidence that <math>S\!</math> is true, and <math>j\!</math> is a piece of evidence that <math>T\!</math> is true, then it follows by these very facts alone that <math>j\!</math> is a piece of evidence that the conjunction <math>S\ \operatorname{and}\ T\!</math> is true. This is analogous to a situation where, if a person <math>j\!</math> draws a set of three lines, <math>AB,\!</math> <math>BC,\!</math> and <math>AC,\!</math> then <math>j\!</math> has drawn a triangle <math>ABC,\!</math> whether <math>j\!</math> recognizes the fact on reflection and further consideration or not.
Some readings of the staging relations are tantamount to statements of (a possibly higher order) model theory. For example, consider the predicate <math>P : J \to \mathbb{B}</math> defined by the following equivalence:
Some readings of the staging relations are tantamount to statements of (a possibly higher order) model theory. For example, consider the predicate <math>P : J \to \mathbb{B}\!</math> defined by the following equivalence:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
| <math>P(j) \quad \Leftrightarrow \quad j\ \text{proposes}\ x\ \text{an instance of}\ y.</math>
| <math>P(j) \quad \Leftrightarrow \quad j\ \text{proposes}\ x\ \text{an instance of}\ y.\!</math>
|}
|}
Consequently, naturally occurring sign relations can be expected to fall into species or natural kinds, and to have special properties that make them keep on occurring in nature. Moreover, cultivated varieties of sign relations, the kinds that have been converted to social purposes and found to be viable in actual practice, will have identifiable and especially effective properties by virtue of which their signs are rendered significant.
Consequently, naturally occurring sign relations can be expected to fall into species or natural kinds, and to have special properties that make them keep on occurring in nature. Moreover, cultivated varieties of sign relations, the kinds that have been converted to social purposes and found to be viable in actual practice, will have identifiable and especially effective properties by virtue of which their signs are rendered significant.
In the pragmatic theory of sign relations, three natural kinds of signs are recognized, under the names of ''icons'', ''indices'', and ''symbols''. Examples of indexical or accessional signs figured significantly in the discussion of <math>A\!</math> and <math>B\!</math>, as illustrated by the pronouns <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}</math> in <math>S\!</math>. Examples of iconic or analogical signs were also present, though keeping to the background, in the very form of the sign relation Tables that were used to schematize the whole activity of each interpreter. Examples of symbolic or conventional signs, of course, abide even more deeply in the background, pervading the whole context and making up the very fabric of this discussion.
In the pragmatic theory of sign relations, three natural kinds of signs are recognized, under the names of ''icons'', ''indices'', and ''symbols''. Examples of indexical or accessional signs figured significantly in the discussion of <math>A\!</math> and <math>B\!</math>, as illustrated by the pronouns <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math> in <math>S\!</math>. Examples of iconic or analogical signs were also present, though keeping to the background, in the very form of the sign relation Tables that were used to schematize the whole activity of each interpreter. Examples of symbolic or conventional signs, of course, abide even more deeply in the background, pervading the whole context and making up the very fabric of this discussion.
In order to deal with the array of issues presented so far in this subsection, all of which have to do with controlling the generative power of sign relations to serve the specific purposes of understanding, I apply the previously introduced concept of an ''objective genre'' (OG). This is intended to be a determinate purpose or a deliberate pattern of analysis and synthesis that one can identify as being active at given moments in a discussion and that affects what one regards as the relevant structural properties of its objects.
In order to deal with the array of issues presented so far in this subsection, all of which have to do with controlling the generative power of sign relations to serve the specific purposes of understanding, I apply the previously introduced concept of an ''objective genre'' (OG). This is intended to be a determinate purpose or a deliberate pattern of analysis and synthesis that one can identify as being active at given moments in a discussion and that affects what one regards as the relevant structural properties of its objects.
Pretty much the same relational structures could be found in the genre or paradigm of ''qualities and examples'', but the use of ''examples'' here is polymorphous enough to include experiential, exegetic, and executable examples. This points the way to a series of related genres, for example, the OGs of ''principles and illustrations'', ''laws and existents'', ''precedents and exercises'', and on to ''lessons and experiences''. All in all, in their turn, these modulations of the basic OG show a way to shift the foundations of ontological hierarchies toward bases in individual and systematic experience, and thus to put existentially dynamic rollers under the blocks of what seem to be essentially invariant pyramids.
Pretty much the same relational structures could be found in the genre or paradigm of ''qualities and examples'', but the use of ''examples'' here is polymorphous enough to include experiential, exegetic, and executable examples. This points the way to a series of related genres, for example, the OGs of ''principles and illustrations'', ''laws and existents'', ''precedents and exercises'', and on to ''lessons and experiences''. All in all, in their turn, these modulations of the basic OG show a way to shift the foundations of ontological hierarchies toward bases in individual and systematic experience, and thus to put existentially dynamic rollers under the blocks of what seem to be essentially invariant pyramids.
Any object of these OGs can be contemplated in the light of two potential relationships, namely, with respect to its chances of being an ''object quality'' or an ''object example'' of something else. In future references, abbreviated notations like <math>\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst})</math> or <math>\operatorname{OG} = (\operatorname{Prop}, \operatorname{Inst})</math> will be used to specify particular genres, giving the intended interpretations of their generating relations <math>\{ \lessdot,\gtrdot \}.</math>
Any object of these OGs can be contemplated in the light of two potential relationships, namely, with respect to its chances of being an ''object quality'' or an ''object example'' of something else. In future references, abbreviated notations like <math>\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst})\!</math> or <math>\operatorname{OG} = (\operatorname{Prop}, \operatorname{Inst})\!</math> will be used to specify particular genres, giving the intended interpretations of their generating relations <math>\{ \lessdot,\gtrdot \}.\!</math>
With respect to this OG, I can now characterize icons and indices. Icons are signs by virtue of being instances of properties of objects. Indices are signs by virtue of being properties of instances of objects.
With respect to this OG, I can now characterize icons and indices. Icons are signs by virtue of being instances of properties of objects. Indices are signs by virtue of being properties of instances of objects.
Turning to the language of ''objective concerns'', what can now be said about the compositional structures of the iconic sign relation <math>M\!</math> and the indexical sign relation <math>N\!</math>? In preparation for this topic, a few additional steps must be taken to continue formalizing the concept of an objective genre and to begin developing a calculus for composing objective motifs.
Turning to the language of ''objective concerns'', what can now be said about the compositional structures of the iconic sign relation <math>M\!</math> and the indexical sign relation <math>N\!</math>? In preparation for this topic, a few additional steps must be taken to continue formalizing the concept of an objective genre and to begin developing a calculus for composing objective motifs.
I recall the objective genre of ''properties and instances'' and re-introduce the symbols <math>\lessdot</math> and <math>\gtrdot</math> for the converse pair of dyadic relations that generate it. Reverting to the convention I employ in formal discussions of applying relational operators on the right, it is convenient to express the relative terms "property of <math>x\!</math>" and "instance of <math>x\!</math>" by means of a case inflection on <math>x\!,</math> that is, as "<math>x\!</math>’s property" and "<math>x\!</math>’s instance", respectively. Described in this way, <math>\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst}) = \langle \lessdot, \gtrdot \rangle,</math> where:
I recall the objective genre of ''properties and instances'' and re-introduce the symbols <math>\lessdot\!</math> and <math>\gtrdot\!</math> for the converse pair of dyadic relations that generate it. Reverting to the convention I employ in formal discussions of applying relational operators on the right, it is convenient to express the relative terms "property of <math>x\!</math>" and "instance of <math>x\!</math>" by means of a case inflection on <math>x,\!</math> that is, as "<math>x\!</math>’s property" and "<math>x\!</math>’s instance", respectively. Described in this way, <math>\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst}) = \langle \lessdot, \gtrdot \rangle,\!</math> where:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
|}
|}
A symbol like <math>^{\backprime\backprime} x \lessdot ^{\prime\prime}</math> or <math>^{\backprime\backprime} x \gtrdot ^{\prime\prime}</math> is called a ''catenation'', where <math>^{\backprime\backprime} x ^{\prime\prime}</math> is the ''catenand'' and <math>^{\backprime\backprime} \lessdot ^{\prime\prime}</math> or <math>^{\backprime\backprime} \gtrdot ^{\prime\prime}</math> is the ''catenator''. Due to the fact that <math>^{\backprime\backprime} \lessdot ^{\prime\prime}</math> and <math>^{\backprime\backprime} \gtrdot ^{\prime\prime}</math> indicate dyadic relations, the significance of these so-called ''unsaturated'' catenations can be rationalized as follows:
A symbol like <math>^{\backprime\backprime} x \lessdot ^{\prime\prime}\!</math> or <math>^{\backprime\backprime} x \gtrdot ^{\prime\prime}\!</math> is called a ''catenation'', where <math>^{\backprime\backprime} x ^{\prime\prime}\!</math> is the ''catenand'' and <math>^{\backprime\backprime} \lessdot ^{\prime\prime}\!</math> or <math>^{\backprime\backprime} \gtrdot ^{\prime\prime}\!</math> is the ''catenator''. Due to the fact that <math>^{\backprime\backprime} \lessdot ^{\prime\prime}\!</math> and <math>^{\backprime\backprime} \gtrdot ^{\prime\prime}\!</math> indicate dyadic relations, the significance of these so-called ''unsaturated'' catenations can be rationalized as follows:
{| align="center" cellpadding="8"
{| align="center" cellpadding="8"
=====1.3.4.15. Application of OF : Motive Level=====
=====1.3.4.15. Application of OF : Motive Level=====
Now that an adequate variety of formal tools have been set in order and the workspace afforded by an objective framework has been rendered reasonably clear, the structural theory of sign relations can be pursued with greater precision. In support of this aim, the concept of an objective genre and the particular example provided by <math>\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst})</math> have served to rough out the basic shapes of the more refined analytic instruments to be developed in this subsection.
Now that an adequate variety of formal tools have been set in order and the workspace afforded by an objective framework has been rendered reasonably clear, the structural theory of sign relations can be pursued with greater precision. In support of this aim, the concept of an objective genre and the particular example provided by <math>\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst})\!</math> have served to rough out the basic shapes of the more refined analytic instruments to be developed in this subsection.
The notion of an ''objective motive'' or ''objective motif'' (OM) is intended to specialize or personalize the application of objective genres to take particular interpreters into account. For example, pursuing the pattern of <math>\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst})</math>, a prospective OM of this genre does not merely tell about the properties and instances that objects can have in general, it recognizes a particular arrangement of objects and supplies them with its own ontology, giving "a local habitation and a name" to the bunch. What matters to an OM is a particular collection of objects (of thought) and a personal selection of links that go from each object (of thought) to higher and lower objects (of thought), all things being relative to a subjective ontology or a live ''hierarchy of thought'', one that is currently known to and actively pursued by a designated interpreter of those thoughts.
The notion of an ''objective motive'' or ''objective motif'' (OM) is intended to specialize or personalize the application of objective genres to take particular interpreters into account. For example, pursuing the pattern of <math>\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst})\!</math>, a prospective OM of this genre does not merely tell about the properties and instances that objects can have in general, it recognizes a particular arrangement of objects and supplies them with its own ontology, giving "a local habitation and a name" to the bunch. What matters to an OM is a particular collection of objects (of thought) and a personal selection of links that go from each object (of thought) to higher and lower objects (of thought), all things being relative to a subjective ontology or a live ''hierarchy of thought'', one that is currently known to and actively pursued by a designated interpreter of those thoughts.
The cautionary details interspersed at critical points in the preceding paragraph are intended to keep this inquiry vigilant against a constant danger of using ontological language, namely, the illusion that one can analyze the being of any real object merely by articulating the grammar of one's own thoughts, that is, simply by parsing signs in the mind. As always, it is best to regard OGs and OMs as ''filters'' and ''reticles'', as transparent templates that are used to view a space, constituting the structures of objects only in one respect at a time, but never with any assurance of totality.
The cautionary details interspersed at critical points in the preceding paragraph are intended to keep this inquiry vigilant against a constant danger of using ontological language, namely, the illusion that one can analyze the being of any real object merely by articulating the grammar of one's own thoughts, that is, simply by parsing signs in the mind. As always, it is best to regard OGs and OMs as ''filters'' and ''reticles'', as transparent templates that are used to view a space, constituting the structures of objects only in one respect at a time, but never with any assurance of totality.
With these refinements, the use of dyadic projections to investigate sign relations can be combined with the perspective of objective motives to ''factor the facets'' or ''decompose the components'' of sign relations in a more systematic fashion. Given a homogeneous sign relation <math>H\!</math> of iconic or indexical type, the dyadic projections <math>H_{OS}\!</math> and <math>H_{OI}\!</math> can be analyzed as compound relations over the basis supplied by the <math>G_j\!</math> in <math>G\!</math>. As an application that is sufficiently important in its own right, the investigation of icons and indices continues to provide a useful testing ground for breaking in likely proposals of concepts and notation.
With these refinements, the use of dyadic projections to investigate sign relations can be combined with the perspective of objective motives to ''factor the facets'' or ''decompose the components'' of sign relations in a more systematic fashion. Given a homogeneous sign relation <math>H\!</math> of iconic or indexical type, the dyadic projections <math>H_{OS}\!</math> and <math>H_{OI}\!</math> can be analyzed as compound relations over the basis supplied by the <math>G_j\!</math> in <math>G\!</math>. As an application that is sufficiently important in its own right, the investigation of icons and indices continues to provide a useful testing ground for breaking in likely proposals of concepts and notation.
To pursue the analysis of icons and indices at the next stage of formalization, fix the OG of this discussion to have the type <math>\langle \lessdot, \gtrdot \rangle</math> and let each sign relation under discussion be articulated in terms of an objective motif that tells what objects and signs, plus what mediating linkages through properties and instances, are assumed to be recognized by its interpreter.
To pursue the analysis of icons and indices at the next stage of formalization, fix the OG of this discussion to have the type <math>\langle \lessdot, \gtrdot \rangle\!</math> and let each sign relation under discussion be articulated in terms of an objective motif that tells what objects and signs, plus what mediating linkages through properties and instances, are assumed to be recognized by its interpreter.
Let <math>X\!</math> collect the objects of thought that fall within a particular OM, and let <math>X\!</math> include the whole world of a sign relation plus everything needed to support and contain it. That is, <math>X\!</math> collects all the types of things that go into a sign relation, <math>O \cup S \cup I = W \subseteq X</math>, plus whatever else in the way of distinct object qualities and object exemplars is discovered or established to be generated out of this basis by the relations of the OM.
Let <math>X\!</math> collect the objects of thought that fall within a particular OM, and let <math>X\!</math> include the whole world of a sign relation plus everything needed to support and contain it. That is, <math>X\!</math> collects all the types of things that go into a sign relation, <math>O \cup S \cup I = W \subseteq X\!</math>, plus whatever else in the way of distinct object qualities and object exemplars is discovered or established to be generated out of this basis by the relations of the OM.
In order to keep this <math>X\!</math> simple enough to contemplate on a single pass but still make it deep enough to cover the issues of interest at present, I limit <math>X\!</math> to having just three disjoint layers of things to worry about:
In order to keep this <math>X\!</math> simple enough to contemplate on a single pass but still make it deep enough to cover the issues of interest at present, I limit <math>X\!</math> to having just three disjoint layers of things to worry about:
| The top layer is the relevant class of object qualities:
| The top layer is the relevant class of object qualities:
|-
|-
| <math>Q = X_0 \lessdot = W \lessdot</math>
| <math>Q = X_0 \lessdot = W \lessdot\!</math>
|-
|-
| The middle layer is the initial collection of objects and signs:
| The middle layer is the initial collection of objects and signs:
| The bottom layer is a suitable set of object exemplars:
| The bottom layer is a suitable set of object exemplars:
|-
|-
| <math>E = X_0 \gtrdot = W \gtrdot</math>
| <math>E = X_0 \gtrdot = W \gtrdot\!</math>
|}
|}
{| align="center" cellpadding="8" style="text-align:center"
{| align="center" cellpadding="8" style="text-align:center"
| <math>h : x \lessdot m</math>
| <math>h : x \lessdot m\!</math>
| <math>\Leftrightarrow</math>
| <math>\Leftrightarrow\!</math>
| <math>h ~\operatorname{regards}~ x ~\operatorname{as~an~instance~of}~ m.</math>
| <math>h ~\operatorname{regards}~ x ~\operatorname{as~an~instance~of}~ m.\!</math>
|-
|-
| <math>h : m \gtrdot y</math>
| <math>h : m \gtrdot y\!</math>
| <math>\Leftrightarrow</math>
| <math>\Leftrightarrow\!</math>
| <math>h ~\operatorname{regards}~ m ~\operatorname{as~a~property~of}~ y.</math>
| <math>h ~\operatorname{regards}~ m ~\operatorname{as~a~property~of}~ y.\!</math>
|-
|-
| <math>h : x \gtrdot n</math>
| <math>h : x \gtrdot n\!</math>
| <math>\Leftrightarrow</math>
| <math>\Leftrightarrow\!</math>
| <math>h ~\operatorname{regards}~ x ~\operatorname{as~a~property~of}~ n.</math>
| <math>h ~\operatorname{regards}~ x ~\operatorname{as~a~property~of}~ n.\!</math>
|-
|-
| <math>h : n \lessdot y</math>
| <math>h : n \lessdot y\!</math>
| <math>\Leftrightarrow</math>
| <math>\Leftrightarrow\!</math>
| <math>h ~\operatorname{regards}~ n ~\operatorname{as~an~instance~of}~ y.</math>
| <math>h ~\operatorname{regards}~ n ~\operatorname{as~an~instance~of}~ y.\!</math>
|}
|}
| <math>M_{OS}\!</math>
| <math>M_{OS}\!</math>
| <math>\colon\!</math>
| <math>\colon\!</math>
| <math>x \lessdot \gtrdot x \operatorname{'s~Sign}.</math>
| <math>x \lessdot \gtrdot x \operatorname{'s~Sign}.\!</math>
|-
|-
| <math>\text{For Indices:}\!</math>
| <math>\text{For Indices:}\!</math>
| <math>N_{OS}\!</math>
| <math>N_{OS}\!</math>
| <math>\colon\!</math>
| <math>\colon\!</math>
| <math>x \gtrdot \lessdot x \operatorname{'s~Sign}.</math>
| <math>x \gtrdot \lessdot x \operatorname{'s~Sign}.\!</math>
|}
|}
|}
|}
| <math>j\!</math>
| <math>j\!</math>
| <math>\colon\!</math>
| <math>\colon\!</math>
| <math>x \lessdot \gtrdot x \operatorname{'s~Sign}.</math>
| <math>x \lessdot \gtrdot x \operatorname{'s~Sign}.\!</math>
|-
|-
| <math>\text{For Indices:}\!</math>
| <math>\text{For Indices:}\!</math>
| <math>k\!</math>
| <math>k\!</math>
| <math>\colon\!</math>
| <math>\colon\!</math>
| <math>x \gtrdot \lessdot x \operatorname{'s~Sign}.</math>
| <math>x \gtrdot \lessdot x \operatorname{'s~Sign}.\!</math>
|}
|}
|}
|}
Readers who object to the anthropomorphism or the approximation of these statements can replace every occurrence of the verb ''thinks'' with the phrase ''interprets … as'', or even the circumlocution ''acts in every formally relevant way as if'', changing what must be changed elsewhere. For the moment, I am not concerned with the exact order of reflective sensitivity that goes into these interpretive linkages, but only with a rough outline of the pragmatic equivalence classes that are afforded by the potential conduct of their agents.
Readers who object to the anthropomorphism or the approximation of these statements can replace every occurrence of the verb ''thinks'' with the phrase ''interprets … as'', or even the circumlocution ''acts in every formally relevant way as if'', changing what must be changed elsewhere. For the moment, I am not concerned with the exact order of reflective sensitivity that goes into these interpretive linkages, but only with a rough outline of the pragmatic equivalence classes that are afforded by the potential conduct of their agents.
In the discussion of the dialogue between <math>\text{A}</math> and <math>\text{B}</math> it was allowed that the same signs <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}</math> could reference the different categories of things they name with a deliberate duality and a systematic ambiguity. Used informally as a part of the peripheral discussion, they indicate the entirety of the sign relations themselves. Used formally within the focal dialogue, they denote the objects of two particular sign relations. In just this way, or an elaboration of it, the signs <math>{}^{\backprime\backprime} j {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} k {}^{\prime\prime}</math> can have their meanings extended to encompass both the objective motifs that inform and regulate experience and the object experiences that fill out and substantiate their forms.
In the discussion of the dialogue between <math>\text{A}\!</math> and <math>\text{B}\!</math> it was allowed that the same signs <math>{}^{\backprime\backprime} \text{A} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \text{B} {}^{\prime\prime}\!</math> could reference the different categories of things they name with a deliberate duality and a systematic ambiguity. Used informally as a part of the peripheral discussion, they indicate the entirety of the sign relations themselves. Used formally within the focal dialogue, they denote the objects of two particular sign relations. In just this way, or an elaboration of it, the signs <math>{}^{\backprime\backprime} j {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} k {}^{\prime\prime}\!</math> can have their meanings extended to encompass both the objective motifs that inform and regulate experience and the object experiences that fill out and substantiate their forms.
=====1.3.4.16. The Integration of Frameworks=====
=====1.3.4.16. The Integration of Frameworks=====
To express the nature of this integration task in logical terms, it combines elements of both proof theory and model theory, interweaving: (1) A phase that develops theories about the symbolic competence or ''knowledge'' of intelligent agents, using abstract formal systems to represent the theories and phenomenological data to constrain them; (2) A phase that seeks concrete models of these theories, looking to the kinds of mathematical structure that have a dynamic or system-theoretic interpretation, and compiling the constraints that a recursive conceptual analysis imposes on the ultimate elements of their construction.
To express the nature of this integration task in logical terms, it combines elements of both proof theory and model theory, interweaving: (1) A phase that develops theories about the symbolic competence or ''knowledge'' of intelligent agents, using abstract formal systems to represent the theories and phenomenological data to constrain them; (2) A phase that seeks concrete models of these theories, looking to the kinds of mathematical structure that have a dynamic or system-theoretic interpretation, and compiling the constraints that a recursive conceptual analysis imposes on the ultimate elements of their construction.
The set of sign relations <math>\{ L_\text{A}, L_\text{B} \}</math> is an example of an extremely simple formal system, encapsulating aspects of the symbolic competence and the pragmatic performance that might be exhibited by potentially intelligent interpretive agents, however abstractly and partially given at this stage of description. The symbols of a formal system like <math>\{ L_\text{A}, L_\text{B} \}</math> can be held subject to abstract constraints, having their meanings in relation to each other determined by definitions and axioms (for example, the laws defining an equivalence relation), making it possible to manipulate the resulting information by means of the inference rules in a proof system. This illustrates the ''proof-theoretic'' aspect of a symbol system.
The set of sign relations <math>\{ L_\text{A}, L_\text{B} \}\!</math> is an example of an extremely simple formal system, encapsulating aspects of the symbolic competence and the pragmatic performance that might be exhibited by potentially intelligent interpretive agents, however abstractly and partially given at this stage of description. The symbols of a formal system like <math>\{ L_\text{A}, L_\text{B} \}\!</math> can be held subject to abstract constraints, having their meanings in relation to each other determined by definitions and axioms (for example, the laws defining an equivalence relation), making it possible to manipulate the resulting information by means of the inference rules in a proof system. This illustrates the ''proof-theoretic'' aspect of a symbol system.
Suppose that a formal system like <math>\{ L_\text{A}, L_\text{B} \}</math> is initially approached from a theoretical direction, in other words, by listing the abstract properties one thinks it ought to have. Then the existence of an extensional model that satisfies these constraints, as exhibited by the sign relation tables, demonstrates that one's theoretical description is logically consistent, even if the models that first come to mind are still a bit too abstractly symbolic and do not have all the dynamic concreteness that is demanded of system-theoretic interpretations. This amounts to the other side of the ledger, the ''model-theoretic'' aspect of a symbol system, at least insofar as the present account has dealt with it.
Suppose that a formal system like <math>\{ L_\text{A}, L_\text{B} \}\!</math> is initially approached from a theoretical direction, in other words, by listing the abstract properties one thinks it ought to have. Then the existence of an extensional model that satisfies these constraints, as exhibited by the sign relation tables, demonstrates that one's theoretical description is logically consistent, even if the models that first come to mind are still a bit too abstractly symbolic and do not have all the dynamic concreteness that is demanded of system-theoretic interpretations. This amounts to the other side of the ledger, the ''model-theoretic'' aspect of a symbol system, at least insofar as the present account has dealt with it.
More is required of the modeler, however, in order to find the desired kinds of system-theoretic models (for example, state transition systems), and this brings the search for realizations of formal systems down to the toughest part of the exercise. Some of the problems that emerge were highlighted in the example of <math>\text{A}</math> and <math>\text{B}</math>. Although it is ordinarily possible to construct state transition systems in which the states of interpreters correspond relatively directly to the acceptations of the primitive signs given, the conflict of interpretations that develops between different interpreters from these prima facie implementations is a sign that there is something superficial about this approach.
More is required of the modeler, however, in order to find the desired kinds of system-theoretic models (for example, state transition systems), and this brings the search for realizations of formal systems down to the toughest part of the exercise. Some of the problems that emerge were highlighted in the example of <math>\text{A}\!</math> and <math>\text{B}\!</math>. Although it is ordinarily possible to construct state transition systems in which the states of interpreters correspond relatively directly to the acceptations of the primitive signs given, the conflict of interpretations that develops between different interpreters from these prima facie implementations is a sign that there is something superficial about this approach.
The integration of model-theoretic and proof-theoretic aspects of ''physical symbol systems'', besides being closely analogous to the integration of denotative and connotative aspects of sign relations, is also relevant to the job of integrating dynamic and symbolic frameworks for intelligent systems. This is so because the search for dynamic realizations of symbol systems is only a more pointed exercise in model theory, where the mathematical materials made available for modeling are further constrained by system-theoretic principles, like being able to say what the states are and how the transitions are determined.
The integration of model-theoretic and proof-theoretic aspects of ''physical symbol systems'', besides being closely analogous to the integration of denotative and connotative aspects of sign relations, is also relevant to the job of integrating dynamic and symbolic frameworks for intelligent systems. This is so because the search for dynamic realizations of symbol systems is only a more pointed exercise in model theory, where the mathematical materials made available for modeling are further constrained by system-theoretic principles, like being able to say what the states are and how the transitions are determined.
With this preamble, I return to develop my own account of formalization, with special attention to the kind of step that leads from the inchoate chaos of casual discourse to a well-founded discussion of formal models. A formalization step, of the incipient kind being considered here, has the peculiar property that one can say with some definiteness where it ends, since it leads precisely to a well-defined formal model, but not with any definiteness where it begins. Any attempt to trace the steps of formalization backward toward their ultimate beginnings can lead to an interminable multiplicity of open-ended explorations. In view of these circumstances, I will limit my attention to the frame of the present inquiry and try to sum up what brings me to this point.
With this preamble, I return to develop my own account of formalization, with special attention to the kind of step that leads from the inchoate chaos of casual discourse to a well-founded discussion of formal models. A formalization step, of the incipient kind being considered here, has the peculiar property that one can say with some definiteness where it ends, since it leads precisely to a well-defined formal model, but not with any definiteness where it begins. Any attempt to trace the steps of formalization backward toward their ultimate beginnings can lead to an interminable multiplicity of open-ended explorations. In view of these circumstances, I will limit my attention to the frame of the present inquiry and try to sum up what brings me to this point.
It begins like this: I ask whether it is possible to reason about inquiry in a way that leads to a productive end. I pose my question as an inquiry into inquiry, and I use the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}</math> to express the relationship between the present inquiry, <math>y_0\!</math>, and a generic inquiry, <math>y\!</math>. Then I propose a couple of components of inquiry, discussion and formalization, that appear to be worth investigating, expressing this proposal in the form <math>y >\!\!= \{ d, f \}</math>. Applying these components to each other, as must be done in the present inquiry, I am led to the current discussion of formalization, <math>y_0 = y \cdot y >\!\!= f \cdot d</math>.
It begins like this: I ask whether it is possible to reason about inquiry in a way that leads to a productive end. I pose my question as an inquiry into inquiry, and I use the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!</math> to express the relationship between the present inquiry, <math>y_0\!</math>, and a generic inquiry, <math>y\!</math>. Then I propose a couple of components of inquiry, discussion and formalization, that appear to be worth investigating, expressing this proposal in the form <math>y >\!\!= \{ d, f \}\!</math>. Applying these components to each other, as must be done in the present inquiry, I am led to the current discussion of formalization, <math>y_0 = y \cdot y >\!\!= f \cdot d\!</math>.
There is already much to question here. At least, so many repetitions of the same mysterious formula are bound to lead the reader to question its meaning. Some of the more obvious issues that arise are these:
There is already much to question here. At least, so many repetitions of the same mysterious formula are bound to lead the reader to question its meaning. Some of the more obvious issues that arise are these:
<ol style="list-style-type:decimal">
<ol style="list-style-type:decimal">
<li> The term ''generic inquiry'' is ambiguous. Its meaning in practice depends on whether the description of an inquiry as being generic is interpreted literally or merely as a figure of speech. In the literal case, the name <math>{}^{\backprime\backprime} y {}^{\prime\prime}</math> denotes a particular inquiry, <math>y \in Y\!</math>, one that is assumed to be prototypical in yet to be specified ways. In the figurative case, the name <math>{}^{\backprime\backprime} y {}^{\prime\prime}</math> is simply a variable that ranges over a collection <math>Y\!</math> of nominally conceivable inquiries.</li>
<li> The term ''generic inquiry'' is ambiguous. Its meaning in practice depends on whether the description of an inquiry as being generic is interpreted literally or merely as a figure of speech. In the literal case, the name <math>{}^{\backprime\backprime} y {}^{\prime\prime}\!</math> denotes a particular inquiry, <math>y \in Y\!</math>, one that is assumed to be prototypical in yet to be specified ways. In the figurative case, the name <math>{}^{\backprime\backprime} y {}^{\prime\prime}\!</math> is simply a variable that ranges over a collection <math>Y\!</math> of nominally conceivable inquiries.</li>
<li> First encountered, the recipe <math>y_0 = y \cdot y</math> appears to specify that the present inquiry is constituted by taking everything denoted by the most general concept of inquiry that the present inquirer can imagine and inquiring into it by means of the most general capacity for inquiry that this same inquirer can muster.</li>
<li> First encountered, the recipe <math>y_0 = y \cdot y\!</math> appears to specify that the present inquiry is constituted by taking everything denoted by the most general concept of inquiry that the present inquirer can imagine and inquiring into it by means of the most general capacity for inquiry that this same inquirer can muster.</li>
<li> Contemplating the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}</math> in the context of the subordination <math>y >\!\!= \{ d, f \}</math> and the successive containments <math>F \subseteq M \subseteq D</math>, the <math>y\!</math> that inquires into <math>y\!</math> is not restricted to examining <math>y \operatorname{'s}</math> immediate subordinates, <math>d\!</math> and <math>f\!</math>, but it can investigate any feature of <math>y \operatorname{'s}</math> overall context, whether objective, syntactic, interpretive, and whether definitive or incidental, and finally it can question any supporting claim of the discussion. Moreover, the question <math>y\!</math> is not limited to the particular claims that are being made here, but applies to the abstract relations and the general concepts that are invoked in making them. Among the many kinds of inquiry that suggest themselves, there are the following possibilities:</li>
<li> Contemplating the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!</math> in the context of the subordination <math>y >\!\!= \{ d, f \}\!</math> and the successive containments <math>F \subseteq M \subseteq D\!</math>, the <math>y\!</math> that inquires into <math>y\!</math> is not restricted to examining <math>y \operatorname{'s}\!</math> immediate subordinates, <math>d\!</math> and <math>f\!</math>, but it can investigate any feature of <math>y \operatorname{'s}\!</math> overall context, whether objective, syntactic, interpretive, and whether definitive or incidental, and finally it can question any supporting claim of the discussion. Moreover, the question <math>y\!</math> is not limited to the particular claims that are being made here, but applies to the abstract relations and the general concepts that are invoked in making them. Among the many kinds of inquiry that suggest themselves, there are the following possibilities:</li>
<ol style="list-style-type:lower-alpha">
<ol style="list-style-type:lower-alpha">
<li> Inquiry into propositions about application and equality. One may well begin with the forms of application and equality that are invoked in the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}</math> itself.</li>
<li> Inquiry into propositions about application and equality. One may well begin with the forms of application and equality that are invoked in the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!</math> itself.</li>
<li> Inquiry into application <math>(\cdot)</math>, for example, the way that the term <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime}</math> indicates the application of <math>y\!</math> to <math>y\!</math> in the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}</math>.</li>
<li> Inquiry into application <math>(\cdot)\!</math>, for example, the way that the term <math>{}^{\backprime\backprime} y \cdot y {}^{\prime\prime}\!</math> indicates the application of <math>y\!</math> to <math>y\!</math> in the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!</math>.</li>
<li> Inquiry into equality <math>(=)\!</math>, for example, the meaning of the equal sign in the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}</math>.</li>
<li> Inquiry into equality <math>(=)\!</math>, for example, the meaning of the equal sign in the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!</math>.</li>
<li> Inquiry into indices, for example, the significance of <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}</math> in <math>{}^{\backprime\backprime} y_0 {}^{\prime\prime}</math>.</li>
<li> Inquiry into indices, for example, the significance of <math>{}^{\backprime\backprime} 0 {}^{\prime\prime}\!</math> in <math>{}^{\backprime\backprime} y_0 {}^{\prime\prime}\!</math>.</li>
<li> Inquiry into terms, specifically, constants and variables. What are the functions of <math>{}^{\backprime\backprime} y {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} y_0 {}^{\prime\prime}</math> in this respect?</li>
<li> Inquiry into terms, specifically, constants and variables. What are the functions of <math>{}^{\backprime\backprime} y {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} y_0 {}^{\prime\prime}\!</math> in this respect?</li>
<li> Inquiry into decomposition or subordination, for example, as invoked by the sign <math>{}^{\backprime\backprime} >\!\!= {}^{\prime\prime}</math> in the formula <math>{}^{\backprime\backprime} y >\!\!= \{ d, f \} {}^{\prime\prime}</math>.</li>
<li> Inquiry into decomposition or subordination, for example, as invoked by the sign <math>{}^{\backprime\backprime} >\!\!= {}^{\prime\prime}\!</math> in the formula <math>{}^{\backprime\backprime} y >\!\!= \{ d, f \} {}^{\prime\prime}\!</math>.</li>
<li> Inquiry into containment or inclusion. In particular, examine the assumption that formalization <math>F</math>, mediation <math>M</math>, and discussion <math>D</math> are ordered as <math>F \subseteq M \subseteq D</math>, a claim that determines the chances that a formalization has an object, the degree to which a formalization can be carried out by means of a discussion, and the extent to which an object of formalization can be conveyed by a form of discussion.</li>
<li> Inquiry into containment or inclusion. In particular, examine the assumption that formalization <math>F\!</math>, mediation <math>M\!</math>, and discussion <math>D\!</math> are ordered as <math>F \subseteq M \subseteq D\!</math>, a claim that determines the chances that a formalization has an object, the degree to which a formalization can be carried out by means of a discussion, and the extent to which an object of formalization can be conveyed by a form of discussion.</li>
</ol></ol>
</ol></ol>
If inquiry begins in doubt, then inquiry into inquiry begins in doubt about doubt. All things considered, the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}</math> has to be taken as the first attempt at a description of the problem, a hypothesis about the nature of inquiry, or an image that is tossed out by way of getting an initial fix on the object in question. Everything in this account so far, and everything else that I am likely to add, can only be reckoned as hypothesis, whose accuracy, pertinence, and usefulness can be tested, judged, and redeemed only after the fact of proposing it and after the facts to which it refers have themselves been gathered up.
If inquiry begins in doubt, then inquiry into inquiry begins in doubt about doubt. All things considered, the formula <math>{}^{\backprime\backprime} y_0 = y \cdot y {}^{\prime\prime}\!</math> has to be taken as the first attempt at a description of the problem, a hypothesis about the nature of inquiry, or an image that is tossed out by way of getting an initial fix on the object in question. Everything in this account so far, and everything else that I am likely to add, can only be reckoned as hypothesis, whose accuracy, pertinence, and usefulness can be tested, judged, and redeemed only after the fact of proposing it and after the facts to which it refers have themselves been gathered up.
A number of problems present themselves due to the context in which the present inquiry is aimed to present itself. The hypothesis that suggests itself to one person, as worth exploring at a particular time, does not always present itself to another person as worth exploring at the same time, or even necessarily to the same person at another time. In a community of inquiry that extends beyond an isolated person and in a process of inquiry that extends beyond a singular moment in time, it is therefore necessary to consider the nature of the communication process that the discussion of inquiry in general and the discussion of formalization in particular need to invoke for their ultimate utility.
A number of problems present themselves due to the context in which the present inquiry is aimed to present itself. The hypothesis that suggests itself to one person, as worth exploring at a particular time, does not always present itself to another person as worth exploring at the same time, or even necessarily to the same person at another time. In a community of inquiry that extends beyond an isolated person and in a process of inquiry that extends beyond a singular moment in time, it is therefore necessary to consider the nature of the communication process that the discussion of inquiry in general and the discussion of formalization in particular need to invoke for their ultimate utility.
<li> A ''problem'' calls for a plan of action to resolve the difficulty that is present in it. This difficulty is associated with a difference between observations and intentions.
<li> A ''problem'' calls for a plan of action to resolve the difficulty that is present in it. This difficulty is associated with a difference between observations and intentions.
To express this diversity in a unified formula, both types of inquiry begin with a ''delta'' <math>(\Delta)</math>, a compact symbol that admits a spectrum of expansions: debt, difference, difficulty, discrepancy, dispersion, distribution, doubt, duplicity, or duty.</li>
To express this diversity in a unified formula, both types of inquiry begin with a ''delta'' <math>(\Delta)\!</math>, a compact symbol that admits a spectrum of expansions: debt, difference, difficulty, discrepancy, dispersion, distribution, doubt, duplicity, or duty.</li>
</ol>
</ol>
|}
|}
<math>\cdots</math>
<math>\cdots\!</math>
{| align="center" cellpadding="0" cellspacing="0" width="90%"
{| align="center" cellpadding="0" cellspacing="0" width="90%"
|}
|}
<math>\cdots</math>
<math>\cdots\!</math>
{| align="center" cellpadding="0" cellspacing="0" width="90%"
{| align="center" cellpadding="0" cellspacing="0" width="90%"
|}
|}
<math>\cdots</math>
<math>\cdots\!</math>
====1.3.7. Processus, Regressus, Progressus====
====1.3.7. Processus, Regressus, Progressus====
|}
|}
<math>\cdots</math>
<math>\cdots\!</math>
{| align="center" cellpadding="0" cellspacing="0" width="90%"
{| align="center" cellpadding="0" cellspacing="0" width="90%"
|}
|}
<math>\cdots</math>
<math>\cdots\!</math>
{| align="center" cellpadding="0" cellspacing="0" width="90%"
{| align="center" cellpadding="0" cellspacing="0" width="90%"
|}
|}
<math>\cdots</math>
<math>\cdots\!</math>
====1.3.8. Rondeau : Tempo di Menuetto====
====1.3.8. Rondeau : Tempo di Menuetto====
