Difference between revisions of "Directory:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1"
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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 | ||
| Line 145: | Line 145: | ||
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>. | ||
| Line 155: | Line 155: | ||
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. | ||
| Line 163: | Line 163: | ||
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==== | ||
| Line 185: | Line 185: | ||
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. | ||
| Line 207: | Line 207: | ||
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. | ||
| Line 221: | Line 221: | ||
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. | ||
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=====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> |
|} | |} | ||
|} | |} | ||
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{| 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> |
|} | |} | ||
|} | |} | ||
| Line 630: | Line 630: | ||
<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===== | ||
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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. | ||
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=====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. | ||
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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. | ||
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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. | ||
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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. | ||
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=====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. | ||
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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. | ||
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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. | ||
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{| 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>. |
|} | |} | ||
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{| 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>. |
|} | |} | ||
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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. | ||
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{| 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>, |
|} | |} | ||
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{| 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>. |
|} | |} | ||
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{| 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>, |
|} | |} | ||
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{| 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>. |
|} | |} | ||
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{| 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>. |
|} | |} | ||
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{| 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>. |
|} | |} | ||
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{| 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>. |
|} | |} | ||
| Line 1,018: | Line 1,018: | ||
{| 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>. |
|} | |} | ||
| Line 1,035: | Line 1,035: | ||
{| 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> |
|} | |} | ||
|} | |} | ||
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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> |
|} | |} | ||
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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. | ||
| Line 1,145: | Line 1,145: | ||
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. | ||
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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\! | + | 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" | ||
| Line 1,228: | Line 1,228: | ||
|} | |} | ||
| − | 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" | ||
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=====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. | ||
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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: | ||
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| 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: | ||
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| 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> |
|} | |} | ||
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{| 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> |
|} | |} | ||
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| <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> | ||
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| <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> |
|} | |} | ||
|} | |} | ||
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| <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> | ||
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| <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> |
|} | |} | ||
|} | |} | ||
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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===== | ||
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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. | ||
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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: | ||
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<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. | ||
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<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> | ||
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====1.3.7. Processus, Regressus, Progressus==== | ====1.3.7. Processus, Regressus, Progressus==== | ||
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====1.3.8. Rondeau : Tempo di Menuetto==== | ====1.3.8. Rondeau : Tempo di Menuetto==== | ||
Revision as of 13:40, 19 April 2012
• Contents • Part 1 • Part 2 • Part 3 • Part 4 • Part 5 • Part 6 • Part 7 • Part 8 • Appendices • References • Document History •
1. Introduction
1.1. Outline of the Project : Inquiry Into Inquiry
1.1.1. Problem
This research is oriented toward a single problem: What is the nature of inquiry? I intend to address crucial questions about the operation, organization, and computational facilitation of inquiry, taking inquiry to encompass the general trend of all forms of reasoning that lead to the features of scientific investigation as their ultimate development.
1.1.2. Method
How will I approach this problem about the nature of inquiry? The simplest answer is this: I will apply the method of inquiry to the problem of inquiry's nature.
This is the most concise and comprehensive answer I know, but it is likely to sound facetious at this point. On the other hand, if I did not actually use the method of inquiry that I describe as inquiry, how could the results possibly be taken seriously? Correspondingly, the questions of methodological self-application and self-referential consistency will be found at the center of this research.
In truth, it is fully possible that every means at inquiry's disposal will ultimately find application in resolving the problem of inquiry's nature. Other than a restraint to valid methods of inquiry — what those are is part of the question — there is no reason to expect a prior limitation on the range of methods that might be required.
This only leads up to the question of priorities: Which methods do I think it wise to apply first? In this project I will give preference to two kinds of technique, one analytic and one synthetic.
The prevailing method of research I will exercise throughout this work involves representing problematic phenomena in a variety of formal systems and then implementing these representations in a computational medium as a way of clarifying the more complex descriptions that evolve.
Aside from its theoretical core, this research is partly empirical and partly heuristic. Therefore, I expect that the various components of methodology will need to be applied in an iterative or even opportunistic fashion, working on any edge of research that appears to be ready at a given time. If forced to anticipate the likely developments, I would sketch the possibilities roughly as follows.
The methodology that underlies this approach has two components: The analytic component involves describing the performance and competence of intelligent agents in the medium of various formal systems. The synthetic component involves implementing these formal systems and the descriptions they express in the form of computational interpreters or language processors.
If everything goes according to the pattern I have observed in previous work, the principal facets of analytic and synthetic procedure will each be prefaced by its own distinctive phase of preparatory activity, where the basic materials needed for further investigation are brought together for comparative study. Taking these initial stages into consideration, I can describe the main modalities of this research in greater detail.
1.1.2.1. The Paradigmatic and Process-Analytic Phase
In this phase I describe the performance and competence of intelligent agents in terms of various formal systems. For aspects of an inquiry process that affect its dynamic or temporal performance I will typically use representations modeled on finite automata and differential systems. For aspects of an inquiry faculty that reflect its formal or symbolic competence I will commonly use representations like formal grammars, logical calculi, constraint-based axiom systems, and rule-based theories in association with different proof styles.
Paradigm. Generic example that reflects significant properties of a target class of phenomena, often derived from a tradition of study.
Analysis. Effective analysis of concepts, capacities, structures, and functions in terms of fundamental operations and computable functions.
Work in this phase typically proceeds according to the following recipe.
- Focus on a problematic phenomenon. This is a generic property or process that attracts one's interest, like intelligence or inquiry.
- Gather under consideration significant examples of concrete systems or agents that exhibit the property or process in question.
- Reflect on their common properties in a search for less obvious traits that might explain their more surprising features.
- Check these accounts of the phenomenon in one of several ways. For example, one might (a) search out other systems or situations in nature that manifest the critical traits, or (b) implement the putative traits in computer simulations. If these hypothesized traits generate (give rise to, provide a basis for) the phenomenon of interest, either in nature or on the computer, then one has reason to consider them further as possible explanations.
The last option of the last step already overlaps with the synthetic phase of work. Viewing this procedure within the frame of experimental research, it is important to recognize that computer programs can fill the role of hypotheses, testable (defeasible or falsifiable) construals of how a process is actually, might be possibly, or ought to be optimally carried out.
1.1.2.2. The Paraphrastic and Faculty-Synthetic Phase
The closely allied techniques of task analysis and software development that are known as step-wise refinement and top-down programming in computer science (Wirth 1976, 49, 303) have a long ancestry in logic and philosophy, going back to a strategy for establishing or discharging contextual definitions known as paraphrasis. All of these methods are founded on the idea of providing meaning for operational specifications, definitions in use, alleged descriptions, or incomplete symbols. No excessive generosity with the resources of meaning is intended, though. In practice, a larger share of the routine is spent detecting meaningless fictions rather than discovering meaningful concepts.
Paraphrasis. "A method of accounting for fictions by explaining various purported terms away" (Quine, in Van Heijenoort, 216). See also (Whitehead and Russell, in Van Heijenoort, 217–223).
Synthesis. Regard computer programs as implementations of hypothetical or postulated faculties. Within the framework of experimental research, programs can serve as descriptive, modal, or normative hypotheses, that is, conjectures about how a process is actually accomplished in nature, speculations as to how it might be done in principle, or explorations of how it might be done better in the medium of technological extensions.
For the purposes of this project, I will take paraphrastic definition to denote the analysis of formal specifications and contextual constraints to derive effective implementations of a process or its faculty. This is carried out by considering what the faculty in question is required to do in the many contexts it is expected to serve, and then by analyzing these formal specifications in order to design computer programs that fulfill them.
1.1.2.3. Reprise of Methods
In summary, the whole array of methods will be typical of the top-down strategies used in artificial intelligence research (AIR), involving the conceptual and operational analysis of higher-order cognitive capacities with an eye toward the modeling, grounding, and support of these faculties in the form of effective computer programs. The toughest part of this discipline is in making sure that one does "come down", that is, in finding guarantees that the analytic reagents and synthetic apparatus that one applies are actually effective, reducing the fat of speculation into something that will wash.
Finally, I ought to observe a hedge against betting too much on this or any neat arrangement of research stages. It should not be forgotten that the flourishing of inquiry evolves its own forms of organic integrity. No matter how one tries to tease them apart, the various tendrils of research tend to interleave and intertwine as they will.
1.1.3. Criterion
When is enough enough? What measure can I use to tell if my effort is working? What information is critical in deciding whether my exercise of the method is advancing my state of knowledge toward a solution of the problem?
Given that the problem is inquiry and the method is inquiry, the test of progress and eventual success is just the measure of any inquiry's performance. According to my current understanding of inquiry, and the tentative model of inquiry that will guide this project, the criterion of an inquiry's competence is how well it succeeds in reducing the uncertainty of its agent about its object.
What are the practical tests of whether the results of inquiry succeed in reducing uncertainty? Two gains are often cited: Successful results of inquiry provide the agent with increased powers of prediction and control as to how the object system will behave in given circumstances. If a common theme is desired, at the price of a finely equivocal thread, it can be said that the agent has gained in its power of determination. Hence, more certainty is exhibited by less hesitation, more determination is manifested by less vacillation.
1.1.4. Application
Where can the results be used? Knowledge about the nature of inquiry can be applied. It can be used to improve our personal competence at inquiry. It can be used to build software support for the tasks involved in inquiry.
If it is desired to articulate the loop of self-application a bit further, computer models of inquiry can be seen as building a two-way bridge between experimental science and software engineering, allowing the results of each to be applied in the furtherance of the other.
In yet another development, computer models of learning and reasoning form a linkage among cognitive psychology (the descriptive study of how we think), artificial intelligence (the prospective study of how we might think), and the logic of operations research (the normative study of how we ought to think in order to achieve the goals of reasoning).
1.2. Onus of the Project : No Way But Inquiry
At the beginning of inquiry there is nothing for me to work with but the actual constellation of doubts and beliefs that I have at the moment. Beliefs that operate at the deepest levels can be so taken for granted that they rarely if ever obtrude on awareness. Doubts that oppress in the most obvious ways are still known only as debits and droughts, as the absence of something, one knows not what, and a desire that obliges one only to try. Obscure forms of oversight provide an impulse to replenish the condition of privation but never out of necessity afford a sense of direction. One senses there ought to be a way out at once, or ordered ways to overcome obstruction, or organized or otherwise ways to obviate one's opacity of omission and rescue a secure motivation from the array of conflicting possibilities. In the roughest sense of the word, any action that does in fact lead out of this onerous state can be regarded as a form of "inquiry". Only later, in moments of more leisurely inquiry, when it comes down to classifying and comparing the manner of escapes that can be recounted, does it become possible to recognize the ways in which certain general patterns of strategy are routinely more successful in the long run than others.
1.2.1. A Modulating Prelude
If I aim to devise the kind of computational support that can give the greatest assistance to inquiry, then it must be able to come in at the very beginning, to be of service in the kinds of formless and negative conditions that I just described, and to help people navigate a way through the constellations of contingent, incomplete, and contradictory indications that they actually find themselves sailing under at present.
In the remainder of this section I will try to indicate as briefly as possible the nature of the problem that must be faced in this particular approach to inquiry, and to explain what a large share of the ensuing fuss will be directed toward clearing up.
Toward the end of this discussion I will be using highly concrete mathematical models, or very specific families of combinatorial objects, to represent the abstract structures of experiential sequences that agents pass through. If these primitive and simplified models are to be regarded as something more than mere toys, and if the relations of particular experiences to particular models, along with the structural relationships that exist within the field of experiences and again within the collection of models, are not to be dismissed as category confusions, then I will need to develop a toolbox of logical techniques that can be used to justify these constructions. The required technology of categorical and relational notions will be developed in the process of addressing its basic task: To show how the same conceptual categories can be applied to materials and models of experience that are radically diverse in their specific contents and peculiar to the states of the particular agents to which they attach.
1.2.2. A Fugitive Canon
The principal difficulties associated with this task appear to spring from two roots.
First, there is the issue of computational mediation. In using the sorts of sequences that computers go through to mediate discussion of the sorts of sequences that people go through, it becomes necessary to re-examine all of the facilitating assumptions that are commonly taken for granted in relating one human experience to another, that is, in describing and building structural relationships among the experiences of human agents.
Second, there is the problem of representing the general in the particular. How is it possible for the most particular imaginable things, namely, the transient experiential states of agents, to represent the most general imaginable things, namely, the agents' own conceptions of the abstract categories of experience?
Finally, not altogether as an afterthought, there is a question that binds these issues together. How does it make sense to apply one's individual conceptions of the abstract categories of experience, not only to the experiences of oneself and others, but in points of form to compare them with the structures present in mathematical models?
1.3. Option of the Project : A Way Up To Inquiry
I begin with an informal examination of the concept of inquiry. This section takes as its subjects the supposed faculty of inquiry in general and the present inquiry into inquiry in particular, and attempts to analyze them in relation to each other on formal principles alone.
The initial set of concepts I need to get discussion started are few. Assuming that a working set of ideas can be understood on informal grounds at the outset, I anticipate being able to formalize them to a greater degree as the project gets under way. Inquiry in general will be described as encompassing particular inquiries. Particular forms of inquiry, regarded as phenomenal processes, will be analyzed in terms of simpler kinds of phenomenal processes.
As a phenomenon, a particular way of doing inquiry is regarded as embodied in a faculty of inquiry, as possessed by an agent of inquiry. As a process, a particular example of inquiry is regarded as extended in time through a sequence of states, as experienced by its ongoing agent. It is envisioned that an agent or faculty of any generically described phenomenal process, inquiry included, could be started off from different initial states and would follow different trajectories of subsequent states, and yet there would be a recognizable quality or abstractable property that justifies invoking the name of the genus.
The steps of this analysis will be annotated below by making use of the following conventions. Lower case letters denote phenomena, processes, or faculties under investigation. Upper case letters denote classes of the same sorts of entities. Special use is made of the following symbols:
- \[Y\!\] = genus of inquiry,
- \[y\!\] = generic inquiry,
- \[y_0\!\] = present inquiry.
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:
- \[f \cdot g\!\]
A notation of the form
- \[f >\!\!= g\]
indicates that \(f\!\) is greater than or equal to \(g\!\) in a decompositional series, in other words, that \(f\!\) possesses \(g\!\) as a component.
The coset notation
- \[F \cdot G\!\]
indicates a class of faculties of the form
- \[f \cdot g\!\],
with \(f\!\) in \(F\!\) and \(g\!\) in \(G\!\).
Notations like
- \[\{ ? \} ~,~ \{ ? , ? \} ~,~ \{ ? , ? , ?\} ~,~ \ldots\!\]
serve as proxies for unknown components and indicate tentative analyses of faculties in question.
1.3.1. Initial Analysis of Inquiry : Allegro Aperto
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.
- \[y = \{ ? \}\!\]
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.
- \[y_0 = y \cdot y = \{ ? \} \{ ? \}\!\]
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.
- \[y >\!\!= \{ ? , ? \}\!\]
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.
- \[y >\!\!= \{ \operatorname{discussion} , \operatorname{formalization} \}\!\]
In accord with this plan, the body of this section is devoted to a discussion of formalization.
- \[y_0 = y \cdot y >\!\!= \{ d , f \} \{ d , f \} >\!\!= \{ f \} \{ d \}\!\]
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.
- \[y_0 = y \cdot y >\!\!= \{ d , f \} \{ d , f \} >\!\!= \{ d \} \{ d \}\!\]
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.
There's a catch here that applies to all living creatures: In order to keep talking one has to keep living. This brings discussion back to its role in inquiry, considered as an adaptation of living creatures designed to help them deal with their not so virtual environments. If discussion is constrained to the envelope of life and required to contribute to the trend of inquiry, instead of representing a kind of internal opposition, then it must be possible to tighten up the loose account and elevate the digressionary narrative into a properly directed inquiry. This brings an end to my initial discussion of discussion.
1.3.3. Discussion of Formalization : General Topics
Because this project makes constant use of formal models of phenomenal processes, it is appropriate at this point to introduce the understanding of formalization that I will use throughout this work and to preview a concrete example of its application.
1.3.3.1. A Formal Charge
An introduction to the topic of formalization, if proper, is obliged to begin informally. But it will be my constant practice to keep a formal eye on the whole proceedings. What this form of observation reveals must be kept silent for the most part at first, but I see no rule against sharing with the reader the general order of this watch:
- Examine every notion of the casual intuition that enters into the informal discussion and inquire into its qualifications as a potential candidate for formalization.
- Pay special attention to the nominal operations that are invoked to substantiate each tentative explanation of a critically important process. Often, but not infallibly, these can be detected appearing in the guise of "-ionized" terms, words ending in "-ion" that typically connote both a process and its result.
- Ask yourself, with regard to each postulant faculty in the current account, explicitly charged or otherwise, whether you can imagine any recipe, any program, any rule of procedure for carrying out the form, if not the substance, of what it does, or an aspect thereof.
1.3.3.2. A Formalization of Formalization?
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.
- \[y_0 = y \cdot y >\!\!= \{ d , f \} \{ d , f \} >\!\!= \{ f \} \{ f \}\!\]
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.
Unlike a mechanical interpreter meeting with the declaration of an undefined term for the very first time, the human reader of this text has the advantage of a prior acquaintance with almost every term that might conceivably enter into informal discussion. And formalization is a stock term widely traded in the forums of ordinary and technical discussion, so the reader is bound to have met with it in the context of practical experience and to have attached a personal concept to it. Therefore, this inquiry into formalization begins with a writer and a reader in a state of limited uncertainty, each attaching a distribution of meanings in practice to the word formalization, but uncertain whether their diverse spectra of associations can presently constitute or eventually converge to compatible arrays of effective meaning.
To review: The concept of formalization itself is an item of informal discussion that might be investigated as a candidate for formalization. For each aspect or component of the formalization process that I plan to transport across the semi-permeable threshold from informal to formal discussion, the reader has permission to challenge it, plus an open invitation to question every further process that I mention as a part of its constitution, and to ask with regard to each item whether its registration has cleared up the account in any measure or merely rung up a higher charge on the running bill of fare.
The reader can follow this example with every concept that I mention in the explanation of formalization, and again in the larger investigation of inquiry, and be assured that it is has not often slipped my attention to at least venture the same, though a delimitation of each exploration in its present state of completion would be far too tedious and tenuous to escape expurgation.
1.3.3.3. A Formalization of Discussion?
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.
- \[F \subseteq D\!\]
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.
- \[F \subseteq M \subseteq D\!\]
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?
- \[y_0 = y \cdot y >\!\!= \{ d , f \} \{d , f \} >\!\!= \{ d \} \{ f \}\!\]
- 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.
In case there is difficulty with the meaning of the word meaning, I replace its use with references to a system of interpretation (SOI), a technical concept that will be increasingly formalized as this project proceeds. Thus, the writer's job description is reformulated as follows.
- The writer's task is not to create a system of interpretation (SOI) from nothing, but to construct a relation from the typical SOIs that are available in ordinary discourse to the particular SOIs that are intended to be the effects of a particular discussion.
This assignment begins with an informal system of interpretation (SOI1), and builds a relation from it to another system of interpretation (SOI2). The first is an informal SOI that amounts to a shared resource of writer and reader. The latter is a system of meanings in practice that is the current object of the writer's intention to recommend for the reader's consideration and, hopefully, edification. In order to have a compact term for highlighting the effects of a discussion that builds a relation between SOI's, I will call this aspect of the process narration.
It is the writer's ethical responsibility to ensure that a discourse is potentially edifying with respect to the reader's current SOI, and the reader's self-interest to evaluate whether a discourse is actually edifying from the perspective of the reader's present SOI.
Formally, the relation that the writer builds from SOI to SOI can always be cast or recast as a three-place relation, one whose staple element of structure is an ordered or indexed triple. One component of each triple is anchored in the interpreter of the moment, and the other two form a connection with the source and target SOIs of the current assignment.
Once this relation is built, a shift in the attention of any interpreter or a change in the present focus of discourse can leave the impression of a transformation taking place from SOI1 to SOI2, but this is more illusory (or allusory) than real. To be more precise, this style of transformation takes place on a virtual basis, and need not have the substantive impact (or import) that a substantial replacement of one SOI by another would imply. For a writer to affect a reader in this way would simply not be polite. A moment's consideration of the kinds of SOI-building worth having leads me to enumerate a few characteristics of polite discourse or considerate discussion.
If this form of SOI-building narrative is truly intended to edify and educate, whether pursued in monologue or dialogue fashion, then its action cannot be forcibly to replace the meanings in practice a sign already has with others of an arbitrary nature, but freely to augment the options for meaning and powers for choice in the resulting SOI.
As conditions for the possibility of considerate but significant narration, there are a couple of requirements placed on the writer and the reader. Considerate narration, constructing a relation from SOI to SOI in a politic fashion, cannot operate in an infectious or addictive manner, invading a SOI like a virus or a trojan horse, but must transfer its communication into the control of the receiving SOI. Significant communication, in which the receiving SOI is augmented by options for meaning and powers for choice that it did not have before, requires a SOI on the reader's part that is extensible in non-trivial ways.
At this point, the discussion has touched on a topic, in one of its manifold aspects, that it will encounter repeatedly, under a variety of aspects, throughout this work. In recognition of this circumstance, and to prepare the way for future discussion, it seems like a good idea to note a few of the aliases that this protean topic can be found lurking under, and to notice the logical relationships that exist among its several different appearances.
On several occasions, this discussion of inquiry will arrive at a form of aesthetic deduction, in general terms, a piece of reasoning that ends with a design recommendation, in this case, where an analysis of the general purposes and interests of inquiry leads to the conclusion that a certain property of discussion is an admirable one, and that the quality in question forms an essential part of the implicit value system that is required to guide inquiry and make it what it is meant to be, a method for advancing toward desired forms of knowledge. After a collection of admirable qualities has been recognized as cohering together into a unity, it becomes natural to ask: What is the underlying reality that inheres in these qualities, and what are the logical relations that bind them together into the qualifications of inquiry and a definition of exactly what is desired for knowledge?
1.3.3.4. A Concept of Formalization
The concept of formalization is intended to cover the whole collection of activities that serve to build a relation between casual discussions, those that take place in the ordinary context of informal discourse, and formal discussions, those that make use of completely formalized models. To make a long story short, formalization is the narrative operation or active relation that construes the situational context in the form of a definite text. The end product that results from the formalization process is analogous to a snapshot or a candid picture, a relational or functional image that captures an aspect of the casual circumstances.
Relations between casual and formal discussion are often treated in terms of a distinction between two languages, the meta-language and the object language, linguistic systems that take complementary roles in filling out the discussion of interest. In the usual approach, issues of formalization are addressed by postulating a distinction between the meta-language, the descriptions and conceptions from ordinary language and technical discourse that can be used without being formalized, and the object language, the domain of structures and processes that can be studied as a completely formalized object.
1.3.3.5. A Formal Approach
I plan to approach the issue of formalization from a slightly different angle, proceeding through an analysis of the medium of interpretation and developing an effective conception of interpretive frameworks or interpretive systems. This concept refers to any organized system of interpretive practice, ranging from those used in everyday speech, to the ones that inform technical discourse, to the kinds of completely formalized symbol systems that one can safely regard as mathematical objects. Depending on the degree of objectification that it possesses from one's point of view, the same system of conduct can be variously described as an interpretive framework (IF), interpretive system (IS), interpretive object (IO), or object system (OS). These terms are merely suggestive — no rigid form of classification is intended.
Many times, it is convenient to personify the interpretive organization as if it were embodied in the actions of a typical user of the framework or a substantive agent of the system. I will call this agent the interpreter of the moment. At other times, it may be necessary to analyze the action of interpretation more carefully. At these times, it is important to remember that this form of personification is itself a figure of speech, one that has no meaning outside a fairly flexible interpretive framework. Thus, the term interpreter can be a cipher analogous to the terms X, unknown, or to whom it may concern appearing in a system of potentially recursive constraints. As such, it serves in the role of an indeterminate symbol, in the end to be solved for a fitting value, but in the mean time conveying an appearance of knowledge in a place where very little is known about the subject itself.
A meta-language corresponds to what I call an interpretive framework. Besides a set of descriptions and conceptions, it embodies the whole collective activity of unexamined structures and automatic processes that are trusted by agents at a given moment to make its employment meaningful in practice. An interpretive framework is best understood as a form of conduct, that is, a comprehensive organization of related activities.
In use, an interpretive framework operates to contain activity and constrain the engagement of agents to certain forms of active involvement and dynamic participation, and manifests itself only incidentally in the manipulation of compact symbols and isolated instruments. In short, though a framework may have pointer dials and portable tools attached to it, it is usually too incumbent and cumbersome to be easily moved on its own grounds, at least, it rests beyond the scope of any local effort to do so.
An interpretive framework (IF) is set to work when an agent or agency becomes involved in its organization and participates in the forms of activity that make it up. Often, an IF is founded and persists in operation long before any participant is able to reflect on its structure or to post a note of its character to the constituting members of the framework. In some cases, the rules of the IF in question forbid the act of reflecting on its form. In practice, to the extent that agents are actively involved in filling out the requisite forms and taking part in the step by step routines of the IF they may have little surplus memory capacity to memorandize the big picture even when it is permitted in principle.
An object language is a special case of the kind of formal system that is so completely formalized that it can be regarded as combinatorial object, an inactive image of a form of activity that is meant for the moment to be studied rather than joined.
The supposition that there is a meaningful and well-defined distinction between object language and meta-language ordinarily goes unexamined. This means that the assumption of a distinction between them is de facto a part of the meta-language and not even an object of discussion in the object language. A slippery slope begins here. A failure to build reflective capacities into an interpretive framework can let go unchallenged the spurious opinion that presumes there can be only one way to draw a distinction between object language and meta-language.
The next natural development is to iterate the supposed distinction. This represents an attempt to formalize and thereby objectify parts of the meta-language, precipitating it like a new layer of pearl or crystal from the resident medium, and thereby preparing the decantation of a still more pervasive and ethereal meta-meta-language. The successive results of this process can have a positivistically intoxicating effect on the human intellect. But a not so happy side-effect leads the not quite mindful cerebration up and down a blind alley, chasing the specious impression that just beyond the realm of objective nature there lies a unique fractionation of permeabilities and a permanent hierarchy of effabilities in language.
The grounds of discussion I am raking over here constellate a rather striking scene, especially for something intended as a neutral backdrop. Unlike other concerns, the points I am making seem obvious to all reasonable people at the outset of discussion, and yet the difficulties that follow as inquiry develops get muddier and more grating the more one probes and stirs them up. A large measure of the blame, I think, can be charged to a misleading directive that people derive from the epithet meta, leading them to search for higher and higher levels of meaning and truth, on beyond language, on beyond any conceivable system of signs, and on beyond sense. Prolonged use of the prefix meta leads people to act as if a meta-language were step outside of ordinary language, or an artificial platform constructed above and beyond natural language, and then they forget that formal models are developments internal to the informal context. For this reason among others, I suggest replacing talk about rigidly stratified object languages and meta-languages with talk about contingent interpretive frameworks.
To avoid the types of cul-de-sac (cultist act) encountered above, I am taking some pains to ensure a reflective capacity for the interpretive frameworks I develop in this project. This is a capacity that natural languages always assume for themselves, instituting specialized discourses as developments that take place within their frame and not as constructs that lie beyond their scope. Any time the levels of recursive discussion become too involved to manage successfully, one needs to keep available the resource of instant wisdom, the modest but indispensable quantum of ready understanding, that restores itself on each return to the ordinary universe.
From this angle of approach, let us try to view afresh the manner of drawing distinctions between various levels of formalization in language. Once again, I begin in the context of ordinary discussion, and if there is any distinction to be drawn between objective and instrumental languages then it must be possible to describe it within the frame of this informally discursive universe.
1.3.3.6. A Formal Development
The point of view I take on the origin and development of formal models is that they arise with agents retracing structures that already exist in the context of informal activity, until gradually the most relevant and frequently reinforced patterns become emphasized and emboldened enough to continue their development as nearly autonomous styles, in brief, as genres growing out of a particular paradigm.
Taking the position that formal models develop within the framework of informal discussion, the questions that become important to ask of a prospective formal model are (1) whether it highlights the structure of its supporting context in a transparent form of emphasis and a relevant reinforcement of salient features, and (2) whether it reveals the active ingredients of its source materials in a critically reflective recapitulation or an analytically representative recipe, or (3) whether it insistently obscures what little fraction of its domain it manages to cover.
1.3.3.7. A Formal Persuasion
An interpretive system can be taken up with very little fanfare, since it does not enjoin one to declare undying allegiance to a particular point of view or to assign each piece of text in view to a sovereign territory, but only to entertain different points of view on the use of symbols. The chief design consideration for an interpretive system is that it must never function as a virus or addiction. Its suggestions must always be, initially and finally, purely optional adjunctions to whatever interpretive framework was already in place before it installed itself on the scene. Interpretive systems are not constituted in the faith that anything nameable will always be dependable, nor articulated in fixed principles that determine what must be doubted and what must not, but rest only in a form of self-knowledge that recognizes the doubts and beliefs that one actually has at each given moment.
Before this project is done I will need to have developed an analytic and computational theory of interpreters and interpretive frameworks. In the aspects of this theory that I can anticipate at this point, an interpreter or interpretive framework is exemplified by a collective activity of symbol-using practices like those that might be found embodied in a person, a community, or a culture. Each one forms a moderately free and independent perspective, with no objective rankings of supremacy in practice that all interpretive frameworks are likely to support at any foreseeable moment in their fields of view. Of course, each interpreter initially enters discussion operating as if its own perspective were meta in comparison to all the others, but a well-developed interpretive framework is likely to have acquired the notion and taken notice of the fact that this is not likely to be a universally shared opinion (USO).
1.3.4. Discussion of Formalization : Concrete Examples
The previous section outlined a variety of general issues surrounding the concept of formalization. The following section will plot the specific objectives of this project in constructing formal models of intellectual processes. In this section I wish to take a breather between these abstract discussions in order to give their main ideas a few points of contact with terra firma. To do this, I examine a selection of concrete examples, artificially constructed to approach the minimum levels of non-trivial complexity, that are intended to illustrate the kinds of mathematical objects I have in mind using as formal models.
1.3.4.1. Formal Models : A Sketch
To sketch the features of the modeling activity that are relevant to the immediate purpose: The modeler begins with a phenomenon of interest or a process of interest (POI) and relates it to a formal model of interest (MOI), the whole while working within a particular interpretive framework (IF) and relating the results from one system of interpretation (SOI) to another, or to a subsequent development of the same SOI.
The POIs that define the intents and the purposes of this project are the closely related processes of inquiry and interpretation, so the MOIs that must be formulated are models of inquiry and interpretation, species of formal systems that are even more intimately bound up than usual with the IFs employed and the SOIs deployed in their ongoing development as models.
Since all of the interpretive systems and all of the process models that are being mentioned here come from the same broad family of mathematical objects, the different roles that they play in this investigation are mainly distinguished by variations in their manner and degree of formalization:
- The typical POI comes from natural sources and casual conduct. It is not formalized in itself but only in the form of its image or model, and just to the extent that aspects of its structure and function are captured by a formal MOI. But the richness of any natural phenomenon or realistic process seldom falls within the metes and bounds of any final or finite formula.
- Beyond the initial stages of investigation, the MOI is postulated as a completely formalized object, or is quickly on its way to becoming one. As such, it serves as a pivotal fulcrum and a point of application poised between the undefined reaches of phenomena and noumena, respectively, terms that serve more as directions of pointing than as denotations of entities. What enables the MOI to grasp these directions is the quite felicitous mathematical circumsatnce that there can be well-defined and finite relations between entities that are infinite and even indefinite in themselves. Indeed, exploiting this handle on infinity is the main trick of all computational models and effective procedures. It is how a finitely informed creature can "make infinite use of finite means". Thus, my reason for calling the MOI cardinal or pivotal is that it forms a model in two senses, loosely analogical and more strictly logical, integrating twin roles of the model concept in a single focus.
- Finally, the IFs and the SOIs always remain partly out of sight, caught up in various stages of explicit notice between casual informality and partial formalization, with no guarantee or even much likelihood of a completely articulate formulation being forthcoming or even possible. Still, it is usually worth the effort to try lifting one edge or another of these frameworks and backdrops into the light, at least for a time.
1.3.4.2. Sign Relations : A Primer
To the extent that their structures and functions can be discussed at all, it is likely that all of the formal entities that are destined to develop in this approach to inquiry will be instances of a class of three-place relations called sign relations. At any rate, all of the formal structures that I have examined so far in this area have turned out to be easily converted to or ultimately grounded in sign relations. This class of triadic relations constitutes the main study of the pragmatic theory of signs, a branch of logical philosophy devoted to understanding all types of symbolic representation and communication.
There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry. In fact, the correspondence between the two studies exhibits so many parallels and coincidences that it is often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve. In other words, inquiry, "thinking" in its best sense, "is a term denoting the various ways in which things acquire significance" (Dewey). Thus, there is an active and intricate form of cooperation that needs to be appreciated and maintained between these converging modes of investigation. Its proper character is best understood by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject which the theory of signs is specialized to treat from structural and comparative points of view.
Because the examples in this section have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Still, these examples have subtleties of their own, and their careful treatment will serve to illustrate important issues in the general theory of signs.
Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their interpretive practice that involves the use of the following nouns and pronouns: “Ann”, “Bob”, “I”, “you”.
- The object domain of this discussion fragment is the set of two people \(\{ \text{Ann}, \text{Bob} \}\!\).
- The syntactic domain or the sign system of their discussion is limited to the set of four signs \(\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}\!\).
In their discussion, Ann and Bob are not only the passive objects of nominative and accusative references but also the active interpreters of the language that they use. The system of interpretation (SOI) associated with each language user can be represented in the form of an individual three-place relation called the sign relation of that interpreter.
Understood in terms of its set-theoretic extension, a sign relation \(L\!\) is a subset of a cartesian product \(O \times S \times I\!\). Here, \(O, S, I\!\) are three sets that are known as the object domain, the sign domain, and the interpretant domain, respectively, of the sign relation \(L \subseteq O \times S \times I\!\).
In general, the three domains of a sign relation can be any sets whatsoever, but the kinds of sign relations that are contemplated in a computational framework are usually constrained to having \(I \subseteq S\!\). In this case, interpretants are just a special variety of signs, and this makes it convenient to lump signs and interpretants together into a single class called the syntactic domain. In the forthcoming examples, \(S\!\) and \(I\!\) are identical as sets, so the very same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains \(O\!\), \(S\!\), \(I\!\) for a given sign relation \(L\!\), one may refer to this set as the World of \(L\!\) and write \(W = W_L = O \cup S \cup I\!\).
To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief as possible as the examples become more complicated, it serves to introduce the following general notations:
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\(\begin{array}{ccl} O & = & \text{Object Domain} \'"`UNIQ-MathJax19-QINU`"'. Looking to the denotative aspects of the present example, various rows of the Tables specify that \(\text{A}\!\) uses \({}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!\) to denote \(\text{A}\!\) and \({}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!\) to denote \(\text{B}\!\), whereas \(\text{B}\!\) uses \({}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!\) to denote \(\text{B}\!\) and \({}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!\) to denote \(\text{A}\!\). It is utterly amazing that even these impoverished remnants of natural language use have properties that quickly bring the usual prospects of formal semantics to a screeching halt. The other dyadic aspects of semantics that might be considered concern the reference that a sign has to its interpretant and the reference that an interpretant has to its object. As before, either type of reference can be multiple, unique, or empty in its collection of terminal points, and both can be formalized as different types of dyadic relations that are obtained as planar projections of the triadic sign relations. The connection that a sign makes to an interpretant is called its connotation. In the general theory of sign relations, this aspect of semantics includes the references that a sign has to affects, concepts, impressions, intentions, mental ideas, and to the whole realm of an agent's mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct. This complex ecosystem of references is unlikely ever to be mapped in much detail, much less completely formalized, but the tangible warp of its accumulated mass is commonly alluded to as the connotative import of language. Given a particular sign relation \(L\!\), the dyadic relation that constitutes the connotative component of \(L\!\) is denoted \(\operatorname{Con}(L)\!\). The bearing that an interpretant has toward a common object of its sign and itself has no standard name. If an interpretant is considered to be a sign in its own right, then its independent reference to an object can be taken as belonging to another moment of denotation, but this omits the mediational character of the whole transaction. Given the service that interpretants supply in furnishing a locus for critical, reflective, and explanatory glosses on objective scenes and their descriptive texts, it is easy to regard them as annotations both of objects and of signs, but this function points in the opposite direction to what is needed in this connection. What does one call the inverse of the annotation function? More generally asked, what is the converse of the annotation relation? In light of these considerations, I find myself still experimenting with terms to suit this last-mentioned dimension of semantics. On a trial basis, I refer to it as the ideational, the intentional, or the canonical component of the sign relation, and I provisionally refer to the reference of an interpretant sign to its object as its ideation, its intention, or its conation. Given a particular sign relation \(L\!\), the dyadic relation that constitutes the intentional component of \(L\!\) is denoted \(\operatorname{Int}(L)\!\). A full consideration of the connotative and intentional aspects of semantics would force a return to difficult questions about the true nature of the interpretant sign in the general theory of sign relations. It is best to defer these issues to a later discussion. Fortunately, omission of this material does not interfere with understanding the purely formal aspects of the present example. Formally, these new aspects of semantics present no additional problem: The connotative component of a sign relation L can be formalized as its dyadic projection on the plane generated by the sign domain and the interpretant domain, defined as follows: \[\operatorname{Con}(L) = \operatorname{proj}_{SI} L = L_{SI} = \{ (s, i) \in S \times I : (o, s, i) \in L ~\text{for some}~ o \in O \}\!\]. The intentional component of semantics for a sign relation \(L\!\), or its second moment of denotation, is adequately captured by its dyadic projection on the plane generated by the object domain and interpretant domain, defined as follows: \[\operatorname{Int}(L) = \operatorname{proj}_{OI} L = L_{OI} = \{ (o, i) \in O \times I : (o, s, i) \in L ~\text{for some}~ s \in S \}\!\]. As it happens, the sign relations \(L_\text{A}\!\) and \(L_\text{B}\!\) in the present example are fully symmetric with respect to exchanging signs and interpretants, so all of the structure of \((L_\text{A})_{OS}\!\) and \((L_\text{B})_{OS}\!\) is merely echoed in \((L_\text{A})_{OI}\!\) and \((L_\text{B})_{OI}\!\), respectively. Note on notation. When there is only one sign relation \(L_J = L(J)\!\) associated with a given interpreter \(J\!\), it is convenient to use the following forms of abbreviation:
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