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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.

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'''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~~).

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'''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.

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=====1.1.2.3. Reprise of Methods=====

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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.

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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.

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===1.2. Onus of the Project : No Way But Inquiry===

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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.

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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====

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# Examine every notion of the casual intuition that enters into the informal discussion and inquire into its qualifications as a potential candidate for formalization.

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# 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.

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# 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.

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:: <math>F \subseteq D\!</math>

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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.

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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>

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# 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.

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# 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.

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# 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.

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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 [[triadic relation|three-place relation]]s called ''[[sign relation]]s''. 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.

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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.

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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.

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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”.

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:* The ''object domain'' of this discussion fragment is the set of two people <math>\{ \text{Ann}, \text{Bob} \}\!</math>.

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:* The ''object domain'' of this discussion fragment is the set of two people <math>\{ \text{Ann}, \text{Bob} \}.\!</math>

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:* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs <math>\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}\!</math>.

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:* The ''syntactic domain'' or the ''sign system'' of their discussion is limited to the set of four signs <math>\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}.\!</math>

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 [[triadic relation|three-place relation]] called the ''[[sign relation]]'' of that interpreter.

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Understood in terms of its ''~~[[set theory|~~set-theoretic~~]] [[~~extension ~~(logic)|extension]]~~'', a sign relation <math>L\!</math> is a ''[[subset]]'' of a ''[[cartesian product]]'' <math>O \times S \times I\!</math>. Here, <math>O, S, I\!</math> are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation <math>L \subseteq O \times S \times I\!</math>.

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Understood in terms of its ''set-theoretic extension'', a sign relation <math>L\!</math> is a ''subset'' of a ''cartesian product'' <math>O \times S \times I.\!</math> Here, <math>O, S, I\!</math> are three sets that are known as the ''object domain'', the ''sign domain'', and the ''interpretant domain'', respectively, of the sign relation <math>L \subseteq O \times S \times I.\!</math>

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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 <math>I \subseteq S\!</math>. 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, <math>S\!</math> and <math>I\!</math> 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 <math>O~~\!</math>~~, ~~<math>~~S~~\!</math>~~, ~~<math>~~I\!</math> for a given sign relation <math>L\!</math>, one may refer to this set as the ''World'' of <math>L\!</math> and write <math>W = W_L = O \cup S \cup I\!</math>.

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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 <math>I \subseteq S\!</math>. 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, <math>S\!</math> and <math>I\!</math> 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 <math>O, S, I\!</math> for a given sign relation <math>L,\!</math> one may refer to this set as the ''World'' of <math>L\!</math> and write <math>W = W_L = O \cup S \cup I.\!</math>

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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In the present example, <math>S = I = \text{Syntactic Domain}\!</math>.

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The sign relation associated with a given interpreter <math>J\!</math> is denoted <math>L_J\!</math> or <math>L(J)\!</math>. Tables 1 and 2 give the sign relations associated with the interpreters <math>\text{A}\!</math> and <math>\text{B}\!</math>, respectively, putting them in the form of ''[[relational ~~database]]s~~''. Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)\!</math> that make up the corresponding sign relations, <math>L_\text{A}, L_\text{B} \subseteq O \times S \times I\!</math>. It is often tempting to use the same names for objects and for relations involving these objects, but it is best to avoid this in a first approach, taking up the issues that this practice raises after the less problematic features of these relations have been treated.

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The sign relation associated with a given interpreter <math>J\!</math> is denoted <math>L_J\!</math> or <math>{L(J)}.\!</math> Tables 1 and 2 give the sign relations associated with the interpreters <math>\text{A}\!</math> and <math>\text{B},\!</math> respectively, putting them in the form of ''relational databases''. Thus, the rows of each Table list the ordered triples of the form <math>(o, s, i)\!</math> that make up the corresponding sign relations, <math>L_\text{A}, L_\text{B} \subseteq O \times S \times I.\!</math> It is often tempting to use the same names for objects and for relations involving these objects, but it is best to avoid this in a first approach, taking up the issues that this practice raises after the less problematic features of these relations have been treated.

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

|+ style="height:30px" | <math>\text{Table 1.} ~~ \text{Sign Relation of Interpreter A}\!</math>

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

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|- style="height:40px; background:ghostwhite"

| width="33%" | <math>\text{Object}\!</math>

| width="33%" | <math>\text{Sign}\!</math>

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

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|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Sign Relation of Interpreter B}\!</math>

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|+ style="height:30px" | <math>\text{Table 2.} ~~ \text{Sign Relation of Interpreter B}~\!</math>

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

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|- style="height:40px; background:ghostwhite"

| width="33%" | <math>\text{Object}\!</math>

| width="33%" | <math>\text{Sign}\!</math>

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One aspect of semantics is concerned with the reference that a sign has to its object, which is called its ''denotation''. For signs in general, neither the existence nor the uniqueness of a denotation is guaranteed. Thus, the denotation of a sign can refer to a plural, a singular, or a vacuous number of objects. In the pragmatic theory of signs, these references are formalized as certain types of dyadic relations that are obtained by projection from the triadic sign relations.

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The dyadic relation that constitutes the ''denotative component'' of a sign relation <math>L\!</math> is denoted <math>\operatorname{Den}(L)\!</math>. Information about the denotative component of semantics can be derived from <math>L\!</math> by taking its ''dyadic projection'' on the plane that is generated by the object domain and the sign domain, indicated by any one of the equivalent forms, <math>\operatorname{proj}_{OS} L\!</math>, <math>L_{OS}\!</math>, or <math>L_{12}\!</math>, and defined as follows:

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The dyadic relation that constitutes the ''denotative component'' of a sign relation <math>L\!</math> is denoted <math>\operatorname{Den}(L).\!</math> Information about the denotative component of semantics can be derived from <math>L\!</math> by taking its ''dyadic projection'' on the plane that is generated by the object domain and the sign domain, indicated by any one of the equivalent forms, <math>\operatorname{proj}_{OS} L,\!</math> <math>L_{OS},\!</math> or <math>L_{12},\!</math> and defined as follows:

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: <math>\operatorname{Den}(L) = \operatorname{proj}_{OS} L = L_{OS} = \{ (o, s) \in O \times S : (o, s, i) \in L ~\text{for some}~ i \in I \}\!</math>.

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{| align="center" cellspacing="6" width="90%"

+

|

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<math>\operatorname{Den}(L) = \operatorname{proj}_{OS} L = L_{OS} = \{ (o, s) \in O \times S : (o, s, i) \in L ~\text{for some}~ i \in I \}.\!</math>

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|}

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Looking to the denotative aspects of the present example, various rows of the Tables specify that <math>\text{A}\!</math> uses <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math> to denote <math>\text{A}\!</math> and <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math> to denote <math>\text{B}\!</math>, whereas <math>\text{B}\!</math> uses <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math> to denote <math>\text{B}\!</math> and <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math> to denote <math>\text{A}\!</math>. 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.

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Looking to the denotative aspects of the present example, various rows of the Tables specify that <math>\text{A}\!</math> uses <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math> to denote <math>\text{A}\!</math> and <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math> to denote <math>\text{B},\!</math> whereas <math>\text{B}\!</math> uses <math>{}^{\backprime\backprime} \text{i} {}^{\prime\prime}\!</math> to denote <math>\text{B}\!</math> and <math>{}^{\backprime\backprime} \text{u} {}^{\prime\prime}\!</math> to denote <math>\text{A}.\!</math> 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.

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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 <math>L\!</math>, the dyadic relation that constitutes the ''connotative component'' of <math>L\!</math> is denoted <math>\operatorname{Con}(L)\!</math>.

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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 <math>L,\!</math> the dyadic relation that constitutes the ''connotative component'' of <math>L\!</math> is denoted <math>\operatorname{Con}(L).\!</math>

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.

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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?

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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 <math>L\!</math>, the dyadic relation that constitutes the ''intentional component'' of <math>L\!</math> is denoted <math>\operatorname{Int}(L)\!</math>.

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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 <math>L,\!</math> the dyadic relation that constitutes the ''intentional component'' of <math>L\!</math> is denoted <math>\operatorname{Int}(L).\!</math>

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.

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Formally, these new aspects of semantics present no additional problem:

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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:

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The connotative component of a sign relation <math>L\!</math> can be formalized as its dyadic projection on the plane generated by the sign domain and the interpretant domain, defined as follows:

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: <math>\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 \}\!</math>.

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{| align="center" cellspacing="6" width="90%"

+

|

+

<math>\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 \}.\!</math>

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|}

The intentional component of semantics for a sign relation <math>L\!</math>, 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:

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: <math>\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 \}\!</math>.

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{| align="center" cellspacing="6" width="90%"

+

|

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<math>\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 \}.~\!</math>

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|}

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As it happens, the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}\!</math> in the present example are fully symmetric with respect to exchanging signs and interpretants, so all of the structure of <math>(L_\text{A})_{OS}\!</math> and <math>(L_\text{B})_{OS}\!</math> is merely echoed in <math>(L_\text{A})_{OI}\!</math> and <math>(L_\text{B})_{OI}\!</math>, respectively.

+

As it happens, the sign relations <math>L_\text{A}\!</math> and <math>L_\text{B}\!</math> in the present example are fully symmetric with respect to exchanging signs and interpretants, so all of the structure of <math>(L_\text{A})_{OS}\!</math> and <math>(L_\text{B})_{OS}\!</math> is merely echoed in <math>(L_\text{A})_{OI}\!</math> and <math>(L_\text{B})_{OI},~\!</math> respectively.

'''Note on notation.''' When there is only one sign relation <math>L_J = L(J)\!</math> associated with a given interpreter <math>J\!</math>, it is convenient to use the following forms of abbreviation:

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& = & (L_J)_{OI}

& = & L(J)_{OI}

−

\end{array}</math>

+

\end{array}\!</math>

|}

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This subsection presents the ''objective project'' (OP) that I plan to take up for investigating the forms of sign relations, and it outlines three ''objective levels'' (OLs) of formulation that guide the analytic and synthetic study of interpretive structure and regulate the prospective stages of implementing this plan in particular cases. The main purpose of these schematic conceptions is organizational, to provide a conceptual architecture for the burgeoning hierarchies of objects that arise in the generative processes of inquiry.

−

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>.

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In this project I would like to treat the difference between construction and deconstruction as being more or less synonymous with the contrast between synthesis and analysis, but doing this without the introduction of too much distortion requires the intervention of a further distinction. Therefore, let it be recognized that all orientations to the constitutions of objects can be pursued in both ''regimented'' and ''radical'' fashions.

+

In the weaker senses of the terms, analysis and synthesis work within a preset and limited regime of objects, construing each object as being composed from a fixed inventory of stock constituents. In the stronger senses, contracting for the application of these terms places a more strenuous demand on the would-be construer.

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Any given OG can appear under the alias of a ''form of analysis'' (FOA) or a ''form of synthesis'' (FOS), depending on the direction of prevailing interest. A notion frequently invoked for the same purpose is that of an ''ontological hierarchy'' (OH), but I will use this only provisionally, and only so long as it is clear that alternative ontologies can always be proposed for the same space of objects.

−

An OG embodies many ''objective motives'' or ''objective motifs'' (OMs). If an OG constitutes a genus, or generic pattern of object structure, then the OMs amount to its specific and individual exemplars. Thus, an OM can appear in the guise of a particular instance, trial, or "run" of the general form of analytic or synthetic procedure that accords with the protocols of a given OG.

+

An OG embodies many ''objective motives'' or ''objective motifs'' (OMs). If an OG constitutes a genus, or generic pattern of object structure, then the OMs amount to its specific and individual exemplars. Thus, an OM can appear in the guise of a particular instance, trial, or “run” of the general form of analytic or synthetic procedure that accords with the protocols of a given OG.

In order to provide a way of talking about objective points of view in general without having to specify a particular level, I will use the term ''objective concern'' (OC) to cover any individual OF, OG, or OM.

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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.

−

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.

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{| 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>

|}

−

Here, <math>J\!</math> is a set of actual (not formal) parameters used to index the OG, while <math>P_j\!</math> and <math>Q_j\!</math> are domains of objects (initially in the informal sense) that enter into the dyadic relations <math>G_j\!</math>.

+

Here, <math>J\!</math> is a set of actual (not formal) parameters used to index the OG, while <math>{P_j}\!</math> and <math>{Q_j}\!</math> are domains of objects (initially in the informal sense) that enter into the dyadic relations <math>{G_j}.\!</math>

−

Aside from their indices, many of the <math>G_j\!</math> in <math>G\!</math> can be abstractly identical to each other. This would earn <math>G\!</math> the designation of a ''multi-family'' or a ''multi-set'', but I prefer to treat the index <math>j\!</math> as a concrete part of the indexed relation <math>G_j\!</math>, in this way distinguishing it from all other members of the indexed family <math>G\!</math>.

+

Aside from their indices, many of the <math>{G_j}~\!</math> in <math>G\!</math> can be abstractly identical to each other. This would earn <math>G\!</math> the designation of a ''multi-family'' or a ''multi-set'', but I prefer to treat the index <math>j\!</math> as a concrete part of the indexed relation <math>{G_j},\!</math> in this way distinguishing it from all other members of the indexed family <math>G\!</math>.

−

Ordinarily, it is desirable to avoid making individual mention of the separately indexed domains, <math>P_j\!</math> and <math>Q_j\!</math> for all <math>j\!</math> in <math>J\!</math>. Common strategies for getting around this trouble involve the introduction of additional domains, designed to encompass all the objects needed in given contexts. Toward this end, an adequate supply of intermediate domains, called the ''rudiments of universal mediation'', can be defined as follows:

+

Ordinarily, it is desirable to avoid making individual mention of the separately indexed domains, <math>{P_j}\!</math> and <math>{Q_j}\!</math> for all <math>j\!</math> in <math>J.\!</math> Common strategies for getting around this trouble involve the introduction of additional domains, designed to encompass all the objects needed in given contexts. Toward this end, an adequate supply of intermediate domains, called the ''rudiments of universal mediation'', can be defined as follows:

{| 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:

−

: 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:

{| 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>

|}

However, it serves a purpose of this project to preserve the individual indexing of relational domains for while longer, or at least to keep this usage available as an alternative formulation. Generally speaking, it is always possible in principle to form the union required by the universal inclusion convention, or without loss of generality to assume the equality imposed by the universal equality convention. The problem is that the unions and equalities invoked by these rubrics may not be effectively definable or testable in a computational context. Further, even when these sets or tests can be constructed or certified by some computational agent or another, the pertinent question at any interpretive moment is whether each collection or constraint is actively being apprehended or warranted by the particular interpreter charged with responsibility for it by the indicated assignment of domains.

−

But an overall purpose of this formalism is to represent the objects and constituencies ''known to'' specific interpreters at definite moments of their interpretive proceedings, in other words, to depict the information about objective existence and constituent structure that is possessed, recognized, responded to, acted on, and followed up by concrete agents as they move through their immediate contexts of activity. Accordingly, keeping individual tabs on the relational domains <math>P_j\!</math> and <math>Q_j\!</math>, though it does not solve this array of problems, does serve to mark the concern with particularity and to keep before the mind the issues of individual attention and responsibility that are appropriate to interpretive agents. In short, whether or not domains appear with explicit subscripts, one should always be ready to answer ''Who subscribes to these domains?''

+

But an overall purpose of this formalism is to represent the objects and constituencies ''known to'' specific interpreters at definite moments of their interpretive proceedings, in other words, to depict the information about objective existence and constituent structure that is possessed, recognized, responded to, acted on, and followed up by concrete agents as they move through their immediate contexts of activity. Accordingly, keeping individual tabs on the relational domains <math>{P_j}\!</math> and <math>{Q_j},\!</math> though it does not solve this array of problems, does serve to mark the concern with particularity and to keep before the mind the issues of individual attention and responsibility that are appropriate to interpretive agents. In short, whether or not domains appear with explicit subscripts, one should always be ready to answer ''Who subscribes to these domains?''

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.

Line 950: Line 960:

{| 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"

−

| <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"

−

| <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"

−

| <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"

−

| <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"

−

| <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>

|}

−

Often one's level of interest in a genre is ''purely generic''. When the relevant genre is regarded as an indexed family of dyadic relations, <math>G = \{ G_j \}\!</math>, then this generic interest is tantamount to having one's concern rest with the union of all the dyadic relations in the genre.

+

Often one's level of interest in a genre is ''purely generic''. When the relevant genre is regarded as an indexed family of dyadic relations, <math>G = \{ G_j \},\!</math> then this generic interest is tantamount to having one's concern rest with the union of all the dyadic relations in the genre.

{| 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"

−

| <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%"

−

| <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>

|}

−

At other times explicit mention needs to be made of the ''interpretive perspective'' or ''individual dyadic relation'' that links two objects. To indicate that a triple consisting of a motive <math>j\!</math> and two objects <math>x\!</math> and <math>y\!</math> belongs to the standing relation of the genre, in symbols, <math>(j, x, y) \in ~ :\!\lessdot</math>, or equally, to indicate that a triple consisting of a motive <math>j\!</math> and two objects <math>y\!</math> and <math>x\!</math> belongs to the propping relation of the genre, in symbols, <math>(j, y, x) \in ~ :\!\gtrdot</math>, all of the following notations are equivalent:

+

At other times explicit mention needs to be made of the ''interpretive perspective'' or ''individual dyadic relation'' that links two objects. To indicate that a triple consisting of a motive <math>j\!</math> and two objects <math>x\!</math> and <math>y\!</math> belongs to the standing relation of the genre, in symbols, <math>{(j, x, y) \in ~ :\!\lessdot},</math> or equally, to indicate that a triple consisting of a motive <math>j\!</math> and two objects <math>y\!</math> and <math>x\!</math> belongs to the propping relation of the genre, in symbols, <math>{(j, y, x) \in ~ :\!\gtrdot},</math> all of the following notations are equivalent:

{| 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 1,031: Line 1,041:

−

{| align="center" border="1" cellpadding="4" cellspacing="2" style="text-align:left; width:~~100~~%"

+

{| align="center" border="1" cellpadding="4" cellspacing="2" style="text-align:left; width:75%"

|

−

{| 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:ghostwhite; text-align:left; width:100%"

|-

| width="50%" | <math>j : x \lessdot y\!</math>

Line 1,056: Line 1,066:

| <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>

Line 1,083: Line 1,093:

| <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>

Line 1,105: Line 1,115:

−

In making these free interpretations of genres and motifs, one needs to read them in a ''logical'' rather than a ''cognitive'' sense. A statement like "<math>j\!</math> thinks <math>x\!</math> an instance of <math>y\!</math>" should be understood as saying that "<math>j\!</math> is a thought with the logical import that <math>x\!</math> is an instance of <math>y\!</math>", and a statement like "<math>j\!</math> proposes <math>y\!</math> a property of <math>x\!</math>" should be taken to mean that "<math>j\!</math> is a proposition to the effect that <math>y\!</math> is a property of <math>x\!</math>".

+

In making these free interpretations of genres and motifs, one needs to read them in a ''logical'' rather than a ''cognitive'' sense. A statement like “<math>j\!</math> thinks <math>x\!</math> an instance of <math>y\!</math>” should be understood as saying that “<math>j\!</math> is a thought with the logical import that <math>x\!</math> is an instance of <math>y\!</math>”, and a statement like “<math>j\!</math> proposes <math>y\!</math> a property of <math>x\!</math>” should be taken to mean that “<math>j\!</math> is a proposition to the effect that <math>y\!</math> is a property of <math>x\!</math>”.

These cautions are necessary to forestall the problems of intentional attitudes and contexts, something I intend to clarify later on in this project. At present, I regard the well-known opacities of this subject as arising from the circumstance that cognitive glosses tend to impute an unspecified order of extra reflection to each construal of the basic predicates. The way I plan to approach this issue is through a detailed analysis of the cognitive capacity for reflective thought, to be developed to the extent possible in formal terms by using sign relational models.

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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:

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:* 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>

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:* 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:

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|}

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Then <math>P\!</math> is a proposition that applies to a domain of propositions, or elements with the evidentiary import of propositions, and its models are therefore conceived to be certain propositional entities in <math>J\!</math>. And yet all of these expressions are just elaborate ways of stating the underlying assertion which says that there exists a triple <math>(j, x, y)\!</math> in the genre <math>G (:\!\lessdot)</math>.

+

Then <math>P\!</math> is a proposition that applies to a domain of propositions, or elements with the evidentiary import of propositions, and its models are therefore conceived to be certain propositional entities in <math>J.\!</math> And yet all of these expressions are just elaborate ways of stating the underlying assertion which says that there exists a triple <math>(j, x, y)\!</math> in the genre <math>G (:\!\lessdot).</math>

=====1.3.4.14. Application of OF : Generic Level=====

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Given an ontological framework that can provide multiple perspectives and moving platforms for dealing with object structure, in other words, that can organize diverse hierarchies and developing orders of objects, attention can now return to the discussion of sign relations as models of intellectual processes.

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A principal aim of using sign relations as formal models is to be capable of analyzing complex activities that arise in nature and human domains. Proceeding by the opportunistic mode of ''analysis by synthesis'', one generates likely constructions from a stock of favored, familiar, and well-understood sign relations, the supply of which hopefully grows with time, constantly matching their formal properties against the structures encountered in the "wilds" of natural phenomena and human conduct. When salient traits of both the freely generated products and the widely gathered phenomena coincide in enough points, then the details of the constructs one has built for oneself can help to articulate a plausible hypothesis as to how the observable appearances might be explained.

+

A principal aim of using sign relations as formal models is to be capable of analyzing complex activities that arise in nature and human domains. Proceeding by the opportunistic mode of ''analysis by synthesis'', one generates likely constructions from a stock of favored, familiar, and well-understood sign relations, the supply of which hopefully grows with time, constantly matching their formal properties against the structures encountered in the “wilds” of natural phenomena and human conduct. When salient traits of both the freely generated products and the widely gathered phenomena coincide in enough points, then the details of the constructs one has built for oneself can help to articulate a plausible hypothesis as to how the observable appearances might be explained.

A principal difficulty of using sign relations for this purpose arises from the very power of productivity they bring to bear in the process, the capacity of triadic relations to generate a welter of what are bound to be mostly arbitrary structures, with only a scattered few hoping to show any promise, but the massive profusion of which exceeds from the outset any reason's ability to sort them out and test them in practice. And yet, as the phenomena of interest become more complex, the chances grow slimmer that adequate explanations will be found in any of the thinner haystacks. In this respect, sign relations inherit the basic proclivities of set theory, which can be so successful and succinct in presenting and clarifying the properties of already found materials and hard won formal insights, and yet so overwhelming to use as a tool of random exploration and discovery.

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The sign relations of <math>A\!</math> and <math>B\!</math>, though natural in themselves as far as they go, were nevertheless introduced in an artificial fashion and presented by means of arbitrary stipulations. Sign relations that arise in more natural settings usually have a rationale, a reason for being as they are, and therefore become amenable to classification on the basis of the distinctive characters that make them what they are.

+

The sign relations of <math>\text{A}\!</math> and <math>\text{B},\!</math> though natural in themselves as far as they go, were nevertheless introduced in an artificial fashion and presented by means of arbitrary stipulations. Sign relations that arise in more natural settings usually have a rationale, a reason for being as they are, and therefore become amenable to classification on the basis of the distinctive characters that make them what they are.

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.

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

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|}

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In spite of the apparent duality between these patterns of composition, there is a significant asymmetry to be observed in the way that the insistent theme of realism interrupts the underlying genre. In order to understand this, it is necessary to note that the strain of pragmatic thinking I am using here takes its definition of ''reality'' from the word's original Scholastic sources, where the adjective ''real'' means ''having properties''. Taken in this sense, reality is necessary but not sufficient to ''actuality'', where ''actual'' means "existing in act and not merely potentially" (Webster's). To reiterate, actuality is sufficient but not necessary to reality. The distinction between the ideas is further pointed up by the fact that a potential can be real, and that its reality can be independent of any particular moment in which the power acts.

+

In spite of the apparent duality between these patterns of composition, there is a significant asymmetry to be observed in the way that the insistent theme of realism interrupts the underlying genre. In order to understand this, it is necessary to note that the strain of pragmatic thinking I am using here takes its definition of ''reality'' from the word's original Scholastic sources, where the adjective ''real'' means ''having properties''. Taken in this sense, reality is necessary but not sufficient to ''actuality'', where ''actual'' means “existing in act and not merely potentially” (Webster's). To reiterate, actuality is sufficient but not necessary to reality. The distinction between the ideas is further pointed up by the fact that a potential can be real, and that its reality can be independent of any particular moment in which the power acts.

These abstract considerations would probably remain distant from the present concern, were it not for two points of connection:

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# This project seeks articulations and implementations of intelligent activity within dynamically realistic systems. The individual stresses placed on articulation, implementation, actuality, dynamics, and reality collectively reinforce the importance of several issues:

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::* Systems theory, consistently pursued, eventually demands for its rationalization a distinct ontology, in which states of being and modes of action form the principal objects of thought, out of which the ordinary sorts of stably extended objects must be constructed. In the "grammar" of process philosophy, verbs and pronouns are more basic than nouns. In its influence on the course of this discussion, the emphasis on systematic action is tantamount to an objective genre that makes dynamic systems, their momentary states and their passing actions, become the ultimate objects of synthesis and analysis. Consequently, the drift of this inquiry will be turned toward conceiving actions, as traced out in the trajectories of systems, to be the primitive elements of construction, more fundamental in this objective genre than stationary objects extended in space. As a corollary, it expects to find that physical objects of the static variety have a derivative status in relation to the activities that orient agents, both organisms and organizations, toward purposeful objectives.

+

::* Systems theory, consistently pursued, eventually demands for its rationalization a distinct ontology, in which states of being and modes of action form the principal objects of thought, out of which the ordinary sorts of stably extended objects must be constructed. In the “grammar” of process philosophy, verbs and pronouns are more basic than nouns. In its influence on the course of this discussion, the emphasis on systematic action is tantamount to an objective genre that makes dynamic systems, their momentary states and their passing actions, become the ultimate objects of synthesis and analysis. Consequently, the drift of this inquiry will be turned toward conceiving actions, as traced out in the trajectories of systems, to be the primitive elements of construction, more fundamental in this objective genre than stationary objects extended in space. As a corollary, it expects to find that physical objects of the static variety have a derivative status in relation to the activities that orient agents, both organisms and organizations, toward purposeful objectives.

::* At root, the notion of ''dynamics'' is concerned with ''power'' in the sense of ''potential''. The brand of pragmatic thinking that I use in this work permits potential entities to be analyzed as real objects and conceptual objects to be constituted by the conception of their actual effects in practical instances. In the attempt to unify symbolic and dynamic approaches to intelligent systems (Upper and Lower Kingdoms?), there remains an insistent need to build conceptual bridges. A facility for relating objects to their actualizing instances and their instantiating actions lends many useful tools to an effort of this nature, in which the search for understanding cannot rest until each object and phenomenon has been reconstructed in terms of active occurrences and ways of being.

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# ''For indices, the existence of a separate reality is obligatory.'' And yet this reality need not affect the object of the sign. In essence, indices are satisfied with a basis in reality that need only reside in an actual object instance, one that establishes a real connection between the object and its index with regard to the OG in question.

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Finally, suppose that <math>M\!</math> and <math>N\!</math> are hypothetical sign relations intended to capture all the iconic and indexical relationships, respectively, that a typical object <math>x\!</math> enjoys within its genre <math>G\!</math>. A sign relation in which every sign has the same kind of relation to its object under an assumed form of analysis is appropriately called a ''homogeneous sign relation''. In particular, if <math>H\!</math> is a homogeneous sign relation in which every sign has either an iconic or an indexical relation to its object, then it is convenient to apply the corresponding adjective to the whole of <math>H\!</math>.

+

Finally, suppose that <math>M\!</math> and <math>N\!</math> are hypothetical sign relations intended to capture all the iconic and indexical relationships, respectively, that a typical object <math>x\!</math> enjoys within its genre <math>G.\!</math> A sign relation in which every sign has the same kind of relation to its object under an assumed form of analysis is appropriately called a ''homogeneous sign relation''. In particular, if <math>H\!</math> is a homogeneous sign relation in which every sign has either an iconic or an indexical relation to its object, then it is convenient to apply the corresponding adjective to the whole of <math>H.\!</math>

−

Typical sign relations of the iconic or indexical kind generate especially simple and remarkably stable sorts of interpretive processes. In arity, they could almost be classified as ''approximately dyadic'', since most of their interesting structure is wrapped up in their denotative aspects, while their connotative functions are relegated to the tangential role of preserving the directions of their denotative axes. In a metaphorical but true sense, iconic and indexical sign relations equip objective frameworks with "gyroscopes", helping them maintain their interpretive perspectives in a persistent orientation toward their objective world.

+

Typical sign relations of the iconic or indexical kind generate especially simple and remarkably stable sorts of interpretive processes. In arity, they could almost be classified as ''approximately dyadic'', since most of their interesting structure is wrapped up in their denotative aspects, while their connotative functions are relegated to the tangential role of preserving the directions of their denotative axes. In a metaphorical but true sense, iconic and indexical sign relations equip objective frameworks with “gyroscopes”, helping them maintain their interpretive perspectives in a persistent orientation toward their objective world.

Of course, every form of sign relation still depends on the agency of a proper interpreter to bring it to life, and every species of sign process stays forever relative to the interpreters that actually bring it to term. But it is a rather special circumstance by means of which the actions of icons and indices are able to turn on the existence of independently meaningful properties and instances, as recognized within an objective framework, and this means that the interpretive associations of these signs are not always as idiosyncratic as they might otherwise be.

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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.

−

I recall the objective genre of ''properties and instances'' and re-introduce the symbols <math>\lessdot\!</math> and <math>\gtrdot\!</math> for the converse pair of dyadic relations that generate it. Reverting to the convention I employ in formal discussions of applying relational operators on the right, it is convenient to express the relative terms "property of <math>x\!</math>" and "instance of <math>x\!</math>" by means of a case inflection on <math>x,\!</math> that is, as "<math>x\!</math>’s property" and "<math>x\!</math>’s instance", respectively. Described in this way, <math>\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst}) = \langle \lessdot, \gtrdot \rangle,\!</math> where:

+

I recall the objective genre of ''properties and instances'' and re-introduce the symbols <math>{\lessdot}\!</math> and <math>{\gtrdot}\!</math> for the converse pair of dyadic relations that generate it. Reverting to the convention I employ in formal discussions of applying relational operators on the right, it is convenient to express the relative terms “property of <math>x\!</math>” and “instance of <math>x\!</math>” by means of a case inflection on <math>x,\!</math> that is, as “<math>x\!</math>’s property” and “<math>x\!</math>’s instance”, respectively. Described in this way, <math>\operatorname{OG} (\operatorname{Prop}, \operatorname{Inst}) = \langle \lessdot, \gtrdot \rangle,\!</math> where:

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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.

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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.

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=====1.3.4.17. Recapitulation : A Brush with Symbols=====

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A common goal of work in artificial intelligence and cognitive simulation is to understand how is it possible for intelligent life to evolve from elements available in the primordial sea. Simply put, the question is: "What's in the brine that ink may character?"

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A common goal of work in artificial intelligence and cognitive simulation is to understand how is it possible for intelligent life to evolve from elements available in the primordial sea. Simply put, the question is: “What's in the brine that ink may character?”

Pursuant to this particular way of setting out on the long-term quest, a more immediate goal of the current project is to understand the action of full-fledged symbols, insofar as they conduct themselves through the media of minds and quasi-minds. At this very point the quest is joined by the pragmatic investigations of signs and inquiry, which share this interest in chasing down symbols to their precursive lairs.

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At last, even with the needed frameworks only partly shored up, I can finally ravel up and tighten one thread of this rambling investigation. All this time, steadily rising to answer the challenge about the identity of the interpreter, ''Who's there?'', and the role of the interpretant, ''Stand and unfold yourself'', has been the ready and abiding state of a certain system of interpretation, developing its character and gradually evolving its meaning through a series of imputations and extensions. Namely, the MOI (the SOI experienced as an object) can answer for the interpreter, to whatever extent that conduct can be formalized, and the IM (the SOI experienced in action, in statu nascendi) can serve as a proxy for the momentary thrust of interpretive dynamics, to whatever degree that process can be explicated.

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To put a finer point on this result I can do no better at this stage of discussion than to recount the "metaphorical argument" that Peirce often used to illustrate the same conclusion.

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To put a finer point on this result I can do no better at this stage of discussion than to recount the “metaphorical argument” that Peirce often used to illustrate the same conclusion.

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|

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"Knowledge" is a referring back: in its essence a regressus in infinitum. That which comes to a standstill (at a supposed causa prima, at something unconditioned, etc.) is laziness, weariness —

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“Knowledge” is a referring back: in its essence a regressus in infinitum. That which comes to a standstill (at a supposed causa prima, at something unconditioned, etc.) is laziness, weariness —

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| align="right" | — Nietzsche, ''The Will to Power'', [Nie, S575, 309]

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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.

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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:

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<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>

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<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>

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<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>

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<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>

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<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>

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<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 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>

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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.

−

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>

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−

The power of form, the will to give form to oneself. "Happiness" admitted as a goal. Much strength and energy behind the emphasis on forms. The delight in looking at a life that seems so easy. — To the French, the Greeks looked like children.

+

The power of form, the will to give form to oneself. “Happiness” admitted as a goal. Much strength and energy behind the emphasis on forms. The delight in looking at a life that seems so easy. — To the French, the Greeks looked like children.

|-

| align="right" | — Nietzsche, ''The Will to Power'', [Nie, S94, 58]

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−

On "logical semblance" — The concepts "individual" and "species" equally false and merely apparent. "Species" expresses only the fact that an abundance of similar creatures appear at the same time and that the tempo of their further growth and change is for a long time slowed down, so actual small continuations and increases are not very much noticed (— a phase of evolution in which the evolution is not visible, so an equilibrium seems to have been attained, making possible the false notion that a goal has been attained — and that evolution has a goal —).

+

On “logical semblance” — The concepts “individual” and “species” equally false and merely apparent. “Species” expresses only the fact that an abundance of similar creatures appear at the same time and that the tempo of their further growth and change is for a long time slowed down, so actual small continuations and increases are not very much noticed (— a phase of evolution in which the evolution is not visible, so an equilibrium seems to have been attained, making possible the false notion that a goal has been attained — and that evolution has a goal —).

|-

| align="right" | — Nietzsche, ''The Will to Power'', [Nie, S521, 282]

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−

The form counts as something enduring and therefore more valuable; but the form has merely been invented by us; and however often "the same form is attained", it does not mean that it is the same form — what appears is always something new, and it is only we, who are always comparing, who include the new, to the extent that it is similar to the old, in the unity of the "form". As if a type should be attained and, as it were, was intended by and inherent in the process of formation.

+

The form counts as something enduring and therefore more valuable; but the form has merely been invented by us; and however often “the same form is attained”, it does not mean that it is the same form — what appears is always something new, and it is only we, who are always comparing, who include the new, to the extent that it is similar to the old, in the unity of the “form”. As if a type should be attained and, as it were, was intended by and inherent in the process of formation.

|-

| align="right" | — Nietzsche, ''The Will to Power'', [Nie, S521, 282]

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−

Form, species, law, idea, purpose — in all these cases the same error is made of giving a false reality to a fiction, as if events were in some way obedient to something — an artificial distinction is made in respect of events between that which acts and that toward which the act is directed (but this "which" and this "toward" are only posited in obedience to our metaphysical-logical dogmatism: they are not "facts").

+

Form, species, law, idea, purpose — in all these cases the same error is made of giving a false reality to a fiction, as if events were in some way obedient to something — an artificial distinction is made in respect of events between that which acts and that toward which the act is directed (but this “which” and this “toward” are only posited in obedience to our metaphysical-logical dogmatism: they are not “facts”).

|-

| align="right" | — Nietzsche, ''The Will to Power'', [Nie, S521, 282]

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−

One should not understand this compulsion to construct concepts, species, forms, purposes, laws ("a world of identical cases") as if they enabled us to fix the real world; but as a compulsion to arrange a world for ourselves in which our existence is made possible: — we thereby create a world which is calculable, simplified, comprehensible, etc., for us.

+

One should not understand this compulsion to construct concepts, species, forms, purposes, laws (“a world of identical cases”) as if they enabled us to fix the real world; but as a compulsion to arrange a world for ourselves in which our existence is made possible: — we thereby create a world which is calculable, simplified, comprehensible, etc., for us.

|-

| align="right" | — Nietzsche, ''The Will to Power''. [Nie, S521, 282]

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−

This same compulsion exists in the sense activities that support reason — by simplification, coarsening, emphasizing, and elaborating, upon which all "recognition", all ability to make oneself intelligible rests. Our needs have made our senses so precise that the "same apparent world" always reappears and has thus acquired the semblance of reality.

+

This same compulsion exists in the sense activities that support reason — by simplification, coarsening, emphasizing, and elaborating, upon which all “recognition”, all ability to make oneself intelligible rests. Our needs have made our senses so precise that the “same apparent world” always reappears and has thus acquired the semblance of reality.

|-

| align="right" | — Nietzsche, ''The Will to Power'', [Nie, S521, 282]

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−

Our subjective compulsion to believe in logic only reveals that, long before logic itself entered our consciousness, we did nothing but introduce its postulates into events: now we discover them in events — we can no longer do otherwise — and imagine that this compulsion guarantees something connected with "truth".

+

Our subjective compulsion to believe in logic only reveals that, long before logic itself entered our consciousness, we did nothing but introduce its postulates into events: now we discover them in events — we can no longer do otherwise — and imagine that this compulsion guarantees something connected with “truth”.

|-

| align="right" | — Nietzsche, ''The Will to Power'', [Nie, S521, 282–283]

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−

It is we who created the "thing", the "identical thing", subject, attribute, activity, object, substance, form, after we had long pursued the process of making identical, coarse and simple. The world seems logical to us because we have made it logical.

+

It is we who created the “thing”, the “identical thing”, subject, attribute, activity, object, substance, form, after we had long pursued the process of making identical, coarse and simple. The world seems logical to us because we have made it logical.

|-

| align="right" | — Nietzsche, ''The Will to Power'', [Nie, S521, 283]

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−

Now we read disharmonies and problems into things because we think only in the form of language — and thus believe in the "eternal truth" of "reason" (e.g., subject, attribute, etc.)

+

Now we read disharmonies and problems into things because we think only in the form of language — and thus believe in the “eternal truth” of “reason” (e.g., subject, attribute, etc.)

|-

| align="right" | — Nietzsche, ''The Will to Power'', [Nie, S522, 283]

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−

Nothing is more erroneous than to make of psychical and physical phenomena the two faces, the two revelations of one and the same substance. Nothing is explained thereby: the concept "substance" is perfectly useless as an explanation. Consciousness in a subsidiary role, almost indifferent, superfluous, perhaps destined to vanish and give way to a perfect automatism —

+

Nothing is more erroneous than to make of psychical and physical phenomena the two faces, the two revelations of one and the same substance. Nothing is explained thereby: the concept “substance” is perfectly useless as an explanation. Consciousness in a subsidiary role, almost indifferent, superfluous, perhaps destined to vanish and give way to a perfect automatism —

|-

| align="right" | — Nietzsche, ''The Will to Power'', [Nie, S523, 283]

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−

Before there is "thought" (gedacht) there must have been "invention" (gedichtet); the construction of identical cases, of the appearance of sameness, is more primitive than the knowledge of sameness.

+

Before there is “thought” (gedacht) there must have been “invention” (gedichtet); the construction of identical cases, of the appearance of sameness, is more primitive than the knowledge of sameness.

|-

| align="right" | Nietzsche, ''The Will to Power'', [Nie, S544, 293]

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Will to truth is a making firm, a making true and durable, an abolition of the false character of things, a reinterpretation of it into beings.

−

"Truth" is therefore not something there, that might be found or discovered — but something that must be created and that gives a name to a process, or rather to a will to overcome that has in itself no end — introducing truth, as a processus in infinitum, an active determining — not a becoming-conscious of something that is in itself firm and determined.

+

“Truth” is therefore not something there, that might be found or discovered — but something that must be created and that gives a name to a process, or rather to a will to overcome that has in itself no end — introducing truth, as a processus in infinitum, an active determining — not a becoming-conscious of something that is in itself firm and determined.

−

It is a word for the "will to power".

+

It is a word for the “will to power”.

|-

| align="right" | — Nietzsche, ''The Will to Power'', [Nie, S552, 298]

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−

Man projects his drive to truth, his "goal" in a certain sense, outside himself as a world that has being, as a metaphysical world, as a "thing-in-itself", as a world already in existence. His needs as creator invent the world upon which he works, anticipate it; this anticipation (this "belief" in truth) is his support.

+

Man projects his drive to truth, his “goal” in a certain sense, outside himself as a world that has being, as a metaphysical world, as a “thing-in-itself”, as a world already in existence. His needs as creator invent the world upon which he works, anticipate it; this anticipation (this “belief” in truth) is his support.

|-

| align="right" | — Nietzsche, ''The Will to Power'', [Nie, S552, 299]

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−

Thus there seems to be in preparation a universal disvaluation: "Nothing has any meaning" — this melancholy sentence means "All meaning lies in intention, and if intention is altogether lacking, then meaning is altogether lacking, too".

+

Thus there seems to be in preparation a universal disvaluation: “Nothing has any meaning” — this melancholy sentence means “All meaning lies in intention, and if intention is altogether lacking, then meaning is altogether lacking, too”.

|-

| align="right" | — Nietzsche, ''The Will to Power'', [Nie, S666, 351]

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−

In accordance with this valuation, one was constrained to transfer the value of life to a "life after death", or to the progressive development of ideas or of mankind or of the people or beyond mankind; but with that one had arrived at a progressus in infinitum of purposes: one was at last constrained to make a place for oneself in the "world process" (perhaps with the dysdaemonistic perspective that it was a process into nothingness).

+

In accordance with this valuation, one was constrained to transfer the value of life to a “life after death”, or to the progressive development of ideas or of mankind or of the people or beyond mankind; but with that one had arrived at a progressus in infinitum of purposes: one was at last constrained to make a place for oneself in the “world process” (perhaps with the dysdaemonistic perspective that it was a process into nothingness).

|-

| align="right" | — Nietzsche, ''The Will to Power'', [Nie, S666, 351]

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And do you know what ~~“the world”~~ is to me? Shall I show it to you in my mirror? This world: a monster of energy, without beginning, without end; a firm, iron magnitude of force that does not grow bigger or smaller, that does not expend itself but only transforms itself; as a whole, of unalterable size, a household without expenses or losses, but likewise without increase or income; enclosed by ~~“nothingness”~~ as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be ~~“empty”~~ here or there, but rather as force throughout, as a play of forces and waves of forces, at the same time one and many, increasing here and at the same time decreasing there; a sea of forces flowing and rushing together, eternally changing, eternally flooding back, with tremendous years of recurrence, with an ebb and a flood of its forms; out of the simplest forms striving toward the most complex, out of the stillest, most rigid, coldest forms toward the hottest, most turbulent, most self-contradictory, and then again returning home to the simple out of this abundance, out of the play of contradictions back to the joy of concord, still affirming itself in this uniformity of its courses and its years, blessing itself as that which must return eternally, as a becoming that knows no satiety, no disgust, no weariness: this, my Dionysian world of the eternally self-creating, the eternally self-destroying, this mystery world of the twofold voluptuous delight, my ~~“beyond~~ good and ~~evil”~~, without goal, unless the joy of the circle is itself a goal; without will, unless a ring feels good will toward itself — do you want a name for this world? A solution for all its riddles? A light for you, too, you best-concealed, strongest, most intrepid, most midnightly men? — This world is the will to power — and nothing besides! And you yourselves are also this will to power — and nothing besides!

+

And do you know what “the world” is to me? Shall I show it to you in my mirror? This world: a monster of energy, without beginning, without end; a firm, iron magnitude of force that does not grow bigger or smaller, that does not expend itself but only transforms itself; as a whole, of unalterable size, a household without expenses or losses, but likewise without increase or income; enclosed by “nothingness” as by a boundary; not something blurry or wasted, not something endlessly extended, but set in a definite space as a definite force, and not a space that might be “empty” here or there, but rather as force throughout, as a play of forces and waves of forces, at the same time one and many, increasing here and at the same time decreasing there; a sea of forces flowing and rushing together, eternally changing, eternally flooding back, with tremendous years of recurrence, with an ebb and a flood of its forms; out of the simplest forms striving toward the most complex, out of the stillest, most rigid, coldest forms toward the hottest, most turbulent, most self-contradictory, and then again returning home to the simple out of this abundance, out of the play of contradictions back to the joy of concord, still affirming itself in this uniformity of its courses and its years, blessing itself as that which must return eternally, as a becoming that knows no satiety, no disgust, no weariness: this, my Dionysian world of the eternally self-creating, the eternally self-destroying, this mystery world of the twofold voluptuous delight, my “beyond good and evil”, without goal, unless the joy of the circle is itself a goal; without will, unless a ring feels good will toward itself — do you want a name for this world? A solution for all its riddles? A light for you, too, you best-concealed, strongest, most intrepid, most midnightly men? — This world is the will to power — and nothing besides! And you yourselves are also this will to power — and nothing besides!

|-

| align="right" | — Nietzsche, ''The Will to Power'', [Nie, S1067, 549–550]

Jon Awbrey

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