Inquiry Driven Systems:
An Inquiry Into Inquiry
Jon Awbrey
Oakland University
1. Research Proposal
1.1 Outline of the Project: Inquiry Driven Systems
1.1.1 Problem
This research is oriented toward a single problem: What is the nature of inquiry? I intend to address
crucial questions about the operation, organization, and computational facilitation of inquiry, taking
inquiry to encompass the general trend of all forms of reasoning that lead to the features of scientific
investigation as their ultimate development.
1.1.2 Method
How will I approach this problem about the nature of inquiry? The simplest answer is this:
I will apply the method of inquiry to the problem of inquiry's nature.
This is the most concise and comprehensive answer I know, but it is likely to sound facetious
at this point. On the other hand, if I did not actually use the method of inquiry that I describe
as inquiry, how could the results possibly be taken seriously? Correspondingly, the questions
of methodological self-application and self-referential consistency will be found at the center
of this research.
In truth, it is fully possible that every means at inquiry's disposal will ultimately find application
in resolving the problem of inquiry's nature. Other than a restraint to valid methods of inquiry --
what those are is part of the question -- there is no reason to expect a prior limitation on the range
of methods that might be required.
This only leads up to the question of priorities: Which methods do I think it wise to apply first?
In this project I will give preference to two kinds of technique, one analytic and one synthetic.
The prevailing method of research that I will exercise throughout this work involves representing
problematic phenomena in a variety of formal systems and then implementing these representations
in a computational medium as a way of clarifying the more complex descriptions that evolve.
Aside from its theoretical core, this research is partly empirical and partly heuristic. Therefore,
I expect that the various components of methodology will need to be applied in an iterative or even
an opportunistic fashion, working on any edge of research that appears to be ready at a given time.
If forced to anticipate the likely developments, I would sketch the possibilities roughly as follows:
The methodology that underlies this approach has two components: The analytic component
involves describing the performance and the competence of intelligent agents in the medium of
various formal systems. The synthetic component involves implementing these formal systems
and the descriptions they express in the form of computational interpreters or language processors.
If everything goes according to the pattern I have observed in previous work, the principal facets
of analytic and synthetic procedure will each be prefaced by its own distinctive phase of preparatory
activity, where the basic materials needed for further investigation are brought together for comparative
study. Taking these initial stages into consideration, I can describe the main modalities of this research
in greater detail, as follows.
1.1.2.1 The Paradigmatic &
Process-Analytic Phase. In this phase I describe the performance and competence of intelligent agents in terms of various formal
systems. For aspects of an inquiry process that affect its dynamic or temporal performance I will typically
use representations modeled on finite automata and differential systems. For aspects of an inquiry faculty
that reflect its formal or symbolic competence I will commonly use representations like formal grammars,
logical calculi, constraint-based axiom systems, and rule-based theories in association with different proof
styles.
Paradigm. Generic example that reflects significant properties of a target class of phenomena,
often derived from a tradition of study.
Analysis. Effective analysis of concepts, capacities, structures, and functions in terms of
fundamental operations and computable functions.
Work in this phase typically proceeds according to the following recipe:
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Focus on a problematic phenomenon. This is a generic property or process that attracts one's
interest, like intelligence or inquiry.
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2.
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Gather under consideration significant examples of concrete systems or agents that exhibit the
property or process in question.
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3.
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Reflect on their common properties in a search for less obvious traits that might explain their
more surprising features.
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4.
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Check these accounts of the phenomenon in one of several ways. For example, one
might (a) search out other systems or situations in nature that manifest the critical traits,
or (b) implement the putative traits in computer simulations. If these hypothesized traits
generate (give rise to, provide a basis for) the phenomenon of interest, either in nature or
on the computer, then one has reason to consider them further as possible explanations.
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The last option of the last step already overlaps with the synthetic phase of work. Viewing this procedure
within the frame of experimental research, it is important to recognize that computer programs can fill the
role of hypotheses, testable (defeasible or falsifiable) construals of how a process is actually, might be
possibly, or ought to be optimally carried out.
1.1.2.2 The Paraphrastic &
Faculty-Synthetic Phase. The closely allied techniques of task analysis and software development that are known as
"step-wise refinement" and "top-down programming" in computer science (Wirth 1976, 49, 303)
have a long ancestry in logic and philosophy, going back to a strategy for establishing or discharging
contextual definitions known as "paraphrasis". All of these methods are founded on the idea of providing
meaning for operational specifications, "definitions in use", alleged descriptions, or "incomplete symbols".
No excessive generosity with the resources of meaning is intended, though. In practice, a larger share of
the routine is spent detecting meaningless fictions rather than discovering meaningful concepts.
Paraphrasis. "A method of accounting for fictions by explaining various purported terms away"
(Quine, in Van Heijenoort, 216). See also (Whitehead & 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, as modal, or as
normative hypotheses, that is, conjectures about how a process is actually accomplished in nature,
speculations as to how it might be done in principle, or explorations of how it might be done better
in the medium of technological extensions.
For the purposes of this project, I will take "paraphrastic definition" to denote the analysis of formal
specifications and contextual constraints to derive effective implementations of a process or its faculty.
This is carried out by considering what the faculty in question is required to do in the many contexts
it is expected to serve, and then by analyzing these formal specifications in order to design computer
programs that fulfill them.
1.1.2.3 Reprise of Methods In summary, the whole array of methods will be typical of the top-down strategies that are used in
"artificial intelligence research" (AIR), involving the conceptual and operational analysis of higher-order
cognitive capacities with an eye toward the modeling, the grounding, and the support of these faculties
in the form of effective computer programs. The toughest part of this discipline is in making sure that
one does indeed "come down", that is, in finding guarantees that the analytic reagents and the 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 other too 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 to intertwine as they will.
1.1.3 Criterion
When is enough enough? What measure can I use to tell if my effort is working? What information
is critical in deciding whether my exercise of the method is advancing my state of knowledge toward
a solution of the problem?
Given that the problem is "inquiry" and the method is "inquiry", the test of progress and eventual success
is just the measure of any inquiry's performance. According to my current understanding of inquiry, and
the tentative model of inquiry that will guide this project, the criterion of an inquiry's competence is how
well it succeeds in reducing the uncertainty of its agent about its object.
What are the practical tests of whether the results of inquiry succeed in reducing uncertainty? Two gains
are often cited: Successful results of inquiry provide the agent with increased powers of prediction and
control as to how the object system will behave in given circumstances. If a common theme is desired, at
the price of a finely equivocal thread, it can be said that the agent has gained in its power of determination.
Hence, more certainty is exhibited by less hesitation, more determination is manifested by less vacillation.
1.1.4 Application
Where can the results be used? Knowledge about the nature of inquiry can be applied. It can be used
to improve our personal competence at inquiry. It can be used to build software support for the tasks
involved in inquiry.
If it is desired to articulate the loop of self-application a bit further, computer models of inquiry can be
seen as building a two-way bridge between experimental science and software engineering, allowing the
results of each to be applied in the furtherance of the other.
In yet another development, computer models of learning and reasoning form a linkage among cognitive
psychology (the descriptive study of how we think), artificial intelligence (the prospective study of how
we might think), and the logic of operations research (the normative study of how we ought to think in
order to achieve the goals of reasoning).
1.2 Onus of the Project: No Way But Inquiry
At the beginning of inquiry there is nothing for me to work with but the actual constellation of doubts
and beliefs that I have at the moment. Beliefs that operate at the deepest levels can be taken so far for
granted that they rarely if ever obtrude on awareness. Doubts that oppress in the most obvious ways are
still known only as debits and droughts, as the absence of something, one knows not what, and a desire
that obliges one only to try. Obscure forms of oversight provide an impulse to replenish the condition of
privation but never out of necessity afford a sense of direction. One senses there ought to be a way out
at once, or ordered ways to overcome obstruction, or organized or otherwise ways to obviate one's opacity
of omission and rescue a secure motivation from the array of conflicting possibilities. In the roughest
sense of the word, any action that does in fact lead out of this onerous state can be regarded as a form
of "inquiry". Only later, in moments of more leisurely inquiry, when it comes down to classifying and
comparing the manner of escapes that can be recounted, does it become possible to recognize the ways
in which certain general patterns of strategy are routinely more successful in the long run than others.
1.2.1 A Modulating Prelude
If I aim to devise the kind of computational support that can give the greatest assistance to inquiry,
then it must be able to come in at the very beginning, to be of service in the kinds of formless and
negative conditions that I just described, and to help people navigate a way through the constellations
of contingent, incomplete, and contradictory indications that they actually find themselves sailing under
at present.
In the remainder of this section I will try to indicate as briefly as possible the nature of the problem
that must be faced in this particular approach to inquiry, and to explain what a large share of the
ensuing fuss will be directed toward clearing up.
Toward the end of this discussion I will be using highly concrete mathematical models, that is, very
specific families of combinatorial objects, to represent the abstract structures of experiential sequences
that agents pass through. If these primitive and simplified models are to be regarded as something more
than mere toys, and if the relations of particular experiences to particular models, along with the structural
relationships that exist within the field of experiences and again within the collection of models, are not to
be dismissed as category confusions, then I will need to develop a toolbox of logical techniques that can
be used to justify these constructions. The required technology of categorical and relational notions will
be developed in the process of addressing its basic task: To show how the same conceptual categories
can be applied to materials and models of experience that are radically diverse in their specific contents
and peculiar to the states of the particular agents to which they attach.
1.2.2 A Fugitive Canon
The principal difficulties associated with this task appear to spring from two roots.
First, there is the issue of "computational mediation". In using the sorts of sequences that computers
go through to mediate discussion of the sorts of sequences that people go through, it becomes necessary
to re-examine all of the facilitating assumptions that are commonly taken for granted in relating one human
experience to another, that is, in describing and building structural relationships among the experiences of
human agents.
Second, there is the problem of "representing the general in the particular". How is it possible for
the most particular imaginable things, namely, the transient experiential states of agents, to represent
the most general imaginable things, namely, the agents' own conceptions of the abstract categories
of experience?
Finally, not altogether as an afterthought, there is a question that binds these issues together. How does it
make sense to apply one's individual conceptions of the abstract categories of experience, not only to the
experiences of oneself and others, but in points of form to compare them with the structures present in
mathematical models?
1.3 Option of the Project: A Way Up To Inquiry
I begin with an informal examination of the concept of inquiry. This section takes as its subjects the
supposed faculty of inquiry in general and the present inquiry into inquiry in particular, and attempts
to analyze them in relation to each other on formal principles alone.
The initial set of concepts I need to get discussion started are few. Assuming that a working set of ideas
can be understood on informal grounds at the outset, I anticipate being able to formalize them to a greater
degree as the project gets under way. Inquiry in general will be described as encompassing particular
inquiries. Particular forms of inquiry, regarded as phenomenal processes, will be analyzed in terms
of simpler kinds of phenomenal processes.
As a phenomenon, a particular way of doing inquiry is regarded as embodied in a faculty of inquiry, as
possessed by an agent of inquiry. As a process, a particular example of inquiry is regarded as extended
in time through a sequence of states, as experienced by its ongoing agent. It is envisioned that an agent
or faculty of any generically described phenomenal process, inquiry included, could be started off from
different initial states and would follow different trajectories of subsequent states, and yet there would be
a recognizable quality or abstractable property that justifies invoking the name of the genus.
The steps of this analysis will be annotated below by making use of the following conventions:
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Lower case letters denote phenomena, processes, or faculties under investigation.
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2.
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Upper case letters denote classes of the same sorts of entities, that is, phenomena, processes,
or faculties of interest in a particular investigation.
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3.
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Special use is made of the following symbols:
Y = genus of inquiry; y = generic inquiry; y0 = present inquiry.
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4.
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Compositions of "faculties" are indicated by concatenating their names, as in "f.g", and
are posed in the sense that the faculty on the right "applies to" the faculty on the left.
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5.
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The notation "f >= g" indicates that f is greater than or equal to g in a decompositional series,
in other words, that f possesses g as a component.
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6.
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The coset notation F.G indicates a class of "faculties" of the form f.g, with f in F and g in G.
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7.
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Notations like "{?}", "{?,?}", and so on, serve as proxies for unknown components and
can be used in an informal way to indicate tentative analyses of the faculties in question.
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1.3.1 Initial Analysis of Inquiry
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Allegro Aperto
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If the faculty of inquiry is a coherent power, then it has an active or instrumental face, a passive or
objective face, and a substantial body of connections between them. y = {?}.
In giving the current inquiry a reflexive cast, as inquiry into inquiry, I have brought inquiry face to face
with itself, inditing it to apply its action in pursuing a knowledge of its passion. y0 = y.y = {?}{?}.
If this juxtaposition of characters is to have a meaningful issue, then the fullness of its instrumental
and objective aspects must have recourse to easier actions and simpler objects. y >= {?,?}.
Looking for an edge on each face of inquiry, as a plausible option for beginning to apply one to the
other, I find what seems a likely pair. I begin with an aspect of instrumental inquiry that is easy to do,
namely "discussion", along with an aspect of objective inquiry that is unavoidable to discuss, namely
"formalization". y >= {disc, form}.
In accord with this plan, the body of this section is devoted to a discussion of formalization.
y0 = y.y >= {d,f}{d,f} >= {f}{d}.
1.3.2 Discussion of Discussion
But first, I nearly skipped a step. Though it might present itself as an interruption, a topic so easy that
I almost omitted it altogether deserves at least a passing notice. y0 = y.y >= {d,f}{d,f} >= {d}{d}.
Discussion is easy in general because its termination criterion is relaxed to the point of becoming otiose.
A discussion of things in general can be pursued as an end in itself, with no consideration of any purpose
but persevering in its current form, and this accounts for the virtually perpetual continuation of many
a familiar and perennial discussion.
There's a catch here that applies to all living creatures: In order to keep talking one has to keep living.
This brings discussion back to its role in inquiry, considered as an adaptation of living creatures designed
to help them deal with their not so virtual environments. If discussion is constrained to the envelope of life
and required to contribute to the trend of inquiry, instead of representing a kind of internal opposition, then
it must be possible to tighten up the loose account and elevate the digressionary narrative into a properly
directed inquiry. This brings an end to my initial discussion of "discussion".
1.3.3 Discussion of Formalization: General Topics
Because this project makes constant use of formal models of phenomenal processes, it is appropriate
at this point to introduce the understanding of formalization that I will use throughout this work and
to preview a concrete example of its application.
An introduction to the topic of formalization, if proper, is obliged to begin informally. But it will be
my constant practice to keep a formal eye on the whole proceedings. What this form of observation
reveals must be kept silent for the most part at first, but I see no rule against sharing with the reader
the general order of this watch:
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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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2.
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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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3.
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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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1.3.3.2 A Formalization of Formalization? An immediate application of the above rules is presented here, in hopes of giving the reader a concrete
illustration of their use in a ready example, but the issues raised can quickly diverge into yet another
distracting digression, one not so easily brought under control as the discussion of discussion, but whose
complexity probably approaches that of the entire task. Therefore, a partial adumbration of its character
will have to suffice for the present. y0 = y.y >= {d,f}{d,f} >= {f}{f}.
To illustrate the formal charge by taking the present matter to task, the word "formalization" is itself
exemplary of the "-ionized" terms falling under the charge, and so it can be lionized as the nominal
head of a prospectively formal discussion. The reader has a right to object at this point that I have not
described what particular action I intend to convey under the heading of "formalization", by no means
enough to begin applying it to any term, much less itself. However, anyone can recognize on syntactic
grounds that the word is an instance of the formal rule, purely from the character of its terminal "-ion",
and this can be done aside from all clues about the particular meaning that I intend it to have at the end
of formalization.
Unlike a mechanical interpreter meeting with the declaration of an undefined term for the very first time,
the human reader of this text has the advantage of a prior acquaintance with almost every term that might
conceivably enter into informal discussion. And "formalization" is a stock term widely traded in the forums
of ordinary and technical discussion, so the reader is bound to have met with it in the context of practical
experience and to have attached a personal concept to it. Therefore, this inquiry into formalization begins
with a writer and a reader in a state of limited uncertainty, each attaching a distribution of meanings in
practice to the word "formalization", but uncertain whether their diverse spectra of associations can
presently constitute or eventually converge to compatible arrays of effective meaning.
To review: The concept of formalization itself is an item of informal discussion that might be investigated
as a candidate for formalization. For each aspect or component of the formalization process that I plan
to transport across the semi-permeable threshold from informal to formal discussion, the reader has
permission to challenge it, plus an open invitation to question every further process that I mention as
a part of its constitution, and to ask with regard to each item whether its registration has cleared up the
account in any measure or merely rung up a higher charge on the running bill of fare.
The reader can follow this example with every concept that I mention in the explanation of formalization,
and again in the larger investigation of inquiry, and be assured that it is has not often slipped my attention
to at least venture the same, though a delimitation of each exploration in its present state of completion
would be far too tedious and tenuous to escape expurgation.
1.3.3.3 A Formalization of Discussion? The previous section took the concept of "formalization" as an example of a topic that a writer might
try to translate from informal to formal discussion, perhaps as a way of clarifying the general concept
to an optimal degree, or perhaps as a way of communicating a particular concept of it to a reader.
In either case the formalization process, that aims to translate a concept from informal to formal
discussion, is itself mediated by a form of discussion: (1) that interpreters conduct as a part of their
ongoing monologue with themselves, or (2) that a writer (speaker) conducts in real or imagined dialogue
with a reader (hearer). In view of this, I see no harm in letting the concept of discussion be stretched to
cover all attempted processes of formalization. F c D.
In this section, I step back from the example of "formalization" and consider the general task of clarifying
and communicating concepts by means of a properly directed discussion. Let this kind of "motivated"
or "measured" discussion be referred to as a "meditation", that is, "a discourse intended to express its
author's reflections or to guide others in contemplation" (Webster's). The motive of a meditation is
to mediate a certain object or intention, namely, the system of concepts intended for clarification or
communication. The measure of a meditation is a system of values that permits its participants to tell
how close they are to achieving its object. The letter "M" will be used to annotate this form of meditation.
F c M c D.
This brings the discussion around to considering the intentional objects of measured discussions and
the qualifications of a writer so motivated. Just what is involved in achieving the object of a motivated
discussion? Can these intentions be formalized? y0 = y.y >= {d,f}{d,f} >= {d}{f}.
The writer's task is not to create meaning from nothing, but to construct a relation from the typical
meanings that are available in ordinary discourse to the particular meanings that are intended to be
the effects of a particular discussion.
In case there is difficulty with the meaning of the word "meaning", I replace its use with references
to a "system of interpretation" (SOI), a technical concept that will be increasingly formalized as this
project proceeds. Thus, the writer's job description is reformulated as follows.
The writer's task is not to create a system of interpretation (SOI) from nothing, but to construct a relation
from the typical SOI's that are available in ordinary discourse to the particular SOI's that are intended to be
the effects of a particular discussion.
This assignment begins with an informal system of interpretation (SOI1), and builds a relation from it to
another system of interpretation (SOI2). The first is an informal SOI that amounts to a shared resource of
writer and reader. The latter is a system of meanings in practice that is the current object of the writer's
intention to recommend for the reader's consideration and, hopefully, edification. In order to have
a compact term for highlighting the effects of a discussion that "builds a relation" between SOI's,
I will refer to this aspect of the overall discussion process by the name of "narration".
It is the writer's ethical responsibility to ensure that a discourse is potentially edifying with respect to the
reader's current SOI, and the reader's self-interest to evaluate whether a discourse is actually edifying from
the perspective of the reader's present SOI.
Formally, the relation that the writer builds from SOI to SOI can always be cast or recast as a three-place
relation, one whose staple element of structure is an ordered or indexed triple. One component of each
triple is anchored in the interpreter of the moment, and the other two form a connection with the source
and target SOI's of the current assignment.
Once this relation is built, a shift in the attention of any interpreter or a change in the present focus
of discourse can leave the impression of a transformation taking place from SOI1 to SOI2, but this is
more illusory (or allusory) than real. To be more precise, this style of transformation takes place on
a virtual basis, and need not have the substantive impact (or import) that a substantial replacement
of one SOI by another would imply. For a writer to affect a reader in this way would simply not be
polite. A moment's consideration of the kinds of SOI-building worth having leads me to enumerate
a few characteristics of "polite discourse" or "considerate discussion".
If this form of SOI-building narrative is truly intended to edify and educate, whether pursued in monologue
or dialogue fashion, then its action cannot be forcibly to replace the meanings in practice a sign already has
with others of an arbitrary nature, but freely to augment the options for meaning and powers for choice in
the resulting SOI.
As conditions for the possibility of considerate but significant narration, there are a couple of requirements
that are placed on the writer and the reader, as appropriate to their respective roles. Considerate narration,
constructing a relation from SOI to SOI in a politic fashion, cannot operate in an infectious or an addictive
manner, invading a SOI like a virus or a trojan horse, but must transfer its communication wholly into the
control of the receiving SOI. Significant communication, in which the receiving SOI is augmented by
options for meaning and powers for choice that it did not have before, requires a SOI on the reader's
part that is truly "extensible" in non-trivial ways.
At this point, the discussion has touched on a topic, in one of its manifold aspects, that it will encounter
repeatedly, under a variety of aspects, throughout this work. In recognition of this circumstance, and
to prepare the way for future discussion, it seems like a good idea to note a few of the aliases that this
protean topic can be found lurking under, and to notice the logical relationships that exist among its
several different appearances.
On several occasions, this discussion of inquiry will arrive at a form of "aesthetic deduction", in general
terms, a piece of reasoning that ends with a design recommendation, in this case, where an analysis of
the general purposes and interests of inquiry leads to the conclusion that a certain property of discussion
is an admirable one, and that the quality in question forms an essential part of the implicit value system
that is required to guide inquiry and make it what it is meant to be, a method for advancing toward desired
forms of knowledge. After a collection of admirable qualities has been recognized as cohering together
into a unity, it becomes natural to ask: What is the underlying reality that inheres in these qualities, and
what are the logical relations that bind them together into the qualifications of inquiry and a definition
of exactly what is desired for knowledge?
1.3.3.4 A Concept of Formalization The concept of formalization is intended to cover the whole collection of activities that serve to build
a relation between casual discussions, those that take place in the ordinary context of informal discourse,
and formal discussions, those that make use of completely formalized models. To make a long story short,
formalization is the narrative operation or active relation that construes the situational context in the form
of a definite text. The end product that results from the formalization process is analogous to a snapshot
or a candid picture, a relational or functional image that captures an aspect of the casual circumstances.
Relations between casual and formal discussion are often treated in terms of a distinction between two
languages, the "meta-language" and the "object language", linguistic systems that take complementary
roles in filling out the discussion of interest. In the usual approach, issues of formalization are addressed
by postulating a distinction between the meta-language, the descriptions and conceptions from ordinary
language and technical discourse that can be used without being formalized, and the object language,
the domain of structures and processes that can be studied as a completely formalized object.
1.3.3.5 A Formal Approach I plan to approach the issue of formalization from a slightly different angle, proceeding through an analysis
of the medium of interpretation and developing an effective conception of an "interpretive framework" or
an "interpretive system". This concept refers to any organized system of interpretive practice, ranging from
those that we use in everyday speech, to the ones that inform technical discourse, to the kinds of completely
formalized symbol systems that are safely regarded as mathematical objects. Depending on the degree of
objectification that it possesses from a particular person's operative point of view, the very same system of
conduct can variously be described as an "interpretive framework" (IF), an "interpretive system" (IS), an
"interpretive object" (IO), or an "object system" (OS). These terms are merely suggestive -- no rigid form
of classification is intended.
Many times, it is convenient to personify the interpretive organization as if it were embodied in the actions
of a typical user of the framework or a substantive agent of the system. I will call this agent the "interpreter"
of the moment. At other times, it may be necessary to analyze the action of interpretation more carefully.
At these times, it is important to remember that this form of personification is itself a figure of speech,
one that has no meaning outside a fairly flexible interpretive framework. Thus, the term "interpreter"
can be a cipher analogous to the terms "X", "unknown", or "to whom it may concern" appearing in
a system of potentially recursive constraints. As such, it serves in the role of an indeterminate symbol,
in the end to be solved for a fitting value, but in the mean time conveying an appearance of knowledge
in a place where very little is known about the subject itself.
A meta-language corresponds to what I call an "interpretive framework". Besides a set of descriptions
and conceptions, this construction embodies the whole collective activity of unexamined structures and
automatic processes that are trusted by agents at a given moment to make its employment meaningful
in practice. An interpretive framework is best understood as a form of conduct, that is, a comprehensive
organization of related activities.
In use, an interpretive framework operates to contain activity and constrain the engagement of agents
to certain forms of active involvement and dynamic participation, and manifests itself only incidentally
in the manipulation of compact symbols and isolated instruments. In short, though a framework may
have "pointer dials" and "portable tools" attached to it, it is usually too incumbent and cumbersome
to be easily moved on its own grounds, at least, it rests beyond the scope of any local effort to do so.
An interpretive framework (IF) is set to work when an agent or agency becomes involved in its
organization and participates in the forms of activity that make it up. Often, an IF is founded and
persists in operation long before any participant is able to reflect on its structure or to post a note
of its character to the constituting members of the framework. In some cases, the rules of the IF
in question forbid the act of reflecting on its form. In practice, to the extent that agents are actively
involved in filling out the requisite forms and taking part in the step by step routines of the IF, they
may have little surplus memory capacity to memorandize the big picture even when it is permitted
in principle.
An object language is a special case of the kind of formal system that is so completely formalized
that it can be regarded as combinatorial object, an inactive image of a form of activity that is meant
for the moment to be studied rather than joined.
The supposition that there is a meaningful and well-defined distinction between object language and
meta-language ordinarily goes unexamined. This means that the assumption of a distinction between
them is de facto a part of the meta-language and not even an object of discussion in the object language.
A slippery slope begins here. A failure to build reflective capacities into an interpretive framework can
let go unchallenged the spurious opinion that presumes there can be only one way to draw a distinction
between object language and meta-language.
The next natural development is to iterate the supposed distinction. This represents an attempt to formalize
and thereby "objectify" parts of the meta-language, precipitating it like a new layer of pearl or crystal from
the resident medium or "mother liquor", and thereby preparing the decantation of a still more pervasive and
ethereal meta-meta-language. The successive results of this process can have a positivistically intoxicating
effect on the human intellect. But a not so happy side-effect leads the not quite mindful cerebration up and
down a blind alley, chasing the specious impression that just beyond the realm of objective nature there lies
a unique fractionation of permeabilities and a permanent hierarchy of effabilities in language.
The grounds of discussion that I am raking over here constellate a rather striking scene, especially for
a level of conceptual architecture that is intended as a neutral backdrop. Unlike other concerns, the points
I am making seem obvious to all reasonable people at the outset of discussion, and yet the difficulties that
follow as inquiry develops get all the muddier and more grating the more one probes and stirs them up.
A large measure of the blame, I think, can be charged to a misleading directive that people derive from
the epithet "meta", leading them to search for higher and higher levels of meaning and truth, on beyond
language, on beyond any conceivable system of signs, and on beyond sense. Prolonged use of the prefix
"meta", without due reflection on its side-effects, leads people to act as if a meta-language were step outside
of ordinary language, or an artificial platform constructed above and beyond natural language, and then they
forget that formal models are developments internal to the informal context. For this reason among others,
I suggest replacing talk about rigidly stratified layers of object languages and meta-languages with talk
about contingent, transcient, and variable forms of interpretive frameworks.
To avoid the types of cul-de-sac (cultist act) encountered above, I am taking some pains to ensure
a reflective capacity for the interpretive frameworks I develop in this project. This is a capacity that
natural languages always assume for themselves, instituting specialized discourses as developments
that take place within their frame and not as constructs that lie beyond their scope. Any time that the
levels of recursive discussion become too involved to manage successfully, one needs to keep available
the resource of "instant wisdom", the modest but indispensable quantum of ready understanding, that
restores itself on each return to the "ordinary universe" (OU).
From this angle of approach, let us try to view afresh the manner of drawing distinctions between
various levels of formalization in language. Once again, I begin in the context of ordinary discussion,
and if there is any distinction to be drawn between objective and instrumental languages then it must
be possible to describe it within the frame of this informally discursive universe.
1.3.3.6 A Formal Development The point of view that I take on the origin and development of formal models is that they arise with
agents retracing structures that already exist in the context of informal activity, until gradually the most
relevant and frequently reinforced patterns become emphasized and emboldened enough to continue their
development as nearly autonomous styles, in brief, as "genres" growing out of a particular "paradigm".
Taking the position that formal models develop within the framework of informal discussion, the questions
that become important to ask of a prospective formal model are (1) whether it highlights the structure of its
supporting context in a transparent form of emphasis and a relevant reinforcement of salient features, and
(2) whether it reveals the active ingredients of its source materials in a critically reflective recapitulation
or an analytically representative recipe, or (3) whether it insistently obscures what little fraction of its
domain it manages to cover.
1.3.3.7 A Formal Persuasion An interpretive system can be taken up with very little fanfare, since it does not enjoin one to
declare undying allegiance to a particular point of view or to assign each piece of text in view
to a sovereign territory, but only to entertain different points of view on the use of the symbols
involved. The chief design consideration for an interpretive system is that it must never function
as a virus or an addiction. Its suggestions must always be, initially and finally, purely optional
adjunctions to whatever interpretive framework was already in place before it installed itself on
the scene. Interpretive systems are thus not constituted in the faith that anything nameable will
always be dependable, nor articulated in fixed principles that determine what must be doubted
and what must not, but rest only in a form of self-knowledge that recognizes the doubts and the
beliefs that one actually has at each given moment.
Before this project is done I will need to have developed an analytic and computational theory
of interpreters and interpretive frameworks. In the aspects of this theory that I can anticipate at
this point, an interpreter or an interpretive framework is exemplified by a collective activity of
symbol-using practices like those that might be found embodied in a person, a community, or
a culture. Each one forms a moderately free and independent perspective, with no objective
rankings of supremacy in practice that all interpretive frameworks are likely to support at any
foreseeable moment in their fields of view. Of course, each interpreter initially enters discussion
operating as if its own perspective were "meta" in comparison to all the others, but a well-developed
interpretive framework is likely to have acquired the notion and taken notice of the fact that this is not
likely to be a universally shared opinion (USO).
1.3.4 Discussion of Formalization: Concrete Examples
The previous section outlined a variety of general issues surrounding the concept of formalization.
The following section will plot the specific objectives of this project in constructing formal models
of intellectual processes. In this section I wish to take a breather between these abstract discussions
in order to give their main ideas a few points of contact with terra firma. To do this, I examine a selection
of concrete examples, artificially constructed to approach the minimum levels of non-trivial complexity,
that are intended to illustrate the kinds of mathematical objects I have in mind using as formal models.
1.3.4.1 Formal Models: A Sketch To sketch the features of the modeling activity that are relevant to the immediate purpose: The modeler
begins with a phenomenon or process of interest (POI) and relates it to a formal model of interest (MOI),
the whole while working within a particular interpretive framework (IF) and relating the results from one
system of interpretation (SOI) to another, or to a subsequent development of the same SOI.
The POI's that define the intents and purposes of this project are the closely related processes of inquiry
and interpretation, so the MOI's that must be formulated are models of inquiry and interpretation, species
of formal systems that are even more intimately bound up than usual with the IF's employed and the SOI's
deployed in their ongoing development as models.
Since all of the process models and interpretive systems mentioned here come from the same broad family
of mathematical objects, the different roles that they play in this investigation are mainly distinguished by
variations in their manner and degree of formalization:
1.
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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 finite or final formula.
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2.
|
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", terms that
serve more as directions of pointing than as denotations of entities. What enables the MOI to
grasp these directions is the mathematical fact 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 (FIC) can "make infinite use of finite means". Thus, my reason for calling the MOI
pivotal or cardinal is that it forms a model in two senses, logical and analogical, integrating twin roles
of the model concept in a single focus.
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3.
|
Finally, the IF's and SOI's 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 a very great
likelihood of a completely articulate formulation being 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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1.3.4.2 Sign Relations: A Primer To the extent that their structures and functions can be discussed at all, it is likely that all of the formal
entities destined to develop in this approach to inquiry will be instances of a class of three-place relations
called "sign relations". At any rate, all of the formal structures that I have examined so far in this area have
turned out to be, if not manifestly sign relations themselves, erither easily convertible to, or else ultimately
grounded in, some variation on sign relations. This class of triadic relations constitutes the main study
of the "pragmatic theory of signs", a branch of logical philosophy that is devoted to understanding
all manners of symbolic representation and all types of significant communication.
There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry.
In fact, the correspondence between the two studies exhibits so many parallels and coincidences that it is
often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the
process by which sign relations come to be established and continue to evolve. In other words, inquiry,
"thinking" in its best sense, "is a term denoting the various ways in which things acquire significance"
(Dewey). Thus, there is an active and intricate form of cooperation that needs to be appreciated and
maintained between these converging modes of investigation. Its proper character is best understood
by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations,
a subject which the theory of signs is specialized to treat from structural and comparative points of view.
Because the examples in this section have been artificially constructed to be a simple as possible,
their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Still,
these examples have subtleties of their own, and their careful treatment will serve to illustrate
important issues in the general theory of signs.
Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their
expressive and interpretive practice that involves the use of the following nouns and pronouns:
"Ann", "Bob", "I", "You".
The "object domain" of this discussion fragment is the set consiting of two people {Ann, Bob}.
The "syntactic domain" or the "sign system" of their discussion is limited to the set of four signs
{"Ann", "Bob", "I", "You"}.
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 they use. The system of interpretation (SOI)
associated with each language user can be represented in the form of an individual three-place relation
called the "sign relation" of that interpreter.
Understood in terms of its set-theoretic extension, a sign relation R is a subset of a cartesian product OxSxI.
Here, O, S, and I are three sets called the "object domain", the "sign domain", and the "interpretant domain",
respectively, of the sign relation R c OxSxI. In general, the three domains of a sign relation can be any
sets whatsoever, but the kinds of sign relation contemplated in a computational framework are usually
constrained to having I c S. In this case, interpretants are just a special type of signs, and this makes
it convenient to lump signs and interpretants together into a "syntactic domain". In the forthcoming
examples, S and I are identical as sets, so the very same elements appear in two distinct roles of the
pertinent sign relations. When it is necessary to refer to the whole set of objects and signs in the
union of the domains O, S, and I for a given sign relation R, I will call this the "world of R" and
write W = W(R) = O U S U I.
To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief
as possible when the examples get more complicated, I introduce the following abbreviations:
O = object domain;
|
S = sign domain;
|
I = interpretant domain.
|
O
|
=
|
{ Ann, Bob }
|
=
|
{ A, B }.
|
S
|
=
|
{"Ann", "Bob", "I", "You"}
|
=
|
{"A", "B", "i", "u"}.
|
In the present examples, S = I = syntactic domain.
Tables 1 and 2 give the sign relations associated with the interpreters A and B, respectively, putting
them in the form of relational databases. Thus, the rows of each Table list the ordered triples <o, s, i>
that make up the corresponding sign relations: A, B c OxSxI. The issue of using the same names for
objects and for relations involving these objects will be taken up later, after the less problematic features
of these relations have been treated.
These Tables codify a rudimentary level of interpretive practice for the agents A and B, and provide
a basis for formalizing the initial semantics appropriate to their common syntactic domain. Each row
of a Table names an object and two co-referent signs, making up an ordered triple <o, s, i> called an
"elementary relation", that is, one element of the relation's set-theoretic extension.
Already in this elementary context, there are several different meanings that might attach to the project
of a "formal semantics". In the process of discussing these alternatives, I will introduce a few terms that
are occasionally used in the philosophy of language to point out the needed distinctions.
Table 1. Sign Relation of Interpreter A
Table 2. Sign Relation of Interpreter B
One aspect of semantics is concerned with the reference that a sign has to its object, which is often
called its "denotation". For signs in the most general type of situation, neither the existence nor the
uniqueness of a denotation is guaranteed. Thus, the denotation of a sign can refer to a plural, to a
singular, or to a vacuous number of objects. In the pragmatic theory of signs, these references of
signs to their objects are formalized as certain types of dyadic sub-relations that are found embedded
in the triadic sign relations. When it comes to dealing with the degenerate cases of signs that do not
denote, it is necessary to introduce what is, strictly speaking, a slightly more general concept than
a sign relation proper, namely, what is called a "sign-relational complex". But that is the subject
of a much later discussion. For now, I shall keep to signs which are known to have one or more
objects among their denotations.
The dyadic relation that constitutes the "denotative component" of a sign relation R is denoted by
"Den (R)". Information about the denotative component of semantics can be derived from R by taking
its "dyadic projection" on the object and sign domains, indicated by any one of the equivalent forms
"ProjOS(R)", "ROS", or "R12", and defined as:
Den (R) = ProjOS(R) = ROS = R12 = {<o, s> C OxS : <o, s, i> C R for some i C I}.
Looking to the denotative aspects of the present example, various rows of the Tables specify that
A uses "i" to denote A and "u" to denote B, whereas B uses "i" to denote B and "u" to denote A.
It is utterly amazing that even these impoverished remnants of natural language use have properties
that quickly bring the usual prospects of formal semantics to a screeching halt.
The other dyadic aspects of semantics that might be considered concern the reference that a sign has to its
interpretant and the reference that an interpretant has to its object. As before, either type of reference can
be multiple, unique, or empty in its collection of terminal points, and both can be formalized as different
kinds of dyadic sub-relations that can be found embedded in the triadic sign relations.
The connection that a sign makes to an interpretant is called its "connotation". In the general theory of
sign relations, this aspect of semantics includes the references that a sign has to ideas, concepts, affects,
intentions, and to the whole realm of an agent's mental states and allied activities, broadly encompassing
intellectual associations, emotional impressions, and motivational impulses. 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 R, the dyadic relation that constitutes the "connotative component" of R is denoted
by "Con (R)".
The bearing that an interpretant has toward a common object of its sign and itself has no standard name.
If an interpretant is considered to be a sign in its own right, then its independent reference to an object
can be taken as belonging to another moment of denotation, but this omits the mediational character
of the whole transaction.
Given the service that interpretants supply in furnishing a locus for critical, reflective, and explanatory
glosses on objective scenes and their descriptive texts, it is easy to regard them as "annotations" of both
objects and signs, but this function points in the opposite direction to what is needed in this connection.
What does one call the inverse of the annotation function? More generally asked, what is the converse
of the annotation relation?
In light of these considerations, I find myself still experimenting with terms to suit this last-mentioned
dimension of semantics. On a trial basis, I will refer to it as the "ideational", "intentional", or "canonical"
component of the sign relation, and I will try calling the reference of an interpretant sign to an object its
"ideation", "intention", or "conation". Given a particular sign relation R, the dyadic relation that constitutes
the "intentional component" of R is denoted by "Int (R)".
A full consideration of the connotative and intentional aspects of semantics would force a return to
difficult questions about the true nature of the interpretant sign in the general theory of sign relations.
It is best to defer these issues to a later discussion. Fortunately, omission of this material does not interfere
with understanding the purely formal aspects of the present example.
Formally, these new aspects of semantics present no additional problem. The connotative component
of a sign relation R can be formalized as its dyadic projection on the sign and interpretant domains,
defined as:
Con (R) = ProjSI(R) = RSI = R23 = {<s, i> C SxI : <o, s, i> C R for some o C O}.
The intentional component of semantics in a sign relation R, or the "second moment of denotation",
is captured by its dyadic projection on the object and interpretant domains, defined as:
Int (R) = ProjOI(R) = ROI = R13 = {<o, i> C OxI : <o, s, i> C R for some s C S}.
Indeed, the sign relations A and B in the present example are fully symmetric with respect to exchanging
signs and interpretants, so all the structure of AOS and BOS is merely echoed in AOI and BOI, respectively.
The concern of this project is not with every conceivable sign relation but only with those that are capable
of supporting inquiry processes. In these, the relationship between the denotational and connotational
aspects of meaning is not wholly arbitrary. Instead, this relationship must be naturally constrained or
deliberately designed in such a way that it (1) supports the achievement of particular purposes that
have intentional value for the agent and (2) represents the embodiment of significant properties that
have objective reality in the agent's domain. Therefore, my attention is directed toward understanding
the forms of correlation, coordination, and cooperation among the various components of sign relations
that form the necessary conditions for carrying out coherent inquiries.
1.3.4.3 Semiotic Equivalence Relations A nice property possessed by the sign relations A and B is that their connotative components ASI and BSI
constitute a pair of equivalence relations on their common syntactic domain S = I. It is convenient to refer
to such structures as "semiotic equivalence relations" (SER's) since they equate signs that mean the same
thing to somebody. Each of these semiotic equivalence relations ASI, BSI c SxI = SxS partitions the whole
collection of signs into "semiotic equivalence classes" (SEC's). This constitution makes for a strong form of
representation in that the structure of the participants' common object domain is reflected or reconstructed,
part for part, in the structure of each of their "semiotic partitions" (SEP's) of the syntactic domain.
The main trouble with this notion of semantics in the present situation is that the two semiotic partitions
for A and B are not the same, indeed, they are orthogonal to each other. This makes it difficult to interpret
either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of
objective structure or invariant reality, independent of the individual interpreter's "point of view" (POV).
Information about the different forms of semiotic equivalence induced by the interpreters A and B
is summarized in Tables 3 and 4. The form of these Tables should suffice to explain what is meant
by saying that the SEP's for A and B are orthogonal to each other.
Table 3. Semiotic Partition of Interpreter A
Table 4. Semiotic Partition of Interpreter B
To discuss this situation further, I introduce the square bracket notation "[x]E" for "the equivalence class
of the element x under the equivalence relation E". A statement that the elements x and y are equivalent
under E is called an "equation". When the particular equivalence relation that qualifies an equation needs
to be made explicit, or cannot otherwise be taken for granted, as being implicitly understood, the equation
can be written in either one of two ways, as "[x]E = [y]E" or as "x =E y".
In the application to sign relations I extend this notation in the following ways. When R is a sign relation
whose "syntactic projection" or connotative component RSI is an equivalence relation on S, I write "[s]R"
for "the equivalence class of s under RSI". A statement that the signs x and y are synonymous under
a semiotic equivalence relation RSI is called a "semiotic equation" (SEQ), and can be written in either
of the forms: "[x]R = [y]R" or "x =R y".
In many situations there is one further adaptation of the square bracket notation that can be useful.
Namely, when there is known to exist a particular triple <o, s, i> C R, it is permissible to use "[o]R"
to mean the same thing as "[s]R". These modifications are designed to make the notation for semiotic
equivalence classes harmonize as well as possible with the frequent use of similar devices for the
denotations of signs and expressions.
In these terms, the SER for interpreter A yields the semiotic equations:
["A"]A
|
=
|
["i"]A
|
,
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["B"]A
|
=
|
["u"]A
|
,
|
"A"
|
=A
|
"i"
|
,
|
"B"
|
=A
|
"u"
|
,
|
and the semiotic partition: {{"A", "i"}, {"B", "u"}}.
In contrast, the SER for interpreter B yields the semiotic equations:
["A"]B
|
=
|
["u"]B
|
,
|
["B"]B
|
=
|
["i"]B
|
,
|
"A"
|
=B
|
"u"
|
,
|
"B"
|
=B
|
"i"
|
,
|
and the semiotic partition: {{"A", "u"}, {"B", "i"}}.
1.3.4.4 Graphical Representations The dyadic components of sign relations can be given graph-theoretic representations, as "digraphs"
(or "directed graphs"), that provide concise pictures of their structural and potential dynamic properties.
By way of terminology, a directed edge <x, y> is called an "arc" from point x to point y, and a self-loop
<x, x> is called a "sling" at x.
The denotative components Den (A) and Den (B) can be represented as directed graphs on the six points
of their common world set W = O U S U I = {A, B, "A", "B", "i", "u"}. The arcs of the corresponding
digraphs are given as follows:
1.
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Den (A) has an arc from each point of {"A", "i"} to A and from each point of {"B", "u"} to B.
|
2.
|
Den (B) has an arc from each point of {"A", "u"} to A and from each point of {"B", "i"} to B.
|
Den (A) and Den (B) can be interpreted as "transition digraphs" that chart the succession of steps or
the connection of states in a computational process. Read this way, the denotational arcs summarize
the "upshots" of the computations that are involved when the interpreters A and B evaluate the signs
in S according to their own lights, that is to say, in line with their own respective frames of reference.
The connotative components Con (A) and Con (B) can be represented as digraphs on the four points
of their common syntactic domain S = I = {"A", "B", "i", "u"}. Since Con (A) and Con (B) are SER's,
their digraphs conform to the generic pattern that is manifested by all digraphs of equivalence relations.
In general, a digraph of an equivalence relation falls into connected components that correspond to the
parts of the associated partition, with a "complete digraph" on the points of each part, and no other arcs.
By way of definition, a "complete digraph" is one that has all of the possible arcs on a given point set.
In the present case, the arcs of the digraphs for Con (A) and Con (B) are given as follows:
1.
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Con (A) has the structure of a SER on S, with a sling on each of the points in S,
two-way arcs between the points of the syntactic subset {"A", "i"}, and
two-way arcs between the points of the syntactic subset {"B", "u"}.
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2.
|
Con (B) has the structure of a SER on S, with a sling on each of the points in S,
two-way arcs between the points of the syntactic subset {"A", "u"}, and
two-way arcs between the points of the syntactic subset {"B", "i"}.
|
Taken as transition digraphs, Con (A) and Con (B) highlight the associations that are permitted
between equivalent signs, as this equivalence is judged by the interpreters A and B, respectively.
The theme running through the last two subsections, that associates different interpreters and
different aspects of interpretation with different kinds of relational structures on the same set
of points, heralds a topic that will be developed extensively in the sequel.
So far, my discussion of the discussion between A and B, in the picture it gives of sign relations and their
connection to the imagined processes of interpretation and inquiry, can best be described as fragmentary.
In the story of A and B, as I have presented it up to this point, a sample of typical language use has been
drawn from the context of informal discussion and partially formalized in the guise of two independent
sign relations, but no unified conception of the commonly understood interpretive practices in such
a situation has yet been drafted.
It seems like a good idea to pause at this point and reflect on the state of understanding that has been
reached. In order to motivate further developments it will be useful to inventory two types of shortfall
in the present state of discussion, the first having to do with the defects of my present narration in revealing
the relevant attributes of even so simple an example as the one I used to begin, the second having to do with
the defects that this species of example exhibits within the genus of sign relations it is intended to illustrate.
As a general schema, I describe these respective limitations as the "rhetorical" and the "objective" defects
that a discussion, a meditation, or a narration can have in addressing its intended object. The immediate
concern is to remedy the insufficiencies of the current analysis that affect the treatment of the present case.
The overarching task is to address the atypically simplistic features of this example as it falls within the class
of sign relations that are relevant to actual, pressing, and somewhat more realistic inquiries.
The next few subsections will be concerned with the most problematic features of the A and B dialogue,
especially with the sorts of difficulties that furnish clues to significant deficits in theory and technique,
and that point out directions for future improvements.
1.3.4.6 The "Meta" Question There is one point of common contention that I finessed from play in my handling of the discussion
between A and B, even though it lies in plain view on both of their Tables. This is that troubling business,
recalcitrant to analysis precisely because its operations race on so heedlessly ahead of thought and grind on
so routinely beneath its notice, that concerns the proper placements of object languages within the frame of
a meta-language.
Numerous bars to insight appear to interlock here. Each one is forged with a good aim in mind, if a bit
single-minded in its coverage of the scene, and the whole gang is set to work innocently enough in the
unavoidable circumstances of informal discussion. But a failure to absorb their amalgamated impact
on the figurative representations and the analytic intentions of sign relations can lead to several types
of false impression, both about the true characters of the tables presented here and about the proper
utilities of their graphical equivalents to be implemented as data structures in the computer. The next
few remarks are put forward in hopes of averting these brands of misreading.
The general character of this question can be expressed in the schematic terms that were used earlier
to give a rough sketch of the modeling activity as a whole. How do the isolated SOI's of A and B relate
to the interpretive framework that I am using to present them, and how does this IF operate, not only to
objectify A and B as "models of interpretation" (MOI's), but simultaneously to embrace the present and
the prospective SOI's of the current narrative, the implicit systems of interpretation that embody in turn
the initial conditions and the final intentions of this whole discussion?
One way to see how this issue arises in the discussion of A and B is to recognize that each inscribed table
of a sign relation is a complex sign in itself, each of whose syntactic constituents plays the role of a simpler
sign. In other words, there is nothing but text to be seen on the page. In comparison to what it represents,
the table is like a sign relation that has undergone a step of "semantic ascent". It is as if the entire contents
of the original sign relation have been transposed up a notch on the scale that registers levels of indirectness
in reference, each item passing from a more objective standing to a more symbolic mode of presentation.
Sign relations themselves, like any real objects of discussion, are either too abstract or else too concrete
to reside in the medium of communication, but can only find themselves represented there. The tokens
of tables and graphs that are used to represent sign relations are themselves complex signs, necessarily
involving a step of denotation to reach the sign relation intended. The intricacies of this step demand
interpretive agents who are able, over and above executing all of the rudimentary steps of denotation,
to orchestrate the requisite kinds of concerted steps. This performance in turn requires a whole array
of interpretive techniques to match the connotations of complex signs and to test their alternative styles
of representation for semiotic equivalence. In a fashion that is not coincidentally analogous to the ways
that matrices represent linear transformations and that multiplication tables represent group operations,
a large part of the usefulness of these complex signs comes from the fact that they are not just another
pretty fine mass of conventional symbols for their objects but iconic representations of their structure.
In the pragmatic theory of signs, an "icon" is a sign that accomplishes its representation, including the
projects of denotation and connotation, by virtue of properties that it shares with its object. In the case
of relational tables and graphs, interpreted as iconic representations or analogous expressions of logical
and mathematical objects, the pivotal properties are formal and abstract in character. Since a uniform
translation through any dimension (of sight, of sound, or the imagination) does not affect the abstract
structural properties of a configuration of signs in relation to each other, this may help to explain how
tables and graphs, in spite of all their semantic shiftiness, can succeed in representing sign relations
without essential distortion.
Taking this unsuspecting introduction of iconic signs as a serendipitous lesson in the art of representation,
an important principle is there to be lifted from the style of their peculiar form of representational success.
They bring the search for models of intellectual processes to look for classes of representation that do not
lean too heavily on local idioms for devising labels but rather suspend their abstract formal structures in
qualities of media that can best be preserved through a wide variety of global transformations. In time
these ventures will lead this project to contemplate various forms of graphical abstraction as supplying
possibly the most solid sites for pouring the foundations of formal expression.
What does appear in one of these Tables? It is not the objects that appear under the "Object" heading,
but only the signs of these objects. It is not even the signs and interpretants themselves that appear under
the "Sign" and "Interpretant" headings, but only the remoter signs of these signs that are formed through
the syntactic offices of quotation marks. The unformalized sign relation in which these signs of objects,
signs of signs, and signs of interpretants have their role as such is not the one Tabled, but another one,
a casually causative sign relation that operates behind the scenes to bring the image and intent of the
thus-raised Table, through the facilitations, the interventions, and the obstructions of the medium,
to the eventual reader.
To understand what the Table is meant to convey the reader has to participate in the informal and more
accessory sign relation in order to follow its indications to the intended and more accessible sign relation.
As logical or mathematical objects, the sign relations A and B do not exist in the medium of their Tables
but are represented there by dint of the relevant structural properties that they share with these Tables.
As fictional characters, the interpretive agents A and B do not exist in a uniquely literal sense but serve
as typical literary figures to convey the intended formal account, standing in for concrete experiences
with everyday language use the likes of which are familiar to writer and reader alike.
The successful formalization of a focal sign relation cannot get around its reliance on prior forms
of understanding, like the raw ability to follow indications whose components of competence are
embodied in the vaster and largely unarticulated context of a peripheral sign relation. But the extent
to which the analysis of a formal sign relation depends on a particular context or a particular interpreter
is the extent to which an opportunity for understanding is undermined by a prior petition of the very
principles to be explained. Thus, there is little satisfaction in special pleadings or ad hoc accounts of
interpretive practice that cannot be transported across a multitude of contexts, media, and interpreters.
What does all of this mean, in concrete form, for the proper appreciation of the present example?
And looking beyond that question, what does it mean in terms of the concrete activities that need
to be tackled by this work?
One task is to eliminate several types of formal confound that currently affect this investigation.
Even though there is an essential tension to be maintained down the lines between casual and formal
discussion, the traffic across these realms needs to be monitored carefully. There are identifiable sources
of confusion that devolve from the context of informal discussion and invade the arena of formal study,
subverting its necessary powers of reflection and undermining its overall effectiveness.
One serious form of contamination can be traced to the accidental circumstance that A and B and I
all use the same proper names for A and B. This renders it is impossible to tell, purely from the tokens
that are being tendered, whether it is a formal or a casual transaction that forms the issue of the moment.
It also means that a formalization of the writer's and the reader's accessory sign relations would have
several portions of its abstract mien, its formal countenance or its idealized visage, that look identical
to various pieces of those Tables that are presently placed under formal review. This is tantamount to
having a sort of "mirror" between the formal and the informal domains, one that reflects certain aspects
of the casual circumstances of discourse into the forms of the formalization itself. Although a device of
this kind can be a useful tool, even, on reflection, indispensable at times, so long as the character and the
nature of its operation are understood by its prospective user, it can also become a source of bewildering
confusions and thoroughly distracting nuisances, if the conditions for its effective employment are not met.
1.3.4.8 The Conflict of Interpretations One discrepancy that needs to be documented can be observed in the conflict of interpretations between
A and B, as reflected in the lack of congruity between their semiotic partitions of their syntactic domain.
This is a problematic but realistic feature of the present example. That is, it represents a type of problem
with the interpretation of pronouns (indexical signs or bound variables) that actually arises in practice
whenever one attempts to formalize the semantics of natural, logical, and programming languages.
On this account, the deficiency resides with the present analysis, and the burden remains to clarify
exactly what is going on here.
Notice, however, that I have deliberately avoided dealing with indexical tokens in the usual ways, namely,
by seeking to eliminate all semantic ambiguities from the initial formalization. Instead, I have preserved
this aspect of interpretive discrepancy as one of the essential phenomena or the inescapable facts in the
realm of pragmatic semantics, tantamount to the irreducible nature of perspective diversity. I believe
that the desired competence at this faculty of language will come, not from any strategy of substitution
that constantly replenishes bound variables with their objective referents on every fixed occasion, but from
a pattern of recognition that keeps indexical signs persistently attached to their due interpreters of reference.
In the pragmatic theory of signs, an "index" is a sign that achieves its representation of its object by virtue
of an actual connection with it. Though real and objective, however, the indexical linkage can be purely
incidental and even a bit accidental. Its effectiveness depends only on the fact that an object in actual
existence has many properties, definitive and derivative, any number of which can serve as its signs.
Indices of an object frequently reside among the more tangential varieties of its attributes, typically,
its accessory traces or its accidental traits, which are really the features of some but not all of the
points in the locus, the orbit, the status, or the trajectory of the object's existential actualization.
Pronouns qualify as indices because their objective references cannot be traced without recovering further
information about their actual context, not just their objective and their syntactic contexts but the pragmatic
context that is involved in the actualizing "situation of use" (SOU) or the realizing "instance of use" (IOU).
To fulfill their duty to sense the reading of indices demands to be supplemented by a suitably determinate
indication of their interpreter of reference, the agent that is responsible for putting them into active use
at the moment of interpretation in question.
Typical examples of indexical signs in programming languages are: (1) "variables", signs that need
to be bound to a syntactic context or an instantiation frame in order to have a determinate meaning,
and (2) "pointers", signs that serve particular interpreters, as they find themselves operating relative
to their locally active contexts and temporally dynamic environments, as accessory addresses of
modifiable memory contents. In any case something extra -- some further information about
the objective, syntactic, or interpretive context -- must be added to the index in order to tell
what it does in fact denote, or even if it does indeed denote anything at all.
If a real object can be regarded as a generic and permanent property that is shared by all of its specific
and momentary instantiations, then it is possible to re-characterize indexical signs in the following terms:
An index of an object is a property of an actual instance of that object. It is in this sense that indices are
said to have actual but not essential connections to what they denote.
Saying that an index is a property of an instance of an object almost makes it sound as though the relation
of an index to what it denotes could be defined in purely objective terms, as a relative product of the two
dyadic relations, "property of" and "instance of", and wholly independently of any particular interpreter.
But jumping to this conclusion would only produce an approximation to the truth, or a likely story, and
thus the kind of a shabby pretension to real-life narrative that would deservedly provoke the rejoinders:
"In whose approach?" or "Likely to whom?"
Taking up these challenges provides a clue as to how a sign relation can appear to be "nearly objective",
"moderately independent", or "relatively composite", all with appropriate respect to the intermediary of
an operative interpreter and all within the medium of a particular framework for analysis and interpretation.
Careful inspection of the context of definition reveals that it is not really the supposedly frame-free relations
of properties and instances that suffice to compose the indexical connection. Indeed, it is not nearly enough
that the separate links exist in principle to make something a property of an instance of something. In order
to constitute a genuine sign relation, indexical or otherwise, each link must be recognized to exist by one
and the same interpreter.
From this point of view, the object is considered to be something in the external world and the index is
considered to be something that touches on the interpreter's experience, both of which subsume, though
perhaps in different senses, the "object instance" (OI) that mediates their actual connection. Although the
respective subsumptions, of OI to object and of OI to index, can appear to fall at first glance only within
the reach of divergent senses, both must appeal for their eventual realization to a common sense, one that
rests within the grasp of a single interpreter. Apparently then, the object instance is the sort of entity that
can contribute to generating both the object and the experience, in this way connecting the diverse forms
of abstraction called "objects" and "indices".
If a suitable framework of object instances can be found to rationalize a particular interpreter's experience
with objects, then the actual connection that subsists between an object and its index becomes in this frame
precisely the connection that exists between two properties of the same object instance, or between two sets
intersecting in a common element. Relative to the appropriate framework, the actual connections needed to
explain a global indexing operation can be identified, point for point, with the collective function of those
joint instances or common elements.
At this stage of analysis, what were originally regarded as real objects have become hypostatic abstractions,
extended as generic entities over classes of more transient objects, their instantiating actualizations. In this
setting, a real object is now analogous to an extended property or a generative predicate, whose extension
generates the trajectory of its momentary instances or the locus of its points in actual existence.
Persisting in this form of analysis appears to lead the discussion toward levels of existence that are in one
way or another more real, more determinate, in a word, more objective than its original objects. If only
a particular way of pursuing this form of analysis could be established as reaching a truly fundamental
level of existence, then reason could not object to speaking of objects of objects, nor even to invoking
the ultimate objects of objects, meaning the unique atoms at the base of the hierarchy that is formed
by the descent of objects.
And yet experience leads me to believe that forms of analysis are too peculiar to persons and communities,
times and mores, too dependent on their particular experiences and traditions, and overall too much bound
to interpretive constitutions of learning and culture to ever be justly established as invariants of nature.
In the end, or rather, by way of appeal to the many courts of final opinion, to invoke any particular
form of analysis, no matter whether it is baseless or well-founded, is just another way of referring
judgment to a particular interpreter, a contingent IF or a self-serving SOI. Consequently, every
form of arbitration retains an irreducibly arbitrary element, and the best policy remains what
it has always been, to maintain an honest index of that fact.
Therefore, I consider any supposed form of "ontological descent" to be, more likely, just one among many
possible forms of "semantic descent", each one of which details a particular way to reformulate objects as
signs of more determinate objects, and every one of which forms of devolution operates with respect to its
own implicit assumption of a form of analysis or its own proper, and probably tacit, analytic framework.
There are moments in the development of an analytic discussion when a thing initially described as a single
object under a single sign needs to be reformulated as a congeries extending over more determinate objects.
If the usage of the original singular sign is preserved, as it often is, then the multitude of new instances that
one comes to fathom beneath the old object's superficial appearance gradually serve to reconstitute the
singular sign's denotation in the fashion of a plural reference.
One such moment was reached in the preceding subsection, where the topics opened up by indexical signs
invited the discussion to begin addressing much wider areas of concern. Eventually, to account for the
effective operation of indexical signs I will have to invoke the concept of a "real object" and pursue the
analysis of ostensible objects in terms of still more objective things. These are the extended multitudes
of increasingly determinate objects that I will variously refer to as the actualizations, the instantiations,
the realizations, etc. of objects, and on occasion (and not without reason) the "objects of objects" (OOO's).
Another such moment will arrive when I turn to developing suitable embodiments of sign relations within
dynamically realistic systems. In order to implement interpreters as state transition systems, I will have to
justify the idea that dynamic states are the "real signs" and proceed to reconstitute the customary types of
signs as abstractions from still more significant tokens. These are the immediate occasions of sign-using
transactions that I will tender as "instances of use" (IOU's) or "situations of use" (SOU's), plus the states
and motions, that is, the projectable evolutions, of dynamic systems that solely are able to realize these
uses and discharge the obligations they incur to reality.
In every case, working within the framework of systems theory will lead this discussion toward systems
and conditions of systems as the ultimate objects of investigation, implicated as the ends of both synthetic
and analytic proceedings. Sign relations, initially formulated as relations among three arbitrary sets, will
gradually have their original substrates replaced with three systems, the object system, the sign system,
and the interpretant system -- not just sets, but systems all -- not just set in their several places, but
actively, dynamically, triadically interacting.
Since the roles of a sign relation are formally and pragmatically defined, they do not overly fixedly depend,
to the exclusion of every conceivable transubstantiation, on the material aspects or the entitative attributes
of the initial domains of their consensual definition or the transient elements of their temporal realization.
Therefore, it is conceivable that the very same system could appear in all three roles, and from the open
chance of this possibility arises much of the ensuing complication of the subject, not to mention many
of its most baffling subtleties.
A related source of conceptual turbulence stems from the circumstance that, even though a certain
aesthetic dynamics attracts the mind toward sign relational systems that are capable of reflecting on,
commenting on, and thus "counter-rolling" their own behavior, it is still important to distinguish in
every active instance the part of the system that is doing the discussing from the part of the system
that is being discussed. To do this, the duly observant interpreter needs two things: (1) the senses
to discern the essential tensions that typically prevail between the formal pole and the informal arena,
and (2) the language to articulate, aside from their potential roles, the moment by moment placement
of dynamic elements and systematic components with respect to this compassing field of polarities.
1.3.4.11 Review & Prospect What has been learned from the foregoing study of icons and indices? The import of this examination
can be sized up in two stages, at first, by reflecting on the action of both the formal and the casual signs
that were found to be operating in and around the discussion of A and B, and then, by taking up the
lessons of this circumscribed arena as a paradigm for future investigation.
In order to explain the operation of sign relations corresponding to the iconic and the indexical signs
in the A and B example, it becomes necessary to refer to potential objects of thought that are located,
if they exist at all, outside the realm of the initial object set, in other words, lying beyond the objects
of thought that are present at the outset of the discussion and that one initially recognizes as objects
of formally identified signs. In the process of doing this, it is incumbent on a satisfying explanation
of the initial objects to invoke the abstract properties of objects and the actual instances of objects,
where all of these properties and all of these instances are usually assumed to be "new" objects of
thought, that is, ones that are normally and typically distinct from the objects to which they relate.
In the pragmatic account of things, thoughts are just signs in the mind of their thinker, so every object
of a thought is the object of a sign, though perhaps in a sign relation that has not been fully formalized.
Considered on these grounds, the search for a satisfactory context in which to explain the actions and
the effects of signs turns into a recursive process that potentially calls on ever higher levels of properties
and ever deeper levels of instances that are found to stem from whatever objects instigated the search.
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