User:Jon Awbrey/IDS

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Inquiry Driven Systems

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IDS.  Inquiry Driven Systems

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Version : May-Jun 2004 [Draft 11.00]

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Chapter 1.
Division 1.1
Section 1.1.1
Subsection 1.1.1.1

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1. Introduction

1.1. Outline of the Project : Inquiry Into Inquiry

1.1.1. Problem

This research is oriented toward a single problem:

What is the nature of inquiry?

I intend to address crucial questions about the operation, organization,
and computational facilitation of inquiry, taking inquiry to encompass
the general trend of all forms of reasoning that lead to the features
of scientific investigation as their ultimate development.

1.1.2. Method

How will I approach this problem about the nature of inquiry?

The simplest answer is this:

I will apply the method of inquiry to the problem of inquiry's nature.

This is the most concise and comprehensive answer that 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?
Accordingly, 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 -- and what those are
is just another 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 give preference to two kinds
of technique, one analytic and one synthetic.

The principal 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 in the study.

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 iterative or even opportunistic
fashions, working on any edge of research that appears to be ready at
a given time.  If forced to anticipate the most 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 that they express in the form of computational
interpreters or language processors.

If everything goes according to the pattern I have observed in previous work,
the principal facets of analytic and synthetic procedure will each be prefaced
by its own distinctive phase of preparatory activity, where the basic materials
needed for further investigation are brought together for comparative study.
Taking these initial stages into consideration, I can describe the main
modalities of this research in greater detail.

1.1.2.1. The Paradigmatic and Process-Analytic Phase

In this phase I describe the performance and the competence
of intelligent agents in terms of a variety of formal systems:

For aspects of an inquiry process that affect its dynamic or
its temporal performance I will typically use representations
that are modeled on finite automata and differential systems.

For aspects of an inquiry faculty that reflect its formal or
its symbolic competence I will generally use representations
like formal grammars, logical calculi, constraint-based axiom
systems, and rule-based theories, all in connection with many
different "proof styles", for example, equational or illative.

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:

1.  Focus on a problematic phenomenon.  This is a general property
    or a generic process that attracts one's interest, for example,
    intelligence or inquiry.

2.  Gather under consideration significant cases of concrete agents
    or systems that exhibit the property or the process in question.

3.  Reflect on the common properties of these systems in a search for the
    less obvious traits that might explain their more surprising features.

4.  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 the hypothesized traits
    generate (give rise to, provide a basis for) the phenomenon of interest,
    either in nature or on the computer, then one has reason to consider them
    further as possible explanations.

The last option of the last step already overlaps with the synthetic phase of work.
Regarding this phase of procedure within the frame of experimental research, it is
important to recognize that a computer program can fill the role of a hypothesis,
that is to say, a testable (defeasible or falsifiable) construal of how a process
is actually, might be possibly, or ought to be optimally carried out.

1.1.2.2. The Paraphrastic and Faculty-Synthetic Phase

The closely allied techniques of task analysis and software development that are
known as "step-wise refinement" and "top-down programming" in computer science
(Wirth 1976, 49, 303) have a long ancestry in logic and philosophy, going back
to a strategy for establishing or for discharging contextual definitions that
is known as "paraphrasis".  All of these methods are founded on the idea of
providing meanings for operational specifications, "definitions in use",
alleged descriptions, or "incomplete symbols".  No excessive generosity
with the resources of meaning is intended, though.  It often happens
in practice that 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, p. 216).
See also (Whitehead & Russell, in Van Heijenoort, pp. 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 to say, as conjectures about how a process is actually accomplished
in nature, as speculations about how it might be done in principle, or
as explorations of how it might be done better in the medium of
technological extensions.

For the purposes of this project, "paraphrastic definition" denotes the
analysis of formal specifications and contextual constraints to derive
effective implementations of a process or its corresponding faculty.
This is carried out by considering what the faculty in question is
required to do in the many contexts that it is required to serve,
and then by analyzing these formal specifications with an eye to
the design of computer programs that can fulfill them, at least,
to whatever extent makes sense with regard to the ends in view.

1.1.2.3. Reprise of Methods

The whole array of methods will be typical of the "top-down" strategies
used in artificial intelligence research, 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 most critical and toughest
part of this discipline is in making sure that one does "come down",
that is, in finding guarantees that the analytic reagents and the
synthetic apparatus that one applies are actually effective,
reducing the excipients of speculation to arrive at active
ingredients and effective principles.

Finally, I ought to observe a hedge against betting too much on this
or any other 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 runners, shoots, and tendrils of research tend to interleave
and intertwine as they will.

1.1.3. Criterion

When is enough enough?  What measure can I use to tell if my effort is working?
What information is critical in deciding whether my exercise of the method is
advancing my state of knowledge toward a solution of the problem?

Given that the problem is "Inquiry" and the method is "Inquiry", the test of
progress and eventual success is just the measure of any inquiry's performance.
According to my current understanding of inquiry, and the provisional model of
inquiry that will guide this project, the criterion of an inquiry's competence
is how well it works 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 advanced:

Successful results of inquiry provide the agent with augmented
powers of (1) control and (2) prediction with respect to how
the object system will behave under the given circumstances.
If a common theme is sought that will cover both of these
goals, even if 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 two-way bridges 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
linkages among cognitive psychology (the descriptive study of how we think),
artificial intelligence (the prospective study of how we might think), and
logic (the normative study of how we ought to think in order to accomplish
the goals of reasoning).

1.2. Onus of the Project : No Way But Inquiry

At the beginning of inquiry there is nothing for me to work with
but the actual constellation of doubts and beliefs that I have at
the moment.  Beliefs that operate at the deepest levels can be so
taken for granted that they rarely if ever obtrude on awareness.
Doubts that oppress in the most obvious ways are still known only
as debits and droughts, as the absence of something, one knows not
what, and a desire that obliges one only to try.  Obscure forms of
oversight provide an impulse to replenish the condition of privation
but never out of necessity afford a sense of direction.  One senses
that 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 their way
through the constellations of contingent, incomplete, and contradictory
indications that they actually find themselves sailing under at present.

In the remainder of this Division (1.2) I will indicate
as briefly as possible the nature of the problem that
must be faced in this particular approach to inquiry,
and try to explain what a large share of the ensuing
effort will be directed toward clearing up.

Toward the end of this discussion I will be using highly concrete
mathematical models, or very specific families of combinatorial
objects, to represent the abstract structures of experiential
sequences that agents pass through.  If these primitive and
simplified models are to be regarded as something more than
mere toys, and if the relations of particular experiences to
particular models, along with the structural relationships
that exist within the field of experiences and again within
the collection of models, are not to be dismissed as category
confusions, then I will need to develop a toolbox of logical
techniques that can be used to justify these constructions.
The required technology of categorical and relational notions
will be developed in the process of addressing its basic task:
To show how the same conceptual categories can be applied to
materials and models of experience that are radically diverse
in their specific contents and peculiar to the states of the
particular agents to which they attach.

1.2.2. A Fugitive Canon

The principal difficulties associated with
this task appear to spring from two roots.

First, there is the issue of "computational mediation".  In using the sorts
of sequences that computers go through to mediate discussion of the sorts
of sequences that people go through, it becomes necessary to re-examine
all of the facilitating assumptions that are commonly taken for granted
in relating one human experience to another, that is, in describing and
building structural relationships among the experiences of human agents.

Second, there is the problem of "representing the general in the particular".
How is it possible for the most particular imaginable things, namely, the
transient experiential states of agents, to represent the most general
imaginable things, namely, the agents' own conceptions of the abstract
categories of experience?

Finally, not altogether as an afterthought, there is a question that binds
these issues together.  How does it make sense to apply one's individual
conceptions of the abstract categories of experience, not only to the
experiences of oneself and others, but in points of form to compare
them with the structures present in mathematical models?

1.3. Option of the Project : A Way Up To Inquiry

I begin with an informal examination of the concept of inquiry.

In this Division I take as subjects the supposed faculty of
inquiry in general and the present inquiry into inquiry in
particular and I attempt to analyze them in relation to
each other on formal principles alone.

The initial set of concepts that I need to get the discussion
started are relatively few in number.  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 into components
that amount to simpler types of phenomenal processes, to the extent
that this is seen possible.

As a phenomenon, a particular way of doing inquiry will here be regarded
as embodied in a faculty of inquiry, as possessed by an agent of inquiry.
As a process, a particular example of inquiry will be regarded as extended
in time through a sequence of states, as experienced by its mediating agent.
In this view of phenomena and processes, it is envisioned that an agent or
a faculty of any generically described phenomenal process, inquiry included,
could be started off from different initial states and would conceivably go
through different trajectories of subsequent states, and yet there would be
a recognizable quality or an abstractable property that justifies invoking
the name of the genus in question, inquiry included.

The steps of this analysis will be annotated below by making use of the
following conventions.  Lower case letters denote phenomena, processes,
or faculties under investigation.  Upper case letters denote classes of
the same sorts of entities.  Special use is made of the following symbols:
Y = genus of inquiry;  y = generic inquiry;  y_0 = present inquiry.

Compositions of "faculties" are indicated by concatenating their names,
as fg, and are posed in the sense that the right "applies to" the left.
The notation "f >= g" indicates that f is greater than or equal to g in
a decompositional series, in other words, f possesses g as a component.
The coset notation FG indicates a class of "faculties" of the form fg,
with f in F and g in G.  Notations like "{?}", "{?, ?}", and so on,
serve as proxies for unknown components and indicate tentative
analyses of faculties in question.

1.3.1. Initial Analysis of Inquiry -- Allegro Aperto

If the faculty of inquiry is a coherent power, then it has
an active or instrumental face, a passive or objective face,
and a substantial body of connections between them.  y = {?}.

In giving the current inquiry a reflexive cast, as inquiry into inquiry,
I have brought inquiry face to face with itself, inditing it to apply
its action in pursuing a knowledge of its passion.  y_0 = y 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 >= {discussion, formalization}.

In accord with this plan, the main body of this Division (1.3) is devoted
to a discussion of formalization.  y_0 = 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, and that is the discussion of
discussion itself.  y_0 = 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 incumbent on me at this
point to introduce the understanding of formalization
that I will use throughout this work and to preview
a concrete example of its application.

1.3.3.1. A Formal Charge

An introduction to the topic of formalization, if proper,
is obliged to begin informally.  But it will be my constant
practice here 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:

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

   2.  Pay special attention to the nominal operations
       that are invoked to substantiate each tentative
       explanation of a critically important process.
       Frequently, but not infallibly, operations of
       this sort can be detected appearing in the
       guise of "-ionized" words, in other words,
       terms ending in the suffix "-ion" that
       typically connote both a process and
       its result.

   3.  Ask yourself, with regard to each postulant faculty
       in the current running account, explicitly charged
       or otherwise, whether you can imagine any recipe,
       any program, any rule of procedure for carrying
       out the form, if not the substance, of what it
       does, or an aspect thereof.

1.3.3.2. A Formalization of Formalization?

An immediate application of the above rules is presented here, in hopes of
giving the reader a concrete illustration of their use in a ready example,
but the issues raised can quickly diverge into yet another distracting
digression, one not so easily brought under control as the discussion
of discussion, but whose complexity probably approaches that of the
entire task.  Therefore, a mere foreshadowing of its character will
have to do for the present.  y_0 = 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
that fall under the charge, and so it can be lionized as the nominal
head of a prospectively formal discussion.  The reader is entitled 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 discussion 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 Subsection took the concept of "formalization" as an example
of a topic that a writer might try to translate from informal discussion
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 monologues with themselves, or
(2) that a writer (speaker) conducts in real or imagined dialogue
with a reader (hearer).  In view of this implicitly discursive
mediation, I see no harm in letting the concept of discussion
be stretched to cover all attempted processes of formalization.
This assumption may be annotated as F c D.

In this Subsection, I step back from the example of "formalization"
and consider the general task of clarifying and communicating concepts
by means of a suitably directed discussion.  Let this kind of "motivated"
or "measured" discussion be referred to as a "meditation", in other words,
"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, allowing the chain of subsumptions 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?  y_0 = 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 cases where 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 (SOI_1),
and builds a relation from it to another system of interpretation (SOI_2).
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 to highlight the effects of a discussion that "builds a relation"
between SOI's, I will call this aspect of the process "narration".

It is the writer's ethical responsibility to ensure that a discourse
is potentially edifying with respect to the reader's current SOI, and
the reader's self-interest to evaluate whether a discourse is actually
edifying from the perspective of the reader's present SOI.

Formally, the relation that the writer builds from SOI to SOI can always
be cast or recast as a three-place relation, one whose staple element of
structure is an indexed or ordered 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 SOI_1 to SOI_2, 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 "considerate discussion" or "polite discourse".

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
the powers for choice in the resulting SOI.

As conditions for the possibility of considerate but significant
narration, there are a few requirements placed on the writer and
the reader.  Considerate narration, constructing a relation from
SOI to SOI in a politic fashion, cannot operate in an infectious
or addictive manner, invading a SOI like a virus or trojan horse,
but ultimately must transfer its communication into the control
of the receiving SOI.  Significant communication, in which the
receiving SOI is augmented by options for meaning and powers
for choice that it did not have before, requires a SOI on
the reader's part that is extensible in non-trivial ways.

At this point, the discussion has touched on a topic, in one of its
manifold aspects, that it will encounter repeatedly, under a variety
of aspects, throughout this work.  In recognition of this circumstance,
and to prepare the way for future discussion, it seems like a good idea
to note a few of the aliases that this protean topic can be found lurking
under, and to notice the logical relationships that exist among its several
different appearances.

On several occasions this discussion of inquiry will arrive at a form
of "aesthetic deduction", in general terms, a piece of reasoning that
results in a design recommendation, and in the immediate case, where
an analysis of the general interests and objectives of inquiry leads
us to conclude 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 what exactly is desired in order to constitute 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
a functional image that captures an aspect of the casual circumstances.

Relations between casual and formal discussion are often treated in
terms of a distinction between two languages, the "meta-language" and
the "object language", linguistic systems that take complementary roles
in filling out the discussion of interest.  In the usual approach, issues
of formalization are addressed by postulating a distinction between the
meta-language, the descriptions and conceptions from ordinary language
and technical discourse that can be used without being formalized, and
the object language, the domain of structures and processes that can be
studied as a completely formalized object.

1.3.3.5. A Formal Approach

I plan to approach the issue of formalization from a slightly different angle,
proceeding through an analysis of the medium of interpretation and developing
an effective conception of "interpretive frameworks" or "interpretive systems".
This concept encompasses any organized system of interpretive practice, ranging
from those used in everyday speech, to the ones that inform technical discourse,
to the kinds of completely formalized symbol systems that one can safely regard
as mathematical objects.  Depending on the degree of objectification that it
possesses from one's point of view, the same system of conduct can variously
be described as an interpretive framework (IF), interpretive system (IS),
interpretive object (IO), or object system (OS).  These terms are merely
suggestive -- no rigid form of classification is intended.

Many times, it is convenient to personify the interpretive organization
as if it were embodied in the actions of a typical user of the framework
or a substantive agent of the system.  I will often refer to an agent of
this kind as the "interpreter" of the moment.  At other times, it may be
necessary to analyze the action of interpretation a bit more carefully.
At these latter 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.  Therefore, 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 meantime a sign
that serves to convey an appearance of knowledge in a place where very
little is known about the subject itself.

A meta-language corresponds to what I call an "interpretive framework".
Besides a set of descriptions and conceptions, it embodies the whole
collective activity of unexamined structures and automatic processes
that are trusted by agents at a given moment to make its employment
meaningful in practice.  An interpretive framework is best understood
as a form of conduct, in other words, as 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
far too incumbent and cumbersome to be easily moved on its own grounds,
at least, it often 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 constituent members of the framework.  In some cases,
the rules of the IF in question proscribe against 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 these acts of reflection and critique are 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 the two
languages is de facto a part of the meta-language and not even
an object of discussion in the object language.  A slippery
slope begins at this step.  A failure to build reflective
capacities into an interpretive framework can let go
unchallenged the spurious opinion that presumes that
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 to "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'm raking over here constellate a rather
striking scene, especially for a setting that is intended to function as
a neutral backdrop.  Departing from the rule that we seek and often find
in other concerns, the points I am making seem obvious to all reasonable
people at the outset of discussion, and yet the difficulties that follow
as inquiry develops get muddier and more grating the more one probes and
stirs them up.  A large measure of the blame, I think, can be charged to
a misleading directive that people tend to derive from the prefix "meta",
leading them to search for higher and higher levels of meaning and truth,
on beyond language, on beyond every conceivable system of signs, thus on
beyond the realm of sense.  Prolonged use of the affix "meta" after this
fashion leads people to act as if the meta-language were step outside of
ordinary language, or an artificial platform built above and beyond our
natural languages, then they forget that formal models are developments
that are internal to the informal context.  For this reason and others,
I recommend replacing allusions to rigidly stratified object languages
and meta-languages with indices of contingent interpretive frameworks.

To avoid the types of cul-de-sac that are outlined above, I am taking 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 own 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.

From this angle of approach, let us try to view afresh the manner
of drawing distinctions between various levels of formalization
in language.  Once again, I begin in the context of ordinary
discussion, and if there is any distinction to be drawn
between objective and instrumental languages then it
must be possible to describe it within the frame
of this informally discursive universe.

1.3.3.6. A Formal Development

The point of view I take on the origin and development of formal models
is that they arise with agents retracing structures that already exist
in the context of informal activity, until gradually the most relevant
and frequently reinforced patterns become emphasized and emboldened
enough to continue their development as nearly autonomous styles,
in brief, as "genres" growing out of a particular "paradigm".

Taking the position that formal models develop within the framework
of informal discussion, the questions that become important to ask of
a prospective formal model are:  (1) whether it highlights the structure
of its supporting context in a transparent form of emphasis and a relevant
reinforcement of salient features, and (2) whether it discloses the active
ingredients of its source materials in a critically reflective recapitulation
or an analytically representative recipe, or (3) whether it instead insistently
obscures what portion of its domain it manages to cover.

1.3.3.7. A Formal Persuasion

An interpretive system can be taken up with very little fanfare, since it
does not enjoin one to declare undying allegiance to a particular point of
view or to assign each piece of text in view to a sovereign territory, but
only to entertain different points of view on the use of symbols.  One of
the chief design considerations for an interpretive system is that it must
never function as a virus or addiction.  Its suggestions must always be,
initially and finally, purely optional adjunctions to the interpretive
framework that was already in place before it installed itself on the
scene.  Interpretive systems are not constituted in the faith that
anything nameable will always be dependable, nor articulated in
fixed principles that determine what must be doubted and what
must not, but rest only in a form of self-knowledge that
recognizes the doubts and beliefs that one actually
has at each given moment.

Before this project is done I will need to have developed
an analytic and computational theory of interpreters and
interpretive frameworks.  In the aspects of this theory
that I can anticipate at this point, an interpreter or
interpretive framework is exemplified by a collective
activity of symbol-using practices like those that
might be found embodied in a person, a community,
or a culture.  Each one forms a moderately free
and independent perspective, with no objective
rankings of supremacy in practice that every
interpretive framework is likely to support
at any foreseeable moment in its field of
view.  Of course, each interpreter enters
discussion initially operating as if its
own perspective were "meta" in relation
to all the others, but a well-developed
interpretive system is likely to have
acquired the notion and taken notice
of the fact that this is not likely
to be a universally shared opinion.

1.3.4. Discussion of Formalization : Concrete Examples

Section 1.3.3 outlined a variety of general issues surrounding the concept
of formalization.  Section 1.3.5 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 as briefly as possible the features of the modeling activity
that are most relevant to our present purpose:  The modeler begins with
a "phenomenon of interest" or a "process of interest" (POI) and relates
it to a formal "model of interest" (MOI), the whole while working within
a given "interpretive framework" (IF) and relating the results from one
"system of interpretation" (SOI) to another, or to a later development
of the same SOI.

The POI's that define the intents and the 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 interpretive systems and all of the process models
that are being mentioned here come from the same broad family of
mathematical objects, the different roles that they play in this
investigation are mainly distinguished by variations in their
manner and degree of formalization:

1.  The typical POI comes from natural sources or from 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.

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 that
    is 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 get a handle on these directions is
    the opportune mathematical circumstance that there can be well-defined
    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".
    In sum, the MOI is pivotal or cardinal in that it constitutes a model
    in two senses, (a) loosely analogical and (b) more strictly logical,
    integrating twin roles of the model concept in a single focus.

3.  Finally, the IF's and the 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 much likelihood of a completely
    articulate formulation being forthcoming.  Still, it is usually worth
    the effort to try lifting one or another edge of these frameworks
    and backdrops into the light, at least for a time.

1.3.4.2. Sign Relations : A Primer

To the extent that their structures and functions can be discussed at all,
it appears likely at this point 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 easily converted to or ultimately grounded in
sign relations.  This class of triadic relations constitutes
the main study of the "pragmatic theory of signs", a branch
of logical philosophy devoted to understanding all types
of symbolic representation and communication.

There is a close relationship between the pragmatic theory of signs and the
pragmatic theory of inquiry.  In fact, the correspondence between the two
studies exhibits so many parallels and coincidences that it is often best
to treat them as integral parts of one and the same subject.  In a very
real sense, inquiry is the process by which sign relations come to be
established and continue to evolve.  In other words, inquiry, "thinking"
in its best sense, "is a term denoting the various ways in which things
acquire significance" (Dewey).  Thus, there is an active and intricate
form of cooperation that needs to be appreciated and maintained between
these converging modes of investigation.  Its proper character is best
understood by realizing that the theory of inquiry is adapted to study
the developmental aspects of sign relations, a subject which the theory
of signs is specialized to treat from structural and comparative points
of view.

Because the examples in this Section (1.3.4) 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.
Despite their simplicity, however, these examples have subtleties of
their own, and their careful treatment will serve to illustrate many
important issues in the general theory of signs.

Example 1.  The Story of A and B

Imagine a discussion between two people, Ann and Bob, and attend only
to that aspect of their interpretive practice that involves the use
of the following nouns and pronouns:  "Ann", "Bob", "I", "you".

The "object domain" of this discussion fragment is the
set of two people {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 that they use.  The "system of interpretation" (SOI)
associated with each language user can be represented in the form of
an individual three-place relation called the "sign relation" of that
interpreter.

Understood in terms of its set-theoretic extension, a sign relation L
is a subset of a cartesian product O x S x I.  Here, O, S, I are three
sets that are known as the "object domain", the "sign domain", and the
"interpretant domain", respectively, of the sign relation L c O x S x I.

Broadly speaking, the three domains of a sign relation can be
any sets whatsoever, but the kinds of sign relations that are
typically contemplated in a computational setting are usually
constrained to having I c S.  In this case, interpretants are
just a special variety of signs, and this makes it convenient
to lump signs and interpretants together into a single class
called the "syntactic domain".   In the forthcoming examples,
S and I are identical as sets, so the same elements manifest
themselves in two different roles of the sign relations in
question.  When it is necessary to refer to the whole set
of objects and signs in the union of the domains O, S, I
for a given sign relation L, one may call this set the
"world of L" and write W = W(L) = O |_| S |_| I.

To facilitate an interest in the abstract structures of sign relations,
and to keep the notations as brief as possible as the examples become
more complicated, I introduce the following general notations:

   O  =  Object Domain

   S  =  Sign Domain

   I  =  Interpretant Domain

Introducing a few abbreviations for use in considering
the present Example, we have the following set of data:

   O  =  {Ann, Bob}  =  {A, B}

   S  =  {"Ann", "Bob", "I", "You"}  =  {"A", "B", "i", "u"}

   I  =  {"Ann", "Bob", "I", "You"}  =  {"A", "B", "i", "u"}

In the present Example, 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 of the form <o, s, i> that make up the
corresponding sign relations:  L(A), L(B) c O x S x I.  It is often tempting
to use the same names for objects and for relations involving these objects,
but I will avoid this here, taking up the issues that this practice raises
after the less problematic features of these relations have been treated.

Table 1.  Sign Relation of Interpreter A
o---------------o---------------o---------------o
| Object        | Sign          | Interpretant  |
o---------------o---------------o---------------o
| A             | "A"           | "A"           |
| A             | "A"           | "i"           |
| A             | "i"           | "A"           |
| A             | "i"           | "i"           |
| B             | "B"           | "B"           |
| B             | "B"           | "u"           |
| B             | "u"           | "B"           |
| B             | "u"           | "u"           |
o---------------o---------------o---------------o

Table 2.  Sign Relation of Interpreter B
o---------------o---------------o---------------o
| Object        | Sign          | Interpretant  |
o---------------o---------------o---------------o
| A             | "A"           | "A"           |
| A             | "A"           | "u"           |
| A             | "u"           | "A"           |
| A             | "u"           | "u"           |
| B             | "B"           | "B"           |
| B             | "B"           | "i"           |
| B             | "i"           | "B"           |
| B             | "i"           | "i"           |
o---------------o---------------o---------------o

These Tables codify a rudimentary level of interpretive practice for the
agents A and B, and provide a basis for formalizing the initial semantics
that is appropriate to their common syntactic domain.  Each row of a Table
names an object and two co-referent signs, making up an ordered triple of
the form <o, s, i> that is 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.

One aspect of the meaning of a sign is concerned with the reference
that a sign has to its objects, which objects are collectively known
as the "denotation" of the sign.

There is a difficulty that needs to be mentioned at this point, though
for the sake of a first approach to the general theory of sign relations
I will need to sidestep detailed discussion of it until later in the game.
The problem is this:  Generally speaking, when it comes to things that are
being contemplated as ostensible or potential signs, neither the existence
nor the uniqueness of any objects in their denotations is guaranteed.  Thus,
the denotation of a putative sign can refer to a singular, a plural, or even
a vacuous number of objects.  A proper treatment of this complication calls
for the conception of something slightly more general than a sign relation
proper, namely, a construct called a "sign relational complex".  In effect,
expressed in the roughest practical terms, this allows for "missing data"
in the columns of the relational database table for the sign relation in
question.  Until this concept can be properly developed, let us operate
on the default assumption that signs actually have objects, but remain
wary enough of the exceptions to deal with them on an ad hoc basis.

In the pragmatic theory of sign relations, denotative references are
formalized as certain types of dyadic relations that are obtained by
projection from the triadic sign relations.

The dyadic relation that constitutes the "denotative component" of
a sign relation L will here be notated as Den(L).  Information about
the denotative component of meaning can be obtained from L by taking
its "dyadic projection" on the object-sign plane, in other words, on
the 2-dimensional space that is generated by the object domain O and
the sign domain S.  This denotative aspect or semantic projection of
a sign relation L will here be notated in any one of the following
equivalent forms, Proj_OS (L), Proj_12 (L), L_OS, or L_12, and it
is defined as follows:

Den(L) = Proj_OS (L) = {<o, s> in O x S : <o, s, i> in L for some i in 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 meaning 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 empty, unique, or multiple in its collection of
terminal points, and both can be formalized as different types
of dyadic relations that are obtained as planar projections
of the triadic sign relations.

The connection that a sign makes to an interpretant will here be referred to
as its "connotation".  In the general theory of sign relations, this aspect
of meaning includes the references that a sign has to affects, concepts,
impressions, intentions, mental ideas, and 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 system 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 sign relation L,
the dyadic relation that forms the "connotative component"
of L will here be denoted as Con(L).

The bearing that an interpretant has toward a common object of its sign
and itself has no standard name.  If an interpretant is considered to be
a sign in its own right, then its independent reference to an object can
be taken as belonging to another moment of denotation, but this neglects
the mediational character of the whole transaction in which this occurs.

In view of the service that interpretants supply in furnishing a locus
for critical, explanatory, and reflective glosses, both with regard to
the objective scenes and also with respect to their descriptive themes,
it is possible to regard interpretant signs as providing "annotations"
for both objects and signs, but this function points in the opposite
direction of the arrow from interpretants to objects that is needed
in the present connection.  What does one call the inverse of the
annotation function?  More generally asked, what does one call
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 meaning.  On a trial basis,
I will describe it as the "ideational", "intentional", or "canonical" aspect
of the sign relation, and I will see how it works in the long run to call the
reference of an interpretant sign to its object by the name of its "ideation",
"intention", or "conation".  For the time being, then, the dyadic relation that
constitutes the "intentional component" of a given sign relation L will here be
notated as Int(L).

A full consideration of the connotative and intentional aspects of meaning
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 speaking, the connotative and intentional
aspects of meaning present no additional difficulty.

The connotative component of a sign relation L can be formulated
as a dyadic projection on the plane of the sign and interpretant
domains, and thus defined as follows:

Con(L) = Proj_SI (L) = {< s, i> in S x I : <o, s, i> in L for some o in O}.

The intentional component of meaning for a sign relation L, or its
"second moment of denotation", is adequately captured as a dyadic
projection on the plane of the object and interpretant domains,
and thus defined as follows:

Int(L) = Proj_OI (L) = {<o, i> in O x I : <o, s, i> in L for some s in S}.

As it happens, the sign relations L(A) and L(B) in the present Example
are fully symmetric with respect to exchanging signs and interpretants,
so all of the data of Proj_OS L(A) is echoed unchanged in Proj_OI L(A)
and all of the data of Proj_OS L(B) is echoed unchanged in Proj_OI L(B).

The principal concern of this project is not with every conceivable
sign relation but chiefly with those that are capable of supporting
inquiry processes.  In these species of sign relation, the relation
between the connotational and the denotational aspects of meaning
is not wholly arbitrary.  Instead, this relationship is naturally
constrained or deliberately designed in such a way that it can
achieve the following aims:

   1.  Represent the embodiment of significant properties
       that have objective reality in the agent's domain.

   2.  Support the achievement of particular purposes
       that have intentional value for the agent.

Therefore, my attention is directed chiefly toward understanding the forms
of correlation, coordination, and cooperation among the various components
of sign relations that form the necessary conditions for achieving these
aims and thus for being able to conduct coherently directed inquiries.

1.3.4.3. Semiotic Equivalence Relations

If one examines the sign relations L(A) and L(B) that are associated with
the interpreters A and B, respectively, one observes that they have many
contingent properties that are not possessed by sign relations in general.

One of the nicest properties possessed by the sign relations L(A) and L(B)
is that their connotative components L(A)_SI and L(B)_SI constitute a pair
of equivalence relations on their common syntactic domain S = I.  There is
reason to call such constructions "semiotic equivalence relations" (SER's),
since they equate signs that mean the same thing to somebody.  Each of the
SER's L(A)_SI and L(B)_SI c S x I = S x S, partitions the whole collection
of signs into "semiotic equivalence classes" (SEC's).  These constructions
make for an especially strong form of representational relationship between
objects and signs in that the structure of the participants' common object
domain is reflected or reconstructed, part for part, in the structure of
each one's "semiotic partition" (SEP) of their shared syntactic domain.

The main trouble with this notion of shared meaning in the present Example
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 that are
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.  A's Semiotic Partition
o-------------------------------o
|      "A"             "i"      |
o-------------------------------o
|      "u"             "B"      |
o-------------------------------o

Table 4.  B's Semiotic Partition
o---------------o---------------o
|      "A"      |      "i"      |
|               |               |
|      "u"      |      "B"      |
o---------------o---------------o

In order to discuss this type of situation further,
I introduce the square bracket notation "[x]_E" to
mean "the equivalence class of the element x under
the equivalence relation E".  A statement that two
elements x and y are equivalent under E is called
an "equation", and it can be expressed in either
of two ways, [x]_E = [y]_E, or x =_E y.

In application to sign relations, I extend the use
of the square bracket notation in the following ways.
When L is a sign relation whose "syntactic projection"
or connotative component L_SI is an equivalence relation
on S, then I write "[s]_L" for "the equivalence class of s
under the equivalence relation L_SI".  A statement that the
two signs x and y are synonymous under a semiotic equivalence
relation L_SI is called a "semiotic equation" (SEQ), and can be
expressed in either one of the forms, [x]_L = [y]_L, or x =_L y.

In many situations there is one further adaptation of the
square bracket notation that is very useful.  Namely, when
there is known to exist a particular triple <o, s, i> in L,
it is permissible to use "[o]_L" to mean the same thing as
"[s]_L".  This modification is 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 the case of our present Example
we have the following information.

The SER for interpreter A yields
the following semiotic equations:

   ["A"]_L(A)  =  ["i"]_L(A)

   ["B"]_L(A)  =  ["u"]_L(A)

Otherwise expressed:

   "A"   =_L(A)   "i"

   "B"   =_L(A)   "u"

This amounts to the semiotic partition:

   {{"A", "i"}, {"B", "u"}}.

The SER for interpreter B yields
the following semiotic equations:

   ["A"]_L(B)  =  ["u"]_L(B)

   ["B"]_L(B)  =  ["i"]_L(B)

Otherwise expressed:

   "A"   =_L(B)   "u"

   "B"   =_L(B)   "i"

This amounts to 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 "directed graphs", or "digraphs" for short, that
provide concise pictures of their structural and potential dynamic
properties.  In graph-theoretic terminology, an ordered pair <x, y>
is called an "arc" or a directed edge from point x to point y, and
in the case that x = y, an arc from x to itself is called a "sling"
or a self-loop at the point x.

The denotative components Den L(A) and Den L(B) can be
pictured as digraphs on the six points of their common
world set W = O |_| S |_| I = {A, B, "A", "B", "i", "u"}.

The arcs are given as follows:

   Den L(A) has an arc
   from each point of {"A", "i"} to A and
   from each point of {"B", "u"} to B.

   Den L(B) has an arc
   from each point of {"A", "u"} to A and
   from each point of {"B", "i"} to B.

Den L(A) and Den L(B) can be interpreted as "transition digraphs"
that chart the succession of steps or the connection of states in
a computational process.  If the graph is read this way, then 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 frames of reference.

The connotative components Con L(A) and Con L(B) can be represented
as digraphs on the four points of their common syntactic domain
S = I = {"A", "B", "i", "u"}.  Con L(A) and Con L(B) are SER's,
and so their digraphs conform to the pattern that is shown 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 arcs between the parts.  In the present Example,
the arcs are given as follows:

   Con L(A) has the structure of a SER on S, that is,
   the digraph has a sling at each of the points in S,
   two-way arcs between the points of {"A", "i"}, and
   two-way arcs between the points of {"B", "u"}.

   Con L(B) has the structure of a SER on S, that is,
   the digraph has a sling at each of the points in S,
   two-way arcs between the points of {"A", "u"}, and
   two-way arcs between the points of {"B", "i"}.

Taken as transition digraphs, Con L(A) and Con L(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 three Subsections, that associates
different interpreters and different aspects of interpretation with
different sorts of relational structures on the same set of points,
heralds a topic that will be developed extensively in the sequel.

1.3.4.5. Taking Stock

So far, my discussion of the discussion between A and B, in the picture that
it gives of sign relations and their connection to the imagined processes of
interpretation and inquiry, can best be described as fragmentary.  By way of
telling the story of A and B, a sample of typical language use was 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 or even attempted.

It seems like a good idea to pause at this point and to reflect on the state of
understanding that has been reached.  In order to motivate further developments
it will be useful to inventory two types of shortfall in the present state of
discussion, the first having to do with the defects of my present discussion
in revealing the relevant attributes of even so simple an example as the one
that I am taking as a nominal beginning, the second having to do with the
defects that this species of example exhibits within the broader genus
of sign relations that it is meant, however nascently, to exemplify.

As a general schema, I describe these respective types of limitations as
the "rhetorical defects" and the "objective defects" that any discussion
can have in addressing its intended object.  My immediate concern is to
remedy the insufficiencies of analysis that affect my treatment of the
present example.  The overarching task is to address the atypically
simplistic features of this example as it falls within the class
of sign relations that may be found relevant to realistic cases
of inquiry.

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 are
clues to significant deficits in theory and technique, and that can serve
to 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 transaction between A and B, even though it
lies in plain view on both of their sign relational Tables.  This
is that troubling business, recalcitrant to analysis precisely on
account of the fact that its dealings race on so heedlessly ahead
of thought and grind on so routinely beneath its notice, in short,
it concerns the placement of object languages within the frame of
a meta-language.

Numerous bars to insight appear to interlock here.  Each one is forged
with a good aim in mind, if a bit single-minded in its coverage of the
scene, and the whole gang is set to work innocently enough on behalf of
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 numerous types of false
impression, both about the true characters of the Tables that are presented
here and about the proper utilities of their graphical equivalents that are
designed to be implemented as data structures in the computer.  The next few
remarks are put forth in hopes of averting the ordinary brands of misreading.

The general character of this question can be expressed in the schematic terms
that I used earlier to give a rough sketch of the modeling activity as a whole.
How do the isolated "systems of interpretation" (SOI's) of the agents A and B
relate to the "interpretive framework" (IF) that I am using to present them,
and how does this IF operate, not only to objectify A and B in the guise of
the coordinated "models of interpretation" (MOI's), but simultaneously to
embrace the present and the prospective SOI's of the current narrative,
namely, 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 Table of a sign relation is a complex sign in itself,
each of whose syntactic constituents is assigned a smaller part and plays
the role of a simpler sign in its makeup.  To put it succinctly, there is
nothing but text to be seen on the page.  Viewed 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 are transposed up a notch on the scale that registers
levels of indirectness in reference, with each item passing from
a more objective to a more symbolic mode of presentation.

Sign relations themselves, like any real objects of discussion,
are either too abstract or too concrete to reside in the medium
of communication, but can only find themselves represented there.
The tables and graphs that are used to represent sign relations
are themselves complex signs, involving a step of denotation to
reach the sign relation intended.  The intricacies of this step
require an order of interpretive performers who are able, over
and above executing all of the rudimentary steps of denotation,
to orchestrate these steps in concerted coordination with each
other.  This performance in its turn requires a whole array of
techniques to match the connotations of complex signs and to
test their alternative styles of representation for semiotic
equivalence.  Analogous to the ways that matrices represent
linear transformations and multiplication tables represent
group operations, a large part of the usefulness of these
complex signs comes from the fact that they are not just
conventional symbols for their objects but fully iconic
representations of their objective operative structure.

1.3.4.7. Iconic Signs

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 tabels and graphs, interpreted as iconic representations or
analogous expressions of logical and mathematical objects, the pivotal
properties are formal and abstract in their character.  Since a uniform
translation through any dimension -- of sight, of sound, or imagination --
does not affect the structural properties of a configuration of signs in
relation to one another, this may help to explain how tables and graphs,
in spite of their semantic shiftiness, can succeed in representing the
forms of sign relations without essential loss or radical distortion.

Taking this unsuspecting introduction of iconic signs as a serendipitous
lesson, an important principle can be lifted from their style of 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 what are
possibly the most solid sites for pouring the foundations of formal expression.

What does appear in one of these sign relational Tables?  It is clearly
not the objects that appear under the "Object" heading, but only the
signs of these objects.  It is not even the signs and interpretants
themselves that appear under the "Sign" and "Interpretant" headings,
but only the remoter signs of them that are formed by quotation or
by other devices that effect a similar function.  The unformalized
sign relation in which these signs of objects, signs of signs, and
signs of interpretants have their role as such is not the one Tabled,
but another one that operates behind the scenes to bring its image and
its intent to the reader.

To understand what the Table is meant to convey the reader is called 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 L(A) and L(B) do
not exist in the literal medium of their Tables but are only represented
there by dint of the formal configurational properties that they happen
to 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 experience with language the likes
of which is familiar to writer and reader alike.

The successful formalization of a focal sign relation cannot get by
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 severity to
which an opportunity for understanding is undermined by prior petitions
of the very principles to be explained.  Consequently, 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, what does it mean in terms of
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 this range of 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 casual or a formal transaction that constitutes the issue of the moment.
And it means that a formalization of the writer's and the reader's accessory
sign relations would have several portions that look identical to pieces of
the very sign relational Tables that are being placed under formal review.

1.3.4.8. The Conflict of Interpretations

One discrepancy that needs to be documented at this point
can be observed in the conflict of interpretations between
A and B, as reflected in the lack of congruity between their
separate semiotic partitions of their shared syntactic domain.
This is a problematic feature of the present Example but also
one its more realistic characters.  That is, it exemplifies
a type of problem with the interpretation of pronouns, more
generally considered, all indexical signs and bound variables,
that actually arises in practice when attempting to formalize
the semantics of natural, logical, and programming languages.
On this account, the deficiency is with the present analysis,
and the burden remains to clarify precisely what is going on 
with indexical signs of all kinds.

Notice, however, that I have deliberately avoided trying to deal with
these types of indexical tokens in all of the more usual ways, namely,
by trying to eliminate all of the incipient semantic ambiguities from
the canonical formalization of the working textual material.  Instead,
I have sought to preserve this quality of interpretive discrepancy as
one of the essential phenomena and one of the inevitable facts in the
realm of pragmatic semantics, tantamount to the irreducible nature of
perspective diversity.  I believe that the desired competence in this
faculty of language must develop, not from a strategy of substitution
that replaces bound variables with their objective referents on every
fixed occasion, but from a pattern of recognizing interpretive context
that keeps indexical signs persistently attached to their interpreters
of reference.

1.3.4.9. Indexical Signs

In the pragmatic theory of signs, an "index" is a sign that achieves its
representation of an object by virtue of an actual connection with it.
Though real and objective, the indexical connection can nevertheless
be purely incidental and even a bit accidental.  Its effectiveness
depends only on the fact that any object in actual existence has
a multitude of properties, definitive and derivative, any number
of which are able to serve as its signs.  Indices of an object
reside among its more tangential varieties of attributes, its
accidental or its accessory features, which are really the
properties of some but not all points in the locus of its
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 syntactic contexts but the pragmatic context
that is involved in their actualizing "situation of use" (SOU) or their
realizing "instance of use" (IOU).  To fulfill its proper duty to sense
the reading of an indexical sign demands to be supplemented by a still
more determinate indication of its interpreter of reference, the agent
that is duly responsible for putting it into active use at the moment
in question.

Typical examples of indexical signs in programming languages are:

   1.  Variables, signs that need to be bound to a syntactic context or
       to an instantiation frame in order to have a determinate meaning.

   2.  Pointers, signs that serve particular interpreters operating
       relative to locally active 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 denotes.

If a real object can be regarded as a generic and permanent property
that is shared by all of its momentary and specific 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 properly
said to have "actual connections" but not of
necessity "essential connections" to whatever
objects they do in fact 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 product of the two
dyadic relations, "property of" and "instance of", and independently
of any particular interpreter.  But jumping to this conclusion would
only produce an approximation to the truth, or a likely story, one that
duly provokes 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 "moderately independent", "nearly objective", or "relatively composite",
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.  It is not 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
of these links 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,
although 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
kind of entity that can contribute to generating both the object and
the experience, in this way connecting the diverse abstractions that
are called "objects" and "indices", respectively.

If a suitable framework of object instances can be found to rationalize
an interpreter's experience with objects, then the actual connection that
subsists between an object and its index becomes in this framework precisely
the connection that exists between two properties of the same object instance,
or between two sets that happen to intersect in a common element.  Relative to
the appropriate framework, the actual connections that are 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 to
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 would not object to speaking of objects of objects,
and 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.

However, experience leads me to believe that forms of analysis are too
peculiar to persons and communities, 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 special 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 operates
with respect to its own presumptuous form of analysis and all of
the circular viscosities of its "tacit analytic framework" (TAF).

1.3.4.10. Sundry Problems

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 tends to be,
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 form of a generic 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 to 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, instantiations,
realizations, and so on, of objects, and on occasion, and not
without sufficient 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 coordination with implementing interpreters as state transition
systems, I will be obliged to justify the idea that dynamic states
of dynamical systems are the "real signs" of concern to us and then
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 tender as "situations of use" (SOU's)
or as "instances of use" (IOU's), plus the states and motions of dynamic
systems that solely are able to realize these uses and to discharge the
obligations that 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, sign, and interpretant systems.

Since the roles of a sign relation are formally and pragmatically defined,
they do not depend on the material aspects or the essential attributes
of elements or domains.  Therefore, it is conceivable that the very
same system could appear in all three pragmatic roles, and from
this possibility arises many of the ensuing complications of
the subject.

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 controlling ("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.  In order to do this, interpretive agents need two things:

   1.  The senses to discern the essential tensions that typically prevail
       between the formal pole and the informal arena of each discussion.

   2.  The language to articulate, over and above their potential roles,
       the moment to moment placement of dynamic elements and systematic
       components with regard to this underlying field of polarities.

1.3.4.11. Review and Prospect

What has been learned from the foregoing study of icons and indices?
The impact of this examination can be sized up in a couple of stages:
in the first instance, by reflecting on the action of both the formal
and the formative signs that were found to be operating in and around
the discussion of A and B, and next, by taking up the lessons of this
radically circumscribed arena as a paradigm for further investigation.

In order to explain the operation of sign relations corresponding to the
iconic signs 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,
that is, 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 particular, it is
incumbent on a satisfying explanation to invoke the abstract
properties of objects and the actual instances of objects,
where these properties and instances are normally assumed
to be new objects of thought that are distinct from the
objects to which they refer.

In the pragmatic account of things, thoughts are just signs in the mind
of their thinker, so every object of a thought is the object of a sign,
though perhaps in a sign relation that has yet to be fully formalized.
Considered on these grounds, the search for a satisfactory context
wherein 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 initially instigated the search.

To make it serve as a paradigm for future developments,
I will reiterate the basic pattern that has just been
observed, but with a slightly different emphasis:

In order to explain the operation of icons and indices in a particular
discussion, it is necessary to invoke the abstract properties of objects
and the actual instances of objects, where by this mention of "objects"
one initially comprehends a limited collection of objects of thought
under discussion.  If these properties and instances are themselves
regarded as potential objects of thought, and if they are conceived
to be distinct from the objects whose properties and instances they
happen to be, then every initial collection of objects is forced to
expand on further consideration, in this way pointing to a world of
objects of thought that extends in two directions beyond the initial
frame of discussion.

Can this manner of recursively searching for explanation be established
as well-founded?  In order to organize the expanding circle of thoughts
and the growing wealth of objects that are envisioned within its scheme,
it helps to introduce a set of organizing conceptions.  Doing this will
be the business of the next four Subsections.

1.3.4.12. Objective Plans and Levels

In accounting for the special characteristics of icons and indices
that arose in previous discussions, it became necessary to open up
the domain of objects coming under formal consideration to include
unforeseen numbers of properties and instances of whatever objects
were originally set down.  This is a general phenomenon, affecting
every motion toward toward explanation whether pursued by analytic
or by synthetic means.  What it calls for in practice is a way of
organizing growing domains of objects, without having to specify
in advance all the objects there are.

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" (OL's) of formulation that guide the analytic
and the synthetic studies of interpretive structure and that 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 a 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 A and B.

At this point in the work, I will need to apologize in advance
for introducing a certain idiosyncrasy of terminology, but the
underlying issue that I can see no other way to address except
by means of it can no longer be avoided.  To be specific, I am
forced to use the word "objective" in a sense that will appear
to conflict with several traditions of interpretation, running
so seriously against the grain of some prevailing connotations
that it may sound like a pun or a joke to many readers.  Still,
it's a definite "motive of consistency" (MOC) that requires me
to do this, as I will try to justify in the end.

As always, my use of the word "object" derives from the stock of the
Greek root "pragma", which captures all of the senses needed to suggest
the stability of concern and the dedication to purpose that are forever
bound up in the constitution of objects and the institution of objectives.
What it implies is that every object, objective, or objectivity is always
somebody's object, objective, or objectivity.

In other words, objectivity is always a matter of interpretation.
It is concerned with and quantified by the magnitude of the consensus
that a matter is bound to have at the end of inquiry, but in no way does
this diminish or dismiss the fact that the fated determination is something
on which any particular collection of current opinions are granted to differ.
In principle, there begins to be a degree of objectivity as soon as something
becomes an object to somebody, and the issue of whether this objective waxes
or wanes in time is bound up with the number of observers that are destined
to concur on it.

The critical question is not whether a thing is an object of discussion
and thought, but what kind of discussion and thought it is an object of.
How does one determine the character of this discussion or this thought?
Should this query be construed as a task of finding or a task of making?
Whether it appeals to arts of acquisition, production, or discernment,
and however one expects to decide or decode the conduct it requires,
the character of the discussion and thought in view is sized up and
riddled out in turn by looking at the whole domain of objects and
the pattern of relations among them that it actively charts and
encompasses.  This makes what is usually called "subjectivity"
a special case of what I must call "objectivity", since the
interpretive and the perspectival elements are ab initio
operative and cannot be eliminated from any conceivable
form of discernment, including their own.

Consequently, analyses of objects and syntheses of objects are always
analyses and syntheses to somebody.  Both of these modes of approaching
the constitutions of objects lead to the sorts of approximation that are
appropriate to particular agents and that are able to be appropriated by
whole communities of interpretation.  By way of relief, on occasions when
this motive of consistency hobbles discussion too severely, I will resort
to using chimeras like "object-analytic" and "object-synthetic", paying the
price of biasing the constitution of objects in one direction or the other.

In this work I would like to treat the two-way street of construction
and deconstruction as parallel to the difference of direction between
synthesis and analysis, respectively.  However, being able to do this
without the introduction of too much distortion demands the mediation
of a further distinction.  Accordingly, let it be recognized that all
orientations to the constitutions of objects can be pursued in either
"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.

A radical form of analysis, in order to discern the contrasting
intentions in everything construed as an object, obliges agents
to leave or at least to re-place objects within the contexts of
their live acquaintance, to reflect on their prevailing motives
or their underlying motifs for construing and employing objects
in the ways that they do, and to deconstruct how their own aims
and biases enter into the form and the use of objects.

A radical form of synthesis, in order to integrate ideas and information
devolving from entirely different "frameworks of interpretation" (FOI's),
requires interpreters to reconstruct isolated concepts and descriptions
on a mutually compatible basis and to use means of composition that can
constitute a medium for common sensibilities.

In sum, the radical project in all of these directions demands
forms of interpretation, analysis, synthesis that can reflect
a measure of light on the initially unstated assumptions of
their prospective agents.

The foregoing considerations lead up to the organizing conception of
an "objective framework" (OF), in which objects can be analyzed into
sets of constituent objects, perhaps proceeding recursively to some
limiting level where the fundamental objects of thought are thought
to rest -- or not.  If an OF is felt to be completely unique and
uniquely complete, then people tend to regard it as constituting
a veritable "ontology", but I will not be able to go that far.
The recognition of plural and fallible perspectives that goes
with pragmatic forms of thinking does not see itself falling
into line any time soon with any one or only one ontology.

On the opposite score, there is no reason to deny the possibility
that a complete and unique OF exists.  Indeed, the hope that such
a "place to stand" does exist, somehow, somewhere, somewhen, often
serves to provide inquiry with a beneficial regulative principle or
a heuristic hypothesis to work on.  But it just so happens, for the
run of "finitely informed creatures" (FIC's) at any rate, that the
existence of an ideal framework is a contingency to be established
after the fact, at least, somewhat nearer toward the ultimate end
of inquiry than the present time is apt to mark.

In the project developed here, an "objective framework" (OF)
embodies one or more "objective genres" (OG's), also called
"forms of analysis" (FOA's) or "forms of synthesis" (FOS's),
each of which genres delivers its own rendition of a great
chain of being for all of the objects that happen to fall
under its purview.  In effect, each OG develops its own
version of an "ontological hierarchy" (OH), designed
independently of the conceivable others to capture
an aspect of structure in its objective domain.

For the moment, the level of an OF operates as a catch-all category,
giving the projected discussion the elbow room that it needs to range
over an emerging variety of different OG's and to place the particular
OG's of active interest in a running context of comparative evaluations
and developmental options.

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 current
interest.  A concept that is 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 many alternative
ontologies can always be proposed for the same collection of objects.

An OG embodies many "objective motives" or "objective motifs" (OM's).
If an OG constitutes a genus, or a generic pattern of object structure,
then the OM's 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 various 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 framework, genre, or motive.

An OG, in its general way, or an OM, in its individual way, begins by
relating each object in its purview to a unique set of further objects,
called the "components", "constituents", "effects", "ingredients", or
"instances" of that object with respect to that "objective concern" (OC).
As long as a discussion remains fixed to what is visible within the scope
of a particular OC, the collected effects of each object in view constitute
its "active ingredients", supplying it with a unique decomposition that fixes
it to a degree sufficient for all purposes conceivable within that discussion.

Contemplated from an outside perspective, however, the status of these effects
as the "defining unique determinants" (DUD's) of each object under examination
is something to be questioned.  The supposed constituents of an object that are
obvious with respect to one OC can be regarded with suspicion from the points of
view of alternative OC's, and their apparent status as rock-bottom substantives
can find itself reconstituted in the guise of provisional placeholders (placebos
or excipients) that precipitately index the potential operation of more subtly
active ingredients.

If a single OG could be unique and the realization of every object
in it could be complete, then there might be some basis for saying
that the elements of objects and the extensions of objects are known,
and thus that the very "objects of objects" (OOO's) are determined by
its plan.  In practice, however, it takes a diversity of overlapping
and not entirely systematic OG's to make up a moderately useful OF.

What gives an OG a definite constitution is the naming of a space
of objects that falls under its purview and the setting down of
a system of axioms that affects its generating relations.

What gives an OM a determinate character from moment to moment
is the particular selection of objects and relational linkages
from its governing OG that it can say it has appropriated,
apprehended, or actualized, that is to say, the portion
of its OG that it can say actually belongs to it, and
whether they make up a lot or a little, the roles
that it can say it has made its own.

In setting out the preceding characterization, I have reiterated what is
likely to seem like an anthropomorphism, prefacing each requirement of the
candidate OM with the qualification "it can say".  This is done in order to
emphasize that an OM's command of a share of its OG is partly a function of
the interpretive effability that it brings to bear on the object domain and
partly a matter of the expressive power that it is able to dictate over its
own development.

1.3.4.13. Formalization of OF : Objective Levels

The three levels of objective detail that I gave a first sketch of above are
described as the objective framework, objective genre, and objective motive
that one finds actively involved in organizing, guiding, and regulating a
particular inquiry.  I will now give a more comprehensive description of
these three objective levels.

   1.  An "objective framework" (OF) consists of one or several
       "objective genres" (OG's), each of which may also be known
       as a "form of analysis" (FOA), a "form of synthesis" (FOS),
       or an "ontological hierarchy" (OH).  Typically, these span
       a diverse spectrum of formal characteristics and intended
       interpretations.

   2.  An OG is made up of one or more "objective motives"
       or "objective motifs" (OM's), each of which may also
       be regarded as a particular "instance of analysis" (IOA)
       or a particular "instance of synthesis" (IOS).  All of the
       OM's that are governed by a particular OG exhibit a kinship
       of structures and intentions, and each OM roughly fits the
       pattern or "follows in the footsteps" of its guiding OG.

   3.  An OM can be identified with a certain moment of interpretation,
       one in which a particular 2-adic relation appears to govern all of
       the objects in its purview.  Initially presented as an abstraction,
       an individual OM is commonly fleshed out by identifying it with its
       interpretive agent.  As this practice amounts to a very loose form
       of personification, it is subject to all of the dangers of its type
       and is bound eventually to engender a multitude of misunderstandings.
       In contexts where more precision is needed it is best to acknowledge
       that the application of an OM is restricted to special instants and
       to limited intervals of time.  This means that an individual OM must
       look to the "interpretive moment" (IM) of its immediate activity to
       find the materials available for both its concrete instantiation and
       its real implementation.  Finally, having come round to the picture
       of an objective motive that is realized in an interpretive moment,
       this discussion has achieved a discrete advance toward the desired
       forms of dynamically realistic models, providing itself with what
       begins to look like the elemental states and dispositions that
       are needed to build fully actualized systems of interpretation.

A major theoretical task that remains outstanding for this project is to
discover a minimally adequate basis for defining the state of uncertainty
that an interpretive system has with respect to the questions it is able to
formulate about the state of an object system.  Achieving this would permit
a measure of definiteness to be brought to the question of inquiry's nature,
since it can already be grasped intuitively that the gist of inquiry is to
reduce an agent's level of uncertainty about its object, its objective,
or its objectivity through appropriate changes of state.

In this regard, 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 meant to formalize the intuitive notions of a generic
mental constitution and a specific mental disposition that
are commonly called into play when 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 "-<-" and "->-".  At the more generic levels of OF's and OG's
the "staging operations" associated with the generators "-<-" and "->-"
involve the application of 2-adic relations analogous to those of class
membership "element of" and its converse, but the increasing amounts of
parametric information that are needed to determine specific motives and
detailed motifs give OM's 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 the context.
In particular, even fundamental properties like the effective "arity"
or "valence" of the relations signified can vary from level to level.

The staging relations divide into two orientations, "-<-" versus "->-",
indicating opposing senses of direction with respect to the distinction
between analytic and synthetic projects:

   1.  The "standing relations", indicated by "-<-", are analogous to
       the "element of", "belongs to", or the membership relation "in".
       Another interpretation of "-<-" is the "instance of" relation.
       At least with respect to the more generic levels of analysis,
       any distinction between these readings is largely immaterial
       to the formal interests and the structural objectives of
       this discussion.

   2.  The "propping relations", indicated by "->-", are analogous to
       the "class of" relation or converse of the membership relation.
       An alternative meaning for "->-" is the "property of" relation.
       Although it is possible to maintain a distinction in this regard,
       the present discussion is mainly concerned with a level of purely
       formal structure to which this difference is largely irrelevant.

Although it is strictly speaking logically redundant to do so,
it turns out to be extremely useful in practice to introduce
efficient symbolic devices for both directions of the staging
relations, "-<-" and "->-", and to maintain a formal calculus
that treats analogous pairs of relations on an equal footing.
Extra measures of convenience come into play if the relations
are used as assignment operations or as "field promotions",
that is, to create titles, to define terms, and to establish
the offices of objects in active contexts of given relations.
Accordingly, I regard these dual relationships as symmetric
primitives and I employ them as the "generating relations"
of all three objective levels.

Next, I present several different ways of formalizing OG's and OM's.
The reason for employing multiple descriptions is simply to capture
the diversity of ways that these patterns of organization appear in
actual practice.

One way to approach the formalization of an objective genre G
is by means of an indexed collection {G_j} of 2-adic relations:

   G  =  {G_j}  =  {G_j : j in J}, with G_j c P_j x Q_j for all j in J.

Here, J is a set of actual (that is, not formal) parameters that
are used to index the OG, while P_j and each Q_j are domains of
objects (initially in the informal sense) that enter into the
2-adic relation G_j.

Aside from their indices, any of the G_j in G can be abstractly identical
to each other.  This would earn G the designation of a "multi-family" or
a "multi-set" according to some customs of usage, but I prefer to treat
the index j as a concrete part of the indexed relation G_j, thereby
distinguishing G_j from all other members of the indexed family G.

Ordinarily, it is desirable to avoid making individual mention
of the separately indexed domains, P_j and Q_j for all j in J.
Common strategies for getting around this nuisance involve the
introduction of additional domains, designed to encompass all
of the objects that may happen to be needed in given contexts.
Toward this end, an adequate supply of intermediate domains,
called the "rudiments of universal mediation" (RUM's), are
defined as follows:

   X_j  =  P_j |_| Q_j, where P = |_|^j P_j and Q = |_|^j Q_j.

Ultimately, all of these "totalitarian" strategies end the same way,
at first, by envisioning a domain X that is big enough to encompass
all of the objects of thought that might demand entry into a given
discussion, and then, by invoking one of the following conventions:

   1.  Rubric of Universal Inclusion (RUI):  X = |_|^j (P_j |_| Q_j).

   2.  Rubric of Universal Equality  (RUE):  X = P_j = Q_j for all j in J.

Working under the aegis of either one of these rubrics,
G can be provided with a simpler style of presentation:

   G  =  {G_j}  =  {G_j : j in J}, with G_j c X x X for all j in J.

Both of these rubrics notwithstanding, it serves a purpose of this project
to preserve the individual indexing of relational domains for while longer,
at minimum keeping the indexing available as an alternative means of access.
Generally speaking, it is always possible in principle to form the union that
is required by the RUI, and to assume without loss of generality the equality
that is imposed by the RUE.  The catch is that the unions and the equalities
that are invoked by these rubrics may not be effectively definable, at least
not testable in a computational context.  Further, even when these sets and
tests can be constructed or certified by one or another computational agent,
the crucial question at any interpretive moment is whether each collection
or constraint is actively apprehended or duly warranted by the particular
interpreter that is charged with responsibility for it in the indicated
assignment of domains.

But an overall purpose of this formalism is to represent the objects and
the constituencies that are "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
actually possessed, recognized, responded to, acted on, and followed up
by concrete agents as they move through their own immediate contexts of
activity.  Therefore, keeping individual tabs on the relational domains
P_j and Q_j, though it does not solve this array of problems, does serve
to mark the pressing concern with particularity and to keep before one's
mind the issues of individual attention and personal responsibility that
are appropriate to interpretive agents.  In short, whether or not domains
appear with explicit subscripts, one should always be ready to answer the
challenge that is gaged by the question "Who subscribes to these domains?"

It is important to emphasize that the index set J and the particular
attachments of indices to 2-adic relations are part and parcel to G,
befitting the concrete character intended for the concept of an OG,
which is expected, in some semblance of a realistic way, to embody
in the character of each G_j both "a local habitation and a name".
For this reason, among others, the G_j can safely be described
as "individual dyadic relations" (IDR's).  Since the classical
notion of an "individual" as a "perfectly determinate entity"
has no application in finite information contexts like these,
it is safe for our sake to recycle the term to distinguish
the "terminally informative particulars" (TIP's) that the
concrete index j adds to its thematic object G_j, whether
read parenthetically or recognized as a clinching paraph.

Depending on the prevailing direction of interest in the genre G,
"-<-" or "->-", the same symbol is used equivocally for all the
relations G_j.  The G_j can be regarded as formalizing the OM's
that make up the genre G, provided it is understood that the
information corresponding to the parameter j constitutes
an integral part of the "motive" or the "motif" of G_j.

In this formulation, G constitutes an "ontological hierarchy" (OH)
of an extremely plenary and potentiating type, one that determines
the complete array of objects and relationships that are conceivably
available and describably "effable" within the discussion in question.
Operating with reference to the global field of possibilities that is
afforded by G, each G_j corresponds to the specialized competence of
a particular agent, selecting out the objects and the links of the
general purpose hierarchy that are known to, owing to, or owned by
the given interpreter j.

Another way to formalize the defining structure of an OG
can be posed in terms of a "relative membership relation"
or a notion of "relative elementhood".  The constitutional
structure of a particular OG can be set up in a flexible
manner by taking it in two stages, starting from a level
of finer detail and working up to the bigger picture:

1.  Each OM is constituted by what it means to be an object within it.
    What constitutes an object in a given OM can be fixed as follows:

    a.  In absolute terms, by specifying the domain of objects that
        fall under its purview.  For the present, I assume that each
        OM inherits the same object domain X from its governing OG.

    b.  In relative terms, by specifying a converse pair of 2-adic
        relations that (redundantly) determine two sets of facts:

        i.   What is an instance, example, member, or element of what,
             relative to the OM in question.

        ii.  What is a property, quality, class, or set of what,
             relative to the OM in question.

2.  The various OM's of a particular OG can be unified under
    its aegis by means of a single triadic relation, one that
    names an OM and a pair of objects, and that holds when one
    object belongs to the other object in the sense identified
    by the relevant OM.  If it becomes absolutely essential to
    emphasize the parametric relativity of elementhood, it is
    permissible to resort, in all good humor, to calling such
    relative elmennts by the name of "relements", in this way
    jostling the interpretive mind to read between the lines,
    and perhaps to ask:  "relement to what, relement to whom?"

The last and most likely the best way that one can choose to follow
by way of forming an objective genre G is to depict it as a triadic
relation, in particular, choosing between one of these two fashions:

   G  =  {<j, p, q>}  c  J x P x Q

   G  =  {<j, x, y>}  c  J x X x X

For some reason the ultimately obvious method seldom presents itself
exactly in this wise without diligent work on the part of the inquirer,
or one who would arrogate the roles of both its former and its follower.
Perhaps this has to do with the problematic role of "synthetic a priori"
truths in constructive mathematics.  Perhaps the mystery lies encrypted,
no doubt buried in some obscure dead letter office, due to the obliterate
indicia on the letters "P", "Q", and "X" inscribed above.  No matter --
for the moment we have far more pressing rounds to make.

Given a genre G whose OM's are indexed by a set J and whose objects
comprise a set X, there is a 3-adic relation among an OM and a pair
of objects that exists when the first object belongs to the second
object according to that OM.  This is called the "standing relation"
of the OG, and it can be taken as one way of defining and establishing
the genre G.  In the way that 3-adic relations ordinarily give rise to
2-adic operations, the associated "standing operation" of the OG can be
regarded as a brand of assignment operation that makes one object belong
to another in a certain sense, namely, in the sense that is indicated by
the designated OM.

There is a "partial converse" of the standing relation that transposes
the order in which the two object domains are mentioned.  This is called
the "propping relation" of the OG, and it can be taken as an alternate way
of defining the genre G, proceeding by way of its converse G^ in one of the
following two manners:

   G^  =  {<j, q, p> in J x Q x P : <j, p, q> in G}

   G^  =  {<j, y, x> in J x X x X : <j, x, y> in G}

The following conventions are useful for discussing the set-theoretic
extensions of the staging relations and staging operations of an OG:

   The standing relation of an OG will be denoted by the symbol ":<-",
   pronounced "set-in".  Thus [:<-] c J x P x Q, or [:<-] c J x X x X.

   The propping relation of an OG will be denoted by the symbol ":>-",
   pronounced "set-on".  Thus [:>-] c J x Q x P, or [:>-] c J x X x X.

Often one's level of interest in a genre is "purely generic".
When the relevant genre is regarded as an indexed family of
2-adic relations, G = {G_j}, then this generic interest is
tantamount to having one's concern rest with the union of
all the 2-adic relations in the genre:

   |_|^J G  =  |_|^j G_j  =  {<x, y> in X x X : <x, y> in G_j for some j in J}.

When the relevant genre is contemplated as a triadic relation, G c J x X x X,
then one is dealing with the projection of G on the object domain dyad X x X.

   G_XX  =  Proj_XX (G)  =  {<x, y> in X x X : <j, x, y> in G for some j in J}.

On these occasions, the assertion that a pair of elements <x, y>
is in the union or projection |_|^J G = G_XX can be indicated by
any one of the following six equivalent expressions:

   G : x -<- y,   x -<G<- y,   x -<- y : G,

   G : y ->- x,   y ->G>- x,   y ->- x : G.

At other times, it necessary to make more explicit mention of the
interpretive perspective or the "individual dyadic relation" (IDR)
that links two objects.  To indicate that the 3-tuple consisting
of the OM j and the ordered pair of objects <x, y> belongs to the
standing relation of the OG, that is to say, <j, x, y> is in [:<-],
or what is the same thing, to indicate that a 3-tuple consisting
of the OM j and the ordered pair of objects <y, z> belongs to the
propping relation of the OG, that is to say, <j, y, x> is in [:>-],
all of the following six notations are equivalent:

   j : x -<- y,   x -<j<- y,   x -<- y : j,

   j : y ->- x,   y ->j>- x,   y ->- x : j.

Depending on the specific context of application, assertions
of these relations can be read in various ways, for instance:

   j sets x in y.
   j sets y on x.

   j makes x an instance of y.
   j makes y a property of x.

   j thinks x an instance of y.
   j thinks y a property of x.

   j attests x an instance of y.
   j attests y a property of x.

   j appoints x an instance of y.
   j appoints y a property of x.

   j witnesses x an instance of y.
   j witnesses y a property of x.

   j interprets x an instance of y.
   j interprets y a property of x.

   j contributes x to y.
   j attributes y to x.

   j determines x an example of y.
   j determines y a quality of x.

   j evaluates x an example of y.
   j evaluates y a quality of x.

   j proposes x an example of y.
   j proposes y a quality of x.

   j musters x under y.
   j marshals y over x.

   j indites x among y.
   j ascribes y about x.

   j imputes x among y.
   j imputes y about x.

   j judges x beneath y.
   j judges y beyond x.

   j finds x preceding y.
   j finds y succeeding x.

   j poses x before y.
   j poses y after x.

   j forms x below y.
   j forms y above x.

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 "j thinks x an instance of y" should be understood as saying that
"j is a thought with the logical import that x is an instance of y",
and a statement like "j proposes y a property of x" should be taken to
mean that "j is a proposition to the effect that y is a property of x".

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 constructing sign relational models.

By way of anticipating the nature of the problem, consider the following
examples to illustrate the contrast between logical and cognitive senses:

   1.  In a cognitive context, if j is a considered opinion that p is true,
       and j is a considered opinion that q is true, then it does not have
       to automatically follow that j is a considered opinion that p and q
       are true, since an extra measure of consideration might conceivably
       be involved in cognizing the conjunction of p and q.

   2.  In a logical context, if j is a piece of evidence that p is true,
       and j is a piece of evidence that q is true, then it necessarily
       follows by these very facts alone that j is a piece of evidence
       that p and q are true.  This is analogous to the situation where,
       if a person j draws a set of three lines AB, BC, and AC, then j
       has drawn a triangle ABC, whether j recognizes the fact at first,
       on reflection, with extended contemplation, or never does at all.

Some readings of the staging relations are tantamount to statements
of (a possibly higher order) model theory.  For example, the predicate
p : J -> %B%, defined by p(j) <=> "j proposes x an instance of y", is a
proposition that applies to a domain of propositions, at least, elements
with the evidentiary import of propositions, and its models are therefore
conceived to be certain propositional entities in J.  And yet, all of these
expressions are just elaborate ways of stating the underlying assertion that
says that there exists a triple <j, x, y> in the relevant genre G.

1.3.4.14. Application of OF : Generic Level

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.

A principal aim of using sign relations as formal models is to be capable
of analyzing complex activities that arise in human domains and in nature
generally.  Proceeding by the opportunistic mode that is commonly known as
"analysis by synthesis" (ABS), one generates plausible 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 that one has built for oneself can help to
articulate plausible hypotheses 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 3-adic 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 steadily 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, that 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.

The sign relations of A and B, though natural in themselves as far as they go,
were nevertheless introduced in an artificial fashion and established 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.  As a consequence, 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 their significant utility.

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
A and B, as illustrated by the pronouns "i" and "u" in S.  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 of discourse and making up the very fabric of this inquiry.

In order to deal with the array of issues presented so far in this Subsection,
all of which have to do with controlling the generative power of sign relations
to serve the specific purposes of understanding, I apply the previously introduced
concept of an "objective genre" (OG).  This is intended to be a determinate purpose
or a deliberate pattern of analysis and synthesis that one can identify as being
active at given moments in a discussion and that affects what one regards as
the relevant structural properties of its objects.

In the remainder of this Subsection the concept of an OG is used informally,
and only to the extent that it is needed for a pressing application, namely,
to rationalize the natural kinds that are claimed for signs and to clarify
an important contrast that exists between icons and indices.

The OG I apply here is called the "genre of properties and instances".
One moves through its space, higher and lower in a particular ontology,
by means of converse 2-adic relations, upward by taking a "property of"
and downward by taking an "instance of" whatever object initially enters
one's focus of attention.  Each object of this OG is reckoned to be the
unique common property of the set of objects that lie one step below it,
objects that are in turn reckoned to be instances of the given object.

Pretty much the same relational patterns could be found in the genre or
the paradigm of "qualities and examples", but the use of "examples" here
is polymorphous enough to include experiential, exegetic, and executable
examples (EXE's).  This points the way to a series of related genres, for
example, the OG's of "principles and illustrations", "laws and existents",
"precedents and exercises", or on to "lessons and experiences".  All in all,
as things work out in their respective turns, these modulations of the same
basic OG show a way to shift the foundations of ontological hierarchies
toward bases in individual and systematic experience, and thus to put
existentially dynamic rollers under the blocks of what all too often
appear to be essentially invariant pyramids, even when they are not.

Any object of these OG's can be contemplated in the light of two
potential relationships, namely, with respect to its chances of
being an "object quality" (OQ) or an "object example" (OE) of
something else.  In future references, abbreviated notations
like "OG(Prop, Inst)" or "OG = <Prop, Inst>" will be used to
specify particular genres, in this sketchy but suggestive
way serving to adumbrate the intended interpretations of
their generating relations {-<- , ->-}.

With respect to the objective genre of properties and instances,
I can now give the following characterization of icons and indices:

   Icons are signs by virtue of being instances of properties of objects.
   Indices are signs by virtue of being properties of instances of objects.

Because the initial discussion seems to flow more smoothly if I apply
2-adic relations on the left, I formulate these depictions as follows:

   For Icons:     Sign(Obj)  =  Inst(Prop(Obj))
   For Indices:   Sign(Obj)  =  Prop(Inst(Obj))

Imagine starting from the sign and retracing steps to reach the object,
in this way finding the converses of these relations to be as follows:

   For Icons:     Obj(Sign)  =  Inst(Prop(Sign))
   For Indices:   Obj(Sign)  =  Prop(Inst(Sign))

In spite of the apparent duality that is manifested between these patterns
of composition, there is nevertheless 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 term'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 difference between
the ideas of actuality and reality 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.

The "angelic doctrines" about the nature of reality
that I alluded to above would probably remain distant
from the present concern, were it not for the following
points of connection.

Pragmatic reality means possessing properties under interpretation.
Relative to the genre of properties and instances, the distinction
of reality, that can be granted to certain objects of thought and
not to others, serves a function analogous to the distinction that
sets apart "sets" from "classes" in modern versions of set theory.
Regarding the membership relation "element of" as the predecessor
relation in a predetermined hierarchy of classes, a class attains
the status of a set, and by dint of this becomes an object of more
determinate discussion, simply if it has successors.  Standing over
and against this parallelism, pragmatic reality is distinguished from
both the medieval and the modern versions by the fact that its reality
is always a reality to somebody, and thus bears information about its
reality in the proportion of the interpretive community to whom its
presence is borne.  This is due to the circumstance that it takes
both an abstract property and a concrete interpreter to evidence
the practical reality of an object.

This project seeks articulations and implementations of intelligent activity
within dynamically realistic systems.  The individual stresses that were just
now placed on articulation, implementation, actuality, dynamics, and reality
serve to reinforce the importance of three further issues:

Acting and Being.  Systems theory, consistently pursued, requires for
its rationalization a distinct ontology, one in which states of being
and modes of action form the principal objects of thought, and out of
which the ordinary species of stably extended but derivative objects
have to be constructed.  In the "grammar" of this process philosophy,
verbs and pronouns are more basic than nouns.  In its impact on the
direction of this discussion, this emphasis on systematic action is
tantamount to the constitution of an objective genre that regards
dynamic systems, their momentary states and their passing actions,
as the ultimate objects of synthesis and analysis.  Consequently,
the course of this work will be turned toward conceiving actions,
as traced out in the trajectories of systems, to be the primitive
elements of construction, in this objective genre more fundamental
than the customary arrays of stationary objects extended in space.
As a corollary, one comes to anticipate that physical objects of
the static variety will be relegated to a derivative status in
relation to the activities that orient agents, both organisms
and organizations, towards the ends of purposeful objectives.

Dynamic and Symbolic.  Taking clues from etymology, the notion of "dynamics"
has to do with "power", but in the original sense of the word that connotes
"potential".  The brand of pragmatic thought that I use in this work allows
potential entities to be treated as real objects and it regards the objects
of concepts to be constituted by the conception of their actual effects in
practical instances.  In the move to unify dynamic and symbolic approaches
to intelligent systems, there remains a need to build conceptual bridges
between these two realms.  A intellectual 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 is
reconstructed in terms of active occurrences and ways of being.

Finding and Making.  In prospect of form, it does not matter whether
one conceives this project as a task of analyzing and articulating
the actualizations of a capacity for inquiry that already exist
in nature, or whether one views it as a task of synthesizing
and artificing the potentials for inquiry that have yet to
be conceived in practice.  From a formal perspective, the
analysis and the synthesis are just reciprocal ways of
tracing and retracing the same generic patterns of
potential structure that determine actual form.

Returning to the examination of icons and indices, and keeping the criterion
of reality in mind, notice the radical difference that comes into play in
recursive settings between the two types of contemplated moves that are
needed to trace the respective signs back to their objects, that is,
to discover their denotations:

1.  Icon -> Object.  Taking the iconic sign as an initial instance,
    try to go up to a property and then down to a different or perhaps
    the same instance.  This form of ascent does not require a distinct
    object, since the reality of the sign is sufficient unto itself.
    That is, if the sign has any properties at all, then it is an
    icon of a real object, even if that object is only itself.

2.  Index -> Object.  Taking the indexical sign as an initial property,
    try to go down to an instance and then up to a different or perhaps
    the same property.  This form of descent requires a real instance to
    substantiate it, but not necessarily a distinct object.  Consequently,
    the index always has a real connection to its object, even if that
    object is only itself.

In sum:

    For icons a separate reality is optional.
    For indices a separate reality is obligatory.

As often happens with a form of analysis, each term under the
above indicated sum immanently verges on unlimited expansions:

1.  For icons, the existence of a separate reality is optional.
    This means that the question of reality in the sign relation
    can depend on nothing more than the reality of each sign itself,
    on whether it has any property with respect to the OG in question.
    In effect, icons can rely on their own reality to faithfully provide
    a real object.

2.  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.

As a last application, for now, of these first ideas about objective levels,
let's consider the collective bearing of icons and indices in a given genre.
Suppose that M and N are hypothetical sign relations that are abstracted to
capture all of the iconic and indexical relationships, respectively, that
a typical object x enjoys within its objective genre G.  A sign relation
in which every sign has the same kind of relation to its object under an
assumed form of analysis is aptly called a "homogeneous sign relation".
In particular, if H 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 H.

Typical sign relations of the iconic or indexical kind generate especially
simple and remarkably stable sorts of interpretive processes.  In regard to
arity or valence, they could almost be classified as "approximately 2-adic",
since the greater share of their interesting structure is wrapt up in their
denotative aspects, while their connotative functions are relegated to the
more 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 the semiotic equivalents of "gyroscopes", helping
them to maintain their interpretive perspectives in a persistent orientation
toward their objective worlds.

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.
Still, 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.

The dispensation of consensual bonds in a common medium leaves room
for many otherwise isolated interpreters to inhabit a shared frame
of reference, and for a diversity of transient interpretive moments
to take up and consolidate a continuing perspective on a world of
mutual interests.  This increases the likelihood that developing
interpreters and differing observers will be able to participate
in coherent views and compatible values that are held in relation
to the aggregate of things, to collate information from a variety
of sources, and to bring concerted action to bear on appreciably
broad distributions of extended realities and intended objectives.
Instead of the disparities due to parallax leading to disorder and
paralysis, accounting for the different points of view behind the
discrepancies can bring about stereoscopic perspectives.  Finally,
in a community of interpretation and inquiry that has all of these
virtues, each individual "try at objectivity" (TAO) is a venture
that all interpreters are nonetheless able to call their own.

Is this prospect a utopian vision?  Perhaps it is exactly that.
But it is the hope that inquiry discovers resting first and last
within itself, quietly guiding every other aim and motive of inquiry.

Turning to the language of objective concerns, what can now be said
about the compositional structures of the iconic sign relation M and
the indexical sign relation N?  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.

Recalling the genre of properties and instances, let us adopt the
symbols "-<-" and "->-" for the converse pair of 2-adic relations
that generate it.  Reverting to the convention I employ in formal
discussions of applying relational operators on the right, I will
express the relative terms "property of x" and "instance of x" by
means of a case inflection on "x", that is, as "x's property" and
as "x's instance", respectively.  Described in this fashion, our
OG(Prop, Inst) is generated by the set {-<- , ->-}, as follows:

   "x -<-"  =  "x's property"  =  "property of x"  =  "object above x"
   "x ->-"  =  "x's instance"  =  "instance of x"  =  "object below x"

A symbol like "x -<-" or "x ->-", by itself or linked together in chains,
with relational application and composition indicated by optional spaces
or by the composition symbol "o", is called a "catenation", where "x" is
the "catenand" and "-<-" or "->-" is the "catenator".  On account of the
fact that "-<-" and "->-" indicate 2-adic relations, the significance of
these so-called "unsaturated" catenations can be rationalized as follows:

   "x -<-"  =  "x is the instance of what?"  =  "x's property"
   "x ->-"  =  "x is the property of what?"  =  "x's instance"

Working in this manner, the definitions of icons and indices
can be reformulated in terms of the following two equations:

   x's icon   =  x's property's instance  =  x -<-->-
   x's index  =  x's instance's property  =  x ->--<-

According to the earlier definitions of the
homogeneous iconic sign relation M and the
homogeneous indexical sign relation N,
we have the following equations:

   x's icon   =  x M_OS
   x's index  =  x N_OS

Equating the results of the corresponding pairs of equations
yields an analysis of M and N as forms of composition within
the genre of properties and instances:

   x's icon   =  x M_OS  =  x -<-->-
   x's index  =  x N_OS  =  x ->--<-

On the assumption (to be examined more closely later) that any object x
can be taken as a sign, the converse relations appear to be manifestly
identical to the obverse relations:

   For Icons:     x's object  =  x M_SO  =  x -<-->-
   For Indices:   x's object  =  x N_SO  =  x ->--<-

Abstracting from the applications to an otiose x delivers the results:

   For Icons:     M_OS  =  M_SO  =  -<-->-
   For Indices:   N_OS  =  N_SO  =  ->--<-

This appears to suggest that icons and their objects are icons of each other,
and that indices and their objects are indices of each other.  Are the results
of these symbolic manipulations really to be trusted?  Given that there is no
mention of the interpretive agent to whom these sign relations are supposed
to appear, one might well suspect that these results can only amount to
approximate truths or potential verities.

1.3.4.15. Application of OF : Motive Level

Now that an adequate variety of formal tools have been set in order and
the virtual workspace afforded by an objective framework has been rendered
reasonably clear, the structural theory of sign relations can now be pursued
with greater precision.  In support of this aim, the concept of an objective
genre and the particular example provided by OG(Prop, Inst) have served to
rough out the basic shapes of the more refined analytic instruments to be
developed in this Subsection.

The notion of an "objective motive" or an "objective motif" (OM) is intended
to personalize and to specialize the application of objective genres to take
particular interpreters into account.  For example, staying with the pattern
of OG(Prop, Inst), 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" (HOT), one that is currently acknowledged by and
actively pursued by the designated interpreter of these very 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 to say, simply parsing signs in the mind.
As always, it is best to regard OG's and OM's as "filters" and "reticles",
as transparent templates that are used to view a space, constituting the
structures of objects only in one respect at a time, but never with any
assurance of totality.

With these refinements at the ready, the use of 2-adic projections to
investigate sign relations can be combined with the vantage points of
objective motives to "factor the facets" or "decompose the components"
of sign relations in a more systematic fashion.  Given a homogeneous
sign relation H of iconic or indexical type, the 2-adic projections
H_OS and H_OI can be analyzed as compound relations over the basis
supplied by the G_j in G.  As an application that is sufficiently
important in its own right, the investigation of icons and indices
continues to provide a useful testing ground for breaking in likely
proposals of concepts and notation.

To pursue the analysis of icons and indices at the next stage of
formalization, fix the OG of this discussion to be of the type
generated by {-<- , ->-}, and let each sign relation under
discussion be articulated in terms of an objective motif
that tells what objects and signs, plus what mediating
linkages through properties and instances, are assumed
to be recognized by its interpreter.

Let X collect the objects of thought that fall within a particular OM,
and let X include the whole world W = O |_| S |_| I of a sign relation
plus everything needed to support and contain it.  Thus, X collects all
of the types of things that go into a sign relation, O |_| S |_| I c X,
plus whatever else in the way of distinct object qualities (OQ's) and
object exemplars (OE's) is discovered or established to be generated
out of this basis by the relations of the OM.

In order to keep this X simple enough to contemplate on a single pass,
but still make it deep enough to cover the issues of interest at present,
I restrict X to having just three disjoint layers of things to worry about:

   The top layer Q is the relevant set of object qualities:

      Q  =  [X_0 -<-]  =  [W -<-]

   The middle layer X_0 is the initial collection of objects and signs:

      X_0  =  W  =  O |_| S |_| I

   The bottom layer E is a suitable set of object exemplars:

      E  =  [X_0 ->-]  =  [W ->-]

Recall the reading of the staging relations:

   "h : x -<- m"  =  "h regards x as an instance of m"
   "h : m ->- y"  =  "h regards m as  a property of y"
   "h : x ->- n"  =  "h regards x as  a property of n"
   "h : n -<- y"  =  "h regards n as an instance of y"

Express the analysis of icons and indices as follows:

   For Icons:    M_OS : x -<-->- x's sign
   For Indices:  N_OS : x ->--<- x's sign

Let j and k be hypothetical interpreters that do the jobs of M and N,
respectively.  In this case we have the following series of equations:

   For Icons:    x's sign  =  x M_OS  =  x-<j<-->j>-
   For Indices:  x's sign  =  x N_OS  =  x->k>--<k<-

Factor out the names of the interpreters j and k to act as
identifiers of objective motifs, and we have the following:

   For Icons:    j : x -<-->- x's sign
   For Indices:  k : x ->--<- x's sign

Finally, the constant motif names j and k can be collected to one
side of a composition or else distributed to its individual links:

   j : x -<-->- y  <=>  j : x -<- m  and  j : m ->- y, for some m in Q.
   k : x ->--<- y  <=>  k : x ->- n  and  k : n -<- y, for some n in E.

These statements can be read to say:

   1.  j thinks x an icon of y if and only if there is an m such that
       j thinks x an instance of m and j thinks m a property of y.

   2.  k thinks x an index of y if and only if there is an n such that
       k thinks x a property of n and k thinks n an instance of y.

Readers who object to the anthropomorphism or the approximation of
these statements can replace every occurrence of the verb "thinks"
with the phrase "interprets ... as", or even less committally with
the circumlocution "acts in every formally significant way as if",
changing what must be changed elsewhere.  For the moment, I am not
concerned with the exact order of reflective sensitivity that goes
into these interpretive linkages, but only with a rough outline of
the pragmatic equivalence classes implied by the potential conduct
of their interpretive agents.

In discussing the Example of A and B, I have been careful so far to
distinguish the interpretive agents A and B from the sign relations
L(A) and L(B) that were selected as samples of the agents' sign use.
In future discussions, I will tend to relax the explicit marking of
this distinction, though not the distinction itself, using the same
names to denote either the agent or the corresponding sign relation,
and trusting the reader to read between the lines of the systematic
ambiguity or the deliberate duality to know which category of thing
is intended.  Used informally as parts of the peripheral discussion,
the names of interpreters will then designate entire sign relations,
but used more formally within the object dialogue under examination
they denote certain objects that occur in particular sign relations.
In precisely this way, or what amounts to an elaboration of it, the
names "j" and "k" can have their senses stretched to encompass both
the objective motifs (OM's) that inform and regulate experience and
also the object experiences (OE's) that fill out and give substance
to the forms of these same motifs.

1.3.4.16. Integration of Frameworks

A large number of the problems arising in this work have to do with the
integration of different interpretive frameworks over a common objective
basis, or the prospective bases that may be provided by shared objectives.
The main concern of this project continues to be the integration of dynamic
and symbolic frameworks for understanding intelligent systems, concentrating
on the kinds of interpretive agents that are capable of involvement in inquiry.

Integrating divergent IF's and reconciling their objectifications is,
generally speaking, a very difficult maneuver to carry out successfully.
Two factors that contribute to the near intractability of this task can be
described and addressed as follows:

1.  The trouble is partly due to the obligatory tactics and
    the ossified taxonomies and that arise through time and
    training to inhabit the conceptual landscapes of agents,
    especially if they have spent the majority of their time
    operating according to a single IF.  The IF informs their
    activities in ways they no longer have to think about, and
    thus rarely find a reason to modify.  But it also inhibits
    their interpretive and practical conduct to the customary
    ways of seeing and doing things that are granted by that
    framework, and it binds them to the "forms of intuition"
    that are suggested and sanctioned by the operative IF.
    Without critical reflection, or mechanisms to make
    amendments to its own constitution, an IF tends to
    operate behind the scenes of observation in such a
    way as to obliterate any inkling of flexibility in
    practice or thinking and to obstruct every hint or
    threat (often so perceived) of conceptual revision.

2.  Apparently it is so much easier to devise techniques for
    taking things apart than it is to find ways of putting them
    back together again that there seem to be but a few heuristic
    strategies of general application that are available to guide
    the work of integration.  A few of the tools and the materials
    that are needed for these constructions have been illustrated
    in concrete form throughout the presentation of examples in
    this Section.  An overall survey of their principles can
    be summed up as follows:

    a.  One integration heuristic is the "lattice" metaphor, frequently
        called the "partial order" or the "common denominator" paradigm.
        When IF's can be objectified as OF's that are organized according
        to the principles of suitably ordered sets, then it may be possible
        to "lift" or extend their order properties to the space of frameworks
        themselves, and thereby to enable construction of the desired kinds of
        integrative frameworks as upper and lower bounds in the higher ordering.

    b.  Another heuristic of integration is the "mosaic" metaphor, also
        known as the "stereoscopic" or the "inverse projection" paradigm.
        This technique is illustrated especially well by the methods used
        throughout this Section to analyze the three-dimensional structures
        of sign relations.  In fact, the picture of any sign relation offers
        a paradigm in microcosm for the macroscopic enterprise of integration,
        showing how reductive aspects of structure are often projected from a
        shared but irreducible reality.  The extent to which the "full-bodied"
        structure of a 3-adic sign relation can be reconstructed from the data
        of its 2-adic projections, although it is a limited extent in general,
        presents a near perfect epitome of the larger task in this situation,
        namely, to find an integrated framework that embodies the diverse
        facets of reality severally observed from inside the individual
        frameworks.  Acting as gnomonic recipes for the higher order
        processes they limn and delimit, sign relations keep before
        the mind the ways in which a higher dimensional structure
        determines its fragmentary aspects but is not in general
        determined by them.
To express the nature of this integration task in logical terms, it combines
aspects of both proof theory and model theory, interweaving these two themes:

   1.  A phase that develops theories about the symbolic competence
       or the "knowledge" that is possessed by intelligent agents,
       using abstract formal systems to represent the theories
       and using phenomenological data to constrain them.

   2.  A phase that seeks concrete and dynamic models of these theories,
       looking to the varieties of mathematical structure that have
       dynamic or system-theoretic interpretations, and compiling
       the constraints that a recursive conceptual analysis
       imposes on the ultimate elements of construction.

The set of sign relations {L(A), L(B)} affords an example of an extremely
simple formal system, encapsulating aspects of the symbolic competence and
the pragmatic performance that might be exhibited by potentially intelligent
interpretive agents, however abstractly and partially given at this stage of
description.  The symbols of a formal system like {L(A), L(B)} can be held
subject to abstract constraints, having their meanings in relation to each
other determined by definitions and axioms, for example, the laws defining
an equivalence relation, making it possible to manipulate the resulting
information by means of the inference rules in a logical proof system.
This illustrates the "proof-theoretic" aspect of a symbol system.

Suppose that a formal system like {L(A), L(B)} is initially approached from
a theoretical direction, in other words, by listing the abstract properties
that one thinks it ought to have.  Then the existence of an extensional model
that satisfies these constraints, as exhibited by way of sign relation tables,
demonstrates that one's theoretical description is logically consistent, even
if the models that first come to mind are still a bit too abstractly symbolic
and lack all of the dynamic concreteness that is demanded of system-theoretic
interpretations.  This account is tantamount to the other side of the ledger,
the "model-theoretic" aspect of a symbol system, at least in so far as the
present discussion has dealt with it.

More is required of the modeler, however, in order to find the desired
kinds of system-theoretic models, for example, state transition systems,
and this brings the search for realizations of formal systems down to the
tougher part of the exercise.  Some of the problems that emerge have already
been highlighted in the story of A and B.  Although it is ordinarily possible
to construct state transition systems in which the states of the interpreters
correspond relatively directly to the acceptance of the primitive signs given,
the conflict of interpretations that develops between different interpreters
from these prima facie implementations is a sign that there may be something
superficial about this approach.

The integration of model-theoretic and proof-theoretic aspects
of "physical symbol systems", besides being closely analogous
to the integration of denotative and connotative aspects of
sign relations, is also relevant to the job of integrating
dynamic and symbolic frameworks for intelligent systems.
This is so because the search for dynamic realizations
of symbol systems is only a more pointed exercise in
model theory, where the mathematical materials made
available for modeling are further constrained by
system-theoretic principles, like being able to
say what the states are and how the transitions
are determined, to the extent that they may be.

1.3.4.17. Recapitulation : A Brush with Symbols

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.

In the pragmatic theory of signs, a "symbol" is a strangely insistent
yet curiously indirect type of sign, one whose accordance with its
object depends sheerly on the real possibility that it will be so
interpreted.  Taking on the nature of a bet, a symbol's prospective
value trades on nothing more than the chance of acquiring the desired
interpretant, and thus it can capitalize on the simple fact that what
it proposes is not impossible.  In this way it is possible to see that
a formal principle is involved in the meaningful successes of symbols.
The elementary conceivability of a particular sign relation, the pure
circumstance that renders it logically or mathematically possible,
means that the formal constraint it places on its domains is always
really and potentially there, awaiting its discovery and exploitation
for the purposes of representation and communication.

In this question about the symbol's capacity for meaning, then,
is found another point of contact between the theory of signs
and the logic of inquiry.  As Charles S. Peirce expressed it:

| Now, I ask, how is it that anything can be done with a symbol,
| without reflecting upon the conception, much less imagining the
| object that belongs to it?  It is simply because the symbol has
| acquired a nature, which may be described thus, that when it is
| brought before the mind certain principles of its use -- whether
| reflected on or not -- by association immediately regulate the
| action of the mind;  and these may be regarded as laws of the
| symbol itself which it cannot 'as a symbol' transgress.
|
| (Peirce, CE 1, 173).

| Inference in general obviously supposes symbolization;
| and all symbolization is inference.  For every symbol
| as we have seen contains information.  And ... we saw
| that all kinds of information involve inference.
| Inference, then, is symbolization.  They are the same
| notions.  Now we have already analyzed the notion of
| a 'symbol', and we have found that it depends upon
| the possibility of representations acquiring a nature,
| that is to say an immediate representative power.
| This principle is therefore the ground of inference
| in general.
|
| (Peirce, CE 1, 280).

| A symbol which has connotation and denotation contains information.
| Whatever symbol contains information contains more connotation than
| is necessary to limit its possible denotation to those things which
| it may denote.  That is, every symbol contains more than is sufficient
| for a principle of selection.
|
| (Peirce, CE 1, 282).

| The information of a term is the measure of its superfluous comprehension.
| That is to say that the proper office of the comprehension is to determine
| the extension of the term.  ...
|
| Every addition to the comprehension of a term, lessens its extension
| up to a certain point, after that further additions increase the
| information instead.  ...
|
| And therefore as every term must have information, every term has
| superfluous comprehension.  And, hence, whenever we make a symbol
| to express any thing or any attribute we cannot make it so empty
| that it shall have no superfluous comprehension.
|
| I am going, next, to show that inference is symbolization and that
| the puzzle of the validity of scientific inference lies merely in
| this superfluous comprehension and is therefore entirely removed
| by a consideration of the laws of 'information'.
|
| (Peirce, CE 1, 467).

A full explanation of these statements, linking scientific inference,
symbolization, and information together in such an integral fashion,
would require an excursion into the pragmatic theory of information
that Peirce was presenting in lectures at Harvard as early as 1865.

For now, let it suffice to say that this anticipation of the information
concept, fully recognizing the reality of its dimension, would not sound
too remote from the varieties of "law abiding constraint exploitation"
that have become increasingly familiar since the dawn of cybernetics.

But more than this, Peirce's notion of information supplies an array
of missing links that joins together in one scheme the logical roles
of terms, propositions, and arguments, the semantic functions of
denotation and connotation, and the practical methodology needed
to address and measure the quantitative dimensions of information.
This is precisely the kind of linkage that I need in this project
to integrate the dynamic and symbolic aspects of inquiry.

Not by sheer coincidence, the task of understanding symbolic action,
working up through icons and indices to the point of tackling symbols,
is one of the aims that the combination of interpretive frameworks and
objective frameworks being proposed in this project is intended to serve.

An OF is a convenient stage for those works that have progressed far enough
to make use of it, but in times of flux it must be remembered that an OF is
only a hypostatic projection, that is, a virtual image, a reified concept,
or a "phantom limb" of the IF that tentatively extends it.

When the IF and the OF being sketched here have been developed far enough,
I hope to tell wherein and whereof a sign may grow able, by virtue of its
very own character, to address itself to a purpose, one determined by its
objective nature and determining, in a measure, that of its duly intended
interpreter, to the extent that it renders the other wiser than the other
would otherwise be.

1.3.4.18. C'est Moi

From the emblem unfurled on a tapestry to tease out the working of its
loom and spindle, a charge to bind these frameworks together is drawn
by necessity from a single request:  "To whom is the sign addressed?"
The easy, all too easy answer comes "To whom it may concern", but this
works more to put off the question than it acts as a genuine response.
To say that a sign relation is intended for the use of its interpreter,
unless one has ready an independent account of that agent's conduct,
only rephrases the initial question about the end of interpretation.

The interpreter is an agency depicted over and above the sign relation,
but in a very real sense it is simply identical with the whole of it.
And so one is led to examine the relationship between the interpreter
and the interpretant, the element falling within the sign relation to
which the sign in actuality tends.  The catch is that the whole of the
intended sign relation is seldom known from the beginning of inquiry,
and so the aimed for interpretant is often just as unknown as the rest.

These eventualities call for the elaboration of interpretive and objective
frameworks in which not just the specious but the speculative purpose of
a sign can be contemplated, permitting extensions of the initial data,
through error and retrial, to satisfy emergent and recurring questions.

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 about the role of the interpretant,
"Stand and unfold yourself!", there 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 extensions and imputations.  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 as
it is 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 in the meantime medium of this discussion.

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

| I think we need to reflect upon the circumstance that every word
| implies some proposition or, what is the same thing, every word,
| concept, symbol has an equivalent term -- or one which has become
| identified with it, -- in short, has an 'interpretant'.
|
| Consider, what a word or symbol is;  it is a sort of representation.
| Now a representation is something which stands for something.  ...
| A thing cannot stand for something without standing 'to' something
| 'for' that something.  Now, what is this that a word stands 'to'?
| Is it a person?
|
| We usually say that the word 'homme' stands to a Frenchman for 'man'.
| It would be a little more precise to say that it stands to the Frenchman's
| mind -- to his memory.  It is still more accurate to say that it addresses
| a particular remembrance or image in that memory.  And what 'image', what
| remembrance?  Plainly, the one which is the mental equivalent of the word
| 'homme' -- in short, its interpretant.  Whatever a word addresses then or
| 'stands to', is its interpretant or identified symbol.  ...
|
| The interpretant of a term, then, and that which it stands to are identical.
| Hence, since it is of the very essence of a symbol that it should stand 'to'
| something, every symbol -- every word and every 'conception' -- must have an
| interpretant -- or what is the same thing, must have information or implication.
|
| (Peirce, CE 1, 466-467).

It will take a while to develop the wealth of information that
a suitably perspicacious and persistent IF would reveal to be
implicit in this unassuming homily.  The main innovations that
the present project can hope to add to the story are as follows:

   1.  To prescribe a "context of effective systems theory" (C'EST), one that
       can provide for the computational formalization of each intuitively given
       interpreter as a determinate "model of interpretation" (MOI).  A suitable
       set of concepts and methods would deal with the generic constitutions of
       interpreters, converting paraphrastic and periphrastic descriptions of
       their interpretive practice into moderately complete and relatively
       concrete specifications of sign relations.

   2.  To prepare a fully dynamic basis for actualizing interpretants.
       This means that an interpretant addressed by the interpretation
       of a sign would not be left in the form of an abstractly-minded
       memory image or a detached token to be processed by a hypothetical
       but largely nondescript interpretive agent, but realized as a fully
       descript type of state configuration in a qualitative dynamic system.
       To fathom what might be the symbolic analogue of a "state with momentum"
       has presented this project with numerous difficulties both conceptual and
       terminological.  So far in this work, I have tried to approach the character
       of an active sign-theoretic state in terms of an "interpretive moment" (IM),
       "information state" (IS), "attended token" (AT), "situation of use" (SOU),
       or "instance of use" (IOU).  A successful candidate for a concept to this
       purpose would capture the transient dispositions that drive interpreters
       to engage in specific forms of inquiry, defining their ongoing state of
       uncertainty with regard to objects and questions of immediate concern.

1.3.4.19. Entr'acte

Have I addressed this problem area from enough different directions to
convey an idea of its location and extent?  Here is one more variation
on the theme:  I believe that our theoretical empire is bare in spots.
There does not exist in the field yet a suitably comprehensive concept
of a dynamic system moving through a developing state of information.
This conceptual gap apparently forces investigators to focus on one
aspect or the other, on the dynamic bearing or the information borne,
but leaves their studies unable to integrate the several perspectives
into a fully-dimensioned outlook on the evolving knowledge system.

It is always possible that the dual aspects of transformation and
information are conceptually complementary and even non-orientable.
That is, there may be no way to arrange our mental apparatus to grasp
both sides at the same time, and the whole appearance that there are
two sides may be an illusion of overly local and myopic perspectives.
Nevertheless, none of this should be taken for granted without proof.

Whatever the case, a unyielding fixation on the restricted aspects of
dynamics adequately covered by currently available concepts leads one
to ignore the growing body of symbolic knowledge that the states of
systems potentially carry.  Conversely, to leap from the relatively
secure grounds of physically based dynamics into the briar patch of
formally defined symbol systems often marks the last time that one
has sufficient footing on the dynamic landscape to contemplate any
form of overarching law, nor any rule to prospectively govern the
evolution of reflective knowledge.  This is one of the reasons
that I continue to strive after the key ideas here.  If straw
is all that one has in reach, then ships and shelters will
have to be built from straw.

1.3.5. Discussion of Formalization : Specific Objects

| "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 --
|
| (Nietzsche, 'The Will to Power', S 575, 309).

With this preamble, I return to develop my own account of formalization,
with special attention to the kind of step that leads from the inchoate
chaos of casual discourse to a well-founded discussion of formal models.
A formalization step, of the incipient kind being considered here, has
the peculiar property that one can say with some definiteness where it
ends, since it leads precisely to a well-defined formal model, but not
with any definiteness where it begins.  Any attempt to trace the steps
of formalization backward toward their ultimate beginnings can lead to
an interminable multiplicity of open-ended explorations.  In view of
these circumstances, I will limit my attention to the frame of the
present inquiry and try to sum up what brings me to this point.

It begins like this:  I ask whether it is possible to reason about inquiry
in a way that leads to a productive end.  I pose my question as an inquiry
into inquiry, and I use the formula "y_0 = y y" to express the relationship
between the present inquiry, y_0, and a generic inquiry, y.  Then I propose
a couple of components of inquiry, discussion and formalization, that appear
to be worth investigating, expressing this proposal in the form "y >= {d, f}".
Applying these components to each other, as must be done in the present inquiry,
I am led to the current discussion of formalization, y_0 = y y >= f d.

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:

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 "y" denotes a particular inquiry, y in Y,
one that is assumed to be plenipotential or prototypical in yet
to be specified ways.  In the figurative case, the name "y" is
simply a variable that ranges over a collection Y of nominally
conceivable inquiries.

First encountered, the recipe "y_0 = y y" seems to specify that
the present inquiry is constituted by taking everything that is
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.

Contemplating the formula "y_0 = y y" in the context of the subordination
y >= {d, f} and the successive containments F c M c D, the y that inquires
into y is not restricted to examining y's immediate subordinates, d and f,
but it can investigate any feature of y's 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 y
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 additional kinds of inquiry that suggest themselves at this point,
I see at least the following possibilities:

   1.  Inquiry into propositions about application and equality.
       Just by way of a first example, one might well begin by
       considering the forms of application and equality that
       are invoked in the formula "y_0 = y y" itself.

   2.  Inquiry into application, for example, the way that
       the term "y y" indicates the application of y to y
       in the formula "y_0 = y y".  

   3.  Inquiry into equality, for example,
       the meaning of "=" in "y_0 = y y".

   4.  Inquiry into indices, for example,
       the significance of "0" in "y_0".

   5.  Inquiry into terms, specifically, constants and variables.
       What are the functions of "y" and "y_0" in this respect?

   6.  Inquiry into decomposition or subordination, for example,
       as invoked by the sign ">=" in the formula "y >= {d, f}".

   7.  Inquiry into containment or inclusion.  In particular, examine the
       claim "F c M c D" that conditions 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.

If inquiry begins in doubt, then inquiry into inquiry begins in
doubt about doubt.  All things considered, the formula "y_0 = y y"
has to be taken as the first attempt at a description of the problem,
a hypothesis about the nature of inquiry, or an image that is tossed out
by way of getting an initial fix on the object in question.  Everything in
this account so far, and everything else that I am likely to add, can only
be reckoned as hypothesis, whose accuracy, pertinence, and usefulness can
be tested, judged, and redeemed only after the fact of proposing it and
after the facts to which it refers have themselves been gathered up.

A number of problems present themselves due to the context in which
the present inquiry is aimed to present itself.  The hypothesis that
suggests itself to one person, as worth exploring at a particular time,
does not always present itself to another person as worth exploring at
the same time, or even necessarily to the same person at another time.
In a community of inquiry that extends beyond an isolated person and
in a process of inquiry that extends beyond a singular moment in time,
it is therefore necessary to consider the nature of the communication
process that the discussion of inquiry in general and the discussion of
formalization in particular need to invoke for their ultimate utility.

Solitude and solipsism are no solution to the problems of community and
communication, since even an isolated individual, if ever there was, is,
or comes to be such a thing, has to maintain the lines of communication
that are required to integrate past, present, and prospective selves --
in other words, translating everything into present terms, the parts of
one's actually present self that involve actual experiences and present
observations, do present expectations as reflective of actual memories,
and do present intentions as reflective of actual hopes.  Consequently,
the dialogue that one holds with oneself is every bit as problematic
as the dialogue that one enters with others.  Others only surprise
one in other ways than one ordinarily surprises oneself.

I recognize inquiry as beginning with a "surprising phenomenon" or
a "problematic situation", more briefly described as a "surprise"
or a "problem", respectively.  These are the types of moments that
try our souls, the instances of events that instigate inquiry as
an effort to achieve their own resolution.  Surprises and problems
are experienced as afflicted with an irritating uncertainty or a
compelling difficulty, one that calls for a response on the part
of the agent in question:

   1.  A "surprise" calls for an explanation to resolve the
       uncertainty that is present in it.  This uncertainty
       is associated with a difference between observations
       and expectations.

   2.  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", a compact term that admits of expansion as a debt,
a difference, a difficulty, a discrepancy, a dispersion, a distribution,
a doubt, a duplicity, or a duty.

Expressed another way, inquiry begins with a doubt about one's object,
whether this means what is true of a case, an object, or a world, what
to do about reaching a goal, or whether the hoped-for goal is really
good for oneself -- with all that these questions lead to in essence,
in deed, or in fact.

Perhaps there is an inexhaustible reality that issues in these
apparent mysteries and recurrent crises, but, by the time I say
this much, I am already indulging in a finite image, a hypothesis
about what is going on.  If nothing else, then, one finds again the
familiar pattern, where the formative relation between the informal
and the formal merely serves to remind one anew of the relationship
between the infinite and the finite.

1.3.5.1. The Will to Form

| 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.
|
| (Nietzsche, 'The Will to Power', S 94, 58).

Let me see if I can summarize as quickly as possible the problem that I see before me.
On each occasion that I try to express my experience, to lend it a form that others
can recognize, to put it in a shape that I myself can later recall, or to store it
in a state that allows me the chance of its re-experience, I generate an image of
the way things are, or at least a description of how things seem to me.  I call
this process "reflection", since it fabricates an image in a medium of signs
that reflects an aspect of experience.  Very often this experience is said
to be "of" -- what? -- something that exists or persists at least partly
outside the immediate experience, some action, event, or object that is
imagined to inform the present experience, or perhaps some conduct of
one's own doing that obtrudes for a moment into the world of others
and meets with a reaction there.  In all of these cases, where the
experience is everted to refer to an object and thus becomes the
attribute of something with an external aspect, something that
is thus supposed to be a prior cause of the experience, the
reflection on experience doubles as a reflection on that
conduct, performance, or transaction that the experience
is an experience "of".  In short, if the experience has
an eversion that makes it an experience of an object,
then its reflection is again a reflection that is
also of this object.

Just at the point where one threatens to become lost in the morass of
words for describing experience and the nuances of their interpretation,
one can adopt a formal perspective, and realize that the relation among
objects, experiences, and reflective images is formally analogous to the
relation among objects, signs, and interpretant signs that is covered by
the pragmatic theory of signs.  One still has the problem:  How are the
expressions of experience everted to form the exterior faces of extended
objects and exploited to embed them in their external circumstances, and
no matter whether this object with an outer face is oneself or another?
Here, one needs to understand that expressions of experience include
the original experiences themselves, at least, to the extent that
they permit themselves to be recognized and reflected in ongoing
experience.  But now, from the formal point of view, "how" means
only:  To describe the formal conditions of a formal possibility.

1.3.5.2. The Forms of Reasoning

| The most valuable insights are arrived at last;
| but the most valuable insights are methods.
|
| (Nietzsche, 'The Will to Power', S 469, 261).

A certain arbitrariness has to be faced in the terms that one uses
to talk about reasoning, to split it up into different parts and
to sort it out into different types.  It is like the arbitrary
choice that one makes in assigning the midpoint of an interval
to the subintervals on its sides.  In setting out the forms of
a nomenclature, in fitting the schemes of my terminology to the
territory that it disturbs in the process of mapping, I cannot
avoid making arbitrary choices, but I can aim for a strategy
that is flexible enough to recognize its own alternatives and
to accommodate the other options that lie within their scope.

If I make the mark of deduction the fact that it reduces the
number of terms, as it moves from the grounds to the end of
an argument, then I am due to devise a name for the process
that augments the number of terms, and thus prepares the
grounds for any account of experience.

What name hints at the many ways that signs arise in regard to things?
What name covers the manifest ways that a map takes over its territory?
What name fits this naming of names, these proceedings that inaugurate
a sign in the first place, that duly install it on the office of a term?
What name suits all these actions of addition, annexation, incursion, and
invention that instigate the initial bearing of signs on an object domain?

In the interests of a "maximal analytic precision" (MAP), it is fitting
that I should try to sharpen this notion to the point where it applies
purely to a simple act, that of entering a new term on the lists, in
effect, of enlisting a new term to the ongoing account of experience.
Thus, let me style this process as "adduction" or "production", in
spite of the fact that the aim of precision is partially blunted
by the circumstance that these words have well-worn uses in other
contexts.  In this way, I can isolate to some degree the singular
step of adding a term, leaving it to a later point to distinguish
the role that it plays in an argument.

As it stands, the words "adduction" and "production" could apply to the
arbitrary addition of terms to a discussion, whether or not these terms
participate in valid forms of argument or contribute to their mediation.
Although there are a number of auxiliary terms, like "factorization",
"mediation", or "resolution", that can help to pin down these meanings,
it is also useful to have a word that can convey the exact sense meant.
Therefore, I coin the term "obduction" to suggest the type of reasoning
process that is opposite or converse to deduction and that introduces
a middle term "in the way" as it passes from a subject to a predicate.

Consider the adjunction to one's vocabulary that is comprised of these three words:
"adduction", "production", "obduction".  In particular, how do they appear in the
light of their mutual applications to each other and especially with respect to
their own reflexivities?  Notice that the terms "adduction" and "production"
apply to the ways that all three terms enter this general discussion, but
that "obduction" applies only to their introduction only in specific
contexts of argument.

Another dimension of variation that needs to be noted among these different types
of processes is their status with regard to determimism.  Given the ordinary case
of a well-formed syllogism, deduction is a fully deterministic process, since the
middle term to be eliminated is clearly marked by its appearance in a couple of
premisses.  But if one is given nothing but the fact that forms this conclusion,
or starts with a fact that is barely suspected to be the conclusion of a possible
deduction, then there are many other middle terms and many other premisses that
might be construed to result in this fact.  Therefore, adduction and production,
for all of their uncontrolled generality, but even obduction, in spite of its
specificity, cannot be treated as deterministic processes.  Only in degenerate
cases, where the number of terms in a discussion is extremely limited, or where
the availability of middle terms is otherwise restricted, can it happen that
these processes become deterministic.

1.3.5.3. A Fork in the Road

| 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 --).
|
| (Nietzsche, 'The Will to Power', S 521, 282).

It is worth trying to discover, as I currently am, how many properties of inquiry
can be derived from the simple fact that it needs to be able to apply to itself.
I find three main ways to approach the problem of inquiry's self-application,
or the question of inquiry's reflexivity:

   1.  One way attempts to continue the derivation in the manner of a
       necessary deduction, perhaps by reasoning in the following vein:
       If self-application is a property of inquiry, then it is sensible
       to inquire into the concept of application that could make this
       conceivable, and not just conceivable, but potentially fruitful.

   2.  Another way breaks off the attempt at a deductive development and puts forth
       a full-scale model of inquiry, one that has enough plausibility to be probated
       in the court of experience and enough specificity to be tested in the context
       of self-application.

   3.  The last way is a bit ambivalent in its indications, seeking as it does
       both the original unity and the ultimate synthesis at one and the same
       time.  Perhaps it goes toward reversing the steps that lead up to this
       juncture, marking it down as an impasse, chalking it up as a learning
       experience, or admitting the failure of the imagined distinction to
       make a difference in reality.  Whether this form of egress is read
       as a backtracking correction or as a leaping forward to the next
       level of integration, it serves to erase the distinction between
       demonstration and exploration.

Without a clear sense of how many properties of inquiry are necessary
consequences of its self-application and how many are merely accessory
to it, or even whether some contradiction still lies lurking within the
notion of reflexivity, I have no choice but to follow all three lines of
inquiry wherever they lead, keeping an eye out for the synchronicities,
the constructive collusions and the destructive collisions that may
happen to occur among them.

The fictions that one devises to shore up a shaky account of experience
can often be discharged at a later stage of development, gradually coming
to be replaced with primitive elements of less and less dubious characters.
Hypostases and hypotheses, the creative terms and the inventive propositions
that one coins to account for otherwise ineffable experiences, are tokens that
are subject to a later account.  Under recurring examination, many such tokens
are found to be ciphers, marks that no one will miss if they are cancelled out
altogether.  The symbolic currencies that tend to survive lend themselves to
being exchanged for stronger and more settled constructions, in other words,
for concrete definitions and explicit demonstrations, gradually leading to
primitive elements of more and more durable utilities.

1.3.5.4. A Forged Bond

| 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.
|
| (Nietzsche, 'The Will to Power', S 521, 282).

A unity can be forged among the methods by noticing the following
connections among them.  All the while that one proceeds deductively,
the primitive elements, the definitions and the axioms, must still be
introduced hypothetically, notwithstanding the support they get from
common sense and widespread assent.  And the whole symbolic system
that is constructed through hypothesis and deduction must still be
tested in experience to see if it serves any purpose to maintain it.

1.3.5.5. A Formal Account

| 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").
|
| (Nietzsche, 'The Will to Power', S 521, 282).

In this Section (1.3.5), I am considering the step of formalization that
takes discussion from a large scale informal inquiry to a well-defined
formal inquiry, establishing a relation between the implicit context
and the explicit text.

In this project as a whole, formalization is used to produce formal models
that represent relevant features of a phenomenon or process of interest.
Thus, the formal model is what constitutes the image of formalization.

The role of formalization splits into two different cases depending on
the intended use of the formal model.  When the phenomenon of interest
is external to the agent that is carrying out the formalization, then
the model of that phenomenon can be developed without doing any great
amount of significant reflection on the formalization process itself.
This is usually a more straightforward operation, since it can avail
itself of automatic competencies that are not themselves in question.
But when the phenomenon of interest is entangled with the conduct of
the agent in question, then the formal modeling of that conduct will
generally involve a more or less difficult component of reflection.

In a recursive context, a principal benefit of the formalization
step is to find constituents of inquiry with reduced complexities,
drawing attention from the context of informal inquiry, whose stock
of questions may not be grasped well enough to ever be fruitful and
the scope of whose questions may not be focused well enough to ever
see an answer, and concentrating effort in an arena of formalized
inquiry, where the questions are posed well enough to have some
hope of bearing productive answers in a finite time.

1.3.5.6. Analogs, Icons, Models, Surrogates

| 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.
|
| (Nietzsche, 'The Will to Power', S 521, 282).

This project makes pivotal use of certain formal models to represent the
conceived structure in a "phenomenon of interest" (POI).  For my purposes,
the phenomenon of interest is typically a process of interpretation or a
process of inquiry, two nominal species of process that will turn out to
evolve from different points of view on the very same form of conduct.

Commonly, a process of interest presents itself as the trajectory
that an agent describes through an extended space of configurations.
The work of conceptualization and formalization is to represent this
process as a conceptual object in terms of a formal model.  Depending
on the point of view that is taken from moment to moment in this work,
the "model of interest" (MOI) may be cast as a model of interpretation
or as a model of inquiry.  As might be anticipated, it will turn out
that both descriptions refer essentially to the same subject, but
this will take some development to become clear.

In this work, the basic structure of each MOI is adopted from the
pragmatic theory of signs and the general account of its operation
is derived from the pragmatic theory of inquiry.  The indispensable
usefulness of these models hinges on the circumstance that each MOI,
whether playing its part in interpretation or in inquiry, is always
a "model" in two important senses of the word.  First, it is a model
in the logical sense that its structure satisfies a formal theory or
an abstract specification.  Second, it is a model in the analogical
sense that it represents an aspect of the structure that is present
in another object or domain.

1.3.5.7. Steps and Tests of Formalization

| 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.
|
| (Nietzsche, 'The Will to Power', S 521, 282).

A step of formalization moves the active focus of discussion from
the "presentational object" or the source domain that constitutes
the phenomenon of interest to the "representational object" or the
target domain that makes up the relevant model of interest.  If the
structure in the source context is already formalized then the step
of formalization can itself be formalized in an especially elegant
and satisfying way as a structure-preserving map, a homomorphism,
or an "arrow" in the sense of mathematical category theory.

The test of a formalization being complete is that a computer program could
in principle carry out the steps of the process being formalized exactly as
represented in the formal model or image.  It needs to be appreciated that
this test is a criterion of sufficiency to formal understanding and not of
necessity directed toward a material re-creation or a concrete simulation
of the formalized process.  The ordinary agents of informal discussion
who address the task of formalization do not disappear in the process
of completing it, since it is precisely for their understanding that
the step is undertaken.  Only if the phenomenon or process at issue
were by its very nature solely a matter of form could its formal
analogue constitute an authentic reproduction.  However, this
potential consideration is far from the ordinary case that
I need to discuss at present.

In ordinary discussion, agents of inquiry and interpretation depend on
the likely interpretations of others to give their common notions and
their shared notations a meaning in practice.  This means that a high
level of implicit understanding is relied on to ground each informal
inquiry in practice.  The entire framework of logical assumptions and
interpretive activities that is needed to shore up this platform will
itself resist analysis, since it is precisely to save the effort of
repeating routine analyses that the whole infrastructure is built.

1.3.5.8. A Puckish Ref

| 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".
|
| (Nietzsche, 'The Will to Power', S 521, 282-283).

In a formal inquiry of the sort projected here, the less the discussants
need to depend on the compliance of understanding interpreters the more
they will necessarily understand at the end of the formalization step.

It might then be thought that the ultimate zero of understanding expected
on the part of the interpreter would correspond to the ultimate height of
understanding demanded on the part of the formalizer, but this assumption
neglects the negative potential of misunderstanding, the sheer perversity
of interpretation that our human creativity can bring to bear on any text.

But computers are initially just as incapable of misunderstanding as they
are of understanding.  Therefore, it actually forms a moderate compromise
to address the task of interpretation to a computational system, a thing
that is known to begin from a moderately neutral intitial condition.

1.3.5.9. Partial Formalizations

| 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.
|
| (Nietzsche, 'The Will to Power', S 521, 283).

In many discussions the source context remains unformalized in itself,
taking form only according to the image it receives in one or another
individual MOI.  In cases like these, the step of formalization does
not amount to a total function but is limited to a partial mapping
from the source to the target.  Such a partial representation is
analogous to a sampling operation.  It is not defined on every
point of the source domain but assigns values only to a proper
selection of source elements.  Thus, a partial formalization
can be regarded as achieving its form of simplification in
a loose way, ignoring elements of the source domain and
collapsing material distinctions in irregular fashions.

1.3.5.10. A Formal Utility

| Ultimate solution. -- We believe in reason:
| this, however, is the philosophy of gray concepts.
| Language depends on the most naive prejudices.
|
| (Nietzsche, 'The Will to Power', S 522, 283).

The usefulness of the MOI as the upshot of the formalization arrow is
that it provides discussion with a compact image of the source domain.
In formalization one strives to extract a simpler image of the larger
inquiry, a context of participatory action that one is too embroiled
in carrying out step by step to see as a whole.  Seen in this light,
the purpose of formalization is to identify a simpler version of the
problematic phenomenon or to fashion a simpler image of the difficult
inquiry, one that is well-defined enough and simple enough to assure
its termination in a finite interval of space and time.  As a result,
one of the main benefits of adopting the objective of formalization
is that it equips discussion with a pre-set termination criterion,
or a "stopping rule".

In the context of the recursive inquiry that I have outlined,
the step of formalization is intended to bring discussion
appreciably closer to a solid base for the operational
definition of inquiry.

1.3.5.11. A Formal Aesthetic

| 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.)
|
| (Nietzsche, 'The Will to Power', S 522, 283).

Recognizing that the Latin word "forma" means not just "form"
but also "beauty" supplies a clue that not all formal models
are equally valuable for a purpose of interest.  There is
a certain quality of formal elegance, or select character,
that is essential to the practical utility of the model.

The virtue of a good formal model is to provide discussion with
a fitting image of the whole phenomenon of interest.  The aim of
formalization is to extract from an informal discussion or locate
within a broader inquiry a clearer and simpler image of the whole
activity.  If the formalized image or precis is unusually apt then
it might be prized as a gnomon or a recapitulation and be said to
capture the essence, the gist, of the nub of the whole affair.

A pragmatic qualification of this virtue requires that the image be
formed quickly enough to take decisive action on.  So the quality of
being a result often takes precedence over the quality of the result.
A definite result, however partial, is frequently reckoned as better
than having to wait for a definitive picture that may never develop.

But an overly narrow or premature formalization, where the nature of
the phenomenon of interest is too much denatured in the formal image,
may result in destroying all interest in the result that does result.

1.3.5.12. A Formal Apology

| We cease to think when we refuse to do so under the constraint of language;
| we barely reach the doubt that sees this limitation as a limitation.
|
| (Nietzsche, 'The Will to Power', S 522, 283).

Seizing the advantage of this formal flexibility makes it possible
to take abstract leaps over a multitude of material obstacles,
to reason about many properties of objects and processes
from a knowledge of their form alone, without having
to know everything about their material content
down to the depths that matter can go.

1.3.5.13. A Formal Suspicion

| Rational thought is interpretation according to a scheme that we cannot throw off.
|
| (Nietzsche, 'The Will to Power', S 522, 283).

I hope that the reader has arrived by now at an independent suspicion that the
process of formalization is a microcosm nearly as complex as the whole subject
of inquiry itself.  Indeed, the initial formulation of a problem is tantamount
to a mode of "representational inquiry".  In many ways this very first effort,
that stirs from the torpor of ineffable unease to seek out any sort of unity
in the manifold of fragmented impressions, is the most difficult, subtle,
and crucial kind of inquiry.  It begins in doubt about even so much as
a fair way to represent the problematic situation, but its result can
predestine whether subsequent inquiry has any hope of success.  There
is very little in this brand of formal engagement and participatory
representation that resembles the simple and disinterested act of
holding a mirror, flat and featureless, up to nature.

If formalization really is a form of inquiry in itself, then
its formulations have deductive consequences that can be tested.
In other words, formal models have logical effects that reflect on
their fitness to qualify as representations, and these effects can
cause them to be rejected merely on the grounds of being a defective
picture or a misleading conception of the source phenomenon.  Therefore,
it should be appreciated that software tailored to this task will probably
need to spend more time in the alterations of backtracking than it will have
occasion to trot out parades of ready-to-wear models.

Impelled by the mass of assembled clues from restarts and refits to the
gathering form of a coherent direction, the inkling may have gradually
accumulated in the reader that something of the same description has
been treated in the pragmatic theory of inquiry under the heading
of "abductive reasoning".  This is distinguished from inductive
reasoning, that goes from the particular to the general, in
that abductive reasoning must work from a mixed collection
of generals and particulars toward a middle term, a formal
intermediary that is more specific than the vague allusions
gathered about its subject and more generic than the elusive
instances fashioned to illustrate its prospective predicates.

In a recursive context, the function of formalization is to relate a
difficult problem to a simpler problem, breaking the original inquiry
into two parts, the step of formalization and the rest of the inquiry,
both of which branches it is hoped will be nearer to solid ground and
easier to grasp than the original question.

1.3.5.14. The Double Aspect of Concepts

| 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 --
|
| (Nietzsche, 'The Will to Power', S 523, 283).

This project is a particular inquiry into the nature of inquiry in general.
As a consequence, every concept that appears in it takes on a double aspect.

To illustrate, let us take the concept of a "sign relation" as an example
of a construct that appears in this work and let me use it to speak about
my own agency in this inquiry.  All I need to say about a sign relation
at this point is that it is a three-place relation, and therefore can
be represented as a relational data-base with three columns, in this
case naming the "object", the "sign", and the "interpretant" of the
relation at each moment in time of the corresponding "sign process".

At any given moment of this inquiry I will be participating in a certain
sign relation that constitutes the informal context of my activity, the
full nature of which I can barely hope to conceptualize in explicitly
formal terms.  At times, the object of this informal sign relation
will itself be a sign relation, typically one that is already
formalized or one that I have a better hope of formalizing,
but it could conceivably be the original sign relation
with which I began.

In such cases, when the object of a sign relation
is also a sign relation, the general concept of
a sign relation takes on a double duty:

   1.  The less formalized sign relation is used to mediate the
       present inquiry.  As a conceptual construct, it is not yet
       fully conceived or not yet fully constructed at the moments
       of inquiry being considered.  Perhaps it is better to regard
       it as a "concept under construction".  Employed as a contextual
       apparatus, this sign relation serves an instrumental role in the
       construal and the study of its designated objective sign relation.

   2.  The more formalized sign relation is mentioned as a substantive object
       to be contemplated and manipulated by the proceedings of this inquiry.
       As a conceptual construct, it exemplifies its intended role best if it
       is already as completely formalized as possible.  It is being engaged
       as a substantive object of inquiry.

I have given this inquiry a reflective or recursive cast, portraying it
as an inquiry into inquiry, and one of the consequences of this picture
is that every concept employed in the work will take on a divided role,
double aspect, or dual purpose.  At any moment, the object inquiry of
the moment is aimed to take on a formal definition, while the active
inquiry need not acknowledge any image that it does not recognize
as reflecting itself, nor is it bound by any horizon that does
not capture its spirit.

1.3.5.15. A Formal Permission

NB.  These sections are still too provisional to share,
but I will record the epitexts that I have in my notes.

| If there are to be synthetic a priori judgments, then reason must
| be in a position to make connections:  connection is a form.
| Reason must possess the capacity of giving form.
|
| (Nietzsche, 'The Will to Power', S 530, 288).

1.3.5.16. A Formal Invention

| 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.
|
| (Nietzsche, 'The Will to Power', S 544, 293).

1.3.6. Recursion in Perpetuity

| 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.
|
| It is a word for the "will to power".
|
| (Nietzsche, 'The Will to Power', S 552, 298).

| Life is founded upon the premise of a belief in enduring
| and regularly recurring things;  the more powerful life is,
| the wider must be the knowable world to which we, as it were,
| attribute being.  Logicizing, rationalizing, systematizing as
| expedients of life.
|
| (Nietzsche, 'The Will to Power', S 552, 298-299).

| 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.
|
| (Nietzsche, 'The Will to Power', S 552, 299).

1.3.7. Processus, Regressus, Progressus

| From time immemorial we have ascribed the value of an action, a character,
| an existence, to the intention, the purpose for the sake of which one has
| acted or lived:  this age-old idiosyncrasy finally takes a dangerous turn --
| provided, that is, that the absence of intention and purpose in events
| comes more and more to the forefront of consciousness.
|
| (Nietzsche, 'The Will to Power', S 666, 351).

| 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".
|
| (Nietzsche, 'The Will to Power', S 666, 351).

| 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).
|
| (Nietzsche, 'The Will to Power', S 666, 351).

1.3.8. Rondeau — Tempo di Menuetto

| 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!
|
| (Nietzsche, 'The Will to Power', S 1067, 549-550).

I have attempted in a narrative form to present an accurate picture
of the formalization process as it develops in practice.  Of course,
accuracy must be distinguished from precision, for there are times
when accuracy is better served by a vague outline that captures the
manner of the subject than it is by a minute account that misses
the mark entirely or catches each detail at the expense of losing
the central point.  Conveying the traffic between chaos and form
under the restraint of an overbearing and excisive taxonomy would
have sheared away half the picture and robbed the whole exchange
of the lion's share of the duty.

At moments I could do no better than to break into metaphor, but
I believe that a certain tolerance for metaphor, especially in the
initial stages of formalization, is a necessary capacity for reaching
beyond the secure boundaries of what is already comfortable to reason.
Plus, a controlled transport of metaphor allows one to draw on the
boundless store of ready analogies and germinal morphisms that
every natural language provides for free.

Finally, it would leave an unfair impression to delete the characters
of narrative and metaphor from the text of the story, and especially
after they have had such a hand in creating it.

Even the most precise of established formulations cannot be protected
from being reused in ways that initially appear as abuses of language.

One of the most difficult questions about the development of intelligent
systems is how the power of abstraction can arise, beginning from the
kinds of formal systems where each symbol has one meaning at most.
I think that the natural pathway of this evolution has to go
through the obscure territory of ambiguity and metaphor.

A critical phase and a crucial step in the development of intelligent systems,
whether biological or technological, is concerned with achieving a certain
power of abstraction, but the real trick is for the budding intelligence
to accomplish this without losing a grip on the material contents of
the abstract categories, the labels and levels of which this power
intercalates and interposes between essence and existence.

If one looks to the surface material of natural languages for signs of
how this power of abstraction might arise, one finds a suggestive set of
potential precursors in the phenomena of ambiguity, anaphora, and metaphor.
Keeping this in mind throughout the project, I aim to pay close attention
to the places where the power of abstraction seems to develop, especially
in the guises of systematic ambiguity and controlled metaphor.

Paradoxically, and a bit ironically, if one's initial attempt to
formalize meaning begins with the goal of stamping out ambiguity,
metaphor, and all forms of figurative language use, then one may
have precluded all hope of developing a capacity for abstraction
at any later stage.

1.3.9. Reconnaissance

| In every sort of project there are two things to consider:
| first, the absolute goodness of the project;  in the second
| place, the facility of execution.
|
| In the first respect it suffices that the project be acceptable
| and practicable in itself, that what is good in it be in the
| nature of the thing;  here, for example, that the proposed
| education be suitable for man and well adapted to the
| human heart.
|
| The second consideration depends on relations given in certain situations --
| relations accidental to the thing, which consequently are not necessary
| and admit of infinite variety.
|
| Rousseau, 'Emile, or On Education', 34-35

This Section provides a glancing introduction to many subjects
that cannot be treated in depth until much later in this work,
but that need to be touched on at this point, if only in order
to "prime the canvass" or to "set the tone" for the remainder
of the work, that is, to suggest the general philosophy, the
implicit assumptions, and the basic conceptions that guide,
limit, and underlie this approach to the subject of inquiry.

In the process of achieving the aims of this preliminary survey,
it is apparently necessary for me, on this occasion, to pick my
way through a densely interwoven web, to wit, a pressing but by
no means a clear context of informal discussion, and to work my
way across and around a nearly invisible warp, that is to say,
a whit less wittingly, a network of not yet fully formalized
thought that nevertheless informs discussion in its own way.

At every stage my work is bound by dint of the necessities that appear,
to me, to occasion it, and thus my initial overture to a more developed
inquiry is bound to continue in an indirect style.  As this venture and
each of its tentative subventures is compelled to try their supervening
and intervening subjects in an array of oblique and incidental manners,
I am constantly forced to detect my likeliest directions of progress by
gently teasing out only the most readily exposed clues from the context
of tangent discourse, and I am consequently obliged to clarify my local
chances of success by provisionally tugging loose only the most roughly
isolated threads from this gradually explicated and formulated network.
Accordingly, a reconnaissance of the immediate surroundings affords but
a minimal opportunity to exercise options for creativity and imagination,
and there is little choice but to pick up each subordinate subject in the
midst of its action and to let go of it again while it is still in progress.

In the process of carrying out the present reconnaissance it is useful
to illustrate the pragmatic theory of signs as it bears on a series of
slightly less impoverished and somewhat more interesting materials, to
demonstrate a few of the ways that the theory of signs can be applied
to a selection of genuinely complex and problematic texts, specifically,
poetic and lyrical texts that are elicited from natural language sources
through the considerable art of creative authors.  In keeping with the
nonchalant provenance of these texts, I let them make their appearance
on the scene of the present discussion in what may seem like a purely
incidental way, and only gradually to acquire an explicit recognition.

1.3.9.1. The Informal Context

| On either side the river lie
| Long fields of barley and of rye,
| That clothe the wold and meet the sky;
| And thro' the field the road runs by
|         To many-tower'd Camelot;
| And up and down the people go,
| Gazing where the lilies blow
| Round an island there below,
|         The island of Shalott.
|
| Tennyson, "The Lady of Shalott", [Ten, 17]

One of the continuing difficulties of this work is the tension between
the formal contexts of representation, where clarity and certainty are
easiest to achieve, and the informal context of applications, where any
degree of insight into the nature of the problems and the structure of
the entanglements affecting it is eagerly awaited and earnestly desired.
This tension is due to the distances that stretch across the expanses
of these contexts, especially if one considers their more extreme poles,
since there is no release given of the necessity to build connections,
conduct negotiations, establish a continuum of reciprocal transactions,
and maintain a community of working relationships that is capable of
uniting their diversity into a coherent whole.  Consequently, it is at
the wide end of the hopper that the real problems of formalization can
be seen to occur, where taking in too resistant and tangled a material
can play havoc with the fragile mechanisms of the formalization process
that the mind has scarcely been able to develop in its time to date.

It may be useful at this point of the discussion to insert a reminder of
why it is apposite to delve into the difficulties of the informal context.
The task of programming is to identify intellectual activities that are
initially carried on in the informal context, especially those that have
obscure aspects in need of clarification or onerous features in need of
facilitation, to analyze the ends and the means of these activities until
formal analogues can be found for some of their parts, thereby devising
suitable surrogates for these components within the formal arena or the
effective sphere, and finally to implement these formalizations within
the efficient arena or the practical sphere.

Inquiry is an activity that still takes place largely in the informal context.
Accordingly, much of what people instinctively and intuitively do in carrying
out an inquiry is done without a fully explicit idea of why they proceed that
way, or even a thorough reflection on what they hope to gain by their efforts.
It may come as a shock to realize this, since most people regard scientific
inquiries, at the very least, as rational proceedings that are founded on
explicit knowledge and that follow a host of established models.  But the
standard of rigor that I have in mind here refers to the kind of utterly
thorough formalization that it would take to create autonomous computer
programs for inquiry, ones that are capable of carrying out significant
branches of full-blown inquiries on their own.  The remoteness of that
goal quickly becomes evident to any programmer who strikes out in the
general direction of trying to achieve it.

| Willows whiten, aspens quiver,
| Little breezes dusk and shiver
| Thro' the wave that runs forever
| By the island in the river
|         Flowing down to Camelot.
| Four gray walls, and four gray towers,
| Overlook a space of flowers,
| And the silent isle imbowers
|         The Lady of Shalott.
|
| Tennyson, "The Lady of Shalott", [Ten, 17]

Nothing says that everything can be formalized.  Nothing says even that
every intellectual process has a formal analogue, at least, nothing yet.
Indeed, one is obliged to formulate the question whether every inquiry
can be formalized, and one has to be prepared for the possibility that
an informal inquiry may lead one to the ultimate conclusion that not
every inquiry has a formalization.  But how can these questions be
any clearer than the terms "inquiry" and "formalization" that they
invoke?  At this point it does not appear that further clarity can
be achieved until specific notions of inquiry and formalization
are set forth.

Although it can be said that a few components of inquiry are partially
formalized in current practice, even this much reference to the parts
of inquiry involves the choice of particular models of inquiry and
specific notions of formalization.  Starting from a sign-theoretic
setting, and with the aim of working toward a system-theoretic
framework, I am led to ask the following questions:

   1.  What is a question,
       for instance, this one?

   2.  How do questions arise,
       for instance, this one?

   3.  How can the formulation of a question,
       for example, as this one happens to be,
       catalyze the formulation of an answer,
       for example, as this happens not to be?

These questions are concerned with the nature, origin, and development,
in turn, of a class of entities called "questions".  One of the first
questions that arises about these "questions" is whether a question
can sensibly refer to a class of entities of which the question is
itself imagined or intended to be a member.  Putting this aside for
a while, I can try to get a handle on the above three questions by
placing them in different lights, that is, by interpreting them in
different contexts:

   a.  To ask these questions in a sign-theoretic context is to ask about
       the nature, the origin, and the development of the entities called
       "questions" as a class of signs, in brief but sufficiently general
       terms, to inquire into the life of a question as a sign.

   b.  To re-pose these questions in a system-theoretic context is to
       inquire into the notion of a "state of question" (SOQ), asking:

       i.    What sort of system is involved in its conception?

       ii.   How does it arise within such a system?

       iii.  How does it evolve over time?

| By the margin, willow-veil'd,
| Slide the heavy barges trail'd
| By slow horses;  and unhail'd
| The shallop flitteth silken-sail'd
|         Skimming down to Camelot:
| But who hath seen her wave her hand?
| Or at the casement seen her stand?
| Or is she known in all the land,
|         The Lady of Shalott?
|
| Tennyson, "The Lady of Shalott", [Ten, 17]

I begin with the idea that a question is an unclear sign.  The question can
express a problematic situation or a surprising phenomenon, but of course it
expresses it only obscurely, or else the inquiry is at an end.  Answering the
question is, generally speaking, a task of converting or replacing the initial
sign with a clearer but logically equivalent or implicit sign, proceeding until
a maximally clear sign or a sufficiently clear sign is achieved, or else until a
sign arises that the first sign has no meaning at all, or no sense worth pursuing.

What gives a person a sense that a sign has meaning, well before its
meaning is clearly known?  What makes one think that a sign leads to
the objects and the ideas that give it meaning, while only a sign is
before the mind?  Are there good and proper ways to test the probable
utility of a sign, short of following its indications out to the end?
And how can one tell if one's sense of meaning is deluded, saving the
resort that suffers the total consequences of belief, faith, or trust
in the sign, namely, of acting on the ostensible meaning of the sign?

An inquiry begins, in general, with an unclear sign that appears to be
indicating an obscure object to an unknown interpreter, that is, to an
interpreter whose own nature is likely to be every bit as mysterious as
the sign that is observed and the object that is indicated put together.

An inquiry viewed as a recursive procedure seeks to compute, to find,
or to generate a satisfactory answer to a hard question by working its
way back to related but easier questions, component questions on which
the whole original question appears to depend, until a set of questions
are found that are so basic and whose answers are so easy, so evident,
or so obvious that the agent of inquiry already knows their answers or
is quickly able to obtain them, whence the agent of the procedure can
continue by building up an adequate answer to the instigating question
in terms of its answers to these fundamental questions.  The couple of
phases that can be distinguished on logical grounds to be taking place
within this process, whether in point of real practice they proceed in
exclusively serial, interactively dialectic, or independently parallel
fashions, are usually described as the "analytic descent" (AD) and the
"synthetic ascent" (SA) of the recursion in question.

| Only reapers, reaping early
| In among the bearded barley,
| Hear a song that echoes cheerly
| From the river winding clearly,
|         Down to tower'd Camelot:
| And by the moon the reaper weary,
| Piling sheaves in uplands airy,
| Listening, whispers, "'T is the fairy
|         Lady of Shalott."
|
| Tennyson, "The Lady of Shalott", [Ten, 17]

One of the continuing claims of this work is that the formal structures
of sign relations are not only adequate to address the needs of building
a basic commerce among objects, signs, and ideas but are ideally suited
to the task of linking vastly different realms of objective realities and
widely disparate realms of interpretive contexts.  What accounts for the
utility that sign relations enjoy as a staple element for this job, not
only for establishing the connectivity and maintaining the integrity of
the mind in the world, but for holding the world and the mind together?

This utility is largely due to the augmented arity of sign relations as
triadic relations.  This endows them with an ability to extend in plural
dimensions at once, to span the distances between the objective and the
interpretive domains that the duties of denotation are likely to demand,
while concurrently expanding the volumes of contextual dispersion that
the courts of connotation are liable to exact in the process of waging
their syntax.  The use of sign relations represents a significant advance
over the more restrictive employments of dyadic relations, which do not
allow of extension in more than one dimension at a time, permitting no
area to be swept out nor any volume to be enclosed.  For these reasons,
sign relations afford an admirable way to distribute the tensions of
the task of inquiry over a space that is adequate to carry the loads.

Incidentally, it needs to be noted that this inquiry into the utility of
sign relations for inquiry is not so much a question of whether the mind
makes use of sign relations, or something that is isomorphic to them by
any other name, since an acquaintance with the comparative strengths of
various arities of relations is enough to make it obvious that no other
way is available for the mind to do the things it does, but it is more
a matter of how aware the mind can be made of its use of sign relations,
and of how explicitly it can learn to express itself in regard to the
structures and the functions of the sign relations in which it works.

In view of this distinction, the issue for this inquiry is not so much a
question about the bare facts of sign relation use themselves as it is a
question about the abilities of sign-using agents to accomplish anything
amounting to, analogous to, or approaching an awareness of these facts.
This is a question about an additional aptitude of sign-bearing agents,
an extra capacity for the articulation and the expression of the facts
and the factors that affect their very bearing as agents, and it amounts
to an aptness for "reflection" on the facilities, the facticities, and the
faculties that factor into making up their own sign use.  If nothing else,
these reflections serve to settle the question of a name, permitting this
ability to be called "reflection", however little else is known about it.

| There she weaves by night and day
| A magic web with colors gay.
| She has heard a whisper say,
| A curse is on her if she stay
|         To look down to Camelot.
| She knows not what the curse may be,
| And so she weaveth steadily,
| And little other care hath she,
|         The Lady of Shalott.
|
| Tennyson, "The Lady of Shalott", [Ten, 17]

The purpose of a sign, for instance, a name, an expression, a program,
or a text, is to denote and possibly to describe an object, for instance,
a thing, a situation, or an activity in the world.  When the reality to be
described is infinitely more complex than the typically finite resources
that one has to describe it, then strategic uses of these resources are
bound to occur.  For example, elliptic, multiple, and repeated uses of
signs are almost bound to be called for, involving the strategies of
approximation, abstraction, and recursion, respectively.

The agent of a system of interpretation that is driven to the point of
distraction by the task of describing an inexhaustibly complex reality
has many strategies, aside from dropping the task altogether, that are
available to it for recovering from a lapse of attention to its object:

1.  The agent can resort to approximation.  This involves accepting
    the limitations of attention and restricting one's intention to
    capturing, describing, or representing merely the most salient
    aspect, facet, fraction, or fragment of the objective reality.

2.  The agent can resort to abstraction.  ...

3.  The agent can resort to recursion.  This stratagem can
    in fact be considered as a special type of abstraction.  ...

A common feature of these techniques is the creation of a formal domain,
a context that contains the conceptually manageable images of objective
reality, a circumscribed arena for thought, one that the mind can range
over without an intolerable fear of being overwhelmed by its complexity.
In short, a formal arena, for all the strife that remains to it and for
all the tension that it maintains with its informal surroundings, still
affords a space for thought in which various forms of complete analysis
and full comprehension are at least conceivable in principle.  For all
their illusory character, these meagre comforts are not to be despised.

| And moving thro' a mirror clear
| That hangs before her all the year,
| Shadows of the world appear.
| There she sees the highway near
|         Winding down to Camelot:
| There the river eddy whirls,
| And there the surly village-churls,
| And the red cloaks of market girls,
|         Pass onward from Shalott.
|
| Tennyson, "The Lady of Shalott", [Ten, 17-18]

The formal plane stands like a mirror in relation to the informal scene.
If it did not reflect the interests and represent the objects that endure
within the informal context, no matter how dimly and slightly it is able
to portray them, then what goes on in a formal domain would lose all its
fascination.  At least, it would have little hold on a healthy mentality.
The various formal domains that an individual agent is able to grasp are
set within the informal sphere like so many myriads of mirrored facets
that are available to be cut on a complex gemstone.  Each formal domain
affords a medium for reflection and transmission, a momentary sliver of
selective clarity that allows an agent who realizes it to reflect and to
represent, if always a bit obscurely and partially, a miniscule share of
the wealth of formal possibilities that is there to be apportioned out.

Each portion of this uncut stone provides a space, and thus supplies
a "formal material", that can be used to embody a few of those aspects
of action that are discerned, designed, desired, or destined to transpire
in the grander setting that is incident on it, in a numinous context that
appears to surround its brief flashes of insight from every side at once.
Each selection of an optional cut precludes a wealth of others possible,
forcing an agent with limited resources to make an existential choice.
To put it succinctly, the original impulses and the ultimate objects of
human activity are all wrapped up in the informal context, and a formal
domain can maintain its peculiar motive and its particular rationale for
existing only as a parasite on this larger host of instinctive reasons.

In other images, aside from a mirror, a formal domain can be compared
to a circus arena, a theatrical stage, a motion picture, television, or
other sort of projective screen, a congressional forum, indeed, to that
greatest of all three-ring circuses, the government of certain republics
that we all know and love.  If the clonish characters, clownish figures,
and other colonial representatives that carry on in the formal arena did
not mimic in variously diverting and enlightenting ways the concerns of
their spectators in the stands, then there would hardly be much reason
for attending to their antics.  Even when the action in a formal arena
appears to be designed as a contrast, more diverting than enlightening,
or a recreation, more a comic relief from their momentary intensity than
a serious resolution of the troubles that prevail in the ordinary realm,
it still amounts to a strategic way of dealing with a problematic tension
in the informal context.

| Sometimes a troop of damsels glad,
| An abbot on an ambling pad,
| Sometimes a curly shepherd-lad,
| Or long-hair'd page in crimson clad,
|         Goes by to tower'd Camelot;
| And sometimes thro' the mirror blue
| The knights come riding two and two:
| She hath no loyal knight and true,
|         The Lady of Shalott.
|
| Tennyson, "The Lady of Shalott", [Ten, 18]

Before I can continue any further, it is necessary to discuss a question
of terminology that continues to bedevil this discussion with ambiguities:
Is a "context" still a "text", and thus composed of signs throughout, or
is it something else again, an object among objects of another order, or
the incidental setting of an interpreter's referent and significant acts?

The reason I have to raise this question is to make its ambiguities, up
til now remaining implicit, at least more explicit in future encounters.
The reason I cannot settle this question is that the array of its answers
is already too fixed in established usage, and so it seems unavoidable to
rely on intelligent interpreters and context-sensitive interpretation to
pick up the option that makes the most sense in and of a given context.
Keeping this degree of flexibility in mind, that allows one to flip back
and forth between the text and the context, and that leaves one all the
while free to cycle through the objective, syntactic, and interpretive
readings of the word "context", it is now possible to make the following
observations about the relation of the formal to the informal context.

All human interests arise in and return to the informal context, an
openly vague region of indefinite duration and ever-expanding scope.
That is to say, all of the objectives that people instinctively value
and all of the phenomena that people genuinely wish to understand are
things that arise in informal conduct, are carried on in pursuit of it,
develop in connection with it, and ultimately have their bearing on it.
Indeed, the wellsprings that nourish a human interest in abstract forms
are never in danger of escaping the watersheds of the informal sphere,
and they promise by dint of their very nature never to totally inundate
nor to wholly overflow the landscape that renders itself visible there.
This fact is apparent from the circumstance that every formal domain is
originally instituted as a flawed inclusion within the informal context,
continues to develop its constitution as a wholly-dependent subsidiary
of it, and sustains itself as worthy of attention only so long as it
remains a sustaining contributor to it.

| But in her web she still delights
| To weave the mirror's magic sights,
| For often thro' the silent nights
| A funeral, with plumes and lights,
|         And music, went to Camelot:
| Or when the moon was overhead,
| Came two young lovers lately wed;
| "I am half-sick of shadows," said
|         The Lady of Shalott.
|
| Tennyson, "The Lady of Shalott", [Ten, 18]

To describe the question that instigates an inquiry in the language of
the pragmatic theory of signs, the original situation of the inquirer is
constituted by an "elementary sign relation", taking the form <o, s, i>.
In other words, the initial state of an inquiry is constellated by an
ordered triple of the form <o, s, i>, a triadic element that is known in
this case to exist as a member of an otherwise unknown sign relation, if
the truth were told, a sign relation that defines the whole conceivable
world of the interpreter along with the nature of the interpreter itself.
Given that the initial situation of an inquiry has this structure, there
are just three different "directions of recursion" (DOR's) that the agent
of the inquiry can take out of it.

On occasion, it is useful to consider a DOR as outlined by two factors:
(1) There is the "line of recursion" (LOR) that extends more generally
in a couple of directions, conventionally referred to as "up" and "down".
(2) There is the "arrow of recursion" (AOR), a binary feature that is
frequently but quite arbitrarily depicted as "positive" or "negative",
and that picks out one of the two possible directions, "up" or "down",
respectively.  Since one is usually more concerned with the devolution
of a complex power, that is, with the direction of analytic descent, the
downward development, or the reductive progress of the recursion, it is
common practice to point to DOR's and to advert to LOR's in a welter of
loosely ambivalent ways, letting context determine the appropriate sense.

| A bow-shot from her bower-eaves,
| He rode between the barley sheaves,
| The sun came dazzling thro' the leaves,
| And flamed upon the brazen greaves
|         Of bold Sir Lancelot.
| A redcross knight forever kneel'd
| To a lady in his shield,
| That sparkled on the yellow field,
|         Beside remote Shalott.
|
| Tennyson, "The Lady of Shalott", [Ten, 18]

A process of interpretation can appear to be working solely and steadily
on the signs that occupy a formal context -- to emblaze it as an emblem:
on an island, in a mirror, and all through the texture of a tapestry --
at least, it can appear this way to an insufficiently attentive onlooker.
But an agent of interpretation is obliged to keep a private counsel, to
maintain a frame that adumbrates the limits of a personal scope, and so
an interpreter recurs in addition to a boundary on, a connection to, or
an interface with the informal context -- returning to the figure blazed:
every interloper on the scene silently resorts to the facile musings and
the potentially delusive inspirations of looking down the road toward the
secret aims of the finished text:  its ideal reader, its eventual critique,
its imagined interest, its hidden intention, and its ultimate importance.
An interpreter keeps at this work within this confine and keeps at this
station within this horizon only so long as the counsel that is kept in
the depths of the self keeps on appearing as a consistent entity in and
of itself and just so long as it comports with continuing to do so.

A recursive quest can lead in many different directions as it develops.
It can lead agents to resources that they set out without knowing that
they bring to the task, to abilities that they start out unaware even of
having or stay oblivious to ever having, and to skills that they possess,
whether they exercise them or not, but do not really know themselves to
be in possession of, at least at first but perhaps forever, though they
automatically, instinctively, and intuitively employ all the appropriate
aptitudes whenever the occasion calls for them.  This happens especially
when learning is first occurring and agents are developing a particular
type of skill, picking it up almost in passing, in conjunction with the
actions that they are learning to exercise on special types of objects.
In a related pattern of development, a recursive quest can lead agents
to resources that they already think they have in their power but that
they are hard pressed to account for when they ask themselves exactly
how they accomplish the corresponding performances.

A recursion can "lead to" a resource in two senses:  (1) It can have
recourse to a resource as power that is meant to be used in carrying
out another action, and merely in the pursuit of a more remote object,
that is, as an ancillary, assumed, implicit, incidental, instrumental,
mediate, or subservient power.  (2) It can be brought face to face with
the fact or the question of this power, as an entity that is explicitly
mentioned or recognized as a problem, and thus be forced to reflect on
the nature of this putative resource in and of itself.

| The gemmy bridle glitter'd free,
| Like to some branch of stars we see
| Hung in the golden Galaxy.
| The bridle bells rang merrily
|         As he rode down to Camelot:
| And from his blazon'd baldric slung
| A mighty silver bugle hung,
| And as he rode his armor rung,
|         Beside remote Shalott.
|
| Tennyson, "The Lady of Shalott", [Ten, 18]

Any attempt to present the informal context in anything approaching its
full detail is likely to lead to so much conflict and confusion that it
begins to appear more akin to a chaotic context or a formless void than
it chances to resemble a merely casual or a purely incidental environ.
For all intents and purposes, the informal context is a coalescence of
many forces and influences and a loose coalition of disparate ambitions.
These forces impact on the individual thinker in what can appear like a
random fashion, especially at the beginnings of individual development.
Broadly speaking, if one considers the "ways of thinking" (WOT's) that
are made available to a thinker, then these factors can be divvied up
according to their bearing on two wide divisons of their full array:

   1.  There are the WOT's that are prevalent in various communities of
       cultural, literary, practical, scientific, and technical discourse.

   2.  There are the WOT's that are peculiar to the individual thinker.

But this division in abstract terms, claiming to separate WOT's communal
from WOT's personal, does not disentangle the synthetic unities that are
fused and woven together in practice, especially in view of the fact that
collective ways of thinking are actualized only by particular individuals.
Indeed, for each established way of thinking there is a further parting
of the ways, collectively speaking, between the ways that it purports to
conduct itself and the ways that it actually conducts itself in practice.
In order to tell the difference, individual thinkers have to participate
in the corresponding forms of practical conduct.

The informal context enfolds a multitude of formal arenas, to selections
of which the particular interpreters usually prefer to attach themselves.
It transforms a space into a medium of reflection, a respite, a retreat,
or a final resort that affords the agent of interpretation a stance from
which to review the action and to reflect on its many possible meanings.
The informal context is so much broader in scope than the formal arenas
of discourse that are located within it that it does not matter if one
styles it with the definite article "the" or the indefinite article "an",
since no one imagines that a unique definition could ever be implied by
the vagueness of its sweeping intension or imposed on the vastness of
its continuing extension.  It is in the informal context that a problem
arising spontaneously is most likely to meet with its first expression,
and if a writer is looking for a common stock of images and signs that
can permit communication with the randomly encountered reader, then it
is here that the author has the best chance of finding such a resource.

| All in the blue unclouded weather
| Thick-jewell'd shone the saddle-leather,
| The helmet and the helmet-feather
| Burn'd like one burning flame together,
|         As he rode down to Camelot.
| As often thro' the purple night,
| Below the starry clusters bright,
| Some bearded meteor, trailing light,
|         Moves over still Shalott.
|
| Tennyson, "The Lady of Shalott", [Ten, 18]

There is a "form of recursion" (FOR) that is a FOR for itself, that seeks
above all to perpetuate itself, that never quite terminates by design and
never quite reaches its end on purpose, but merely seizes the occasional
diaeresis to pause for a while while a state of dynamic equilibrium or a
moment of dialectical equipoise is achieved between its formal focus and
the informal context.  The FOR for itself recurs not to an absolute state
or a static absolute but to a relationship between the ego and the entire
world, between the fictional character or the hypostatic personality that
is hypothesized to explain the occurrence of specific localized phenomena
and something else again, a whole that is larger, more global, and better
integrated, however elusive and undifferentiated it is in its integrity.
This "inclusive other" can be referred to as "nature", so long as this
nature is understood as a form of being that is not alien to the ego and
not wholly external to the agent, and it can be identified as the "self",
so long as this identity is understood as a relation that is not alone
a property of the ego and not wholly internal to the mind of the agent.

| His broad clear brow in sunlight glow'd;
| On burnish'd hooves his war-horse trode;
| From underneath his helmet flow'd
| His coal-black curls as on he rode,
|         As he rode down to Camelot.
| From the bank and from the river
| He flash'd into the crystal mirror,
| "Tirra lirra," by the river
|         Sang Sir Lancelot.
|
| Tennyson, "The Lady of Shalott", [Ten, 18]

There is a FOR for another whose nature is never to quit in its quest
until its aim is within its clasp, though it knows how much chance there
is for success, and it knows the reason why its reach exceeds its grasp.
This FOR, too, never rests in and of itself, but unlike the FOR for itself
it can be satisfied by achieving a particular alternative state that is
distinct from its initial condition, by reaching another besides itself.
This FOR, too, short of reaching its specific end, never quite terminates
in its own right, not of its essence, nor by its intent, nor does it relent
through any deliberate purpose of its own, but only by accident of an
unforseen circumstance or by dint of an incidental mischance.

It needs to be examined whether this state of dynamic equilibrium,
this condition of balance, equanimity, harmony, and peace can be
described as an aim, an end, a goal, or a good that even the
FOR for itself can take for itself.

| She left the web, she left the loom,
| She made three paces thro' the room,
| She saw the water-lily bloom,
| She saw the helmet and the plume,
|         She look'd down to Camelot.
| Out flew the web and floated wide;
| The mirror crack'd from side to side;
| "The curse is come upon me," cried
|         The Lady of Shalott.
|
| Tennyson, "The Lady of Shalott", [Ten, 18]

In stepping back from a "formally engaged existence" (FEE) to reflect
on the activities that normally take place within its formal arena, in
stepping away from the peculiar concerns that normally take precedence
within its jurisdiction to those that prevail in more ordinary contexts --
and unless one is empowered by some miracle of discursive transport to
jump from one charmed circle of discussion to another without entailing
the usual repercussions:  of causing a considerable loss of continuity,
or of suffering a significant shock of dissociation -- then one commonly
enters on, as an intervening stage of discourse, and passes through, as
a transitional phase of discussion, a context that is convenient to call
a "higher order level of discourse" (HOLOD).  This new level of discussion
allows for a fresh supply of signs and ideas that can serve to reinforce
an agent's inherent but transient capacity for reflection, qualifying an
observant agent as a deliberate interpreter of the events under survey.

Opening up a HOLOD affords an agent an almost blank book, constituted
within the boundless contents of the informal context, for noting what
appears in the formal arena that formally incited its initial formation.
This actuates a barely biased count and a basically broader context for
keeping track of what goes on in a target domain.  In other words that
can be used to hint at its potential, it provides an uncarved block and
an ungraven image, an unsullied field and an untrod plain, an unfilled
frame and an unsigned space, a grander sphere and a greater unity, a
higher and a wider plateau, all in all, just the kind of global staging
ground that is needed for reflection on the initial arena of discourse.
It comes already equipped with a "higher order level of syntax" (HOLOS)
that is needed for referring to the objects and the procedures of many
different formal arenas, at least, it presents a generative promise of
creating enough signs and articulating enough expressions to denote the
more important aspects of the formal businesses that it is responsible
for reflecting on, and it generally has all the other accoutrements that
are appropriate to an expanded context of interpretation or an elevated
level of discourse.

In forming a HOLOD one reaches into the informal context for the images
and the methods to do so.  As long as one is restricted by availability
or habit to dyadic relations one tends to pay attention to either one of
two complementary features of the situation at the expense of the other.
One can attend to either (1) the transitions that occur between entities
at a single level of discourse, or (2) the distinctions that exist between
entities at different levels of discourse.

| In the stormy east-wind straining,
| The pale yellow woods were waning,
| The broad stream in his banks complaining,
| Heavily the low sky raining
|         Over tower'd Camelot;
| Down she came and found a boat
| Beneath a willow left afloat,
| And round about the prow she wrote
|         'The Lady of Shalott'.
|
| Tennyson, "The Lady of Shalott", [Ten, 18]

An "ostensibly recursive text" (ORT) is a text that cites itself by title
at some site within its body.  A "wholly ostensibly recursive literature"
(WORL) is a litany, a liturgy, or any other body of texts that names its
entire collective corpus at some locus of citation within its interior.

I am using the words "cite" and "site" to emphasize the superficially
syntactic character of these definitions, where the title of the text
is conventionally marked by capitals, by italics, by quotation, or by
underscoring.  If the text has a definite subject or an explicit theme,
that is, an object or a state of affairs to which it makes a denotative
reference, then it is not unusual for this reference to be reused as the
title of the text, but this is only the rudimentary beginnings of a true
self-reference in the text.  Although a genuine self-reference can take
its inspiration from a text being named after something that it denotes,
the reference in the text to the text itself becomes complete only when
the name of the subject or the title of the theme is stretched to serve
as the explicit denoter of the entire text.

The sort of ostentation that I make conspicuous in these definitions is
neither necessary nor sufficient for an actual recursion to take place,
since the actuality of the recursive circumstance depends on the action
of the interpreter, one who is always free in principle to ignore or to
subvert the suggestions of the text, who has the power to override the
ostensible instructions that go with the territory of any ORT, and who
is potentially invited to invent whatever innovations of interpretation
are conceivably able to come to mind.

In reading the signs of ostensible recursion that appear within a text of
this sort, an interpreter is empowered, if not always explicitly entitled,
to pick out a personal way of refining their implications from among the
plenitude of possible options:  to gloss them over or to read them anew,
to reform the masses of their solid associations into a manifold body of
interpenetrating interpretations or to refuse the resplendence of their
canonical suggestions in the fires of freshly refulgent convictions and
by dint of the impressions that redound from a host of novel directions,
to regard their indications in the light of wholly familiar conventions
or to regale their invitations in the hopes of a rather more sumptuous
symposium, to reinforce their established denominations with a ruthless
redundancy or to riddle their resorts to the rarefied reaches of rhyme
and reason with repeated petitions for their reconciliation and restless
researches to reconstruct the rationales of their resources until they
are honeycombed with an array of rich connotations, to subtilize or to
subvert, in short, to choose between thoroughly undermining or more
thoroughly understanding the suggestions of its WORL.

| And down the river's dim expanse --
| Like some bold seer in a trance,
| Seeing all his own mischance --
| With a glassy countenance
|         Did she look to Camelot.
| And at the closing of the day
| She loosed the chain, and down she lay;
| The broad stream bore her far away,
|         The Lady of Shalott.
|
| Tennyson, "The Lady of Shalott", [Ten, 18]

Given the benefit of hindsight, or with some measure of due reflection,
it is perhaps fair to say that no one should ever have expected that a
property which is delimited solely on syntactic grounds would turn out
to be anything more than ultimately shallow.  But this recognition only
leaves the true nature of recursion yet to be described.  This is a task
that can be duly inaugurated here but that has to be left unfinished in
its present shape, as it occupies the greater body of the current work.

Unless a text calls for some sort of action on the part of the interpreter
then the appearance of an ostensible recursion or a syntactic repetition
also has little import for action, with the possible exception of making
the reading a bit redundant or imparting a rhyme to its reverberations.
Taken fully in the light that a general freedom of interpretation sheds
on the subject of recursion, a syntactic resonance could just as easily
be read to announce the occasion of a break from an automatic routine,
to afford a rest from rote repetition, rather than heralding the advent
of yet another ritual compulsion to repeat.  This is the form of recall,
the kind of recognition or recollection of the self, that is always patent
amid the potential confusion of the reflected image, that is always open
to the intelligent interpreter.

If one can establish the suggestion that an intelligent interpreter does
not have to follow the suggestions of a text -- establish it in the sense
that most people recognize this principle of freedom in their own action,
however stinting they are in granting it to their fellow interpreters and
however skeptical they remain in extending the scope of its application
to machines -- then one is likely to feel more free to pursue the signs
that a text spells out and to explore the actions that they suggest.

Now there is a form of conduct or a pattern of activity that naturally
accompanies a text, no matter how inert its images may be, and this is
the action of reading.  If the act of reading can be led to induce work
on a larger scale, then reading becomes akin to heeding.  In the medium
of an active interpretation a reading can inspire a form of performance,
and legislative declarations acquire the executive force that is needed
to constitute commands, injunctions, instructions, prescriptions, recipes,
and programs.  Under these conditions an ostensible recursion, the mere
repetition of a sign in a context subordinate to its initial appearance,
as in a title role, can serve to codify a perpetual process, a potential
infinitude of action, all in a finite text, where only the details of a
determinate application and the discretion of an individual interpreter
can bring the perennating roots of life to bear fruit in a finite time.

| Lying, robed in snowy white
| That loosely flew to left and right --
| The leaves upon her falling light --
| Thro' the noises of the night
|         She floated down to Camelot:
| And as the boat-head wound along
| The willowy hills and fields among,
| They heard her singing her last song,
|         The Lady of Shalott.
|
| Tennyson, "The Lady of Shalott", [Ten, 18-19]

It is time to discuss a text of a type that bears a kinship to the ORT,
whose cut as a whole is likened to the reclusive cousins of this caste,
each one lying just within reach of a related ORT but keeping itself a
pace away, staying at a discreet remove, reserving the full implications
of its potential recursion against the day of a suitable interpretation,
and all in all residing in similar manors of meaning to the ORT, though
not so ostentatiously.  Even if the manifold ways of reading the senses
of such a text are not as conspicuous as those of an ORT, and if it is a
fair complaint to say that the deliberate design that keeps it from being
obvious can also keep it from ever becoming clear, there is in principle
a key to unlocking its meaning, and the ulterior purpose of the text is
simply to pass on this key.

For the lack of a better name, let the type of text that devolves in
evidence here be called a "pseud-ORT" (PORT) or a "quasi-ORT" (QORT).
These acronyms inherit the hedge word "ostensibly" from the ORT's that
their individual namesakes beget, once they are interpreted as doing so.
It is the main qualification of the indicated PORT's or QORT's, and the
one that continues to be borne by them as the sole inherent property of
their bearing.  As before, this qualification is intended to serve as a
caution to the reader that the properties ordinarily imputed to the text
do not actually belong to the matter of the text, but that they properly
belong to the agent and the process of the active interpretation, namely,
the one that is actually carried out on the material supplied by the text.
The adjoined pair of weasel words "pseudo" and "quasi" are intended to
remind the reader that a PORT or a QORT falls short of even the order
of specious recursion that is afforded by an ORT, but has to be nudged
in the general direction of this development or this evolution through
the intercession of artificial distortions or specialized modulations of
the semantics that is applied to the text.  Whether these extra epithets
exacerbate the spurious character of the putative recursion or whether
they take the edge off the order of ostentation that already occurs in
an ORT is a question that can be deferred to a future time.

| Heard a carol, mournful, holy,
| Chanted loudly, chanted lowly,
| Till her blood was frozen slowly,
| And her eyes were darken'd wholly,
|         Turn'd to tower'd Camelot;
| For ere she reach'd upon the tide
| The first house by the water-side,
| Singing in her song she died,
|         The Lady of Shalott.
|
| Tennyson, "The Lady of Shalott", [Ten, 19]

If its ways are kept in the way intended, lacking only a fitting key to
be unlocked, then the PORT or the QORT in question leads an interloper
into a recursion only whenever the significance of certain analogies,
comparisons, metaphors, or similes is recognized by that interpreter.
Generally speaking, this happens only when the interpreter discovers
that a set of "semiotic equations" (SEQ's), applying to signs that can
be picked out from the text in specific senses, is conceivably in force.
Expressed another way, the recursive or self-referent interpretation is
actualized when the interpreter hypothesizes that the text in question
bears up under a certain kind of additional intention, namely, that a
system of "qualified identifications" (QUI's) ought to be applied to
selected signs in the text.

These analogies and equations have the effect of creating novel forms of
"semiotic equivalence relations" (SER's) that overlay the ostensible text.
These relations generate further layers of "semiotic partitions" (SEP's),
or families of "semiotic equivalence classes" (SEC's), that are typically
restricted in their application to a specially selected sample of symbols
in the text.  Since these classes are generally of an abstract sort and
frequently of a recondite kind, and since they are usually intended for
the purposes of a specialized interpretation, their collective import on
the sense of a text is conveniently summarized under the designation of
an "abstract", "abstruse", "arcane", or "analogical recursion key" (ARK).

By way of summary, a PORT or a QORT is a type of text that approaches
a definite ORT subject to the recognition of an ARK, and thus affords
the opportunity of leading its reader to a recursive interpretation.

The writer borrows a vehicle from the informal context, adapts its forms
to the current conditions, adopts the guises appurtenant to it, and aims
to appropriate to a private advantage what appears as if it is asking to
assist or is long ago abandoned along a public way.  The writer instills
this open form with a living significance, invests it with a new lease of
meaning, inscribes it perhaps with a personal title or a suitable envoi,
and sends it on its way, through whatever medium avails itself and to
whatever party awaits it, without knowing how the sense of the message
is destined to be appreciated when life in the ordinary sense is passed
from its limbs and long after the flashes of its creation are frozen in
the shapes of its reception.  All in all, the writer has no choice but to
assume the good graces of eventually finding a charitable interpretation.

| Under tower and balcony,
| By garden-wall and gallery,
| A gleaming shape she floated by,
| A corse between the houses high,
|         Silent into Camelot.
| Out upon the wharfs they came,
| Knight and burgher, lord and dame,
| And round the prow they read her name,
|         'The Lady of Shalott'.
|
| Tennyson, "The Lady of Shalott", [Ten, 19]

I assume that the reader has gleaned the existence of something beyond
a purely accidental relation that runs between the text and the epitext,
between the prose discussion and the succession of epigraphs, that are
interwoven with each other throughout the course of this presentation.
It is generally best to let these incidental counterpoints develop in a
loosely parallel but rough independence from each other, and to let them
run through their corresponding paces not too strenuously interlocked.
The rule is thus to lay out the principal lines of their generic motives,
their arguments, plans, plots, and themes, without incurring the fear of
inadvertent intersections looming near, and thus to string the beads of
their selective articulations along the strands of their envisioned text
without invoking the undue force of a final collusion among their mass.
In spite of all that, I take the chance of bringing the various threads
together at this point, in order to sound out their accords and discords,
and to make a bolder exegesis of the relationships that they display.

Tennyson's poem "The Lady of Shalott" is akin to an ORT, but a bit more
remote, since the name styled as "The Lady of Shalott", that the author
invokes over the course of the text, is not at first sight the title of
a poem, but a title that its character adopts and afterwards adapts as
the name of a boat.  It is only on a deeper reading that this text can
be related to or transformed into a proper ORT.  Operating on a general
principle of interpretation, the reader is entitled to suspect that the
author is trying to say something about himself, his life, and his work,
and that he is likely to be exploiting for this purpose the figure of his
ostensible character and the vehicle of his manifest text.  If this is an
aspect of the author's intention, whether conscious or unconscious, then
the reader has a right to expect that several forms of analogy are key
to understanding the full intention of the text.

Given the complexity and the subtlety of the epitext in this Subsection,
it makes sense to begin the detailed analysis of ORT's and their ilk with
a much simpler example, and one that exemplifies a straightforward ORT.
These preparations are undertaken at the beginning of the next Section,
after which it is feasible to return to the present example, to consider
the formal analysis of PORT's and QORT's, to explain how the effects of
meaning that are achieved in this general type of text are supported by
its sign-theoretic structure, and to discuss how these semantic intents
are facilitated by the infrastructure of the language that is employed.

| Who is this?  and what is here?
| And in the lighted palace near
| Died the sound of royal cheer;
| And they cross'd themselves for fear,
|         All the knights at Camelot:
| But Lancelot mused a little space;
| He said, "She has a lovely face;
| God in his mercy lend her grace,
|         The Lady of Shalott."
|
| Tennyson, "The Lady of Shalott", [Ten, 19]

As it happens, many a text in literature or science that concerns itself
with hypothetical creatures, mythical entities, or speculative figures,
that contents itself with idealized models of actual situations, indulges
itself with idle idylls that barely allude to the serious threats against
human peace and social well-being that they betray, or satisfies itself
with romantic images of real enough but unknown perils of the soul --
none of these would hold the level of interest that it actually has if it
did not make itself available to many different levels of interpretation,
readings that go far beyond the levels of discourse where it ostensibly
presents itself at first sight.

Although it is easy to pick out examples of sign relations that are
already completely formalized, and thus to study them as combinatorial
objects of a more or less independent interest, this tactic makes it all
the more difficult to see what ties these impoverished examples to the
kinds of sign relations that freely develop in the unformed environment
and that inform all the more natural problems that one might encounter.
Thus, in this Section I make an effort to catch the formalization process
in its very first steps, as it begins to dehisce the very seeds of its
future development from the security of their enveloping integuments.

The form of initiatory task that a certain turn of mind arrives at only
toward the end of its quest is not so much to describe the tensions that
exist among contexts -- those between the formal arenas, bowers, courts
and the informal context that surrounds them all -- as it is to exhibit
these forces in action and to bear up under their influences on inquiry.
The task is not so much to talk about the informal context, to the point
of trying to exhaust it with words, as it is to anchor one's activity in
the infinitudes of its unclaimed resources, to the depth that it allows
this importunity, and to buoy the significant points of one's discussion,
its channels, shallows, shoals, and shores, for the time that the tide
permits this opportunity.

1.3.9.2. The Epitext

It is time to render more explicit a feature of the text in the previous
Subsection, to abstract the form that it realizes from the materials that
it appropriates to fill out its pattern, to extract the generic structure
of its devices as a style of presentation or a standard technique, and to
make this formal resource available for use as future occasions warrant.
To this end, let a succession of epigraphs, incidental to a main text but
having a consistent purpose all their own, and illustrating the points of
the main text in an exemplary, poignant, or succinct way, be referred to
as an "epitext".

What is the point of this poem, or what kind of example do I make of it?
It seems designed to touch on a point that is very near the heart of the
inquiry into inquiry:  This is the question of self-referential integrity,
indeed, the very possibility of referential self-consistency.  The point
is whether a writer can produce a text that says something significant
about the process that produces it.  What "significant" means is open
for discussion.  Its scope is usually taken to encompass the general
properties and the generic powers of the process in question.  And
from there the inquiry, if its double focus allows the drawing of
a hasty inference, is thrown back into its elliptical orbit.

It is not for long that the agent of inquiry remains in the possession
of the inquiry itself, since the very purpose of inquiry is to escape
from the throes of the uncertainty that threw it into action.  And the
writer does not expect to find a reader in the transits of the very same
flux.  So when the inquiry is done, all that one has to remember it by,
and all that another has to reconstruct it from, is the text of inquiry
that came to be produced in the process -- as afterthought, by-product,
end, or unintended consequence.  The text is only an imago, an inactive
image of a living process that does not live wholly in any of its works.
The text is only a parable, a likely story about an action that ended,
for all intents and purposes, a long time before or a short while ago.
And the text is particular, finite, and discrete.  So the problem is not
insignificant, for the text of inquiry to say something of consequence,
not just about its own small self, but about the process of inquiry
that is capable of generating a modest array of texts of its kind.

Nothing says that a text has to be constituted solely at a single level of
discourse, that signs of novel, mysterious, and wholly altered characters
have to be adduced in order to give it multiple levels of interpretation,
or that an interpretive agent has to remain forever chained in the first
tower of syntax that is needed to establish a provisional point of view.
This signifies something weirder than the simple circumstance that texts
intended at different levels of discourse can be laced, mixed, spliced,
and woven together in an indiscriminate style.  It means that each piece
of text and each bit of subtext, in short, each sign that participates in
the whole of a text, is potentially subject to multiple interpretations,
coherent or not with the modes of interpretation that are applied to the
contexts surrounding the sign.

Of course, there are challenges to be faced in leaving a single-minded
perspective, as there are troubles that arise in first rising above the
flat lack of any perspective at all.  If the perversity of polymorphism,
that permits terms to be interpreted under many types, and the curse of
recursion, that permits texts to have recourse to signifying themselves,
could in fact be avoided in practice, then perhaps it would be better to
disallow their mention and their use altogether.  Alas, the complexities
are not so quickly dismissed, not if computers are meant to help people
make use of their formal calculi and to ply their symbolic languages
in all of the ways that people are actually accustomed to use them.

There is an order of interaction that occurs between the issues of
polymorphism and recursion that needs to be noted at this juncture.
It is not always the text that hits its interpreter over the head
with the glaring conceits of its subject and the obvious vanities
of its self-reference that contains the subtlest form of recursion.
As long as its signs are susceptible to allegorical and metaphorical
interpretations then it is always possible that some of the readings
of a text can refer to the process of writing itself, to the nature
of the relationship that is craft or draft from the writer to the
reader, and to all of the adventitious uncertainties that affect
any attempt at achieving a measure of understanding between them.

In order for a text to refer to itself it need not take on any name for
itself nor call itself by any given title.  In order for a text to make
reference to the interpreter who writes it, the interpreter who reads it,
the means, the ends, or any other medium or party to its interpretation,
it need not characterize any of these roles, scenes, or stages in a literal
fashion within the measure of its lines, nor refer to any portion of their
number under the assumptions of aliases, disguises, secret identities, or
cryptic titles, whether put off or put on.  Indeed, all of the signs that
are chained together within the body of the text -— the kind of a body,
by the way, that appears to be able to absorb all of the signs that are
applied to it -— are constrained by the very nature of signs.  They can
do little more than ease the way toward a potential meaning, facilitate
a desired understanding, or hint at a given interpretation among their
manifold conceivable senses.

There is no property of the text itself that is capable of constraining
the freedom of interpretation.  There is nothing at all that constrains
the freedom of interpretation, nothing but the nature of the interpreter.
Of course, I am referring to absolutes here, and disclaiming the force of
absolute constraints.  If it is in the nature of a particular interpreter,
as all of the sensible ones are, to let the interpretation be constrained,
moderately and relatively speaking, by the character of the signs within
a well delimited text, then so be it.  I am merely pointing out that the
degrees of potential freedom are usually much greater than one is likely
initially to think.

When it comes to recursion the freedom of interpretation is a two-edged
sword, or perhaps a two-headed axe.  It allows an interpreter to ignore
the signs of ostensible recursion, and thus to escape the confines of a
labyrinth whose blueprint develops from a compulsion to repeat.  But it
also makes it possible to see reflections of the self where none appear
to be obvious, and thus to encounter a host of recursions where none is
dictated by the text.

It is useful to sum up in the following way the nature of the potentially
explosive interaction that falls out between polymorphism and recursion:

   In order for writers by means of their texts to refer to themselves, and
   in order for readers in terms of these texts to recognize themselves, it
   need only occur to an interpreter that a self-referent interpretation is
   conceivable, whether or not this is the obvious, original, or ostensible
   interpretation of the text.

It is due to this "freedom of interpretation" (FOI), that individualizes
itself in identification with a particular "form of interpretation" (FOI),
that every "liberty of interpretation" (LOI) is practically equivalent to
its very own "law of interpretation" (LOI).  In the end, it is the middle
terms here, form and liberty, that give the only grounds for making sense.
When all is said and done, it is the middle grounds that leave the only
room for practical action, since absolute freedom and absolute law are
indiscernible from the absolute constraint of absolute chaos.

Let me emphasize what this means by developing its implications for the
use of certain phrases in common use and by detecting the bearing that
it has on reforming the fashions of their understanding.  References to
"reflexive signs" and "recursive texts" are misnomers, useful as a way
of pointing out obvious forms of potential self-reference, but neither
sufficient nor necessary to determine whether a self-reference of signs
or their users actually occurs.  Like other properties that one is often
tempted to make the mistake of attributing to signs in fashions that are
absolutely exclusive rather than relatively independent of their users,
reflexivity and recursivity are not properly properties that these signs
possess all by themselves but features that they manifest in particular
exercises of their active senses and their live interpretations.  To the
extent that the course of interpretation and the direction of reference
are under the control of a particular interpreter, the words "recursive",
"reflexive", and "self-referent" do not describe any properties that are
essential to signs or texts, codes or programs, but refer to the manner
of their regard, in other words, to a feature of their interpreter.

This says that a recursive interpretation of a sign or a text can recur
just so long as its interpreter has an interest in pursuing it.  It can
terminate, not just with the absolute extremes of an ideal object or an
objective limit, that is, with states of perfect certainty or tokens of
ultimate clarity, but also in the interpretive direction, that is, with
forms of self-recognition and a conduct that arises from self-knowledge.
In the meantime, between these points of final termination, a recursive
interpretation can also pause on a temporary basis at any time that the
degree of involvement of the interpreter is pushed beyond the limits of
moderation, or any time that the level of interest for the interpreter
drifts beyond or is driven outside the band of personal toleration.

1.3.9.3. The Formative Tension

The incidental arena or the informal context is presently described in casual, derivative, and negative terms, simply as the "not yet formal", and so this admittedly unruly region is currently depicted in ways that suggest a purely unformed and a wholly formless chaos, which it is not. But increasing experience with the formalization process can help one to develop a better appreciation of the informal context, and in time one can argue for a more positive characterization of this realm as a truly "formative context". The formal domain is where risks are contemplated, but the formative context is where risks are taken.

In this view, the informal context is more clearly seen as the off-stage staging ground where everything that appears on the formal scene is first assembled for a formal presentation. In taking this view, one steps back a bit in one's imagination from the scene that presses on one's attention, gets a sense of its frame and its stage, and becomes accustomed to see what appears in ever dimmer lights, in effect, one is learning to reflect on the more obvious actions, to read their pretexts, and to detect the motives that end in them.

It is fair to assume that an agent of inquiry possesses a faculty of inquiry that is available for exercise in the informal context, that is, without the agent being required to formalize its properties prior to their initial use. If this faculty of inquiry is a unity, then it appears as a whole on both sides of the "glass", that is, on both sides of the imaginary line that one pretends to draw between a formal arena and its informal context.

Recognizing the positive value of an informal context as an open forum or a formative space, it is possible to form the alignments of capacities that are indicated in Table 5.

Table 5.  Alignments of Capacities
o-------------------o-----------------------------o
|      Formal       |          Formative          |
o-------------------o-----------------------------o
|     Objective     |        Instrumental         |
|      Passive      |           Active            |
o-------------------o--------------o--------------o
|     Afforded      |  Possessed   |  Exercised   |
o-------------------o--------------o--------------o

This arrangement of capacities, based on the distinction between possession and exercise that arises so naturally in this context, stems from a root that is old indeed. In this connection, it is instructive to compare these alignments with those that we find in Aristotle's treatise On the Soul, a germinal textbook of psychology that ventures to analyze the concept of the mind, psyche, or soul to the point of arriving at a definition. The alignments of capacites, analogous correspondences, and illustrative materials outlined by Aristotle are summarized in Table 6.

Table 6.  Alignments of Capacities in Aristotle
o-------------------o-----------------------------o
|      Matter       |            Form             |
o-------------------o-----------------------------o
|   Potentiality    |          Actuality          |
|    Receptivity    |  Possession  |   Exercise   |
|       Life        |    Sleep     |    Waking    |
|        Wax        |         Impression          |
|        Axe        |    Edge      |   Cutting    |
|        Eye        |   Vision     |    Seeing    |
|       Body        |            Soul             |
o-------------------o-----------------------------o
|       Ship?       |           Sailor?           |
o-------------------o-----------------------------o

An attempt to synthesize the materials and the schemes that are given in Tables 5 and 6 leads to the alignments of capacities that are shown in Table 7. I do not pretend that the resulting alignments are perfect, since there is clearly some sort of twist taking place between the top and the bottom of this synthetic arrangement. Perhaps this is due to the modifications of case, tense, and grammatical category that occur throughout the paradigm, or perhaps it has to do with the fact that the relations through the middle of the Table are more analogical than categorical. For the moment I am content to leave all of these paradoxes intact, taking the pattern of tensions and torsions as a puzzle for future study.

Table 7.  Synthesis of Alignments
o-------------------o-----------------------------o
|      Formal       |          Formative          |
o-------------------o-----------------------------o
|     Objective     |        Instrumental         |
|      Passive      |           Active            |
|     Afforded      |  Possessed   |  Exercised   |
|      To Hold      |   To Have    |    To Use    |
|    Receptivity    |  Possession  |   Exercise   |
|   Potentiality    |          Actuality          |
|      Matter       |            Form             |
o-------------------o-----------------------------o

Due to the importance of Aristotle's account for every discussion that follows it, not to mention for those that follow it without knowing it, and because the issues that it raises arise repeatedly throughout this project, I am going to cite an extended extract from the relevant text (Aristotle, Peri Psyche, 2.1), breaking up the argument into a number of individual premisses, stages, and examples.

Aristotle wrote (W.S. Hett translation):

| a.  The theories of the soul (psyche)
|     handed down by our predecessors have
|     been sufficiently discussed;  now let
|     us start afresh, as it were, and try to
|     determine (diorisai) what the soul is,
|     and what definition (logos) of it will
|     be most comprehensive (koinotatos).
|
| b.  We describe one class of existing things as
|     substance (ousia), and this we subdivide into
|     three:  (1) matter (hyle), which in itself is
|     not an individual thing, (2) shape (morphe) or
|     form (eidos), in virtue of which individuality
|     is directly attributed, and (3) the compound
|     of the two.
| 
| c.  Matter is potentiality (dynamis), while form is
|     realization or actuality (entelecheia), and the
|     word actuality is used in two senses, illustrated
|     by the possession of knowledge (episteme) and the
|     exercise of it (theorein).
|
| d.  Bodies (somata) seem to be pre-eminently
|     substances, and most particularly those
|     which are of natural origin (physica),
|     for these are the sources (archai)
|     from which the rest are derived.
|
| e.  But of natural bodies some have life (zoe)
|     and some have not;  by life we mean the
|     capacity for self-sustenance, growth,
|     and decay.
|
| f.  Every natural body (soma physikon), then,
|     which possesses life must be substance, and
|     substance of the compound type (synthete).
|
| g.  But since it is a body of a definite kind, viz.,
|     having life, the body (soma) cannot be soul (psyche),
|     for the body is not something predicated of a subject,
|     but rather is itself to be regarded as a subject,
|     i.e., as matter.
|
| h.  So the soul must be substance in the sense of being
|     the form of a natural body, which potentially has life.
|     And substance in this sense is actuality.
|
| i.  The soul, then, is the actuality of the kind of body we
|     have described.  But actuality has two senses, analogous
|     to the possession of knowledge and the exercise of it.
|
| j.  Clearly (phaneron), actuality in our present sense
|     is analogous to the possession of knowledge;  for both
|     sleep (hypnos) and waking (egregorsis) depend upon the
|     presence of the soul, and waking is analogous to the
|     exercise of knowledge, sleep to its possession (echein)
|     but not its exercise (energein).
|
| k.  Now in one and the same person the
|     possession of knowledge comes first.
|
| l.  The soul may therefore be defined as the first actuality
|     of a natural body potentially possessing life;  and such
|     will be any body which possesses organs (organikon).
|
| m.  The parts of plants are organs too, though very
|     simple ones:  e.g., the leaf protects the pericarp,
|     and the pericarp protects the seed;  the roots are
|     analogous to the mouth, for both these absorb food.
|
| n.  If then one is to find a definition which will apply
|     to every soul, it will be "the first actuality of
|     a natural body possessed of organs".
|
| o.  So one need no more ask (zetein) whether body and
|     soul are one than whether the wax (keros) and the
|     impression (schema) it receives are one, or in
|     general whether the matter of each thing is
|     the same as that of which it is the matter;
|     for admitting that the terms unity and being
|     are used in many senses, the paramount (kyrios)
|     sense is that of actuality.
|
| p.  We have, then, given a general definition
|     of what the soul is:  it is substance in
|     the sense of formula (logos), i.e., the
|     essence of such-and-such a body.
|
| q.  Suppose that an implement (organon), e.g. an axe,
|     were a natural body;  the substance of the axe
|     would be that which makes it an axe, and this
|     would be its soul;  suppose this removed, and
|     it would no longer be an axe, except equivocally.
|     As it is, it remains an axe, because it is not of
|     this kind of body that the soul is the essence or
|     formula, but only of a certain kind of natural body
|     which has in itself a principle of movement and rest.
|
| r.  We must, however, investigate our definition
|     in relation to the parts of the body.
|
| s.  If the eye were a living creature, its soul would be
|     its vision;  for this is the substance in the sense
|     of formula of the eye.  But the eye is the matter
|     of vision, and if vision fails there is no eye,
|     except in an equivocal sense, as for instance
|     a stone or painted eye.
|
| t.  Now we must apply what we have found true of the part
|     to the whole living body.  For the same relation must
|     hold good of the whole of sensation to the whole sentient
|     body qua sentient as obtains between their respective parts.
|
| u.  That which has the capacity to live is not the body
|     which has lost its soul, but that which possesses
|     its soul;  so seed and fruit are potentially bodies
|     of this kind.
|
| v.  The waking state is actuality in the same sense
|     as the cutting of the axe or the seeing of the eye,
|     while the soul is actuality in the same sense as the
|     faculty of the eye for seeing, or of the implement for
|     doing its work.
|
| w.  The body is that which exists potentially;  but just as
|     the pupil and the faculty of seeing make an eye, so in
|     the other case the soul and body make a living creature.
|
| x.  It is quite clear, then, that neither the soul nor
|     certain parts of it, if it has parts, can be separated
|     from the body;  for in some cases the actuality belongs
|     to the parts themselves.  Not but what there is nothing
|     to prevent some parts being separated, because they are
|     not actualities of any body.
|
| y.  It is also uncertain (adelon) whether the soul as an
|     actuality bears the same relation to the body as the
|     sailor (ploter) to the ship (ploion).
|
| z.  This must suffice as an attempt to determine
|     in rough outline the nature of the soul.

1.3.10. Recurring Themes

The overall purpose of the next several Sections is threefold:

   1.  To continue to illustrate the salient properties of
       sign relations in the medium of selected examples.

   2.  To demonstrate the use of sign relations to analyze and to clarify
       a particular order of difficult symbols and complex texts, namely,
       those that involve recursive, reflective, or reflexive features.

   3.  To begin to suggest the implausibility of understanding this order
       of phenomena without using sign relations or something like them,
       namely, concepts with the power of triadic relations.

The prospective lines of an inquiry into inquiry cannot help but meet at
various points, where a certain entanglement of the subjects of interest
repeatedly has to be faced.  The present discussion of sign relations is
currently approaching one of these points.  As the work progresses, the
formal tools of logic and set theory become more and more indispensable
to say anything significant or to produce any meaningful results in the
study of sign relations.  And yet it appears, at least from the vantage
of the pragmatic perspective, that the best way to formalize, to justify,
and to sharpen the use of these tools is by means of the sign relations
that they involve.  And so the investigation shuffles forward on two or
more feet, shifting from a stance that fixes on a certain level of logic
and set theory, using it to advance the understanding of sign relations,
and then exploits the leverage of this new pivot to consider variations,
and hopefully improvements, in the very language of concepts and terms
that one uses to express questions about logic and sets, in all of its
aspects, from syntax, to semantics, to the pragmatics of both human and
computational interpreters.

The main goals of this Section (1.3.10) are as follows:

   1.  To introduce a basic logical notation and a naive theory of sets,
       just enough to treat sign relations as the set-theoretic extensions
       of logically expressible concepts.

   2.  To use this modicum of formalism to define a number of conceptual
       constructs, useful in the analysis of more general sign relations.

   3.  To develop a proof format that is suitable for deriving facts about
       these constructs in a careful and potentially computational manner.

   4.  More incidentally, but increasingly effectively, to get a sense of
       how sign relations can be used to clarify the very languages that
       are used to talk about them.

1.3.10. Recurring Themes (CFR Version)

The overall purpose of the next sixteen Subsections is threefold:

1.  To continue to illustrate the salient properties of sign relations
    in the medium of selected examples.

2.  To demonstrate the use of sign relations to analyze and to clarify
    a particular order of difficult symbols and complex texts, namely,
    those that involve recursive, reflective, or reflexive features.

3.  To begin to suggest the implausibility of understanding this order
    of phenomena without using sign relations or something like them,
    namely, concepts with the power of 3-adic relations.

The prospective lines of an inquiry into inquiry cannot help but meet at
various points, where a certain entanglement of the subjects of interest
repeatedly has to be faced.  The present discussion of sign relations is
currently approaching one of these points.  As the work progresses, the
formal tools of logic and set theory become more and more indispensable
to say anything significant or to produce any meaningful results in the
study of sign relations.  And yet it appears, at least from the vantage
of the pragmatic perspective, that the best way to formalize, to justify,
and to sharpen the use of these tools is by means of the sign relations
that they involve.  And so the investigation shuffles forward on two or
more feet, shifting from a stance that fixes on a certain level of logic
and set theory, using it to advance the understanding of sign relations,
and then exploits the leverage of this new pivot to consider variations,
and hopefully improvements, in the very language of concepts and terms
that one uses to express questions about logic and sets, in all of its
aspects, from syntax, to semantics, to the pragmatics of both human and
computational interpreters.

The main goals of this Section are as follows:

1.  To introduce a basic logical notation and a naive theory of sets,
    just enough to treat sign relations as the set-theoretic extensions
    of logically expressible concepts.

2.  To use this modicum of formalism to define a number of conceptual
    constructs, useful in the analysis of more general sign relations.

3.  To develop a proof format that is amenable to deriving facts about
    these constructs in careful and potentially computational fashions.

4.  More incidentally, but increasingly effectively, to get a sense
    of how sign relations can be used to clarify the very languages
    that are used to talk about them.

1.3.10.1. Preliminary Notions

The discussion in this Section (1.3.10) proceeds by recalling a series of
basic definitions, refining them to deal with more specialized situations,
and refitting them as necessary to cover larger families of sign relations.

In this discussion the word "semantic" is being used as a generic
adjective to describe anything concerned with or related to meaning,
whether denotative, connotative, or pragmatic, and without regard to
how these different aspects of meaning are correlated with each other.
The word "semiotic" is being used, more specifically, to indicate the
connotative relationships that exist between signs, in particular, to
stress the aspects of process and of potential for progress that are
involved in the transitions between signs and their interpretants.
Whenever the focus fails to be clear from the context of discussion,
the modifiers "denotative" and "referential" are available to pinpoint
the relationships that exist between signs and their objects.  Finally,
there is a common usage of the term "pragmatic" to highlight aspects of
meaning that have to do with the context of use and the language user,
but I reserve the use of this term to refer to the interpreter as an
agent with a purpose, and thus to imply that all three aspects of
sign relations are involved in the subject under discussion.

Recall the definitions of "semiotic equivalence classes" (SEC's),
"semiotic partitions" (SEP's), "semiotic equations" (SEQ's), and
"semiotic equivalence relations" (SER's) from Subsection 1.3.4.3.

The discussion up to this point is partial to examples of sign relations
that enjoy especially nice properties, in particular, whose connotative
components form equivalence relations and whose denotative components
conform to these equivalences, in the sense that all of the signs in
a single equivalence class always denote one and the same object.
By way of liberalizing the discussion to more general cases of
sign relations, this Section develops a number of additional
concepts for describing the internal relationships within
sign relations, and it makes a set of definitions that
do not take the aforementioned features for granted.

The complete sign relation involved in a situation encompasses
all of the things that one thinks about and all of the thoughts
that one thinks about them while engaged in that situation, in
other words, all of the signs and ideas that flit through one's
mind in relation to a given domain of objects.  Only a rarefied
sample of this complete sign relation is bound to avail itself
to reflective awareness, still less of it is likely to inspire
a common interest in the community of inquiry at large, and
only bits and pieces of it can be expected to suit themselves
to a formal analysis.  In view of these considerations, it is
useful to have a general idea of the "sampling relation" that
an investigator, oneself in particular, is likely to form
between two sign relations:  (1) the whole sign relation
that one intends to study, and (2) the selective portion
of it that one is able to pin down for formal examination.

It is important to realize that a "sampling relation", to express it
roughly, is a special case of a sign relation.  Aside from acting on
sign relations and creating an association between sign relations, a
sampling relation is also involved in a larger sign relation, at least,
it can be subsumed within a general order of sign relations that allows
sign relations themselves to be taken as the objects, the signs, and the
interpretants of what can be called a "higher order" (HO) sign relation.
Considered with respect to its full potential, its use, and its purpose,
a sampling relation does not fall outside the closure of sign relations.
To be precise, a sampling relation falls within the denotative component
of a higher order sign relation, since the sign relation sampled is the
object of study and the sample is taken as a sign of it.

Out of the general variety of sampling relations there are a number
of specific conceptions that are likely to be useful in this study,
a few of which can now be discussed.

For the sake of this discussion, a "fragment" of a sign relation
is defined to be any subset of its extension, in other words,
an arbitrary selection from the set of its ordered triples.

Considered in relation to sampling relations, a fragment of a
sign relation is just the most arbitrary possible sample of it,
and thus its occurring in discussion or thought implies the most
general form of sampling relation to be in effect.  In essence,
it is just as if a fragment of a sign relation, by dint of its
appearing in evidence, can always be interpreted as a piece of
evidence that some sort of sampling relation is being applied.

1.3.10.1. Preliminary Notions (CFR Version)

The discussion in this Subsection proceeds by recalling a series of basic
definitions, refining them to deal with more specialized situations, and
refitting them as necessary to cover larger families of sign relations.

In this discussion the word "semantic" is being used as a generic
adjective to describe anything concerned with or related to meaning,
whether denotative, connotative, or pragmatic, and without regard to
how these different aspects of meaning are correlated with each other.
The word "semiotic" is being used, more specifically, to indicate the
connotative relationships that exist between signs, in particular, to
stress the aspects of process and of potential for progress that are
involved in the transitions between signs and their interpretants.
Whenever the focus fails to be clear from the context of discussion,
the modifiers "denotative" and "referential" are available to pinpoint
the relationships that exist between signs and their objects.  Finally,
there is a common usage of the term "pragmatic" to highlight aspects of
meaning that have to do with the context of use and the language user,
but I reserve the use of this term to refer to the interpreter as an
agent with a purpose, and thus to imply that all three aspects of
sign relations are involved in the subject under discussion.

Recall the definitions of "semiotic equivalence classes" (SEC's),
"semiotic partitions" (SEP's), "semiotic equations" (SEQ's), and
"semiotic equivalence relations" (SER's) from Subsection 1.3.4.3.

The discussion of sign relations up to this point has been centered around
and remained partial to examples of sign relations that enjoy especially
nice properties, in particular, its focus has been on sign relations
whose connotative components form equivalence relations and whose
denotative components conform to these equivalences, in the sense
that all of the signs in each semiotic equivalence class always
denote one and the same object.  By way of liberalizing the
discussion to more general cases of sign relations, this
Subsection develops a number of additional concepts for
describing the internal structures of sign relations
and it lays out a set of definitions that do not
take the aforementioned features for granted.

The complete sign relation involved in a given situation encompasses
all of the things that one thinks about and all of the thoughts that
one thinks about them while engaged in that particular situation, in
other words, all of the signs and ideas that flit through one's mind
in relation to a given domain of objects.  Only a rarefied sample of
this rarely completed sign relation is bound or even likely to avail
itself to any reflective awareness, still less of it has much chance
to inspire a concerted interest in the community of inquiry at large,
and only bits and pieces of it can be expected to suit themselves to
a formal analysis.  In view of these considerations, it is useful to
have a general idea of the "sampling relation" that any investigator,
oneself in particular, is likely to forge between two sign relations:

1.  The whole sign relation that one intends to study.

2.  The selective portion of it that one is able to
    pin down for the sake of a formal investigation.

It is important to realize that a "sampling relation", to express it
roughly, is a special case of a sign relation.  Aside from acting on
sign relations and creating an association between sign relations, a
sampling relation is also involved in a larger sign relation, at least,
it can be subsumed within a general order of sign relations that allows
sign relations themselves to be taken as the objects, the signs, and the
interpretants of what can be called a "higher order" (HO) sign relation.
Considered with respect to its full potential, its use, and its purpose,
a sampling relation does not fall outside the closure of sign relations.
To be precise, a sampling relation falls within the denotative component
of a HO sign relation, since the sign relation sampled is the object of
study and the sample is taken as a sign of it.

Out of the general variety of sampling relations one can pick out
a number of specific conceptions that are likely to be useful in
our study, a few of which can now be discussed.  I close out the
current Subsection with a concept of very general application in
the world of sign relations, and dedicate the next Subsection to
a collection of more specialized concepts.

A "piece" of a sign relation is defined to be any subset of its extension,
that is, a wholly arbitrary selection from the set of its ordered 3-tuples.

Described in relation to sampling relations, a piece of a sign relation
is just the most arbitrary possible sample of it, and thus its occurring
to mind implies the most general form of sampling relation to be in effect.
In essence, it is just as if a piece of a sign relation, by virtue of its
appearing in evidence, can always be interpreted as a piece of evidence
that some sort of sampling relation is being applied.

1.3.10.2. Intermediary Notions

A number of additional definitions are relevant to sign relations whose
connotative components constitute equivalence relations, if only in part.

A "dyadic relation on a single set" (DROSS) is a non-empty set of points
plus a set of ordered pairs on these points.  Until further notice, any
reference to a "dyadic relation" is intended to be taken in this sense,
in other words, as a reference to a DROSS.

When the maximum precision of notation is needed, a dyadic relation !G!
will be given in the form !G! = <G(1), G(2)>, where G(1) is a non-empty
set of points and G(2) c G(1) x G(1) is a set of ordered pairs from G(1).

At other times, a dyadic relation may be specified in the form <X, G>, where
X is the set of points and where G c X x X is the set of ordered pairs that
go together to define the relation.  This option is often used in contexts
where the set of points is understood, and thus it becomes convenient to
call the whole relation <X, G> by the name of its second set G c X x X.

A "subrelation" of a dyadic relation !G! = <X, G> = <G(1), G(2)>
is a dyadic relation !H! = <Y, H> = <H(1), H(2)> that has all of
its points and pairs in !G!, more precisely, that has all of its
point-set Y c X and all of its pair-set H c G.

The "induced subrelation on a subset" (ISOS), taken with respect to
the dyadic relation G c X x X and the subset Y c X, is the maximal
subrelation of G whose points belong to Y.  In other words, it is
the dyadic relation on Y whose extension contains all of the pairs
of Y x Y that appear in G.  Since the construction of an ISOS is
uniquely determined by the data of G and Y, it can be represented
as a function of those arguments, as in the notation ISOS(G, Y),
which can be denoted more briefly as !G!_Y.  Using the symbol
"|^|" to indicate the intersection of sets, the construction
of !G!_Y = ISOS(G, Y) can be defined as follows:

   !G!_Y   =   <Y, G_Y>   =   <G_Y (1), G_Y (2)>

           =   <Y, {<x, y> in Y x Y : <x, y> in G(2)}>

           =   <Y, Y x Y |^| G(2)>

These definitions for dyadic relations can now be applied in a context where
each fragment of a sign relation that is being considered satisfies a special
set of conditions.  Namely, if F is the fragment under consideration, we have:

   1.  Syntactic Domain X  =  Sign Domain S  =  Interpretant Domain I.

   2.  Connotative Component  =  F_XX  =  F_SI  =  Equivalence Relation E.

With regard to fragments of sign relations that satisfy these conditions,
it is useful to consider further selections of a specialized sort, namely,
those that keep equivalent signs synonymous.  Here is a first description:

An "arbit" of a sign relation is a more judicious fragment of it, preserving
a semblance of whatever SEP happens to rule over its signs, and respecting the
semiotic parts of the sampled sign relation, when it has such parts.  That is,
an arbit suggests an act of selection that represents the parts of the original
SEP by means of the parts of the resulting SEP, that extracts an ISOS of each
clique in the SER that it bothers to select any points at all from, and that
manages to portray in at least this partial fashion all or none of every SEC
that appears in the original sign relation.

The use of these ideas will become clear when we
meet with concrete examples of their application.

1.3.10.2. Intermediary Notions (CFR Version)

A number of additional definitions are relevant to sign relations whose
connotative components constitute equivalence relations, if only in part.

A "dyadic relation on a single set" (DROSS) is a non-empty set of points
plus a set of ordered pairs on these points.  Until further notice, any
reference to a "dyadic relation" or to a "2-adic relation" is intended
to be taken in this sense, in other words, as a reference to a DROSS.

In a typical notation, the 2-adic relation !G! = <Y, G> = <!G!^(1), !G!^(2)>
is given by the set Y = G^(1) of its points and the set G = !G!^(2) c YxY of
its ordered pairs that go together to define the relation.  In contexts where
the underlying set of points is understood, it is customary to call the entire
2-adic relation !G! by the name of the set G, that is, the set of its 2-tuples.

A "subrelation" of a 2-adic relation !G! = <Y, G> = <!G!^(1), !G!^(2)>
is a 2-adic relation !H! = <Z, H> = <!H!^(1), !H!^(2)> that has all of
its points and all of its pairs in !G!, more precisely, that has all of
its points Z c Y and all of its pairs H c G.

The "induced subrelation on a subset" (ISOS), taken with respect to the
2-adic relation G c YxY and the subset Z c Y, is the maximal subrelation
of G whose points belong to Z.  In other words, it is the 2-adic relation
on Z whose extension contains all of the pairs of ZxZ that appear in G.
Since the construction of an ISOS is uniquely determined by the data of
G and Z, it can be represented as a function of these arguments, as in
the notation "Isos(G, Z)", which can be written more briefly as "!G!_Z".
Using the symbol "|^|" to indicate the intersection of a pair of sets,
the construction of !G!_Z = Isos(G, Z) can be defined as follows:

| !G!_Z  =  <Z, G_Z>  =  <(!G!_Z)^(1), (!G!_Z)^(2)>
|
|        =  <Z, {<z, z'> in ZxZ  :  <z, z'> in !G!^(2)}>
|
|        =  <Z, ZxZ |^| !G!^(2)>.

These definitions for 2-adic relations can now be applied in a context
where each piece of a sign relation that is being considered satisfies
a special set of conditions, to wit, if M is the piece under the scope:

| Syntactic Domain !Y!   =  Sign Domain !S!  =  Interpretant Domain !I!
|
| Connotative Component  =  M_YY   =   M_SI  =  Equivalence Relation E

Under these assumptions, and with regard to pieces of sign relations that
satisfy these conditions, it is useful to consider further selections of
a specialized sort, namely, those that keep equivalent signs synonymous.

An "arbit" of a sign relation is a decidedly more judicious piece of it,
preserving a semblance of whatever SEP actually and objectively happens
to rule over its signs and respecting the semiotic parts of the sampled
sign relation, when and if it has such parts.  In regard to its effects,
an arbit suggests a deliberate act of selection that fairly represents
the parts of the sampled SEP by means of the parts of the sample SEP,
that extracts an ISOS of each clique in the SER from which it exerts
to select any points at all, and that manages to portray in at least
this partial fashion either all or none of every SEC that appears in
the initial, sampled source, or soi-disant "objective" sign relation.

1.3.10.3. Propositions and Sentences

The concept of a sign relation is typically extended as a set L c O x S x I.
Because this extensional representation of a sign relation is one of the most
natural forms that it can take up, along with being one of the most important
forms that it is likely to be encountered in, a good amount of set-theoretic
machinery is necessary in order to carry out a reasonably detailed analysis
of sign relations in general.

For the purposes of this discussion, let it be supposed that each set Q,
that comprises a subject of interest in a particular discussion or that
constitutes a topic of interest in a particular moment of discussion,
is a subset of a set X, one that is sufficiently universal relative
to that discussion or big enough to cover everything that is being
talked about in that moment.  In this setting it is possible to
make a number of useful definitions, to which I now turn.

The "negation" of a sentence z, written as "(z)" and read as "not z",
is a sentence that is true when z is false, and false when z is true.

The "complement" of a set Q with respect to the universe X
is denoted by "X - Q", or simply by "~Q" if the universe X
is understood from context, and it is defined as the set of
elements in X that do not belong to Q.  In symbols, we have:

   ~Q  =  X - Q  =  {x in X : (x in Q)}.

The "relative complement" of P in Q, for two sets P, Q c X,
is denoted by "Q - P" and defined as the set of elements in
Q that do not belong to P.  In symbols:

   Q - P  =  {x in X : x in Q and (x in P)}.

The "intersection" of P and Q, for two sets P, Q c X, is denoted
by "P |^| Q" and defined as the set of elements in X that belong
to both P and Q.  In symbols:

   P |^| Q  =  {x in X : x in P and x in Q}.

The "union" of P and Q, for two sets P, Q c X, is denoted
by "P |_| Q" and defined as the set of elements in X that
belong to at least one of P or Q.  In symbols:

   P |_| Q  =  {x in X : x in P or x in Q}.

The "symmetric difference" of P and Q, for two sets P, Q c X,
is denoted by "P + Q" and defined as the set of elements in X
that belong to just one of P or Q.  In symbols:

   P + Q  =  {x in X : x in P - Q or x in Q - P}.

The preceding "definitions" are the bare essentials that are needed to
get the rest of this discussion moving, but they have to be regarded as
almost purely informal in character, at least, at this stage of the game.
In particular, these definitions all invoke the undefined notion of what
a "sentence" is, they all rely on the reader's native intuition of what
a "set" is, and they all derive their coherence and their meaning from
the common understanding, but the equally casual use and unreflective
acquaintance that just about everybody has of the logical connectives
"and", "or", "not", as these are expressed in natural language terms.

As formative definitions, these initial postulations neither acquire
the privileged status of untouchable axioms and infallible intuitions
nor do they deserve any special suspicion, at least, nothing over and
above the reflective critique that one ought to apply to all important
definitions.  As the dim beginnings of anything that approaches genuine
definitions they also serve to accustom the mind's eye to one particular
style of observation, namely, that of seeing informal concepts presented
in a formal frame, in a way that demands their increasing clarification.
In this style of examination, the frame of the set-builder expression
"{x in X : ... }" functions like the eye of the needle through which
one is trying to transport a suitably rich import of mathematics.

Much of the task of the remaining discussion is to formalize the promissory notes
that are represented by the foregoing terms and stipulations and to see whether
their informal comprehension can be converted into an explicit subject matter,
one that depends on grasping an array of increasingly formalized concepts.

| NB.  In the following asciification of a pre-existing text,
| markups like "!...!" indicate singly-underlined text, and
| markups like "%...%" indicate doubly-underlined text.  

The "binary domain" is the set !B! = {!0!, !1!} of two algebraic values,
whose arithmetic operations obey the rules of GF(2), the "galois field"
of exactly two elements, whose addition and multiplication tables are
tantamount to addition and multiplication of integers "modulo 2".

The "boolean domain" is the set %B% = {%0%, %1%} of two logical values,
whose elements are read as "false" and "true", or as "falsity" and "truth",
respectively.

At this point, I cannot tell whether the distinction between these two
domains is slight or significant, and so this question must evolve its
own answer, while I pursue a larger inquiry by means of its hypothesis.
The weight of the matter appears to increase as the investigation moves
from abstract, algebraic, and formal settings to contexts where logical
semantics, natural language syntax, and concrete categories of grammar
are compelling considerations.  Speaking abstractly and roughly enough,
it is often acceptable to identify these two domains, and up until this
point there has rarely appeared to be a sufficient reason to keep their
concepts separately in mind.  The boolean domain %B% comes with at least
two operations, though often under different names and always included
in a number of others, that are analogous to the field operations of the
binary domain !B!, and operations that are isomorphic to the rest of the
boolean operations in %B% can always be built on the binary basis of !B!.

As sets of the same cardinality, the domains !B! and %B%, along with
all of the structures that can be built on them, are isomorphic at a
high enough level of abstraction.  But the main reason for preserving
their distinction in the present context appears to be more a matter
of natural language grammar than an issue of logical or mathematical
substance, namely, just so that the signs "%0%" and "%1%" can appear
with a semblance of syntactic legitimacy in linguistic contexts that
call for grammatical sentences to represent the classes of sentences
that are "always false" and "always true", respectively.  The signs
"0" and "1", that are customarily read as nouns but not as sentences,
fail to be suitable for this purpose.  Whether these scruples, that
are needed to conform to a natural language context, are ultimately
important or not, is a thing that I just do not know at this point.

The "negation" of x, for x in %B%, written as "(x)"
and read as "not x", is the boolean value (x) in %B%
that is %1% when x is %0%, and %0% when x is %1%.

Thus, negation is a monadic operation on boolean
values, a function of the form (_) : %B% -> %B%.

It is convenient to transport the product and the sum operations of !B!
into the logical setting of %B%, where they can be symbolized by signs
of the same character, doubly underlined as necessary to avoid confusion.
This yields the following definitions of a "product" and a "sum" in %B%
and leads to the following forms of multiplication and addition tables.

The "product" of x and y, for x, y in %B%, is given by Table 8.

Table 8.  Boolean Product
o---------o---------o---------o
|   %*%   %   %0%   |   %1%   |
o=========o=========o=========o
|   %0%   %   %0%   |   %0%   |
o---------o---------o---------o
|   %1%   %   %0%   |   %1%   |
o---------o---------o---------o

Viewed as a function on logical values, %*% : %B% x %B% -> %B%, the
product corresponds to the logical operation that is commonly called
"conjunction" and that is otherwise expressed as "x and y".  In accord
with common practice, the multiplication sign "*", doubly underlined or
otherwise, is frequently omitted from written expressions of the product.

The "sum" of x and y, for x, y in %B%, is given by Table 9.

Table 9.  Boolean Sum 
o---------o---------o---------o
|   %+%   %   %0%   |   %1%   |
o=========o=========o=========o
|   %0%   %   %0%   |   %1%   |
o---------o---------o---------o
|   %1%   %   %1%   |   %0%   |
o---------o---------o---------o

Viewed as a function on logical values, %+% : %B% x %B% -> %B%,
the sum corresponds to the logical operation that is generally
called "exclusive disjunction" and that is otherwise expressed
as "x or y, but not both".  Depending on the context, a couple
of other signs and readings that can invoke this operation are:

   1.  "x =/= y", read "x is not equal to y", or "exactly one of x and y".

   2.  "x <=/=> y", read "x is not equivalent to y", or "x opposes y".

For sentences, the signs of equality ("=") and inequality ("=/=")
are reserved to signify the syntactic identity and the syntactic
non-identity, respectively, of the literal strings of characters
that make up the sentences, while the signs of equivalence ("<=>")
and inequivalence ("<=/=>") refer to the logical values, if any,
that these strings may conceivably bear, and thus they serve to
signify the equality or the inequality, respectively, of their
conceivable boolean values.  For the logical values themselves,
the two pairs of symbols collapse in their meanings to a single
pair, signifying a single form of coincidence or a single form
of distinction, respectively, between the boolean values of the
entities in question.

In logical studies, one tends to be interested in all of the
operations or all of the functions of a given type, at least,
to the extent that their totalities and their individualities
can be comprehended, and not just the specialized collections
that define particular algebraic structures.

Although the rest of the conceivably possible dyadic operations
on boolean values, in other words, the remainder of the sixteen
functions f : %B% x %B% -> %B%, could be presented in the same
way as the multiplication and addition tables, it is better to
look for a more efficient style of representation, one that is
able to express all of the boolean functions on the same number
of variables on a roughly equal basis, and with a bit of luck,
affords us with a calculus for computing with these functions.

The utility of a suitable calculus would involve, among other things:

   1.  Finding the values of given functions for given arguments.

   2.  Inverting boolean functions, that is, "finding the fibers"
       of boolean functions, or solving logical equations that
       are expressed in terms of boolean functions.

   3.  Facilitating the recognition of invariant forms that
       take boolean functions as their functional components.

The whole point of formal logic, the reason for doing logic formally and
the measure that determines how far it is possible to reason abstractly,
is to discover functions that do not vary as much as their variables do,
in other words, to identify forms of logical functions that, though they
express a dependence on the values of their constituent arguments, do not
vary as much as possible, but approach the way of being a function that
constant functions enjoy.  Thus, the recognition of a logical law amounts
to identifying a logical function, that, though it ostensibly depends on
the values of its putative arguments, is not as variable in its values as
the values of its variables are allowed to be.

The "indicator function" or the "characteristic function" of a set Q c X,
written "f_Q", is the map from X to the boolean domain %B% = {%0%, %1%}
that is defined in the following ways:

   1.  Considered in extensional form, f_Q is the subset of X x %B%
       that is given by the following formula:

       f_Q  =  {<x, b> in X x %B% : b = %1% <=> x in Q}.

   2.  Considered in functional form, f_Q is the map from X to %B%
       that is given by the following condition:

       f_Q (x) = %1%  <=>  x in Q.

A "proposition about things in the universe", for short, a "proposition",
is the same thing as an indicator function, that is, a function of the
form f : X -> %B%.  The convenience of this seemingly redundant usage
is that it allows one to refer to an indicator function without having
to specify right away, as a part of its designated subscript, exactly
what set it indicates, even though a proposition always indicates some
subset of its designated universe, and even though one will eventually,
most likely, want to know exactly what subset that is.

According to the stated understandings, a proposition is a function that
indicates a set, in the sense that a function associates values with the
elements of a domain, some of which values can be interpreted to mark out
for special consideration a subset of that domain.  The way in which an
indicator function is interpreted to "indicate" a set can be expressed
in terms of the following concepts.

The "fiber" of a codomain element y in Y under a function f : X -> Y is the
subset of the domain X that is mapped onto y, in short, it is f^(-1)(y) c X.

In other language that is often used, the fiber of y under f is
called the "antecedent set", the "inverse image", the "level set",
or the "pre-image" of y under f.  All of these equivalent concepts
can be defined as follows:

   Fiber of y under f  =  f^(-1)(y)  =  {x in X : f(x) = y}.

In the special case where f is the indicator function f_Q of the set Q c X,
the fiber of %1% under f_Q is just the set Q back again:

   Fiber of %1% under f_Q = (f_Q)^(-1)(%1%) = {x in X : f_Q (x) = %1%} = Q.

In this specifically boolean setting, as in the more generally logical
context, where "truth" under any name is especially valued, it is worth
devoting a specialized notation to the "fiber of truth" in a proposition.
For this purpose, I introduce the use of "fiber bars" or "ground signs",
written as "[| ... |]" around a sentence, in other words, around the
sign of a proposition, and whose application is defined as follows:

   Given f : X -> %B%, define [|f|] c X as follows:

   [| f |]  =  f^(-1)(%1%)  =  {x in X : f(x) = %1%}.

The definition of a fiber, in either the general or the boolean case,
is a purely nominal convenience for referring to the antecedent subset,
the inverse image under a function, or the pre-image of a functional value.
The definition of the fiber operator on propositions, signified by framing
the signs of propositions with fiber bars or ground signs, remains a purely
notational device, and yet the use of the fiber concept in a logical context
raises a number of problematic issues.  By way of example, consider the fact
that it is legitimate to rewrite the above definition in the following form:

   Given  f : X -> %B%, define [|f|] c X as follows:

   [| f |]  =  f^(-1)(%1%)  =  {x in X : f(x)}.

The set-builder frame "{x in X : ... }" requires a grammatical sentence or
a sentential clause to fill in the blank, as with the sentence "f(x) = %1%"
that serves to fill the frame in the initial definition of a logical fiber.
And what is a sentence but the expression of a proposition, in other words,
the name of an indicator function?  As it happens, the sign "f(x)" and the
sentence "f(x) = %1%" represent the very same value to this context, for
all x in X.  That is to say, the two expressions will appear to be equal
in their truth or falsity to any reasonable interpreter of sentences in
this context, and so either one of them can be tendered for the other,
in effect, exchanged for the other, within this context.

Given f : X -> %B%, the sign "f(x)" manifestly names the value f(x).
The value f(x) can in turn be interpreted in many different lights.
Just to enumerate a few of them, the value f(x) can be taken as:

   1.  The value that the proposition f has at the point x,
       in other words, the value that f bears at the point x
       where f is being evaluated, the value that f takes on
       with respect to the argument or the object x that the
       whole proposition f is taken to be about.

   2.  The value that the proposition f not only takes up at
       the point x, but that it carries, conveys, transfers,
       or transports into the setting "{x in X : ... }", or
       into any other context of discourse where f is meant
       to be evaluated.

   3.  The value that the sign "f(x)" has in the context where it is
       placed, that it stands for in the context where it stands, and
       that it continues to stand for in this context just so long as
       the same proposition f and the same object x are borne in mind.

   4.  The value that the sign "f(x)" represents to its complete
       interpretive context as being its own logical interpretant,
       in other words, the value that it signifies as its canonical
       connotation to any interpreter who is cognizant of the context
       in which the sign "f(x)" appears.

The sentence "f(x) = %1%" indirectly names what the sign "f(x)"
more directly names, that is, the value f(x).  In other words,
the sentence "f(x) = %1%" has the same value to its interpretive
context that the sign "f(x)" imparts to any comparable context,
each by way of its respective evaluation for the same x in X.

What is the relation among connoting, denoting, and "evaluing", where
the last term is coined to describe all the ways of bearing, conveying,
developing, or evolving a value in, to, or into an interpretive context?
In other words, when a sign is evaluated to a particular value, one can
say that the sign "evalues" that value, using the verb in a way that is
categorically analogous or grammatically conjugate to the times when one
says that a sign "connotes" an idea or that a sign "denotes" an object.
This does little more than provide the discussion with a "weasel word",
a term that is designed to avoid the main issue, to put off deciding the
exact relation between formal signs and formal values, and ultimately to
finesse the question about the nature of formal values, whether they are
more akin to conceptual signs and figurative ideas or to the kinds of
literal objects and platonic ideas that are independent of the mind.

These questions are confounded by the presence of certain peculiarities in
formal discussions, especially by the fact that an equivalence class of signs
is tantamount to a formal object.  This has the effect of allowing an abstract
connotation to work as a formal denotation.  In other words, if the purpose of
a sign is merely to lead its interpreter up to a sign in an equivalence class
of signs, then it follows that this equivalence class is the object of the
sign, that connotation can achieve denotation, at least, to some degree,
and that the interpretant domain collapses with the object domain,
at least, in some respect, all things being relative to the
sign relation that embeds the discussion.

Introducing the realm of "values" is a stopgap measure that temporarily
permits the discussion to avoid certain singularities in the embedding
sign relation, and allowing the process of "evaluation" as a compromise
mode of signification between connotation and denotation only manages to
steer around a topic that eventually has to be mapped in full, but these
strategies do allow the discussion to proceed a little further without
having to answer questions that are too difficult to be settled fully
or even tackled directly at this point.  As far as the relations among
connoting, denoting, and evaluing are concerned, it is possible that
all of these constitute independent dimensions of significance that
a sign might be able to enjoy, but since the notion of connotation
is already generic enough to contain multitudes of subspecies, I am
going to subsume, on a tentative basis, all of the conceivable modes
of "evaluing" within the broader concept of connotation.

With this degree of flexibility in mind, one can say that the sentence
"f(x) = %1%" latently connotes what the sign "f(x)" patently connotes.
Taken in abstraction, both syntactic entities fall into an equivalence
class of signs that constitutes an abstract object, a thing of value
that is identified by the sign "f(x)", and thus an object that might
as well be identified with the value f(x).

The upshot of this whole discussion of evaluation is that it allows one to
rewrite the definitions of indicator functions and their fibers as follows:

The "indicator function" or the "characteristic function" of a set Q c X,
written "f_Q", is the map from X to the boolean domain %B% = {%0%, %1%}
that is defined in the following ways:

   1.  Considered in its extensional form, f_Q is the subset of X x %B%
       that is given by the following formula:

       f_Q  =  {<x, b> in X x %B%  :  b  <=>  x in Q}.

   2.  Considered in its functional form, f_Q is the map from X to %B%
       that is given by the following condition:

       f_Q (x)  <=>  x in Q.

The "fibers" of truth and falsity under a proposition f : X -> %B%
are subsets of X that are variously described as follows:

   1.  The fiber of %1% under f  =  [| f |]  =  f^(-1)(%1%)

                                 =  {x in X  :  f(x) = %1%}

                                 =  {x in X  :  f(x) }.

   2.  The fiber of %0% under f  =  ~[| f |]  =  f^(-1)(%0%)

                                 =  {x in X  :  f(x) = %0%}

                                 =  {x in X  :  (f(x)) }.

Perhaps this looks like a lot of work for the sake of what seems to be
such a trivial form of syntactic transformation, but it is an important
step in loosening up the syntactic privileges that are held by the sign
of logical equivalence "<=>", as written between logical sentences, and
by the sign of equality "=", as written between their logical values, or
else between propositions and their boolean values.  Doing this removes
a longstanding but wholly unnecessary conceptual confound between the
idea of an "assertion" and notion of an "equation", and it allows one
to treat logical equality on a par with the other logical operations.

As a purely informal aid to interpretation, I frequently use the letters
"p", "q" to denote propositions.  This can serve to tip off the reader
that a function is intended as the indicator function of a set, and
it saves us the trouble of declaring the type f : X -> %B% each
time that a function is introduced as a proposition.

Another convention of use in this context is to let boldface letters
stand for k-tuples, lists, or sequences of objects.  Typically, the
elements of the k-tuple, list, or sequence are all of one type, and
typically the boldface letter is of the same basic character as the
indexed or subscripted letters that are used denote the components
of the k-tuple, list, or sequence.  When the dimension of elements
and functions is clear from the context, we may elect to drop the
bolding of characters that name k-tuples, lists, and sequences.

For example:

   1.  If x_1, ..., x_k in X,       then #x# = <x_1, ..., x_k> in X' = X^k.

   2.  If x_1, ..., x_k  : X,       then #x# = <x_1, ..., x_k>  : X' = X^k.

   3.  If f_1, ..., f_k  : X -> Y,  then #f# = <f_1, ..., f_k>  : (X -> Y)^k.

There is usually felt to be a slight but significant distinction between
the "membership statement" that uses the sign "in" as in Example (1) and
the "type statement" that uses the sign ":" as in examples (2) and (3).
The difference that appears to be perceived in categorical statements,
when those of the form "x in X" and those of the form "x : X" are set
in side by side comparisons with each other, is that a multitude of
objects can be said to have the same type without having to posit
the existence of a set to which they all belong.  Without trying
to decide whether I share this feeling or even fully understand
the distinction in question, I can only try to maintain a style
of notation that respects it to some degree.  It is conceivable
that the question of belonging to a set is rightly sensed to be
the more serious matter, one that has to do with the reality of
an object and the substance of a predicate, than the question of
falling under a type, that may have more to do with the way that
a sign is interpreted and the way that information about an object
is organized.  When it comes to the kinds of hypothetical statements
that appear in these Examples, those of the form "x in X => #x# in X'"
and "x : X => #x# : X'", these are usually read as implying some order
of synthetic construction, one whose contingent consequences involve the
constitution of a new space to contain the elements being compounded and
the recognition of a new type to characterize the elements being moulded,
respectively.  In these applications, the statement about types is again
taken to be less presumptive than the corresponding statement about sets,
since the apodosis is intended to do nothing more than to abbreviate and
to summarize what is already stated in the protasis.

A "boolean connection" of degree k, also known as a "boolean function"
on k variables, is a map of the form F : %B%^k -> %B%.  In other words,
a boolean connection of degree k is a proposition about things in the
universe X = %B%^k.

An "imagination" of degree k on X is a k-tuple of propositions about things
in the universe X.  By way of displaying the various kinds of notation that
are used to express this idea, the imagination #f# = <f_1, ..., f_k> is given
as a sequence of indicator functions f_j : X -> %B%, for j = 1 to k.  All of
these features of the typical imagination #f# can be summed up in either one
of two ways:  either in the form of a membership statement, to the effect that
#f# is in (X -> %B%)^k, or in the form of a type statement, to the effect that
#f# : (X -> %B%)^k, though perhaps the latter form is slightly more precise than
the former.

The "play of images" that is determined by #f# and x, more specifically,
the play of the imagination #f# = <f_1, ..., f_k> that has to with the
element x in X, is the k-tuple #b# = <b_1, ..., b_k> of values in %B%
that satisfies the equations b_j = f_j (x), for all j = 1 to k.

A "projection" of %B%^k, typically denoted by "p_j" or "pr_j",
is one of the maps p_j : %B%^k -> %B%, for j = 1 to k, that is
defined as follows:

   For all #b# = <b_1, ..., b_k> in %B%^k we have:

   p_j (#b#) = p_j (<b_1, ..., b_k>) = b_j in %B%.

The "projective imagination" of %B%^k is the imagination <p_1, ..., p_k>.

A "sentence about things in the universe", for short, a "sentence",
is a sign that denotes a proposition.  In other words, a sentence is
any sign that denotes an indicator function, any sign whose object is
a function of the form f : X -> %B%.

To emphasize the empirical contingency of this definition, one can say
that a sentence is any sign that is interpreted as naming a proposition,
any sign that is taken to denote an indicator function, or any sign whose
object happens to be a function of the form f : X -> %B%.

An "expression" is a type of sign, for instance, a term or a sentence,
that has a value.  In forming this conception of an expression, I am
deliberately leaving a number of options open, for example, whether
the expression amounts to a term or to a sentence and whether it
ought to be accounted as denoting a value or as connoting a value.
Perhaps the expression has different values under different lights,
and perhaps it relates to them differently in different respects.
In the end, what one calls an expression matters less than where
its value lies.  Of course, no matter whether one chooses to call
an expression a "term" or a "sentence", if the value is an element
of %B%, then the expression affords the option of being treated as
a sentence, meaning that it is subject to assertion and composition
in the same way that any sentence is, having its value figure into
the values of larger expressions through the linkages of sentential
connectives, and affording us the consideration of what things in
what universe the corresponding proposition happens to indicate.

Expressions with this degree of flexibility in the types under
which they can be interpreted are difficult to translate from
their formal settings into more natural contexts.  Indeed,
the whole issue can be difficult to talk about, or even
to think about, since the grammatical categories of
sentential clauses and noun phrases are rarely so
fluid in natural language settings as they can
be rendered in artificially formal arenas.

To finesse the issue of whether an expression denotes or connotes its value,
or else to create a general term that covers what both possibilities have
in common, one can say that an expression "evalues" its value.

An "assertion" is just a sentence that is being used in a certain way,
namely, to indicate the indication of the indicator function that the
sentence is usually used to denote.  In other words, an assertion is
a sentence that is being converted to a certain use or that is being
interpreted in a certain role, and one whose immediate denotation is
being pursued to its substantive indication, specifically, the fiber
of truth of the proposition that the sentence potentially denotes.
Thus, an assertion is a sentence that is held to denote the set of
things in the universe for which the sentence is held to be true.

Taken in a context of communication, an assertion is basically a request
that the interpreter consider the things for which the sentence is true,
in other words, to find the fiber of truth in the associated proposition,
or to invert the indicator function that is denoted by the sentence with
respect to its possible value of truth.

A "denial" of a sentence z is an assertion of its negation -(z)-.
The denial acts as a request to think about the things for which the
sentence is false, in other words, to find the fiber of falsity in the
indicted proposition, or to invert the indicator function that is being
denoted by the sentence with respect to its possible value of falsity.

According to this manner of definition, any sign that happens to denote
a proposition, any sign that is taken as denoting an indicator function,
by that very fact alone successfully qualifies as a sentence.  That is,
a sentence is any sign that actually succeeds in denoting a proposition,
any sign that one way or another brings to mind, as its actual object,
a function of the form f : X -> %B%.

There are many features of this definition that need to be understood.
Indeed, there are problems involved in this whole style of definition
that need to be discussed, and doing this requires a slight excursion.

1.3.10.4. Empirical Types and Rational Types

In this Subsection, I want to examine the style of definition that I used
to define a sentence as a type of sign, to adapt its application to other
problems of defining types, and to draw a lesson of general significance.

Notice that I am defining a sentence in terms of what it denotes, and not
in terms of its structure as a sign.  In this way of reckoning, a sign is
not a sentence on account of any property that it has in itself, but only
due to the sign relation that actually works to interpret it.  This makes
the property of being a sentence a question of actualities and contingent
relations, not merely a question of potentialities and absolute categories.
This does nothing to alter the level of interest that one is bound to have
in the structures of signs, it merely shifts the axis of the question from
the logical plane of definition to the pragmatic plane of effective action.
As a practical matter, of course, some signs are better for a given purpose
than others, more conducive to a particular result than others, and turn out
to be more effective in achieving an assigned objective than others, and the
reasons for this are at least partly explained by the relationships that can
be found to exist among a sign's structure, its object, and the sign relation
that fits the sign and its object to each other.

Notice the general character of this development.  I start by
defining a type of sign according to the type of object that it
happens to denote, ignoring at first the structural potential that
it brings to the task.  According to this mode of definition, a type
of sign is singled out from other signs in terms of the type of object
that it actually denotes and not according to the type of object that it
is designed or destined to denote, nor in terms of the type of structure
that it possesses in itself.  This puts the empirical categories, the
classes based on actualities, at odds with the rational categories,
the classes based on intentionalities.  In hopes that this much
explanation is enough to rationalize the account of types that
I am using, I break off the digression at this point and
return to the main discussion.

1.3.10.5. Articulate Sentences

A sentence is called "articulate" if:

   1.  It has a significant form, a compound construction,
       a multi-part constitution, a well-developed composition,
       or a non-trivial structure as a sign.

   2.  There is an informative relationship that exists
       between its structure as a sign and the content
       of the proposition that it happens to denote.

A sentence of the articulate kind is typically given in the form of
a "description", an "expression", or a "formula", in other words, as
an articulated sign or a well-structured element of a formal language.
As a general rule, the category of sentences that one will be willing to
contemplate is compiled from a particular selection of complex signs and
syntactic strings, those that are assembled from the basic building blocks
of a formal language and held in especial esteem for the roles that they
play within its grammar.  Still, even if the typical sentence is a sign
that is generated by a formal regimen, having its form, its meaning,
and its use governed by the principles of a comprehensive grammar,
the class of sentences that one has a mind to contemplate can also
include among its number many other signs of an arbitrary nature.

Frequently this "formula" has a "variable" in it that "ranges over" the
universe X.  A "variable" is an ambiguous or equivocal sign that can be
interpreted as denoting any element of the set that it "ranges over".

If a sentence denotes a proposition f : X -> %B%, then the "value" of the
sentence with regard to x in X is the value f(x) of the proposition at x,
where "%0%" is interpreted as "false" and "%1%" is interpreted as "true".

Since the value of a sentence or a proposition depends on the universe of discourse
to which it is "referred", and since it also depends on the element of the universe
with regard to which it is evaluated, it is conventional to say that a sentence or
a proposition "refers" to a universe of discourse and to its elements, though often
in a variety  of different senses.  Furthermore, a proposition, acting in the guise
of an indicator function, "refers" to the elements that it "indicates", namely, the
elements on which it takes a positive value.  In order to sort out the potential
confusions that are capable of arising here, I need to examine how these various
notions of reference are related to the notion of denotation that is used in the
pragmatic theory of sign relations.

One way to resolve the various and sundry senses of "reference" that arise
in this setting is to make the following brands of distinctions among them:

   1.  Let the reference of a sentence or a proposition to a universe of discourse,
       the one that it acquires by way of taking on any interpretation at all, be
       taken as its "general reference", the kind of reference that one can safely
       ignore as irrelevant, at least, so long as one stays immersed in only one
       context of discourse or only one moment of discussion.

   2.  Let the references that an indicator function f has to the elements
       on which it evaluates to %0% be called its "negative references".

   3.  Let the references that an indicator function f has to the elements
       on which it evaluates to %1% be called its "positive references"
       or its "indications".

Finally, unspecified references to the "references" of a sentence,
a proposition, or an indicator function can all be taken by default
as references to their specific, positive references.

The universe of discourse for a sentence, the set whose elements the
sentence is interpreted to be about, is not a property of the sentence
by itself, but of the sentence in the presence of its interpretation.
Independently of how many explicit variables a sentence contains, its
value can always be interpreted as depending on any number of implicit
variables.  For instance, even a sentence with no explicit variable,
a constant expression like "%0%" or "%1%", can be taken to denote
a constant proposition of the form c : X -> %B%.  Whether or not it
has an explicit variable, I always take a sentence as referring to
a proposition, one whose values refer to elements of a universe X.

Notice that the letters "p" and "q", interpreted as signs that denote
the indicator functions p, q : X -> %B%, have the character of sentences
in relation to propositions, at least, they have the same status in this
abstract discussion as genuine sentences have in concrete applications.
This illustrates the relation between sentences and propositions as
a special case of the relation between signs and objects.

To assist the reading of informal examples, I frequently use the letters
"t", "u", "v", "z" to denote sentences.  Thus, it is conceivable to have
a situation where z = "q" and where q : X -> %B%.  Altogether, this means
that the sign "z" denotes the sentence z, that the sentence z is the same
thing as the sentence "q", and that the sentence "q" denotes the proposition,
characteristic function, or indicator function q : X -> %B%.  In settings where
it is necessary to keep track of a large number of sentences, I use subscripted
letters like "e_1", ..., "e_n" to refer to the various expressions in question.

A "sentential connective" is a sign, a coordinated sequence of signs,
a syntactic pattern of contextual arrangement, or any other syntactic
device that can be used to connect a number of sentences together in
order to form a single sentence.  If k is the number of sentences that
are thereby connected, then the connective is said to be of "order k".
If the sentences acquire a logical relationship through this mechanism,
and are not just strung together by this device, then the connective
is called a "logical connective".  If the value of the constructed
sentence depends on the values of the component sentences in such
a way that the value of the whole is a boolean function of the
values of the parts, then the connective earns the title of
a "propositional connective".

1.3.10.6. Stretching Principles

There is a principle of constant use in this work that needs
to be made explicit at this point.  In order to give it a name,
I will refer to it as the "stretching principle".  Expressed in
a variety of different ways, it can be taken to say any one of
the following things:

   1.  Any relation of values extends
       to a relation of what is valued.

   2.  Any statement about values says something
       about the things that are given these values.

   3.  Any association among a range of values
       establishes an association among the
       domains of things that these values
       are the values of.

   4.  Any connection between two values can be stretched
       to create a connection, of analogous form, between
       the objects, persons, qualities, or relationships
       that are valued in these connections.

   5.  For every operation on values, there is a corresponding operation
       on the actions, conducts, functions, procedures, or processes that
       lead to these values, as well as there being analogous operations
       on the objects that instigate all of these various proceedings.

Nothing about the application of the stretching principle guarantees that
the analogues it generates will be as useful as the material it works on.
It is another question entirely whether the links that are forged in this
fashion are equal in their strength and apposite in their bearing to the
tried and true utilities of the original ties, but in principle they are
always there.

The purpose of this exercise is to illuminate how a sentence,
a sign constituted as a string of characters, can be enfused
with a proposition, an object of no slight abstraction, in a
way that can speak about an external universe of discourse X.

To complete the general discussion of stretching principles,
we will need to call back to mind the following definitions:

A "boolean connection" of degree k, also known as a "boolean function"
on k variables, is a map of the form F : %B%^k -> %B%.  In other words,
a boolean connection of degree k is a proposition about things in the
universe of discourse X = %B%^k.

An "imagination" of degree k on X is a k-tuple of propositions about things
in the universe X.  By way of displaying the various brands of notation that
are used to express this idea, the imagination #f# = <f_1, ..., f_k> is given
as a sequence of indicator functions f_j : X -> %B%, for j = 1 to k.  All of
these features of the typical imagination #f# can be summed up in either one
of two ways:  either in the form of a membership statement, to the effect that
#f# is in (X -> %B%)^k, or in the form of a type statement, to the effect that
#f# : (X -> %B%)^k, though perhaps the latter form is slightly more precise
than the former.

The "play of images" that is determined by #f# and x, more specifically,
the play of the imagination #f# = <f_1, ..., f_k> that has to with the
element x in X, is the k-tuple #b# = <b_1, ..., b_k> of values in %B%
that satisfies the equations b_j = f_j (x), for all j = 1 to k.

A "projection" of %B%^k, typically denoted by "p_j" or "pr_j",
is one of the maps p_j : %B%^k -> %B%, for j = 1 to k, that is
defined as follows:

   If         #b#   =       <b_1, ..., b_k>           in  %B%^k,

   then  p_j (#b#)  =  p_j (<b_1, ..., b_k>)  =  b_j  in  %B%.

The "projective imagination" of %B%^k is the imagination <p_1, ..., p_k>.

As an application of the stretching principle, a connection F : %B%^k -> %B%
can be understood to indicate a relation among boolean values, namely, the
k-ary relation L = F^(-1)(%1%) c %B%^k.  If these k values are values of
things in a universe X, that is, if one imagines each value in a k-tuple
of values to be the functional image that results from evaluating an
element of X under one of its possible aspects of value, then one
has in mind the k propositions f_j : X -> %B%, for j = 1 to k,
in sum, one embodies the imagination #f# = <f_1, ..., f_k>.
Together, the imagination #f# in (X -> %B%)^k and the
connection F : %B%^k -> %B% stretch each other to
cover the universe X, yielding a new proposition
q : X -> %B%.

To encapsulate the form of this general result, I define a scheme of composition
that takes an imagination #f# = <f_1, ..., f_k> in (X -> %B%)^k and a boolean
connection F : %B%^k -> %B% and gives a proposition q : X -> %B%.  Depending
on the situation, specifically, according to whether many F and many #f#,
a single F and many #f#, or many F and a single #f# are being considered,
I refer to the resultant q under one of three descriptions, respectively:

   1.  In a general setting, where the connection F and the imagination #f#
       are both permitted to take up a variety of concrete possibilities,
       call q the "stretch of F and #f# from X to %B%", and write it in
       the style of a composition as "F $ #f#".  This is meant to suggest
       that the symbol "$", here read as "stretch", denotes an operator
       of the form $ : (%B%^k -> %B%) x (X -> %B%)^k -> (X -> %B%).

   2.  In a setting where the connection F is fixed but the imagination #f#
       is allowed to vary over a wide range of possibilities, call q the
       "stretch of F to #f# on X", and write it in the style "F^$ #f#",
       as if "F^$" denotes an operator F^$ : (X -> %B%)^k -> (X -> %B%)
       that is derived from F and applied to #f#, ultimately yielding
       a proposition F^$ #f# : X -> %B%.

   3.  In a setting where the imagination #f# is fixed but the connection F
       is allowed to range over a wide variety of possibilities, call q the
       "stretch of #f# by F to %B%", and write it in the fashion "#f#^$ F",
       as if "#f#^$" denotes an operator #f#^$ : (%B%^k -> %B%) -> (X -> %B%)
       that is derived from #f# and applied to F, ultimately yielding
       a proposition #f#^$ F : X -> %B%.

Because the stretch notation is used only in settings
where the imagination #f# : (X -> %B%)^k and the
connection F : %B%^k -> %B% are distinguished
by their types, it does not really matter
whether one writes "F $ #f#" or "#f# $ F"
for the initial form of composition.

Just as a sentence is a sign that denotes a proposition,
which thereby serves to indicate a set, a propositional
connective is a provision of syntax whose mediate effect
is to denote an operation on propositions, which thereby
manages to indicate the result of an operation on sets.
In order to see how these compound forms of indication
can be defined, it is useful to go through the steps
that are needed to construct them.  In general terms,
the ingredients of the construction are as follows:

   1.  An imagination of degree k on X, in other words, a k-tuple
       of propositions f_j : X -> %B%, for j = 1 to k, or an object
       of the form #f# = <f_1, ..., f_k> : (X -> %B%)^k.

   2.  A connection of degree k, in other words, a proposition
       about things in %B%^k, or a boolean function of the form
       F : %B%^k -> %B%.

From this 2-ply of material, it is required to construct a proposition
q : X -> %B% such that q(x) = F(f_1(x), ..., f_k(x)), for all x in X.
The desired construction can be developed in the following manner:

The cartesian power %B%^k, as a cartesian product, is characterized
by the possession of a projective imagination #p# = <p_1, ..., p_k>
of degree k on %B%^k, along with the property that any imagination
#f# = <f_1, ..., f_k> of degree k on an arbitrary set W determines
a unique map !f! : W -> %B%^k, the play of whose projective images
<p_1(!f!(w)), ..., p_k(!f!(w))> on the functional image !f!(w)
matches the play of images <f_1(w), ..., f_k(w)> under #f#,
term for term and at every element w in W.

Just to be on the safe side, I state this again in more standard terms.
The cartesian power %B%^k, as a cartesian product, is characterized by
the possession of k projection maps p_j : %B%^k -> %B%, for j = 1 to k,
along with the property that any k maps f_j : W -> %B%, from an arbitrary
set W to %B%, determine a unique map !f! : W -> %B%^k satisfying the system
of equations p_j(!f!(w)) = f_j(w), for all j = 1 to k, and for all w in W.

Now suppose that the arbitrary set W in this construction is just
the relevant universe X.  Given that the function !f! : X -> %B%^k
is uniquely determined by the imagination #f# : (X -> %B%)^k, or what
is the same thing, by the k-tuple of propositions #f# = <f_1, ..., f_k>,
it is safe to identify !f! and #f# as being a single function, and this
makes it convenient on many occasions to refer to the identified function
by means of its explicitly descriptive name "<f_1, ..., f_k>".  This facility
of address is especially appropriate whenever a concrete term or a constructive
precision is demanded by the context of discussion.

1.3.10.7. Stretching Operations

The preceding discussion of stretch operations is slightly more general
than is called for in the present context, and so it is probably a good
idea to draw out the particular implications that are needed right away.

If F : %B%^k -> %B% is a boolean function on k variables, then it is possible
to define a mapping F^$ : (X -> %B%)^k -> (X -> %B%), in effect, an operation
that takes k propositions into a single proposition, where F^$ satisfies the
following conditions:

   F^$ (f_1, ..., f_k)  :  X -> %B%

   such that:

   F^$ (f_1, ..., f_k)(x)  =  F(#f#(x))

                           =  F((f_1, ..., f_k)(x))

                           =  F(f_1(x), ..., f_k(x)).

Thus, F^$ is just the sort of entity that a propositional connective denotes,
a particular way of connecting the propositions that are denoted by a number
of sentences into a proposition that is denoted by a single sentence.

Now "f_X" is sign that denotes the proposition f_X,
and it certainly seems like a sufficient sign for it.
Why would we need to recognize any other signs of it?

If one takes a sentence as a type of sign that denotes a proposition and
a proposition as a type of function whose values serve to indicate a set,
then one needs a way to grasp the overall relation between the sentence
and the set as taking place within a higher order sign relation.

The various relationships of denotation and indication that exist
among sets, propositions, sentences, and values in this situation
are illustrated very roughly by the array of materials in Table 10.

Table 10.  Levels of Indication
o-------------------o-------------------o-------------------o
| Object            | Sign              | Higher Order Sign |
o-------------------o-------------------o-------------------o
|                   |                   |                   |
| Set               | Proposition       | Sentence          |
|                   |                   |                   |
| f^(-1)(b)         | f                 | "f"               |
|                   |                   |                   |
o-------------------o-------------------o-------------------o
|                   |                   |                   |
| Q                 | %1%               | "%1%"             |
|                   |                   |                   |
| X - Q             | %0%               | "%0%"             |
|                   |                   |                   |
o-------------------o-------------------o-------------------o

Strictly speaking, propositions are too abstract to be signs, hence the
contents of Table 10 have to be taken with the indicated grains of salt.
Propositions, as indicator functions, are abstract mathematical objects,
not any kinds of syntactic elements, thus propositions cannot literally
constitute the orders of concrete signs that remain of ultimate interest
in the pragmatic theory of signs, or in any theory of effective meaning.

Therefore, it needs to be understood that a proposition f can be said
to "indicate" the set Q only insofar as the values of %1% and %0% that
it assigns to the elements of the universe X are positive and negative
indications, respectively, of the elements in Q, and thus indications
of the set Q and of its complement ~X = X - Q, respectively.  It is
these logical values, when rendered by a concrete implementation of
the indicator function f, that are the actual signs of the objects
inside the set Q and the objects outside the set Q, respectively.

In order to deal with the higher order sign relations
that are involved in the present setting, I introduce
a couple of new notations:

   1.  To mark the relation of denotation between a sentence z and the
       proposition that it denotes, the "spiny bracket" notation "-[z]-"
       will be used for "the indicator function denoted by the sentence z".

   2.  To mark the relation of denotation between a proposition q and
       the set that it indicates, the "spiny brace" notation "-{Q}-"
       will be used for "the indicator function of the set Q".

Notice that the spiny bracket operator "-[ ]-" takes one "downstream",
confluent with the direction of denotation, from a sign to its object,
whereas the spiny brace operator "-{ }-" takes one "upstream", against
the usual direction of denotation, and thus from an object to its sign.

In order to make these notations useful in practice, it is necessary to note
a couple of their finer points, points that might otherwise seem too fine to
take much trouble over.  For the sake their ultimate utility, nevertheless,
I will describe their usage a bit more carefully as follows:

   1.  Let "spiny brackets", like "-[ ]-", be placed around a name
       of a sentence z, as in the expression "-[z]-", or else around
       a token appearance of the sentence itself, to serve as a name
       for the proposition that z denotes.

   2.  Let "spiny braces", like "-{ }-", be placed around a name of
       a set Q, as in the expression "-{Q}-", to serve as a name for
       the indicator function f_Q.

In passing, let us recall the use of the "fiber bars"
or the "ground marker"  "[| ... |]" as an alternate
notation for the fiber of truth in a proposition q,
as follows:

   [| q |]  =  q^(-1)(%1%).

Table 11 illustrates the use of this notation, listing in each Column
several different but equivalent ways of referring to the same entity.

Table 11.  Illustrations of Notation
o-------------------o-------------------o-------------------o
|      Object       |       Sign        | Higher Order Sign |
o-------------------o-------------------o-------------------o
|                   |                   |                   |
|        Set        |    Proposition    |     Sentence      |
|                   |                   |                   |
|         Q         |         q         |         z         |
|                   |                   |                   |
|    [| -[z]- |]    |       -[z]-       |         z         |
|                   |                   |                   |
|      [| q |]      |         q         |        "q"        |
|                   |                   |                   |
|     [| f_Q |]     |        f_Q        |       "f_Q"       |
|                   |                   |                   |
|         Q         |       -{Q}-       |      "-{Q}-"      |
|                   |                   |                   |
o-------------------o-------------------o-------------------o

In effect, one can observe the following relations
and formulas, all of a purely notational character:

   1.  If the sentence z denotes the proposition q : X -> %B%,

       then   -[z]-  =  q.

   2.  If the sentence z denotes the proposition q : X -> %B%,

       hence  [|q|]  =  q^(-1)(%1%)  =  Q  c  X,

       then   -[z]-  =  q  =  f_Q  =  -{Q}-.

   3.  Q      =  {x in X  :  x in Q}

              =  [| -{Q}- |]  =  -{Q}-^(-1)(%1%)

              =  [|  f_Q  |]  =  (f_Q)^(-1)(%1%).

   4.  -{Q}-  =  -{ {x in X  :  x in Q} }-

              =  -[x in Q]-

              =  f_Q.

If a sentence z really denotes a proposition q, and if the notation "-[z]-"
is merely meant to supply another name for the proposition that z already
denotes, then why is there any need for all of this additional notation?
It is because the interpretive mind habitually races from the sentence z,
through the proposition q that it denotes, and on to the set Q = [|q|]
that the proposition indicates, often jumping to the conclusion that
the set Q is the only thing that the sentence z is meant to denote.
The momentum of this type of higher order sign situation, together
with the mind's inclination when placed within its setting, calls
for a linguistic mechanism or a notational device that is capable
of analyzing the compound action and controlling its articulate
performance, and this requires a way to interrupt the flow of
assertion that runs its unreflective course from z to q to Q.

1.3.11. The Cactus Patch

| Thus, what looks to us like a sphere of scientific knowledge more accurately
| should be represented as the inside of a highly irregular and spiky object,
| like a pincushion or porcupine, with very sharp extensions in certain
| directions, and virtually no knowledge in immediately adjacent areas.
| If our intellectual gaze could shift slightly, it would alter each
| quill's direction, and suddenly our entire reality would change.
|
| Herbert J. Bernstein, "Idols", p. 38.
|
| Herbert J. Bernstein,
|"Idols of Modern Science and the Reconstruction of Knowledge", pp. 37-68 in:
|
| Marcus G. Raskin & Herbert J. Bernstein,
|'New Ways of Knowing:  The Sciences, Society, and Reconstructive Knowledge',
| Rowman & Littlefield, Totowa, NJ, 1987.

In this and the four Subsections that follow, I describe a calculus for
representing propositions as sentences, in other words, as syntactically
defined sequences of signs, and for manipulating these sentences chiefly
in the light of their semantically defined contents, in other words, with
respect to their logical values as propositions.  In their computational
representation, the expressions of this calculus parse into a class of
tree-like data structures called "painted cacti".  This is a family of
graph-theoretic data structures that can be observed to have especially
nice properties, turning out to be not only useful from a computational
standpoint but also quite interesting from a theoretical point of view.
The rest of this subsection serves to motivate the development of this
calculus and treats a number of general issues that surround the topic.

In order to facilitate the use of propositions as indicator functions
it helps to acquire a flexible notation for referring to propositions
in that light, for interpreting sentences in a corresponding role, and
for negotiating the requirements of mutual sense between the two domains.
If none of the formalisms that are readily available or in common use are
able to meet all of the design requirements that come to mind, then it is
necessary to contemplate the design of a new language that is especially
tailored to the purpose.  In the present application, there is a pressing
need to devise a general calculus for composing propositions, computing
their values on particular arguments, and inverting their indications to
arrive at the sets of things in the universe that are indicated by them.

For computational purposes, it is convenient to have a middle ground or
an intermediate language for negotiating between the koine of sentences
regarded as strings of literal characters and the realm of propositions
regarded as objects of logical value, even if this renders it necessary
to introduce an artificial medium of exchange between these two domains.
If one envisions these computations to be carried out in any organized
fashion, and ultimately or partially by means of the familiar sorts of
machines, then the strings that express these logical propositions are
likely to find themselves parsed into tree-like data structures at some
stage of the game.  With regard to their abstract structures as graphs,
there are several species of graph-theoretic data structures that can be
used to accomplish this job in a reasonably effective and efficient way.

Over the course of this project, I plan to use two species of graphs:

   1.  "Painted And Rooted Cacti" (PARCAI).

   2.  "Painted And Rooted Conifers" (PARCOI).

For now, it is enough to discuss the former class of data structures,
leaving the consideration of the latter class to a part of the project
where their distinctive features are key to developments at that stage.
Accordingly, within the context of the current patch of discussion, or
until it becomes necessary to attach further notice to the conceivable
varieties of parse graphs, the acronym "PARC" is sufficient to indicate
the pertinent genus of abstract graphs that are under consideration.

By way of making these tasks feasible to carry out on a regular basis,
a prospective language designer is required not only to supply a fluent
medium for the expression of propositions, but further to accompany the
assertions of their sentences with a canonical mechanism for teasing out
the fibers of their indicator functions.  Accordingly, with regard to a
body of conceivable propositions, one needs to furnish a standard array
of techniques for following the threads of their indications from their
objective universe to their values for the mind and back again, that is,
for tracing the clues that sentences provide from the universe of their
objects to the signs of their values, and, in turn, from signs to objects.
Ultimately, one seeks to render propositions so functional as indicators
of sets and so essential for examining the equality of sets that they can
constitute a veritable criterion for the practical conceivability of sets.
Tackling this task requires me to introduce a number of new definitions
and a collection of additional notational devices, to which I now turn.

Depending on whether a formal language is called by the type of sign
that makes it up or whether it is named after the type of object that
its signs are intended to denote, one may refer to this cactus language
as a "sentential calculus" or as a "propositional calculus", respectively.

When the syntactic definition of the language is well enough understood,
then the language can begin to acquire a semantic function.  In natural
circumstances, the syntax and the semantics are likely to be engaged in
a process of co-evolution, whether in ontogeny or in phylogeny, that is,
the two developments probably form parallel sides of a single bootstrap.
But this is not always the easiest way, at least, at first, to formally
comprehend the nature of their action or the power of their interaction.

According to the customary mode of formal reconstruction, the language
is first presented in terms of its syntax, in other words, as a formal
language of strings called "sentences", amounting to a particular subset
of the possible strings that can be formed on a finite alphabet of signs.
A syntactic definition of the "cactus language", one that proceeds along
purely formal lines, is carried out in the next Subsection.  After that,
the development of the language's more concrete aspects can be seen as
a matter of defining two functions:

   1.  The first is a function that takes each sentence of the language
       into a computational data structure, in this particular setting,
       a tree-like parse graph called a "painted cactus".

   2.  The second is a function that takes each sentence of the language,
       by way of its corresponding parse graph, into a logical proposition,
       in effect, ending up with an indicator function as the object denoted
       by the sentence.

The discussion of syntax brings up a number of associated issues that
have to be clarified before going on.  These are questions of "style",
that is, the sort of description, "grammar", or theory of the language
that one finds available or chooses as preferable for a given language.
These issues are discussed in Subsection 1.3.10.10.

There is an aspect of syntax that is so schematic in its basic character
that it can be conveyed by computational data structures, so algorithmic
in its uses that it can be automated by routine mechanisms, and so fixed
in its nature that its practical exploitation can be served by the usual
devices of computation.  Because it involves the transformation of signs
it can be recognized as an aspect of semiotics.  But given the fact that
these transformations can be carried out in abstraction from meaning, it
does not rise to the level of semantics, much less a complete pragmatics,
although it does involve the "pragmatic" aspects of computation that are
auxiliary to, incidental to, or tangent to the use of language by humans.
In light of these characteristics, I will refer to this aspect of formal
language use as the "algorithmics" or "mechanics" of language processing.
An algorithmic conversion of the cactus language into its corresponding
data structures is discussed in Subsection 1.3.10.11.

In the usual way of proceeding on formal grounds, meaning is added by giving
each "grammatical sentence", or each syntactically distinguished string, an
interpretation as a logically meaningful sentence, in effect, equipping or
providing each abstractly well-formed sentence with a logical proposition
for it to denote.  A semantic interpretation of the "cactus language",
just one of at least two classical interpretations, is carried out
in Subsection 1.3.10.12.

1.3.11.1. The Cactus Language : Syntax

| Picture two different configurations of such an irregular shape, superimposed
| on each other in space, like a double exposure photograph.  Of the two images,
| the only part which coincides is the body.  The two different sets of quills
| stick out into very different regions of space.  The objective reality we
| see from within the first position, seemingly so full and spherical,
| actually agrees with the shifted reality only in the body of common
| knowledge.  In every direction in which we look at all deeply, the
| realm of discovered scientific truth could be quite different.
| Yet in each of those two different situations, we would have
| thought the world complete, firmly known, and rather round
| in its penetration of the space of possible knowledge.
|
| Herbert J. Bernstein, "Idols", p. 38.
|
| Herbert J. Bernstein,
|"Idols of Modern Science and the Reconstruction of Knowledge", pp. 37-68 in:
|
| Marcus G. Raskin & Herbert J. Bernstein,
|'New Ways of Knowing:  The Sciences, Society, and Reconstructive Knowledge',
| Rowman & Littlefield, Totowa, NJ, 1987.

In this Subsection, I describe the syntax of a family of formal languages
that I intend to use as a sentential calculus, and thus to interpret for
the purpose of reasoning about propositions and their logical relations.
In order to carry out the discussion, I need a way of referring to signs
as if they were objects like any others, in other words, as the sorts of
things that are subject to being named, indicated, described, discussed,
and renamed if necessary, that can be placed, arranged, and rearranged
within a suitable medium of expression, or else manipulated in the mind,
that can be articulated and decomposed into their elementary signs, and
that can be strung together in sequences to form complex signs.  Signs
that have signs as their objects are called "higher order" (HO) signs,
and this is a topic that demands an apt formalization, but in due time.
The present discussion requires a quicker way to get into this subject,
even if it takes informal means that cannot be made absolutely precise.

As a temporary notation, let the relationship between a particular sign z
and a particular object o, namely, the fact that z denotes o or the fact
that o is denoted by z, be symbolized in one of the following two ways:

   1.  z  >->  o,

       z  den  o.

   2.  o  <-<  z,

       o  ned  z.

Now consider the following paradigm:

   1.  If        "A"  >->  Ann,

       i.e.      "A"  den  Ann,

       then       A    =   Ann,

       thus      "Ann"  >->  A,

       i.e.      "Ann"  den  A.

   2.  If         Bob  <-<  "B",

       i.e.       Bob  ned  "B",

       then       Bob   =    B,

       thus       B  <-<  "Bob",

       i.e.       B  ned  "Bob".

When I say that the sign "blank" denotes the sign " ",
it means that the string of characters inside the first
pair of quotation marks can be used as another name for
the string of characters inside the second pair of quotes.
In other words, "blank" is a HO sign whose object is " ",
and the string of five characters inside the first pair of
quotation marks is a sign at a higher level of signification
than the string of one character inside the second pair of
quotation marks.  This relationship can be abbreviated in
either one of the following ways:

   " "      <-<  "blank"

   "blank"  >->  " "

Using the raised dot "·" as a sign to mark the articulation of a
quoted string into a sequence of possibly shorter quoted strings,
and thus to mark the concatenation of a sequence of quoted strings
into a possibly larger quoted string, one can write:

   " "  <-<  "blank"   =   "b"·"l"·"a"·"n"·"k"

This usage allows us to refer to the blank as a type of character, and
also to refer any blank we choose as a token of this type, referring to
either of them in a marked way, but without the use of quotation marks,
as I just did.  Now, since a blank is just what the name "blank" names,
it is possible to represent the denotation of the sign " " by the name
"blank" in the form of an identity between the named objects, thus:

   " "   =   blank

With these kinds of identity in mind, it is possible to extend the use of
the "·" sign to mark the articulation of either named or quoted strings
into both named and quoted strings.  For example:

   "  "       =   " "·" "       =   blank·blank

   " blank"   =   " "·"blank"   =   blank·"blank"

   "blank "   =   "blank"·" "   =   "blank"·blank

A few definitions from formal language theory are required at this point.

An "alphabet" is a finite set of signs, typically, !A! = {a_1, ..., a_n}.

A "string" over an alphabet !A! is a finite sequence of signs from !A!.

The "length" of a string is just its length as a sequence of signs.
A sequence of length 0 yields the "empty string", here presented as "".
A sequence of length k > 0 is typically presented in the concatenated forms:

   s_1 s_2 ... s_(k-1) s_k,

   or:

   s_1 · s_2 · ... · s_(k-1) · s_k,

with s_j in !A!, for all j = 1 to k.

Two alternative notations are often useful:

   1.  !e!  =   ""    =  the empty string.

   2.  %e%  =  {!e!}  =  the language consisting of a single empty string.

The "kleene star" !A!* of alphabet !A! is the set of all strings over !A!.
In particular, !A!* includes among its elements the empty string !e!.

The "surplus" !A!^+ of an alphabet !A! is the set of all positive length
strings over !A!, in other words, everything in !A!* but the empty string.

A "formal language" !L! over an alphabet !A! is a subset !L! c !A!*.
If z is a string over !A! and if z is an element of !L!, then it is
customary to call z a "sentence" of !L!.  Thus, a formal language !L!
is defined by specifying its elements, which amounts to saying what it
means to be a sentence of !L!.

One last device turns out to be useful in this connection.
If z is a string that ends with a sign t, then z · t^-1 is
the string that results by "deleting" from z the terminal t.

In this context, I make the following distinction:

   1.  By "deleting" an appearance of a sign,
       I mean replacing it with an appearance
       of the empty string "".

   2.  By "erasing" an appearance of a sign, 
       I mean replacing it with an appearance
       of the blank symbol " ".

A "token" is a particular appearance of a sign.

The informal mechanisms that have been illustrated in the immediately preceding
discussion are enough to equip the rest of this discussion with a moderately
exact description of the so-called "cactus language" that I intend to use
in both my conceptual and my computational representations of the minimal
formal logical system that is variously known to sundry communities of
interpretation as "propositional logic", "sentential calculus", or
more inclusively, "zeroth order logic" (ZOL).

The "painted cactus language" !C! is actually a parameterized
family of languages, consisting of one language !C!(!P!) for
each set !P! of "paints".

The alphabet !A!  =  !M! |_| !P! is the disjoint union of two sets of symbols:

   1.  !M! is the alphabet of "measures", the set of "punctuation marks",
       or the collection of "syntactic constants" that is common to all
       of the languages !C!(!P!).  This set of signs is given as follows:

       !M!  =  {m_1, m_2, m_3, m_4}

            =  {" ", "-(", ",", ")-"}

            =  {blank, links, comma, right}.

   2.  !P! is the "palette", the alphabet of "paints", or the collection
       of "syntactic variables" that is peculiar to the language !C!(!P!).
       This set of signs is given as follows:

       !P!  =  {p_j  :  j in J}.

The easiest way to define the language !C!(!P!) is to indicate the general sorts
of operations that suffice to construct the greater share of its sentences from
the specified few of its sentences that require a special election.  In accord
with this manner of proceeding, I introduce a family of operations on strings
of !A!* that are called "syntactic connectives".  If the strings on which
they operate are exclusively sentences of !C!(!P!), then these operations
are tantamount to "sentential connectives", and if the syntactic sentences,
considered as abstract strings of meaningless signs, are given a semantics
in which they denote propositions, considered as indicator functions over
some universe, then these operations amount to "propositional connectives".

NB.  In this transcription, the symbols "-(" and ")-"
will serve for the logically significant parentheses.

The discussion that follows is intended to serve a dual purpose,
in its specific focus presenting the family of cactus languages
with some degree of detail, but more generally and peripherally
developing the subject material and demonstrating the technical
methodology of formal languages and grammars.  I will do this by
taking up a particular method of "stepwise refinement" and using
it to extract a rigorous formal grammar for the cactus language,
starting with little more than a rough description of the target
language and applying a systematic analysis to develop a series
of increasingly more effective and more exact approximations to
the desired form of grammar.

Rather than presenting the most concise description of these languages
right from the beginning, it serves comprehension to develop a picture
of their forms in gradual stages, starting from the most natural ways
of viewing their elements, if somewhat at a distance, and working
through the most easily grasped impressions of their structures,
if not always the sharpest acquaintances with their details.

The first step is to define two sets of basic operations on strings of !A!*.

   1.  The "concatenation" of one string z_1 is just the string z_1.

       The "concatenation" of two strings z_1, z_2 is the string z_1 · z_2.

       The "concatenation" of the k strings z_j, for j = 1 to k,

       is the string of the form z_1 · ... · z_k.

   2.  The "surcatenation" of one string z_1 is the string "-(" · z_1 · ")-".

       The "surcatenation" of two strings z_1, z_2 is "-(" · z_1 · "," · z_2 · ")-".

       The "surcatenation" of k strings z_j, for j = 1 to k,

       is the string of the form "-(" · z_1 · "," · ... · "," · z_k · ")-".

These definitions can be rendered a little more succinct by
defining the following set of generic operators on strings:

   1.  The "concatenation" Conc^k of the k strings z_j,
       for j = 1 to k, is defined recursively as follows:

       a.  Conc^1_j  z_j  =  z_1.

       b.  For k > 1,

           Conc^k_j  z_j  =  (Conc^(k-1)_j  z_j) · z_k.

   2.  The "surcatenation" Surc^k of the k strings z_j,
       for j = 1 to k, is defined recursively as follows:

       a.  Surc^1_j  z_j  =  "-(" · z_1 · ")-".

       b.  For k > 1,

           Surc^k_j  z_j  =  (Surc^(k-1)_j  z_j) · ")-"^(-1) · "," · z_k · ")-".

The definitions of the foregoing syntactic operations can now be organized in
a slightly better fashion, for the sake of both conceptual and computational
purposes, by making a few additional conventions and auxiliary definitions.

   1.  The conception of the k-place concatenation operation
       can be extended to include its natural "prequel":

       Conc^0  =  ""  =  the empty string.

       Next, the construction of the k-place concatenation can be
       broken into stages by means of the following conceptions:

       a.  The "precatenation" Prec(z_1, z_2) of the two strings
           z_1, z_2 is the string that is defined as follows:

           Prec(z_1, z_2)  =  z_1 · z_2.

       b.  The "concatenation" of the k strings z_1, ..., z_k can now be
           defined as an iterated precatenation over the sequence of k+1
           strings that begins with the string z_0 = Conc^0 = "" and then
           continues on through the other k strings:

           i.   Conc^0_j  z_j  =  Conc^0  =  "".

           ii.  For k > 0,

                Conc^k_j  z_j  =  Prec(Conc^(k-1)_j  z_j, z_k).

   2.  The conception of the k-place surcatenation operation
       can be extended to include its natural "prequel":

       Surc^0  =  "-()-".

       Finally, the construction of the k-place surcatenation can be
       broken into stages by means of the following conceptions:

       a.  A "subclause" in !A!* is a string that ends with a ")-".

       b.  The "subcatenation" Subc(z_1, z_2)
           of a subclause z_1 by a string z_2 is
           the string that is defined as follows:

           Subc(z_1, z_2)  =  z_1 · ")-"^(-1) · "," · z_2 · ")-".

       c.  The "surcatenation" of the k strings z_1, ..., z_k can now be
           defined as an iterated subcatenation over the sequence of k+1
           strings that starts with the string z_0 = Surc^0 = "-()-" and
           then continues on through the other k strings:

           i.   Surc^0_j  z_j  =  Surc^0  =  "-()-".

           ii.  For k > 0,

                Surc^k_j  z_j  =  Subc(Surc^(k-1)_j  z_j, z_k).

Notice that the expressions Conc^0_j z_j and Surc^0_j z_j
are defined in such a way that the respective operators
Conc^0 and Surc^0 basically "ignore", in the manner of
constant functions, whatever sequences of strings z_j
may happen to be listed as their ostensible arguments.

Having defined the basic operations of concatenation and surcatenation
on arbitrary strings, in effect, giving them operational meaning for
the all-inclusive language !L! = !A!*, it is time to adjoin the
notion of a more discriminating grammaticality, in other words,
a more properly restrictive concept of a sentence.

If !L! is an arbitrary formal language over an alphabet of the sort that
we are talking about, that is, an alphabet of the form !A! = !M! |_| !P!,
then there are a number of basic structural relations that can be defined
on the strings of !L!.

   1.  z is the "concatenation" of z_1 and z_2 in !L! if and only if

       z_1 is a sentence of !L!, z_2 is a sentence of !L!, and

       z  =  z_1 · z_2.

   2.  z is the "concatenation" of the k strings z1, ..., z_k in !L!,

       if and only if z_j is a sentence of !L!, for all j = 1 to k, and

       z  =  Conc^k_j  z_j  =  z_1 · ... · z_k.

   3.  z is the "discatenation" of z_1 by t if and only if

       z_1 is a sentence of !L!, t is an element of !A!, and

       z_1  =  z · t.

       When this is the case, one more commonly writes:

       z  =  z_1 · t^-1.

   4.  z is a "subclause" of !L! if and only if

       z is a sentence of !L! and z ends with a ")-".

   5.  z is the "subcatenation" of z_1 by z_2 if and only if

       z_1 is a subclause of !L!, z_2 is a sentence of !L!, and

       z  =  z_1 · ")-"^(-1) · "," · z_2 · ")-".

   6.  z is the "surcatenation" of the k strings z_1, ..., z_k in !L!,

       if and only if z_j is a sentence of !L!, for all j = 1 to k, and

       z  =  Surc^k_j  z_j  =  "-(" · z_1 · "," · ... · "," · z_k · ")-".

The converses of these decomposition relations are tantamount to the
corresponding forms of composition operations, making it possible for
these complementary forms of analysis and synthesis to articulate the
structures of strings and sentences in two directions.

The "painted cactus language" with paints in the
set !P! = {p_j : j in J} is the formal language
!L! = !C!(!P!) c !A!* = (!M! |_| !P!)* that is
defined as follows:

   PC 1.  The blank symbol m_1 is a sentence.

   PC 2.  The paint p_j is a sentence, for each j in J.

   PC 3.  Conc^0 and Surc^0 are sentences.

   PC 4.  For each positive integer k,

          if    z_1, ..., z_k are sentences,

          then  Conc^k_j  z_j is a sentence,

          and   Surc^k_j  z_j is a sentence.

As usual, saying that z is a sentence is just a conventional way of
stating that the string z belongs to the relevant formal language !L!.
An individual sentence of !C!(!P!), for any palette !P!, is referred to
as a "painted and rooted cactus expression" (PARCE) on the palette !P!,
or a "cactus expression", for short.  Anticipating the forms that the
parse graphs of these PARCE's will take, to be described in the next
Subsection, the language !L! = !C!(!P!) is also described as the
set PARCE(!P!) of PARCE's on the palette !P!, more generically,
as the PARCE's that constitute the language PARCE.

A "bare" PARCE, a bit loosely referred to as a "bare cactus expression",
is a PARCE on the empty palette !P! = {}.  A bare PARCE is a sentence
in the "bare cactus language", !C!^0 = !C!({}) = PARCE^0 = PARCE({}).
This set of strings, regarded as a formal language in its own right,
is a sublanguage of every cactus language !C!(!P!).  A bare cactus
expression is commonly encountered in practice when one has occasion
to start with an arbitrary PARCE and then finds a reason to delete or
to erase all of its paints.

Only one thing remains to cast this description of the cactus language
into a form that is commonly found acceptable.  As presently formulated,
the principle PC 4 appears to be attempting to define an infinite number
of new concepts all in a single step, at least, it appears to invoke the
indefinitely long sequences of operators, Conc^k and Surc^k, for all k > 0.
As a general rule, one prefers to have an effectively finite description of
conceptual objects, and this means restricting the description to a finite
number of schematic principles, each of which involves a finite number of
schematic effects, that is, a finite number of schemata that explicitly
relate conditions to results.

A start in this direction, taking steps toward an effective description
of the cactus language, a finitary conception of its membership conditions,
and a bounded characterization of a typical sentence in the language, can be
made by recasting the present description of these expressions into the pattern
of what is called, more or less roughly, a "formal grammar".

A notation in the style of "S :> T" is now introduced,
to be read among many others in this manifold of ways:

   S covers T

   S governs T

   S rules T

   S subsumes T

   S types over T

The form "S :> T" is here recruited for polymorphic
employment in at least the following types of roles:

   1.  To signify that an individually named or quoted string T is
       being typed as a sentence S of the language of interest !L!.

   2.  To express the fact or to make the assertion that each member
       of a specified set of strings T c !A!* also belongs to the
       syntactic category S, the one that qualifies a string as
       being a sentence in the relevant formal language !L!.

   3.  To specify the intension or to signify the intention that every
       string that fits the conditions of the abstract type T must also
       fall under the grammatical heading of a sentence, as indicated by
       the type name "S", all within the target language !L!.

In these types of situation the letter "S", that signifies the type of
a sentence in the language of interest, is called the "initial symbol"
or the "sentence symbol" of a candidate formal grammar for the language,
while any number of letters like "T", signifying other types of strings
that are necessary to a reasonable account or a rational reconstruction
of the sentences that belong to the language, are collectively referred
to as "intermediate symbols".

Combining the singleton set {"S"} whose sole member is the initial symbol
with the set !Q! that assembles together all of the intermediate symbols
results in the set {"S"} |_| !Q! of "non-terminal symbols".  Completing
the package, the alphabet !A! of the language is also known as the set
of "terminal symbols".  In this discussion, I will adopt the convention
that !Q! is the set of intermediate symbols, but I will often use "q"
as a typical variable that ranges over all of the non-terminal symbols,
q in {"S"} |_| !Q!.  Finally, it is convenient to refer to all of the
symbols in {"S"} |_| !Q! |_| !A! as the "augmented alphabet" of the
prospective grammar for the language, and accordingly to describe
the strings in ({"S"} |_| !Q! |_| !A!)* as the "augmented strings",
in effect, expressing the forms that are superimposed on a language
by one of its conceivable grammars.  In certain settings it becomes
desirable to separate the augmented strings that contain the symbol
"S" from all other sorts of augmented strings.  In these situations,
the strings in the disjoint union {"S"} |_| (!Q! |_| !A!)* are known
as the "sentential forms" of the associated grammar.

In forming a grammar for a language, statements of the form W :> W',
where W and W' are augmented strings or sentential forms of specified
types that depend on the style of the grammar that is being sought, are
variously known as "characterizations", "covering rules", "productions",
"rewrite rules", "subsumptions", "transformations", or "typing rules".
These are collected together into a set !K! that serves to complete
the definition of the formal grammar in question.

Correlative with the use of this notation, an expression of the
form "T <: S", read as "T is covered by S", can be interpreted
as saying that T is of the type S.  Depending on the context,
this can be taken in either one of two ways:

   1.  Treating "T" as a string variable, it means
       that the individual string T is typed as S.

   2.  Treating "T" as a type name, it means that any
       instance of the type T also falls under the type S.

In accordance with these interpretations, an expression like "t <: T" can be
read in all of the ways that one typically reads an expression like "t : T".

There are several abuses of notation that commonly tolerated in the use
of covering relations.  The worst offense is that of allowing symbols to
stand equivocally either for individual strings or else for their types.
There is a measure of consistency to this practice, considering the fact
that perfectly individual entities are rarely if ever grasped by means of
signs and finite expressions, which entails that every appearance of an
apparent token is only a type of more particular tokens, and meaning in
the end that there is never any recourse but to the sort of discerning
interpretation that can decide just how each sign is intended.  In view
of all this, I continue to permit expressions like "t <: T" and "T <: S",
where any of the symbols "t", "T", "S" can be taken to signify either the
tokens or the subtypes of their covering types.

Employing the notion of a covering relation it becomes possible to
redescribe the cactus language !L! = !C!(!P!) in the following way.

Grammar 1 is something of a misnomer.  It is nowhere near exemplifying
any kind of a standard form and it is only intended as a starting point
for the initiation of more respectable grammars.  Such as it is, it uses
the terminal alphabet !A! = !M! |_| !P! that comes with the territory of
the cactus language !C!(!P!), it specifies !Q! = {}, in other words, it
employs no intermediate symbols, and it embodies the "covering set" !K!
as listed in the following display.

o-------------------------------------------------o
| !C!(!P!).  Grammar 1                  !Q! = {}  |
o-------------------------------------------------o
|                                                 |
| 1.  S  :>  m_1  =  " "                          |
|                                                 |
| 2.  S  :>  p_j, for each j in J                 |
|                                                 |
| 3.  S  :>  Conc^0  =  ""                        |
|                                                 |
| 4.  S  :>  Surc^0  =  "-()-"                    |
|                                                 |
| 5.  S  :>  S*                                   |
|                                                 |
| 6.  S  :>  "-(" · S · ("," · S)* · ")-"         |
|                                                 |
o-------------------------------------------------o

In this formulation, the last two lines specify that:

   5.  The concept of a sentence in !L! covers any
       concatenation of sentences in !L!, in effect,
       any number of freely chosen sentences that are
       available to be concatenated one after another.

   6.  The concept of a sentence in !L! covers any
       surcatenation of sentences in !L!, in effect,
       any string that opens with a "-(", continues
       with a sentence, possibly empty, follows with
       a finite number of phrases of the form "," · S,
       and closes with a ")-".

This appears to be just about the most concise description
of the cactus language !C!(!P!) that one can imagine, but
there exist a couple of problems that are commonly felt
to afflict this style of presentation and to make it
less than completely acceptable.  Briefly stated,
these problems turn on the following properties
of the presentation:

   a.  The invocation of the kleene star operation
       is not reduced to a manifestly finitary form.

   b.  The type of a sentence S is allowed to cover
       not only itself but also the empty string.

I will discuss these issues at first in general, and especially in regard to
how the two features interact with one another, and then I return to address
in further detail the questions that they engender on their individual bases.

In the process of developing a grammar for a language, it is possible
to notice a number of organizational, pragmatic, and stylistic questions,
whose moment to moment answers appear to decide the ongoing direction of the
grammar that develops and the impact of whose considerations work in tandem
to determine, or at least to influence, the sort of grammar that turns out.
The issues that I can see arising at this point I can give the following
prospective names, putting off the discussion of their natures and the
treatment of their details to the points in the development of the
present example where they evolve their full import.

   1.  The "degree of intermediate organization" in a grammar.

   2.  The "distinction between empty and significant strings", and thus
       the "distinction between empty and significant types of strings".

   3.  The "principle of intermediate significance".  This is a constraint
       on the grammar that arises from considering the interaction of the
       first two issues.

In responding to these issues, it is advisable at first to proceed in
a stepwise fashion, all the better thereby to accommodate the chances
of pursuing a series of parallel developments in the grammar, to allow
for the possibility of reversing many steps in its development, indeed,
to take into account the near certain necessity of having to revisit,
to revise, and to reverse many decisions about how to proceed toward
an optimal description or a satisfactory grammar for the language.
Doing all this means exploring the effects of various alterations
and innovations as independently from each other as possible.

The degree of intermediate organization in a grammar is measured by how many
intermediate symbols it has and by how they interact with each other by means
of its productions.  With respect to this issue, Grammar 1 has no intermediate
symbols at all, !Q! = {}, and therefore remains at an ostensibly trivial degree
of intermediate organization.  Some additions to the list of intermediate symbols
are practically obligatory in order to arrive at any reasonable grammar at all,
other inclusions appear to have a more optional character, though obviously
useful from the standpoints of clarity and ease of comprehension.

One of the troubles that is perceived to affect Grammar 1 is that it wastes
so much of the available potential for efficient description in recounting
over and over again the simple fact that the empty string is present in
the language.  This arises in part from the statement that S :> S*,
which implies that:

   S  :>  S*  =  %e% |_| S |_| S · S |_| S · S · S |_| ...

There is nothing wrong with the more expansive pan of the covered equation,
since it follows straightforwardly from the definition of the kleene star
operation, but the covering statement, to the effect that S :> S*, is not
necessarily a very productive piece of information, to the extent that it
does always tell us very much about the language that is being supposed to
fall under the type of a sentence S.  In particular, since it implies that
S :> %e%, and since !L!  =  %e%·!L!  =  !L!·%e%, for any formal language !L!,
the empty string !e! = "" is counted over and over in every term of the union,
and every non-empty sentence under S appears again and again in every term of
the union that follows the initial appearance of S.  As a result, this style
of characterization has to be classified as "true but not very informative".
If at all possible, one prefers to partition the language of interest into
a disjoint union of subsets, thereby accounting for each sentence under
its proper term, and one whose place under the sum serves as a useful
parameter of its character or its complexity.  In general, this form
of description is not always possible to achieve, but it is usually
worth the trouble to actualize it whenever it is.

Suppose that one tries to deal with this problem by eliminating each use of
the kleene star operation, by reducing it to a purely finitary set of steps,
or by finding an alternative way to cover the sublanguage that it is used to
generate.  This amounts, in effect, to "recognizing a type", a complex process
that involves the following steps:

   1.  Noticing a category of strings that
       is generated by iteration or recursion.

   2.  Acknowledging the circumstance that the noted category
       of strings needs to be covered by a non-terminal symbol.

   3.  Making a note of it by declaring and instituting
       an explicitly and even expressively named category.

In sum, one introduces a non-terminal symbol for each type of sentence and
each "part of speech" or sentential component that is generated by means of
iteration or recursion under the ruling constraints of the grammar.  In order
to do this one needs to analyze the iteration of each grammatical operation in
a way that is analogous to a mathematically inductive definition, but further in
a way that is not forced explicitly to recognize a distinct and separate type of
expression merely to account for and to recount every increment in the parameter
of iteration.

Returning to the case of the cactus language, the process of recognizing an
iterative type or a recursive type can be illustrated in the following way.
The operative phrases in the simplest sort of recursive definition are its
initial part and its generic part.  For the cactus language !C!(!P!), one
has the following definitions of concatenation as iterated precatenation
and of surcatenation as iterated subcatenation, respectively:

   1.  Conc^0        =  ""

       Conc^k_j S_j  =  Prec(Conc^(k-1)_j S_j, S_k)

   2.  Surc^0        =  "-()-"

       Surc^k_j S_j  =  Subc(Surc^(k-1)_j S_j, S_k)

In order to transform these recursive definitions into grammar rules,
one introduces a new pair of intermediate symbols, "Conc" and "Surc",
corresponding to the operations that share the same names but ignoring
the inflexions of their individual parameters j and k.  Recognizing the
type of a sentence by means of the initial symbol "S", and interpreting
"Conc" and "Surc" as names for the types of strings that are generated
by concatenation and by surcatenation, respectively, one arrives at
the following transformation of the ruling operator definitions
into the form of covering grammar rules:

   1.  Conc  :>  ""

       Conc  :>  Conc · S

   2.  Surc  :>  "-()-"

       Surc  :>  "-(" · S · ")-"

       Surc  :>  Surc ")-"^(-1) · "," · S · ")-"

As given, this particular fragment of the intended grammar
contains a couple of features that are desirable to amend.

   1.  Given the covering S :> Conc, the covering rule Conc :> Conc · S
       says no more than the covering rule Conc :> S · S.  Consequently,
       all of the information contained in these two covering rules is
       already covered by the statement that S :> S · S.

   2.  A grammar rule that invokes a notion of decatenation, deletion, erasure,
       or any other sort of retrograde production, is frequently considered to
       be lacking in elegance, and a there is a style of critique for grammars
       that holds it preferable to avoid these types of operations if it is at
       all possible to do so.  Accordingly, contingent on the prescriptions of
       the informal rule in question, and pursuing the stylistic dictates that
       are writ in the realm of its aesthetic regime, it becomes necessary for
       us to backtrack a little bit, to temporarily withdraw the suggestion of
       employing these elliptical types of operations, but without, of course,
       eliding the record of doing so.

One way to analyze the surcatenation of any number of sentences is to
introduce an auxiliary type of string, not in general a sentence, but
a proper component of any sentence that is formed by surcatenation.
Doing this brings one to the following definition:

A "tract" is a concatenation of a finite sequence of sentences, with a
literal comma "," interpolated between each pair of adjacent sentences.
Thus, a typical tract T takes the form:

   T  =  S_1 · "," · ...  · "," · S_k

A tract must be distinguished from the abstract sequence of sentences,
S_1, ..., S_k, where the commas that appear to come to mind, as if being
called up to separate the successive sentences of the sequence, remain as
partially abstract conceptions, or as signs that retain a disengaged status
on the borderline between the text and the mind.  In effect, the types of
commas that appear to follow in the abstract sequence continue to exist
as conceptual abstractions and fail to be cognized in a wholly explicit
fashion, whether as concrete tokens in the object language, or as marks
in the strings of signs that are able to engage one's parsing attention.

Returning to the case of the painted cactus language !L! = !C!(!P!),
it is possible to put the currently assembled pieces of a grammar
together in the light of the presently adopted canons of style,
to arrive a more refined analysis of the fact that the concept
of a sentence covers any concatenation of sentences and any
surcatenation of sentences, and so to obtain the following
form of a grammar:

o-------------------------------------------------o
| !C!(!P!).  Grammar 2               !Q! = {"T"}  |
o-------------------------------------------------o
|                                                 |
| 1.  S  :>  !e!                                  |
|                                                 |
| 2.  S  :>  m_1                                  |
|                                                 |
| 3.  S  :>  p_j, for each j in J                 |
|                                                 |
| 4.  S  :>  S · S                                |
|                                                 |
| 5.  S  :>  "-(" · T · ")-"                      |
|                                                 |
| 6.  T  :>  S                                    |
|                                                 |
| 7.  T  :>  T · "," · S                          |
|                                                 |
o-------------------------------------------------o

In this rendition, a string of type T is not in general
a sentence itself but a proper "part of speech", that is,
a strictly "lesser" component of a sentence in any suitable
ordering of sentences and their components.  In order to see
how the grammatical category T gets off the ground, that is,
to detect its minimal strings and to discover how its ensuing
generations gets started from these, it is useful to observe
that the covering rule T :> S means that T "inherits" all of
the initial conditions of S, namely, T  :>  !e!, m_1, p_j.
In accord with these simple beginnings it comes to parse
that the rule T :> T · "," · S, with the substitutions
T = !e! and S = !e! on the covered side of the rule,
bears the germinal implication that T :> ",".

Grammar 2 achieves a portion of its success through a higher degree of
intermediate organization.  Roughly speaking, the level of organization
can be seen as reflected in the cardinality of the intermediate alphabet
!Q! = {"T"}, but it is clearly not explained by this simple circumstance
alone, since it is taken for granted that the intermediate symbols serve
a purpose, a purpose that is easily recognizable but that may not be so
easy to pin down and to specify exactly.  Nevertheless, it is worth the
trouble of exploring this aspect of organization and this direction of
development a little further.  Although it is not strictly necessary
to do so, it is possible to organize the materials of the present
grammar in a slightly better fashion by recognizing two recurrent
types of strings that appear in the typical cactus expression.
In doing this, one arrives at the following two definitions:

A "rune" is a string of blanks and paints concatenated together.
Thus, a typical rune R is a string over {m_1} |_| !P!, possibly
the empty string.

   R  in  ({m_1} |_| !P!)*.

When there is no possibility of confusion, the letter "R" can be used
either as a string variable that ranges over the set of runes or else
as a type name for the class of runes.  The latter reading amounts to
the enlistment of a fresh intermediate symbol, "R" in !Q!, as a part
of a new grammar for !C!(!P!).  In effect, "R" affords a grammatical
recognition for any rune that forms a part of a sentence in !C!(!P!).
In situations where these variant usages are likely to be confused,
the types of strings can be indicated by means of expressions like
"r <: R" and "W <: R".

A "foil" is a string of the form "-(" · T · ")-", where T is a tract.
Thus, a typical foil F has the form:

   F  =  "-(" · S_1 · "," · ... · "," · S_k · ")-".

This is just the surcatenation of the sentences S_1, ..., S_k.
Given the possibility that this sequence of sentences is empty,
and thus that the tract T is the empty string, the minimum foil
F is the expression "-()-".  Explicitly marking each foil F that
is embodied in a cactus expression is tantamount to recognizing
another intermediate symbol, "F" in !Q!, further articulating the
structures of sentences and expanding the grammar for the language
!C!(!P!).  All of the same remarks about the versatile uses of the
intermediate symbols, as string variables and as type names, apply
again to the letter "F".

o-------------------------------------------------o
| !C!(!P!).  Grammar 3     !Q! = {"F", "R", "T"}  |
o-------------------------------------------------o
|                                                 |
|  1.  S  :>  R                                   |
|                                                 |
|  2.  S  :>  F                                   |
|                                                 |
|  3.  S  :>  S · S                               |
|                                                 |
|  4.  R  :>  !e!                                 |
|                                                 |
|  5.  R  :>  m_1                                 |
|                                                 |
|  6.  R  :>  p_j, for each j in J                |
|                                                 |
|  7.  R  :>  R · R                               |
|                                                 |
|  8.  F  :>  "-(" · T · ")-"                     |
|                                                 |
|  9.  T  :>  S                                   |
|                                                 |
| 10.  T  :>  T · "," · S                         |
|                                                 |
o-------------------------------------------------o

In Grammar 3, the first three Rules say that a sentence (a string of type S),
is a rune (a string of type R), a foil (a string of type F), or an arbitrary
concatenation of strings of these two types.  Rules 4 through 7 specify that
a rune R is an empty string !e! = "", a blank symbol m_1 = " ", a paint p_j,
for j in J, or any concatenation of strings of these three types.  Rule 8
characterizes a foil F as a string of the form "-(" · T · ")-", where T is
a tract.  The last two Rules say that a tract T is either a sentence S or
else the concatenation of a tract, a comma, and a sentence, in that order.

At this point in the succession of grammars for !C!(!P!), the explicit
uses of indefinite iterations, like the kleene star operator, are now
completely reduced to finite forms of concatenation, but the problems
that some styles of analysis have with allowing non-terminal symbols
to cover both themselves and the empty string are still present.

Any degree of reflection on this difficulty raises the general question:
What is a practical strategy for accounting for the empty string in the
organization of any formal language that counts it among its sentences?
One answer that presents itself is this:  If the empty string belongs to
a formal language, it suffices to count it once at the beginning of the
formal account that enumerates its sentences and then to move on to more
interesting materials.

Returning to the case of the cactus language !C!(!P!), that is,
the formal language of "painted and rooted cactus expressions",
it serves the purpose of efficient accounting to partition the
language PARCE into the following couple of sublanguages:

   1.  The "emptily painted and rooted cactus expressions"
       make up the language EPARCE that consists of
       a single empty string as its only sentence.
       In short:

       EPARCE  =  {""}.

   2.  The "significantly painted and rooted cactus expressions"
       make up the language SPARCE that consists of everything else,
       namely, all of the non-empty strings in the language PARCE.
       In sum:

       SPARCE  =  PARCE \ "".

As a result of marking the distinction between empty and significant sentences,
that is, by categorizing each of these three classes of strings as an entity
unto itself and by conceptualizing the whole of its membership as falling
under a distinctive symbol, one obtains an equation of sets that connects
the three languages being marked:

   SPARCE  =  PARCE - EPARCE.

In sum, one has the disjoint union:

   PARCE   =  EPARCE |_| SPARCE.

For brevity in the present case, and to serve as a generic device
in any similar array of situations, let the symbol "S" be used to
signify the type of an arbitrary sentence, possibly empty, whereas
the symbol "S'" is reserved to designate the type of a specifically
non-empty sentence.  In addition, let the symbol "%e%" be employed
to indicate the type of the empty sentence, in effect, the language
%e% = {""} that contains a single empty string, and let a plus sign
"+" signify a disjoint union of types.  In the most general type of
situation, where the type S is permitted to include the empty string,
one notes the following relation among types:

   S  =  %e%  +  S'.

Consequences of the distinction between empty expressions and
significant expressions are taken up for discussion next time.

With the distinction between empty and significant expressions in mind,
I return to the grasp of the cactus language !L! = !C!(!P!) = PARCE(!P!)
that is afforded by Grammar 2, and, taking that as a point of departure,
explore other avenues of possible improvement in the comprehension of
these expressions.  In order to observe the effects of this alteration
as clearly as possible, in isolation from any other potential factors,
it is useful to strip away the higher levels intermediate organization
that are present in Grammar 3, and start again with a single intermediate
symbol, as used in Grammar 2.  One way of carrying out this strategy leads
on to a grammar of the variety that will be articulated next.

If one imposes the distinction between empty and significant types on
each non-terminal symbol in Grammar 2, then the non-terminal symbols
"S" and "T" give rise to the non-terminal symbols "S", "S'", "T", "T'",
leaving the last three of these to form the new intermediate alphabet.
Grammar 4 has the intermediate alphabet !Q! = {"S'", "T", "T'"}, with
the set !K! of covering production rules as listed in the next display.

o-------------------------------------------------o
| !C!(!P!).  Grammar 4   !Q! = {"S'", "T", "T'"}  |
o-------------------------------------------------o
|                                                 |
| 1.  S   :>  !e!                                 |
|                                                 |
| 2.  S   :>  S'                                  |
|                                                 |
| 3.  S'  :>  m_1                                 |
|                                                 |
| 4.  S'  :>  p_j, for each j in J                |
|                                                 |
| 5.  S'  :>  "-(" · T · ")-"                     |
|                                                 |
| 6.  S'  :>  S' · S'                             |
|                                                 |
| 7.  T   :>  !e!                                 |
|                                                 |
| 8.  T   :>  T'                                  |
|                                                 |
| 9.  T'  :>  T · "," · S                         |
|                                                 |
o-------------------------------------------------o

In this version of a grammar for !L! = !C!(!P!), the intermediate type T
is partitioned as T = %e% + T', thereby parsing the intermediate symbol T
in parallel fashion with the division of its overlying type as S = %e% + S'.
This is an option that I will choose to close off for now, but leave it open
to consider at a later point.  Thus, it suffices to give a brief discussion
of what it involves, in the process of moving on to its chief alternative.

There does not appear to be anything radically wrong with trying this
approach to types.  It is reasonable and consistent in its underlying
principle, and it provides a rational and a homogeneous strategy toward
all parts of speech, but it does require an extra amount of conceptual
overhead, in that every non-trivial type has to be split into two parts
and comprehended in two stages.  Consequently, in view of the largely
practical difficulties of making the requisite distinctions for every
intermediate symbol, it is a common convention, whenever possible, to
restrict intermediate types to covering exclusively non-empty strings.

For the sake of future reference, it is convenient to refer to this restriction
on intermediate symbols as the "intermediate significance" constraint.  It can
be stated in a compact form as a condition on the relations between non-terminal
symbols q in {"S"} |_| !Q! and sentential forms W in {"S"} |_| (!Q! |_| !A!)*.

o-------------------------------------------------o
| Condition On Intermediate Significance          |
o-------------------------------------------------o
|                                                 |
| If    q  :>  W                                  |
|                                                 |
| and   W  =  !e!                                 |
|                                                 |
| then  q  =  "S"                                 |
|                                                 |
o-------------------------------------------------o

If this is beginning to sound like a monotone condition, then it is
not absurd to sharpen the resemblance and render the likeness more
acute.  This is done by declaring a couple of ordering relations,
denoting them under variant interpretations by the same sign "<".

   1.  The ordering "<" on the set of non-terminal symbols,
       q in {"S"} |_| !Q!, ordains the initial symbol "S"
       to be strictly prior to every intermediate symbol.
       This is tantamount to the axiom that "S" < q,
       for all q in !Q!.

   2.  The ordering "<" on the collection of sentential forms,
       W in {"S"} |_| (!Q! |_| !A!)*, ordains the empty string
       to be strictly minor to every other sentential form.
       This is stipulated in the axiom that !e! < W,
       for every non-empty sentential form W.

Given these two orderings, the constraint in question
on intermediate significance can be stated as follows:

o-------------------------------------------------o
| Condition Of Intermediate Significance          |
o-------------------------------------------------o
|                                                 |
| If    q  :>  W                                  |
|                                                 |
| and   q  >  "S"                                 |
|                                                 |
| then  W  >  !e!                                 |
|                                                 |
o-------------------------------------------------o

Achieving a grammar that respects this convention typically requires a more
detailed account of the initial setting of a type, both with regard to the
type of context that incites its appearance and also with respect to the
minimal strings that arise under the type in question.  In order to find
covering productions that satisfy the intermediate significance condition,
one must be prepared to consider a wider variety of calling contexts or
inciting situations that can be noted to surround each recognized type,
and also to enumerate a larger number of the smallest cases that can
be observed to fall under each significant type.

With the array of foregoing considerations in mind,
one is gradually led to a grammar for !L! = !C!(!P!)
in which all of the covering productions have either
one of the following two forms:

   1.  S  :>  !e!

   2.  q  :>   W,  with  q in {"S"} |_| !Q!,  and  W in (!Q! |_| !A!)^+

A grammar that fits into this mold is called a "context-free" grammar.
The first type of rewrite rule is referred to as a "special production",
while the second type of rewrite rule is called an "ordinary production".
An "ordinary derivation" is one that employs only ordinary productions.
In ordinary productions, those that have the form q :> W, the replacement
string W is never the empty string, and so the lengths of the augmented
strings or the sentential forms that follow one another in an ordinary
derivation, on account of using the ordinary types of rewrite rules,
never decrease at any stage of the process, up to and including the
terminal string that is finally generated by the grammar.  This type
of feature is known as the "non-contracting property" of productions,
derivations, and grammars.  A grammar is said to have the property if
all of its covering productions, with the possible exception of S :> e,
are non-contracting.  In particular, context-free grammars are special
cases of non-contracting grammars.  The presence of the non-contracting
property within a formal grammar makes the length of the augmented string
available as a parameter that can figure into mathematical inductions and
motivate recursive proofs, and this handle on the generative process makes
it possible to establish the kinds of results about the generated language
that are not easy to achieve in more general cases, nor by any other means
even in these brands of special cases.

Grammar 5 is a context-free grammar for the painted cactus language
that uses !Q! = {"S'", "T"}, with !K! as listed in the next display.

o-------------------------------------------------o
| !C!(!P!).  Grammar 5         !Q! = {"S'", "T"}  |
o-------------------------------------------------o
|                                                 |
|  1.  S   :>  !e!                                |
|                                                 |
|  2.  S   :>  S'                                 |
|                                                 |
|  3.  S'  :>  m_1                                |
|                                                 |
|  4.  S'  :>  p_j, for each j in J               |
|                                                 |
|  5.  S'  :>  S' · S'                            |
|                                                 |
|  6.  S'  :>  "-()-"                             |
|                                                 |
|  7.  S'  :>  "-(" · T · ")-"                    |
|                                                 |
|  8.  T   :>  ","                                |
|                                                 |
|  9.  T   :>  S'                                 |
|                                                 |
| 10.  T   :>  T · ","                            |
|                                                 |
| 11.  T   :>  T · "," · S'                       |
|                                                 |
o-------------------------------------------------o

Finally, it is worth trying to bring together the advantages of these
diverse styles of grammar, to whatever extent that they are compatible.
To do this, a prospective grammar must be capable of maintaining a high
level of intermediate organization, like that arrived at in Grammar 2,
while respecting the principle of intermediate significance, and thus
accumulating all the benefits of the context-free format in Grammar 5.
A plausible synthesis of most of these features is given in Grammar 6.

o-----------------------------------------------------------o
| !C!(!P!).  Grammar 6         !Q! = {"S'", "R", "F", "T"}  |
o-----------------------------------------------------------o
|                                                           |
|  1.  S   :>  !e!                                          |
|                                                           |
|  2.  S   :>  S'                                           |
|                                                           |
|  3.  S'  :>  R                                            |
|                                                           |
|  4.  S'  :>  F                                            |
|                                                           |
|  5.  S'  :>  S' · S'                                      |
|                                                           |
|  6.  R   :>  m_1                                          |
|                                                           |
|  7.  R   :>  p_j, for each j in J                         |
|                                                           |
|  8.  R   :>  R · R                                        |
|                                                           |
|  9.  F   :>  "-()-"                                       |
|                                                           |
| 10.  F   :>  "-(" · T · ")-"                              |
|                                                           |
| 11.  T   :>  ","                                          |
|                                                           |
| 12.  T   :>  S'                                           |
|                                                           |
| 13.  T   :>  T · ","                                      |
|                                                           |
| 14.  T   :>  T · "," · S'                                 |
|                                                           |
o-----------------------------------------------------------o

The preceding development provides a typical example of how an initially
effective and conceptually succinct description of a formal language, but
one that is terse to the point of allowing its prospective interpreter to
waste exorbitant amounts of energy in trying to unravel its implications,
can be converted into a form that is more efficient from the operational
point of view, even if slightly more ungainly in regard to its elegance.

The basic idea behind all of this grammatical machinery remains the same:
Aside from the select body of formulas introduced as boundary conditions,
a grammar for the cactus language is nothing more or less than a device
that institutes the following general rule:

   If    the strings S_1, ..., S_k are sentences,

   then  their concatenation in the form

         Conc^k_j S_j  =  S_1 · ... · S_k

         is a sentence,

   and   their surcatenation in the form

         Surc^k_j S_j  =  "-(" · S_1 · "," · ... · "," · S_k · ")-"

         is a sentence.

It is fitting to wrap up the foregoing developments by summarizing the
notion of a formal grammar that appeared to evolve in the present case.
For the sake of future reference and the chance of a wider application,
it is also useful to try to extract the scheme of a formalization that
potentially holds for any formal language.  The following presentation
of the notion of a formal grammar is adapted, with minor modifications,
from the treatment in (DDQ, 60-61).

A "formal grammar" !G! is given by a four-tuple !G! = ("S", !Q!, !A!, !K!)
that takes the following form of description:

   1.  "S" is the "initial", "special", "start", or "sentence symbol".
       Since the letter "S" serves this function only in a special setting,
       its employment in this role need not create any confusion with its
       other typical uses as a string variable or as a sentence variable.

   2.  !Q! = {q_1, ..., q_m} is a finite set of "intermediate symbols",
       all distinct from "S".

   3.  !A! = {a_1, ..., a_n} is a finite set of "terminal symbols",
       also known as the "alphabet" of !G!, all distinct from "S" and
       disjoint from !Q!.  Depending on the particular conception of the
       language !L! that is "covered", "generated", "governed", or "ruled"
       by the grammar !G!, that is, whether !L! is conceived to be a set of
       words, sentences, paragraphs, or more extended structures of discourse,
       it is usual to describe !A! as the "alphabet", "lexicon", "vocabulary",
       "liturgy", or "phrase book" of both the grammar !G! and the language !L!
       that it regulates.

   4.  !K! is a finite set of "characterizations".  Depending on how they
       come into play, these are variously described as "covering rules",
       "formations", "productions", "rewrite rules", "subsumptions",
       "transformations", or "typing rules".

To describe the elements of !K! it helps to define some additional terms:

   a.  The symbols in {"S"} |_| !Q! |_| !A! form the "augmented alphabet" of !G!.

   b.  The symbols in {"S"} |_| !Q! are the "non-terminal symbols" of !G!.

   c.  The symbols in !Q! |_| !A! are the "non-initial symbols" of !G!.

   d.  The strings in ({"S"} |_| !Q! |_| !A!)*  are the "augmented strings" for G.

   e.  The strings in {"S"} |_| (!Q! |_| !A!)* are the "sentential forms" for G.

Each characterization in !K! is an ordered pair of strings (S_1, S_2)
that takes the following form:

   S_1  =  Q_1 · q · Q_2

   S_2  =  Q_1 · W · Q_2

In this scheme, S_1 and S_2 are members of the augmented strings for !G!,
more precisely, S_1 is a non-empty string and a sentential form over !G!,
while S_2 is a possibly empty string and also a sentential form over !G!.

Here also, q is a non-terminal symbol, that is, q is in {"S"} |_| !Q!,
while Q_1, Q_2, and W are possibly empty strings of non-initial symbols,
a fact that can be expressed in the form:  Q_1, Q_2, W in (!Q! |_| !A!)*.

In practice, the ordered pairs of strings in !K! are used to "derive",
to "generate", or to "produce" sentences of the language !L! = <!G!>
that is then said to be "governed" or "regulated" by the grammar !G!.
In order to facilitate this active employment of the grammar, it is
conventional to write the characterization (S_1, S_2) in either one
of the next two forms, where the more generic form is followed by
the more specific form:

   S_1             :>   S_2

   Q_1 · q · Q_2   :>   Q_1 · W · Q_2

In this usage, the characterization S_1 :> S_2 is tantamount to a grammatical
license to transform a string of the form Q_1 · q · Q_2 into a string of the
form Q1 · W · Q2, in effect, replacing the non-terminal symbol q with the
non-initial string W in any selected, preserved, and closely adjoining
context of the form Q1 · ... · Q2.  Accordingly, in this application
the notation "S_1 :> S_2" can be read as "S_1 produces S_2" or as
"S_1 transforms into S_2".

An "immediate derivation" in !G! is an ordered pair (W, W')
of sentential forms in !G! such that:

   W   =  Q_1 · X · Q_2

   W'  =  Q_1 · Y · Q_2

   and  (X, Y)   in !K!

   i.e.  X :> Y  in !G!

This relation is indicated by saying that W "immediately derives" W',
that W' is "immediately derived" from W in !G!, and also by writing:

   W  ::>  W'

A "derivation" in !G! is a finite sequence (W_1, ..., W_k)
of sentential forms over !G! such that each adjacent pair
(W_j, W_(j+1)) of sentential forms in the sequence is an
immediate derivation in !G!, in other words, such that:

   W_j  ::>  W_(j+1),  for all j = 1 to k-1

If there exists a derivation (W_1, ..., W_k) in !G!,
one says that W_1 "derives" W_k in !G!, conversely,
that W_k is "derivable" from W_1 in !G!, and one
typically summarizes the derivation by writing:

   W_1  :*:>  W_k

The language !L! = !L!(!G!) = <!G!> that is "generated"
by the formal grammar !G! = ("S", !Q!, !A!, !K!) is the
set of strings over the terminal alphabet !A! that are
derivable from the initial symbol "S" by way of the
intermediate symbols in !Q! according to the
characterizations in K.  In sum:

   !L!(!G!)  =  <!G!>  =  {W in !A!*  :  "S" :*:> W}

Finally, a string W is called a "word", a "sentence", or so on,
of the language generated by !G! if and only if W is in !L!(!G!).

Reference:

| Denning, P.J., Dennis, J.B., Qualitz, J.E.,
|'Machines, Languages, and Computation',
| Prentice-Hall, Englewood Cliffs, NJ, 1978.

1.3.11.3. The Cactus Language : Stylistics

| As a result, we can hardly conceive of how many possibilities there are for what
| we call objective reality.  Our sharp quills of knowledge are so narrow and so
| concentrated in particular directions that with science there are myriads of
| totally different real worlds, each one accessible from the next simply by
| slight alterations -- shifts of gaze -- of every particular discipline
| and subspecialty.
|
| Herbert J. Bernstein, "Idols", p. 38.
|
| Herbert J. Bernstein,
|"Idols of Modern Science and the Reconstruction of Knowledge", pp. 37-68 in:
|
| Marcus G. Raskin & Herbert J. Bernstein,
|'New Ways of Knowing:  The Sciences, Society, and Reconstructive Knowledge',
| Rowman & Littlefield, Totowa, NJ, 1987.

This Subsection highlights an issue of "style" that arises in describing
a formal language.  In broad terms, I use the word "style" to refer to a
loosely specified class of formal systems, typically ones that have a set
of distinctive features in common.  For instance, a style of proof system
usually dictates one or more rules of inference that are acknowledged as
conforming to that style.  In the present context, the word "style" is a
natural choice to characterize the varieties of formal grammars, or any
other sorts of formal systems that can be contemplated for deriving the
sentences of a formal language.

In looking at what seems like an incidental issue, the discussion arrives
at a critical point.  The question is:  What decides the issue of style?
Taking a given language as the object of discussion, what factors enter
into and determine the choice of a style for its presentation, that is,
a particular way of arranging and selecting the materials that come to
be involved in a description, a grammar, or a theory of the language?
To what degree is the determination accidental, empirical, pragmatic,
rhetorical, or stylistic, and to what extent is the choice essential,
logical, and necessary?  For that matter, what determines the order
of signs in a word, a sentence, a text, or a discussion?  All of
the corresponding parallel questions about the character of this
choice can be posed with regard to the constituent part as well
as with regard to the main constitution of the formal language.

In order to answer this sort of question, at any level of articulation,
one has to inquire into the type of distinction that it invokes, between
arrangements and orders that are essential, logical, and necessary and
orders and arrangements that are accidental, rhetorical, and stylistic.
As a rough guide to its comprehension, a "logical order", if it resides
in the subject at all, can be approached by considering all of the ways
of saying the same things, in all of the languages that are capable of
saying roughly the same things about that subject.  Of course, the "all"
that appears in this rule of thumb has to be interpreted as a reasonably
qualified type of universal.  For all practical purposes, it simply means
"all of the ways that a person can think of" and "all of the languages
that a person can conceive of", with all things being relative to the
particular moment of investigation.  For all of these reasons, the rule
must stand as little more than a rough idea of how to approach its object.

If it is demonstrated that a given formal language can be presented in
any one of several styles of formal grammar, then the choice of a format
is accidental, optional, and stylistic to the very extent that it is free.
But if it can be shown that a particular language cannot be successfully
presented in a particular style of grammar, then the issue of style is
no longer free and rhetorical, but becomes to that very degree essential,
necessary, and obligatory, in other words, a question of the objective
logical order that can be found to reside in the object language.

As a rough illustration of the difference between logical and rhetorical
orders, consider the kinds of order that are expressed and exhibited in
the following conjunction of implications:

   X => Y  and  Y => Z

Here, there is a happy conformity between the logical content and the
rhetorical form, indeed, to such a degree that one hardly notices the
difference between them.  The rhetorical form is given by the order
of sentences in the two implications and the order of implications
in the conjunction.  The logical content is given by the order of
propositions in the extended implicational sequence:

   X  =<  Y  =<  Z

To see the difference between form and content, or manner and matter,
it is enough to observe a few of the ways that the expression can be
varied without changing its meaning, for example:

   Z <= Y  and  Y <= X

Any style of declarative programming, also called "logic programming",
depends on a capacity, as embodied in a programming language or other
formal system, to describe the relation between problems and solutions
in logical terms.  A recurring problem in building this capacity is in
bridging the gap between ostensibly non-logical orders and the logical
orders that are used to describe and to represent them.  For instance,
to mention just a couple of the most pressing cases, and the ones that
are currently proving to be the most resistant to a complete analysis,
one has the orders of dynamic evolution and rhetorical transition that
manifest themselves in the process of inquiry and in the communication
of its results.

This patch of the ongoing discussion is concerned with describing a
particular variety of formal languages, whose typical representative
is the painted cactus language !L! = !C!(!P!).  It is the intention of
this work to interpret this language for propositional logic, and thus
to use it as a sentential calculus, an order of reasoning that forms an
active ingredient and a significant component of all logical reasoning.
To describe this language, the standard devices of formal grammars and
formal language theory are more than adequate, but this only raises the
next question:  What sorts of devices are exactly adequate, and fit the
task to a "T"?  The ultimate desire is to turn the tables on the order
of description, and so begins a process of eversion that evolves to the
point of asking:  To what extent can the language capture the essential
features and laws of its own grammar and describe the active principles
of its own generation?  In other words:  How well can the language be
described by using the language itself to do so?

In order to speak to these questions, I have to express what a grammar says
about a language in terms of what a language can say on its own.  In effect,
it is necessary to analyze the kinds of meaningful statements that grammars
are capable of making about languages in general and to relate them to the
kinds of meaningful statements that the syntactic "sentences" of the cactus
language might be interpreted as making about the very same topics.  So far
in the present discussion, the sentences of the cactus language do not make
any meaningful statements at all, much less any meaningful statements about
languages and their constitutions.  As of yet, these sentences subsist in the
form of purely abstract, formal, and uninterpreted combinatorial constructions.

Before the capacity of a language to describe itself can be evaluated,
the missing link to meaning has to be supplied for each of its strings.
This calls for a dimension of semantics and a notion of interpretation,
topics that are taken up for the case of the cactus language !C!(!P!)
in Subsection 1.3.10.12.  Once a plausible semantics is prescribed for
this language it will be possible to return to these questions and to
address them in a meaningful way.

The prominent issue at this point is the distinct placements of formal
languages and formal grammars with respect to the question of meaning.
The sentences of a formal language are merely the abstract strings of
abstract signs that happen to belong to a certain set.  They do not by
themselves make any meaningful statements at all, not without mounting
a separate effort of interpretation, but the rules of a formal grammar
make meaningful statements about a formal language, to the extent that
they say what strings belong to it and what strings do not.  Thus, the
formal grammar, a formalism that appears to be even more skeletal than
the formal language, still has bits and pieces of meaning attached to it.
In a sense, the question of meaning is factored into two parts, structure
and value, leaving the aspect of value reduced in complexity and subtlety
to the simple question of belonging.  Whether this single bit of meaningful
value is enough to encompass all of the dimensions of meaning that we require,
and whether it can be compounded to cover the complexity that actually exists
in the realm of meaning -- these are questions for an extended future inquiry.

Perhaps I ought to comment on the differences between the present and
the standard definition of a formal grammar, since I am attempting to
strike a compromise with several alternative conventions of usage, and
thus to leave certain options open for future exploration.  All of the
changes are minor, in the sense that they are not intended to alter the
classes of languages that are able to be generated, but only to clear up
various ambiguities and sundry obscurities that affect their conception.

Primarily, the conventional scope of non-terminal symbols was expanded
to encompass the sentence symbol, mainly on account of all the contexts
where the initial and the intermediate symbols are naturally invoked in
the same breath.  By way of compensating for the usual exclusion of the
sentence symbol from the non-terminal class, an equivalent distinction
was introduced in the fashion of a distinction between the initial and
the intermediate symbols, and this serves its purpose in all of those
contexts where the two kind of symbols need to be treated separately.

At the present point, I remain a bit worried about the motivations
and the justifications for introducing this distinction, under any
name, in the first place.  It is purportedly designed to guarantee
that the process of derivation at least gets started in a definite
direction, while the real questions have to do with how it all ends.
The excuses of efficiency and expediency that I offered as plausible
and sufficient reasons for distinguishing between empty and significant
sentences are likely to be ephemeral, if not entirely illusory, since
intermediate symbols are still permitted to characterize or to cover
themselves, not to mention being allowed to cover the empty string,
and so the very types of traps that one exerts oneself to avoid at
the outset are always there to afflict the process at all of the
intervening times.

If one reflects on the form of grammar that is being prescribed here,
it looks as if one sought, rather futilely, to avoid the problems of
recursion by proscribing the main program from calling itself, while
allowing any subprogram to do so.  But any trouble that is avoidable
in the part is also avoidable in the main, while any trouble that is
inevitable in the part is also inevitable in the main.  Consequently,
I am reserving the right to change my mind at a later stage, perhaps
to permit the initial symbol to characterize, to cover, to regenerate,
or to produce itself, if that turns out to be the best way in the end.

Before I leave this Subsection, I need to say a few things about
the manner in which the abstract theory of formal languages and
the pragmatic theory of sign relations interact with each other.

Formal language theory can seem like an awfully picky subject at times,
treating every symbol as a thing in itself the way it does, sorting out
the nominal types of symbols as objects in themselves, and singling out
the passing tokens of symbols as distinct entities in their own rights.
It has to continue doing this, if not for any better reason than to aid
in clarifying the kinds of languages that people are accustomed to use,
to assist in writing computer programs that are capable of parsing real
sentences, and to serve in designing programming languages that people
would like to become accustomed to use.  As a matter of fact, the only
time that formal language theory becomes too picky, or a bit too myopic
in its focus, is when it leads one to think that one is dealing with the
thing itself and not just with the sign of it, in other words, when the
people who use the tools of formal language theory forget that they are
dealing with the mere signs of more interesting objects and not with the
objects of ultimate interest in and of themselves.

As a result, there a number of deleterious effects that can arise from
the extreme pickiness of formal language theory, arising, as is often the
case, when formal theorists forget the practical context of theorization.
It frequently happens that the exacting task of defining the membership
of a formal language leads one to think that this object and this object
alone is the justifiable end of the whole exercise.  The distractions of
this mediate objective render one liable to forget that one's penultimate
interest lies always with various kinds of equivalence classes of signs,
not entirely or exclusively with their more meticulous representatives.

When this happens, one typically goes on working oblivious to the fact
that many details about what transpires in the meantime do not matter
at all in the end, and one is likely to remain in blissful ignorance
of the circumstance that many special details of language membership
are bound, destined, and pre-determined to be glossed over with some
measure of indifference, especially when it comes down to the final
constitution of those equivalence classes of signs that are able to
answer for the genuine objects of the whole enterprise of language.
When any form of theory, against its initial and its best intentions,
leads to this kind of absence of mind that is no longer beneficial in
all of its main effects, the situation calls for an antidotal form of
theory, one that can restore the presence of mind that all forms of
theory are meant to augment.

The pragmatic theory of sign relations is called for in settings where
everything that can be named has many other names, that is to say, in
the usual case.  Of course, one would like to replace this superfluous
multiplicity of signs with an organized system of canonical signs, one
for each object that needs to be denoted, but reducing the redundancy
too far, beyond what is necessary to eliminate the factor of "noise" in
the language, that is, to clear up its effectively useless distractions,
can destroy the very utility of a typical language, which is intended to
provide a ready means to express a present situation, clear or not, and
to describe an ongoing condition of experience in just the way that it
seems to present itself.  Within this fleshed out framework of language,
moreover, the process of transforming the manifestations of a sign from
its ordinary appearance to its canonical aspect is the whole problem of
computation in a nutshell.

It is a well-known truth, but an often forgotten fact, that nobody
computes with numbers, but solely with numerals in respect of numbers,
and numerals themselves are symbols.  Among other things, this renders
all discussion of numeric versus symbolic computation a bit beside the
point, since it is only a question of what kinds of symbols are best for
one's immediate application or for one's selection of ongoing objectives.
The numerals that everybody knows best are just the canonical symbols,
the standard signs or the normal terms for numbers, and the process of
computation is a matter of getting from the arbitrarily obscure signs
that the data of a situation are capable of throwing one's way to the
indications of its character that are clear enough to motivate action.

Having broached the distinction between propositions and sentences, one
can see its similarity to the distinction between numbers and numerals.
What are the implications of the foregoing considerations for reasoning
about propositions and for the realm of reckonings in sentential logic?
If the purpose of a sentence is just to denote a proposition, then the
proposition is just the object of whatever sign is taken for a sentence.
This means that the computational manifestation of a piece of reasoning
about propositions amounts to a process that takes place entirely within
a language of sentences, a procedure that can rationalize its account by
referring to the denominations of these sentences among propositions.

The application of these considerations in the immediate setting is this:
Do not worry too much about what roles the empty string "" and the blank
symbol " " are supposed to play in a given species of formal languages.
As it happens, it is far less important to wonder whether these types
of formal tokens actually constitute genuine sentences than it is to
decide what equivalence classes it makes sense to form over all of
the sentences in the resulting language, and only then to bother
about what equivalence classes these limiting cases of sentences
are most conveniently taken to represent.

These concerns about boundary conditions betray a more general issue.
Already by this point in discussion the limits of the purely syntactic
approach to a language are beginning to be visible.  It is not that one
cannot go a whole lot further by this road in the analysis of a particular
language and in the study of languages in general, but when it comes to the
questions of understanding the purpose of a language, of extending its usage
in a chosen direction, or of designing a language for a particular set of uses,
what matters above all else are the "pragmatic equivalence classes" of signs that
are demanded by the application and intended by the designer, and not so much the
peculiar characters of the signs that represent these classes of practical meaning.

Any description of a language is bound to have alternative descriptions.
More precisely, a circumscribed description of a formal language, as any
effectively finite description is bound to be, is certain to suggest the
equally likely existence and the possible utility of other descriptions.
A single formal grammar describes but a single formal language, but any
formal language is described by many different formal grammars, not all
of which afford the same grasp of its structure, provide an equivalent
comprehension of its character, or yield an interchangeable view of its
aspects.  Consequently, even with respect to the same formal language,
different formal grammars are typically better for different purposes.

With the distinctions that evolve among the different styles of grammar,
and with the preferences that different observers display toward them,
there naturally comes the question:  What is the root of this evolution?

One dimension of variation in the styles of formal grammars can be seen
by treating the union of languages, and especially the disjoint union of
languages, as a "sum", by treating the concatenation of languages as a
"product", and then by distinguishing the styles of analysis that favor
"sums of products" from those that favor "products of sums" as their
canonical forms of description.  If one examines the relation between
languages and grammars carefully enough to see the presence and the
influence of these different styles, and when one comes to appreciate
the ways that different styles of grammars can be used with different
degrees of success for different purposes, then one begins to see the
possibility that alternative styles of description can be based on
altogether different linguistic and logical operations.

It possible to trace this divergence of styles to an even more primitive
division, one that distinguishes the "additive" or the "parallel" styles
from the "multiplicative" or the "serial" styles.  The issue is somewhat
confused by the fact that an "additive" analysis is typically expressed
in the form of a "series", in other words, a disjoint union of sets or a
linear sum of their independent effects.  But it is easy enough to sort
this out if one observes the more telling connection between "parallel"
and "independent".  Another way to keep the right associations straight
is to employ the term "sequential" in preference to the more misleading
term "serial".  Whatever one calls this broad division of styles, the
scope and sweep of their dimensions of variation can be delineated in
the following way:

   1.  The "additive" or "parallel" styles favor "sums of products" as
       canonical forms of expression, pulling sums, unions, co-products,
       and logical disjunctions to the outermost layers of analysis and
       synthesis, while pushing products, intersections, concatenations,
       and logical conjunctions to the innermost levels of articulation
       and generation.  In propositional logic, this style leads to the
       "disjunctive normal form" (DNF).

   2.  The "multiplicative" or "serial" styles favor "products of sums"
       as canonical forms of expression, pulling products, intersections,
       concatenations, and logical conjunctions to the outermost layers of
       analysis and synthesis, while pushing sums, unions, co-products,
       and logical disjunctions to the innermost levels of articulation
       and generation.  In propositional logic, this style leads to the
       "conjunctive normal form" (CNF).

There is a curious sort of diagnostic clue, a veritable shibboleth,
that often serves to reveal the dominance of one mode or the other
within an individual thinker's cognitive style.  Examined on the
question of what constitutes the "natural numbers", an "additive"
thinker tends to start the sequence at 0, while a "multiplicative"
thinker tends to regard it as beginning at 1.

In any style of description, grammar, or theory of a language, it is
usually possible to tease out the influence of these contrasting traits,
namely, the "additive" attitude versus the "mutiplicative" tendency that
go to make up the particular style in question, and even to determine the
dominant inclination or point of view that establishes its perspective on
the target domain.

In each style of formal grammar, the "multiplicative" aspect is present
in the sequential concatenation of signs, both in the augmented strings
and in the terminal strings.  In settings where the non-terminal symbols
classify types of strings, the concatenation of the non-terminal symbols
signifies the cartesian product over the corresponding sets of strings.

In the context-free style of formal grammar, the "additive" aspect is
easy enough to spot.  It is signaled by the parallel covering of many
augmented strings or sentential forms by the same non-terminal symbol.
Expressed in active terms, this calls for the independent rewriting
of that non-terminal symbol by a number of different successors,
as in the following scheme:

   q     :>    W_1

   q     :>    W_2

   ...   ...   ...

   q     :>    W_k

It is useful to examine the relationship between the grammatical covering
or production relation ":>" and the logical relation of implication "=>",
with one eye to what they have in common and one eye to how they differ.
The production "q :> W" says that the appearance of the symbol "q" in
a sentential form implies the possibility of exchanging it for "W".
Although this sounds like a "possible implication", to the extent
that "q implies a possible W" or that "q possibly implies W", the
qualifiers "possible" and "possibly" are the critical elements in
these statements, and they are crucial to the meaning of what is
actually being implied.  In effect, these qualifications reverse
the direction of implication, yielding "q <= W" as the best
analogue for the sense of the production.

One way to sum this up is to say that non-terminal symbols have the
significance of hypotheses.  The terminal strings form the empirical
matter of a language, while the non-terminal symbols mark the patterns
or the types of substrings that can be noticed in the profusion of data.
If one observes a portion of a terminal string that falls into the pattern
of the sentential form W, then it is an admissable hypothesis, according to
the theory of the language that is constituted by the formal grammar, that
this piece not only fits the type q but even comes to be generated under
the auspices of the non-terminal symbol "q".

A moment's reflection on the issue of style, giving due consideration to the
received array of stylistic choices, ought to inspire at least the question:
"Are these the only choices there are?"  In the present setting, there are
abundant indications that other options, more differentiated varieties of
description and more integrated ways of approaching individual languages,
are likely to be conceivable, feasible, and even more ultimately viable.
If a suitably generic style, one that incorporates the full scope of
logical combinations and operations, is broadly available, then it
would no longer be necessary, or even apt, to argue in universal
terms about "which style is best", but more useful to investigate
how we might adapt the local styles to the local requirements.
The medium of a generic style would yield a viable compromise
between "additive" and "multiplicative" canons, and render the
choice between "parallel" and "serial" a false alternative,
at least, when expressed in the globally exclusive terms
that are currently most commonly adopted for posing it.

One set of indications comes from the study of machines, languages, and
computation, especially the theories of their structures and relations.
The forms of composition and decomposition that are generally known as
"parallel" and "serial" are merely the extreme special cases, in variant
directions of specialization, of a more generic form, usually called the
"cascade" form of combination.  This is a well-known fact in the theories
that deal with automata and their associated formal languages, but its
implications do not seem to be widely appreciated outside these fields.
In particular, it dispells the need to choose one extreme or the other,
since most of the natural cases are likely to exist somewhere in between.

Another set of indications appears in algebra and category theory,
where forms of composition and decomposition related to the cascade
combination, namely, the "semi-direct product" and its special case,
the "wreath product", are encountered at higher levels of generality
than the cartesian products of sets or the direct products of spaces.

In these domains of operation, one finds it necessary to consider also
the "co-product" of sets and spaces, a construction that artificially
creates a disjoint union of sets, that is, a union of spaces that are
being treated as independent.  It does this, in effect, by "indexing",
"coloring", or "preparing" the otherwise possibly overlapping domains
that are being combined.  What renders this a "chimera" or a "hybrid"
form of combination is the fact that this indexing is tantamount to a
cartesian product of a singleton set, namely, the conventional "index",
"color", or "affix" in question, with the individual domain that is
entering as a factor, a term, or a participant in the final result.

One of the insights that arises out of Peirce's logical work is that
the set operations of complementation, intersection, and union, along
with the logical operations of negation, conjunction, and disjunction
that operate in isomorphic tandem with them, are not as fundamental as
they first appear.  This is because all of them can be constructed from
or derived from a smaller set of operations, in fact, taking the logical
side of things, from either one of two "solely sufficient" operators,
called "amphecks" by Peirce, "strokes" by those who re-discovered them
later, and known in computer science as the NAND and the NNOR operators.
For this reason, that is, by virtue of their precedence in the orders
of construction and derivation, these operations have to be regarded
as the simplest and the most primitive in principle, even if they are
scarcely recognized as lying among the more familiar elements of logic.

I am throwing together a wide variety of different operations into
the bins labeled "additive" and "multiplicative", but it is easy to
observe a natural organization and even some relations that approach
the level of isomorphisms among and between the members of each class.

The relation between logical disjunction and set-theoretic union and
the relation between logical conjunction and set-theoretic intersection
are most likely clear enough for the purposes of the immediately present
discussion.  At any rate, all of these relations are scheduled to receive
a thorough examination in a subsequent discussion (Subsection 1.3.10.13).
But the relation of set-theoretic union to category-theoretic co-product
and the relation of set-theoretic intersection to syntactic concatenation
deserve a closer look at this point.

The effect of a co-product as a "disjointed union", in other words, that
creates an object tantamount to a disjoint union of sets in the resulting
co-product even if some of these sets intersect non-trivially and even if
some of them are identical "in reality", can be achieved in several ways.
The most usual conception is that of making a "separate copy", for each
part of the intended co-product, of the set that is intended to go there.
Often one thinks of the set that is assigned to a particular part of the
co-product as being distinguished by a particular "color", in other words,
by the attachment of a distinct "index", "label", or "tag", being a marker
that is inherited by and passed on to every element of the set in that part.
A concrete image of this construction can be achieved by imagining that each
set and each element of each set is placed in an ordered pair with the sign
of its color, index, label, or tag.  One describes this as the "injection"
of each set into the corresponding "part" of the co-product.

For example, given the sets P and Q, overlapping or not, one can define
the "indexed" sets or the "marked" sets P_[1] and Q_[2], amounting to the
copy of P into the first part of the co-product and the copy of Q into the
second part of the co-product, in the following manner:

   P_[1]  =  <P, 1>  =  {<x, 1> : x in P},

   Q_[2]  =  <Q, 2>  =  {<x, 2> : x in Q}.

Using the sign "]_[" for this construction, the "sum", the "co-product",
or the "disjointed union" of P and Q in that order can be represented as
the ordinary disjoint union of P_[1] and Q_[2], as follows:

   P ]_[ Q   =   P_[1] |_| Q_[2].

The concatenation L_1 · L_2 of the formal languages L_1 and L_2 is just
the cartesian product of sets L_1 x L_2 without the extra x's, but the
relation of cartesian products to set-theoretic intersections and thus
to logical conjunctions is far from being clear.

One way of seeing a type of relation in this setting is to focus on the
information that is needed to specify each construction, and thereby to
reflect on the signs that are used to carry this information.  As a way
of making a first approach to the topic of information, in accord with
a strategy that seeks to be as elementary and as informal as possible,
I introduce the following collection of ideas, intended to be taken
in a very provisional way.

A "stricture" is syntactic specification of a certain set in a certain place,
relative to a number of other sets, yet to be specified.  It is assumed that
one knows enough about the general form of the specifications in question to
tell if two strictures are equivalent as pieces of information, but any more
determinate indications, like names for the places that are mentioned in the
stricture, or bounds on the number of places that are involved, are regarded
as being extraneous impositions, outside the chief concern of the definition,
no matter how convenient they are found to be within a particular discussion.
As a schematic form of illustration, a stricture can be pictured in this way:

   "... x X x Q x X x ..."

A "strait" is the object that is specified by a stricture, in effect,
a certain set in a certain place of an otherwise yet to be specified
relation.  Somewhat sketchily, the strait that corresponds to the
stricture just given can be pictured in the following shape:

    ... x X x Q x X x ...

In this picture, Q is a certain set, and X is the universe of discourse that is
pertinent to a given discussion.  Since a stricture does not, by itself, contain
a sufficient amount of information to specify the number of sets that it intends
to set in place, or even to specify the absolute location of the set that it does
set in place, it appears to place an unspecified number of unspecified sets in a
vague and uncertain strait.  Taken out of its interpretive context, the residual
information that a stricture can convey makes all of the following potentially
equivalent as strictures:

   "Q",  "X x Q x X",  "X x X x Q x X x X",   ...

With respect to what these strictures specify, this
leaves all of the following equivalent as straits:

    Q  =  X x Q x X  =  X x X x Q x X x X  =  ...

Within the framework of a particular discussion, it is customary to
set a bound on the number of places and to limit the variety of sets
that are regarded as being under active consideration, and it is also
convenient to index the places of the indicated relations, and of their
encompassing cartesian products, in some fixed way.  But the whole idea
of a stricture is to specify a strait that is capable of extending through
and beyond any fixed frame of discussion.  In other words, a stricture is
conceived to constrain a strait at a certain point, and then to leave it
literally embedded, if tacitly expressed, in a yet to be fully specified
relation, one that involves an unspecified number of unspecified domains.

A quantity of information is a measure of constraint.  In this respect,
a set of comparable strictures is ordered on account of the information
that each one conveys, and a system of comparable straits is ordered in
accord with the amount of information that it takes to pin each one of
them down.  Strictures that are more constraining and straits that are
more constrained are placed at higher levels of information than those
that are less so.  In other language that is often used, entities of
either kind that involve more information are said to have a greater
"complexity" in relation to comparable entities which involve less
information, the latter being said to have a greater "simplicity".

In order to create a concrete example, let me now institute a frame of
discussion where the number of places in a relation is bounded at two,
and where the variety of sets under active consideration is limited to
the typical subsets P and Q of a universe X.  Under these conditions,
one can use the following sorts of expression as schematic strictures:

     "X"       "P"       "Q"

   "X x X"   "X x P"   "X x Q"

   "P x X"   "P x P"   "P x Q"

   "Q x X"   "Q x P"   "Q x Q"

These strictures and their corresponding straits are stratified according
to their amounts of information, or their levels of constraint, as follows:

   High:    "P x P"   "P x Q"   "Q x P"   "Q x Q"

   Medium:    "P"     "X x P"   "P x X"

   Medium:    "Q"     "X x Q"   "Q x X"

   Low:       "X"     "X x X"

Within this framework, the more complex strait P x Q can be expressed
in terms of the simpler straits, P x X and X x Q.  More specifically,
it lends itself to being analyzed as their intersection, as follows:

   P x Q  =  P x X  |^|  X x Q

From here it is easy to see the relation of concatenation, by virtue of
these types of intersection, to the logical conjunction of propositions.
A cartesian product P x Q is described by a conjunction of propositions,
namely, "P_<1> and Q_<2>", subject to the following interpretation:

   1.  "P_<1>" asserts that there is an element from
       the set P in the first place of the product.

   2.  "Q_<2>" asserts that there is an element from
       the set Q in the second place of the product.

The integration of these two pieces of information can be taken
in that measure to specify a yet to be fully determined relation.

In a corresponding fashion at the level of the elements,
the ordered pair <p, q> is described by a conjunction
of propositions, namely, "p_<1> and q_<2>", subject
to the following interpretation:

   1.  "p_<1>" says that p is in the first place
       of the product element under construction.

   2.  "q_<2>" says that q is in the second place
       of the product element under construction.

Notice that, in construing the cartesian product of the sets P and Q or the
concatenation of the languages L_1 and L_2 in this way, one shifts the level
of the active construction from the tupling of the elements in P and Q or the
concatenation of the strings that are internal to the languages L_1 and L_2 to
the concatenation of the external signs that it takes to indicate these sets or
these languages, in other words, passing to a conjunction of indexed propositions,
"P_<1> and Q_<2>", or to a conjunction of assertions, "L_1_<1> and L_2_<2>", that
marks the sets or the languages in question for insertion in the indicated places
of a product set or a product language, respectively.  In effect, the subscripting
by the indices "<1>" and "<2>" can be recognized as a special case of concatenation,
albeit through the posting of editorial remarks from an external "mark-up" language.

In order to systematize the relationships that strictures and straits
placed at higher levels of complexity, constraint, information, and
organization bear toward strictures and straits that are placed at
the corresponding lower levels of these measures, I introduce the
following pair of definitions:

The j^th "excerpt" of a stricture of the form "S_1 x ... x S_k", regarded
within a frame of discussion where the number of places is limited to k,
is the stricture of the form "X x ... x S_j x ... x X".  In the proper
context, this can be written more succinctly as the stricture "S_j_<j>",
an assertion that places the j^th set in the j^th place of the product.

The j^th "extract" of a strait of the form S_1 x ... x S_k, constrained
to a frame of discussion where the number of places is restricted to k,
is the strait of the form X x ... x S_j x ... x X.  In the appropriate
context, this can be denoted more succinctly by the stricture "S_j_<j>",
an assertion that places the j^th set in the j^th place of the product.

In these terms, a stricture of the form "S_1 x ... x S_k"
can be expressed in terms of simpler strictures, namely,
as a conjunction of its k excerpts:

   "S_1 x ... x S_k"   =   "S_1_<1>" &  ...  & "S_k_<k>".

In a similar vein, a strait of the form S_1 x ... x S_k
can be expressed in terms of simpler straits, namely,
as an intersection of its k extracts:

    S_1 x ... x S_k    =    S_1_<1> |^| ... |^| S_k_<k>.

There is a measure of ambiguity that remains in this formulation,
but it is the best that I can do in the present informal context.

1.3.11.4. The Cactus Language : Mechanics

| We are only now beginning to see how this works.  Clearly one of the
| mechanisms for picking a reality is the sociohistorical sense of what
| is important -- which research program, with all its particularity of
| knowledge, seems most fundamental, most productive, most penetrating.
| The very judgments which make us push narrowly forward simultaneously
| make us forget how little we know.  And when we look back at history,
| where the lesson is plain to find, we often fail to imagine ourselves
| in a parallel situation.  We ascribe the differences in world view
| to error, rather than to unexamined but consistent and internally
| justified choice.
|
| Herbert J. Bernstein, "Idols", p. 38.
|
| Herbert J. Bernstein,
|"Idols of Modern Science and the Reconstruction of Knowledge", pp. 37-68 in:
|
| Marcus G. Raskin & Herbert J. Bernstein,
|'New Ways of Knowing:  The Sciences, Society, and Reconstructive Knowledge',
| Rowman & Littlefield, Totowa, NJ, 1987.

In this Subsection, I discuss the "mechanics" of parsing the
cactus language into the corresponding class of computational
data structures.  This provides each sentence of the language
with a translation into a computational form that articulates
its syntactic structure and prepares it for automated modes of
processing and evaluation.  For this purpose, it is necessary
to describe the target data structures at a fairly high level
of abstraction only, ignoring the details of address pointers
and record structures and leaving the more operational aspects
of implementation to the imagination of prospective programmers.
In this way, I can put off to another stage of elaboration and
refinement the description of the program that constructs these
pointers and operates on these graph-theoretic data structures.

The structure of a "painted cactus", insofar as it presents itself
to the visual imagination, can be described as follows.  The overall
structure, as given by its underlying graph, falls within the species
of graph that is commonly known as a "rooted cactus", and the only novel
feature that it adds to this is that each of its nodes can be "painted"
with a finite sequence of "paints", chosen from a "palette" that is given
by the parametric set {" "} |_| !P!  =  {m_1} |_| {p_1, ..., p_k}.

It is conceivable, from a purely graph-theoretical point of view, to have
a class of cacti that are painted but not rooted, and so it is frequently
necessary, for the sake of precision, to more exactly pinpoint the target
species of graphical structure as a "painted and rooted cactus" (PARC).

A painted cactus, as a rooted graph, has a distinguished "node" that is
called its "root".  By starting from the root and working recursively,
the rest of its structure can be described in the following fashion.

Each "node" of a PARC consists of a graphical "point" or "vertex" plus
a finite sequence of "attachments", described in relative terms as the
attachments "at" or "to" that node.  An empty sequence of attachments
defines the "empty node".  Otherwise, each attachment is one of three
kinds:  a blank, a paint, or a type of PARC that is called a "lobe".

Each "lobe" of a PARC consists of a directed graphical "cycle" plus a
finite sequence of "accoutrements", described in relative terms as the
accoutrements "of" or "on" that lobe.  Recalling the circumstance that
every lobe that comes under consideration comes already attached to a
particular node, exactly one vertex of the corresponding cycle is the
vertex that comes from that very node.  The remaining vertices of the
cycle have their definitions filled out according to the accoutrements
of the lobe in question.  An empty sequence of accoutrements is taken
to be tantamount to a sequence that contains a single empty node as its
unique accoutrement, and either one of these ways of approaching it can
be regarded as defining a graphical structure that is called a "needle"
or a "terminal edge".  Otherwise, each accoutrement of a lobe is itself
an arbitrary PARC.

Although this definition of a lobe in terms of its intrinsic structural
components is logically sufficient, it is also useful to characterize the
structure of a lobe in comparative terms, that is, to view the structure
that typifies a lobe in relation to the structures of other PARC's and to
mark the inclusion of this special type within the general run of PARC's.
This approach to the question of types results in a form of description
that appears to be a bit more analytic, at least, in mnemonic or prima
facie terms, if not ultimately more revealing.  Working in this vein,
a "lobe" can be characterized as a special type of PARC that is called
an "unpainted root plant" (UR-plant).

An "UR-plant" is a PARC of a simpler sort, at least, with respect to the
recursive ordering of structures that is being followed here.  As a type,
it is defined by the presence of two properties, that of being "planted"
and that of having an "unpainted root".  These are defined as follows:

   1.  A PARC is "planted" if its list of attachments has just one PARC.

   2.  A PARC is "UR" if its list of attachments has no blanks or paints.

In short, an UR-planted PARC has a single PARC as its only attachment,
and since this attachment is prevented from being a blank or a paint,
the single attachment at its root has to be another sort of structure,
that which we call a "lobe".

To express the description of a PARC in terms of its nodes, each node
can be specified in the fashion of a functional expression, letting a
citation of the generic function name "Node" be followed by a list of
arguments that enumerates the attachments of the node in question, and
letting a citation of the generic function name "Lobe" be followed by a
list of arguments that details the accoutrements of the lobe in question.
Thus, one can write expressions of the following forms:

   1.  Node^0         =  Node()

                      =  a node with no attachments.

       Node^k_j  C_j  =  Node(C_1, ..., C_k)

                      =  a node with the attachments C_1, ..., C_k.

   2.  Lobe^0         =  Lobe()

                      =  a lobe with no accoutrements.

       Lobe^k_j  C_j  =  Lobe(C_1, ..., C_k)

                      =  a lobe with the accoutrements C_1, ..., C_k.

Working from a structural description of the cactus language,
or any suitable formal grammar for !C!(!P!), it is possible to
give a recursive definition of the function called "Parse" that
maps each sentence in PARCE(!P!) to the corresponding graph in
PARC(!P!).  One way to do this proceeds as follows:

   1.  The parse of the concatenation Conc^k of the k sentences S_j,
       for j = 1 to k, is defined recursively as follows:

       a.  Parse(Conc^0)        =  Node^0.

       b.  For k > 0,

           Parse(Conc^k_j S_j)  =  Node^k_j Parse(S_j).

   2.  The parse of the surcatenation Surc^k of the k sentences S_j,
       for j = 1 to k, is defined recursively as follows:

       a.  Parse(Surc^0)        =  Lobe^0.

       b.  For k > 0,

           Parse(Surc^k_j S_j)  =  Lobe^k_j Parse(S_j).

For ease of reference, Table 12 summarizes the mechanics of these parsing rules.

Table 12.  Algorithmic Translation Rules
o------------------------o---------o------------------------o
|                        |  Parse  |                        |
| Sentence in PARCE      |   -->   | Graph in PARC          |
o------------------------o---------o------------------------o
|                        |         |                        |
| Conc^0                 |   -->   | Node^0                 |
|                        |         |                        |
| Conc^k_j  S_j          |   -->   | Node^k_j  Parse(S_j)   |
|                        |         |                        |
| Surc^0                 |   -->   | Lobe^0                 |
|                        |         |                        |
| Surc^k_j  S_j          |   -->   | Lobe^k_j  Parse(S_j)   |
|                        |         |                        |
o------------------------o---------o------------------------o

A "substructure" of a PARC is defined recursively as follows.  Starting
at the root node of the cactus C, any attachment is a substructure of C.
If a substructure is a blank or a paint, then it constitutes a minimal
substructure, meaning that no further substructures of C arise from it.
If a substructure is a lobe, then each one of its accoutrements is also
a substructure of C, and has to be examined for further substructures.

The concept of substructure can be used to define varieties of deletion
and erasure operations that respect the structure of the abstract graph.
For the purposes of this depiction, a blank symbol " " is treated as
a "primer", in other words, as a "clear paint", a "neutral tint", or
a "white wash".  In effect, one is letting m_1 = p_0.  In this frame
of discussion, it is useful to make the following distinction:

   1.  To "delete" a substructure is to replace it with an empty node,
       in effect, to reduce the whole structure to a trivial point.

   2.  To "erase" a substructure is to replace it with a blank symbol,
       in effect, to paint it out of the picture or to overwrite it.

A "bare" PARC, loosely referred to as a "bare cactus", is a PARC on the
empty palette !P! = {}.  In other veins, a bare cactus can be described
in several different ways, depending on how the form arises in practice.

   1.  Leaning on the definition of a bare PARCE, a bare PARC can be
       described as the kind of a parse graph that results from parsing
       a bare cactus expression, in other words, as the kind of a graph
       that issues from the requirements of processing a sentence of
       the bare cactus language !C!^0 = PARCE^0.

   2.  To express it more in its own terms, a bare PARC can be defined
       by tracing the recursive definition of a generic PARC, but then
       by detaching an independent form of description from the source
       of that analogy.  The method is sufficiently sketched as follows:

       a.  A "bare PARC" is a PARC whose attachments
           are limited to blanks and "bare lobes".

       b.  A "bare lobe" is a lobe whose accoutrements
           are limited to bare PARC's.

   3.  In practice, a bare cactus is usually encountered in the process
       of analyzing or handling an arbitrary PARC, the circumstances of
       which frequently call for deleting or erasing all of its paints.
       In particular, this generally makes it easier to observe the
       various properties of its underlying graphical structure.

1.3.11.5. The Cactus Language : Semantics

| Alas, and yet what 'are' you, my written and painted thoughts!
| It is not long ago that you were still so many-coloured,
| young and malicious, so full of thorns and hidden
| spices you made me sneeze and laugh -- and now?
| You have already taken off your novelty and
| some of you, I fear, are on the point of
| becoming truths:  they already look so
| immortal, so pathetically righteous,
| so boring!
|
| Friedrich Nietzsche, 'Beyond Good and Evil', Paragraph 296.
|
| Friedrich Nietzsche,
|'Beyond Good and Evil:  Prelude to a Philosophy of the Future',
| trans. by R.J. Hollingdale, intro. by Michael Tanner,
| Penguin Books, London, UK, 1973, 1990.

In this Subsection, I describe a particular semantics for the
painted cactus language, telling what meanings I aim to attach
to its bare syntactic forms.  This supplies an "interpretation"
for this parametric family of formal languages, but it is good
to remember that it forms just one of many such interpretations
that are conceivable and even viable.  In deed, the distinction
between the object domain and the sign domain can be observed in
the fact that many languages can be deployed to depict the same
set of objects and that any language worth its salt is bound to
to give rise to many different forms of interpretive saliency.

In formal settings, it is common to speak of "interpretation" as if it
created a direct connection between the signs of a formal language and
the objects of the intended domain, in other words, as if it determined
the denotative component of a sign relation.  But a closer attention to
what goes on reveals that the process of interpretation is more indirect,
that what it does is to provide each sign of a prospectively meaningful
source language with a translation into an already established target
language, where "already established" means that its relationship to
pragmatic objects is taken for granted at the moment in question.

With this in mind, it is clear that interpretation is an affair of signs
that at best respects the objects of all of the signs that enter into it,
and so it is the connotative aspect of semiotics that is at stake here.
There is nothing wrong with my saying that I interpret a sentence of a
formal language as a sign that refers to a function or to a proposition,
so long as you understand that this reference is likely to be achieved
by way of more familiar and perhaps less formal signs that you already
take to denote those objects.

On entering a context where a logical interpretation is intended for the
sentences of a formal language there are a few conventions that make it
easier to make the translation from abstract syntactic forms to their
intended semantic senses.  Although these conventions are expressed in
unnecessarily colorful terms, from a purely abstract point of view, they
do provide a useful array of connotations that help to negotiate what is
otherwise a difficult transition.  This terminology is introduced as the
need for it arises in the process of interpreting the cactus language.

The task of this Subsection is to specify a "semantic function" for
the sentences of the cactus language !L! = !C!(!P!), in other words,
to define a mapping that "interprets" each sentence of !C!(!P!) as
a sentence that says something, as a sentence that bears a meaning,
in short, as a sentence that denotes a proposition, and thus as a
sign of an indicator function.  When the syntactic sentences of a
formal language are given a referent significance in logical terms,
for example, as denoting propositions or indicator functions, then
each form of syntactic combination takes on a corresponding form
of logical significance.

By way of providing a logical interpretation for the cactus language,
I introduce a family of operators on indicator functions that are
called "propositional connectives", and I distinguish these from
the associated family of syntactic combinations that are called
"sentential connectives", where the relationship between these
two realms of connection is exactly that between objects and
their signs.  A propositional connective, as an entity of a
well-defined functional and operational type, can be treated
in every way as a logical or a mathematical object, and thus
as the type of object that can be denoted by the corresponding
form of syntactic entity, namely, the sentential connective that
is appropriate to the case in question.

There are two basic types of connectives, called the "blank connectives"
and the "bound connectives", respectively, with one connective of each
type for each natural number k = 0, 1, 2, 3, ... .

   1.  The "blank connective" of k places is signified by the
       concatenation of the k sentences that fill those places.

       For the special case of k = 0, the "blank connective" is taken to
       be an empty string or a blank symbol -- it does not matter which,
       since both are assigned the same denotation among propositions.
       For the generic case of k > 0, the "blank connective" takes
       the form "S_1 · ... · S_k".  In the type of data that is
       called a "text", the raised dots "·" are usually omitted,
       supplanted by whatever number of spaces and line breaks
       serve to improve the readability of the resulting text.

   2.  The "bound connective" of k places is signified by the
       surcatenation of the k sentences that fill those places.

       For the special case of k = 0, the "bound connective" is taken to
       be an expression of the form "-()-", "-( )-", "-(  )-", and so on,
       with any number of blank symbols between the parentheses, all of
       which are assigned the same logical denotation among propositions.
       For the generic case of k > 0, the "bound connective" takes the
       form "-(S_1, ..., S_k)-".

At this point, there are actually two different "dialects", "scripts",
or "modes" of presentation for the cactus language that need to be
interpreted, in other words, that need to have a semantic function
defined on their domains.

   a.  There is the literal formal language of strings in PARCE(!P!),
       the "painted and rooted cactus expressions" that constitute
       the langauge !L! = !C!(!P!) c !A!* = (!M! |_| !P!)*.

   b.  There is the figurative formal language of graphs in PARC(!P!),
       the "painted and rooted cacti" themselves, a parametric family
       of graphs or a species of computational data structures that
       is graphically analogous to the language of literal strings.

Of course, these two modalities of formal language, like written and
spoken natural languages, are meant to have compatible interpretations,
and so it is usually sufficient to give just the meanings of either one.
All that remains is to provide a "codomain" or a "target space" for the
intended semantic function, in other words, to supply a suitable range
of logical meanings for the memberships of these languages to map into.
Out of the many interpretations that are formally possible to arrange,
one way of doing this proceeds by making the following definitions:

   1.  The "conjunction" Conj^J_j Q_j of a set of propositions, {Q_j : j in J},
       is a proposition that is true if and only if each one of the Q_j is true.

       Conj^J_j Q_j is true  <=>  Q_j is true for every j in J.

   2.  The "surjunction" Surj^J_j Q_j of a set of propositions, {Q_j : j in J},
       is a proposition that is true if and only if just one of the Q_j is untrue.

       Surj^J_j Q_j is true  <=>  Q_j is untrue for unique j in J.

If the number of propositions that are being joined together is finite,
then the conjunction and the surjunction can be represented by means of
sentential connectives, incorporating the sentences that represent these
propositions into finite strings of symbols.

If J is finite, for instance, if J constitutes the interval j = 1 to k,
and if each proposition Q_j is represented by a sentence S_j, then the
following strategies of expression are open:

   1.  The conjunction Conj^J_j Q_j can be represented by a sentence that
       is constructed by concatenating the S_j in the following fashion:

       Conj^J_j Q_j   <-<   S_1 S_2 ... S_k.

   2.  The surjunction Surj^J_j Q_j can be represented by a sentence that
       is constructed by surcatenating the S_j in the following fashion:

       Surj^J_j Q_j   <-<   -(S_1, S_2, ..., S_k)-.

If one opts for a mode of interpretation that moves more directly from
the parse graph of a sentence to the potential logical meaning of both
the PARC and the PARCE, then the following specifications are in order:

A cactus rooted at a particular node is taken to represent what that
node denotes, its logical denotation or its logical interpretation.

   1.  The logical denotation of a node is the logical conjunction of that node's
       "arguments", which are defined as the logical denotations of that node's
       attachments.  The logical denotation of either a blank symbol or an empty
       node is the boolean value %1% = "true".  The logical denotation of the
       paint p_j is the proposition P_j, a proposition that is regarded as
       "primitive", at least, with respect to the level of analysis that
       is represented in the current instance of !C!(!P!).

   2.  The logical denotation of a lobe is the logical surjunction of that lobe's
       "arguments", which are defined as the logical denotations of that lobe's
       accoutrements.  As a corollary, the logical denotation of the parse graph
       of "-()-", otherwise called a "needle", is the boolean value %0% = "false".

If one takes the point of view that PARC's and PARCE's amount to a
pair of intertranslatable languages for the same domain of objects,
then the "spiny bracket" notation, as in "-[C_j]-" or "-[S_j]-",
can be used on either domain of signs to indicate the logical
denotation of a cactus C_j or the logical denotation of
a sentence S_j, respectively.

Tables 13.1 and 13.2 summarize the relations that serve to connect the
formal language of sentences with the logical language of propositions.
Between these two realms of expression there is a family of graphical
data structures that arise in parsing the sentences and that serve to
facilitate the performance of computations on the indicator functions.
The graphical language supplies an intermediate form of representation
between the formal sentences and the indicator functions, and the form
of mediation that it provides is very useful in rendering the possible
connections between the other two languages conceivable in fact, not to
mention in carrying out the necessary translations on a practical basis.
These Tables include this intermediate domain in their Central Columns.
Between their First and Middle Columns they illustrate the mechanics of
parsing the abstract sentences of the cactus language into the graphical
data structures of the corresponding species.  Between their Middle and
Final Columns they summarize the semantics of interpreting the graphical
forms of representation for the purposes of reasoning with propositions.

Table 13.1  Semantic Translations:  Functional Form
o-------------------o-----o-------------------o-----o-------------------o
|                   | Par |                   | Den |                   |
| Sentence          | --> | Graph             | --> | Proposition       |
o-------------------o-----o-------------------o-----o-------------------o
|                   |     |                   |     |                   |
| S_j               | --> | C_j               | --> | Q_j               |
|                   |     |                   |     |                   |
o-------------------o-----o-------------------o-----o-------------------o
|                   |     |                   |     |                   |
| Conc^0            | --> | Node^0            | --> | %1%               |
|                   |     |                   |     |                   |
| Conc^k_j  S_j     | --> | Node^k_j  C_j     | --> | Conj^k_j  Q_j     |
|                   |     |                   |     |                   |
o-------------------o-----o-------------------o-----o-------------------o
|                   |     |                   |     |                   |
| Surc^0            | --> | Lobe^0            | --> | %0%               |
|                   |     |                   |     |                   |
| Surc^k_j  S_j     | --> | Lobe^k_j  C_j     | --> | Surj^k_j  Q_j     |
|                   |     |                   |     |                   |
o-------------------o-----o-------------------o-----o-------------------o

Table 13.2  Semantic Translations:  Equational Form
o-------------------o-----o-------------------o-----o-------------------o
|                   | Par |                   | Den |                   |
| -[Sentence]-      |  =  | -[Graph]-         |  =  | Proposition       |
o-------------------o-----o-------------------o-----o-------------------o
|                   |     |                   |     |                   |
| -[S_j]-           |  =  | -[C_j]-           |  =  | Q_j               |
|                   |     |                   |     |                   |
o-------------------o-----o-------------------o-----o-------------------o
|                   |     |                   |     |                   |
| -[Conc^0]-        |  =  | -[Node^0]-        |  =  | %1%               |
|                   |     |                   |     |                   |
| -[Conc^k_j  S_j]- |  =  | -[Node^k_j  C_j]- |  =  | Conj^k_j  Q_j     |
|                   |     |                   |     |                   |
o-------------------o-----o-------------------o-----o-------------------o
|                   |     |                   |     |                   |
| -[Surc^0]-        |  =  | -[Lobe^0]-        |  =  | %0%               |
|                   |     |                   |     |                   |
| -[Surc^k_j  S_j]- |  =  | -[Lobe^k_j  C_j]- |  =  | Surj^k_j  Q_j     |
|                   |     |                   |     |                   |
o-------------------o-----o-------------------o-----o-------------------o

Aside from their common topic, the two Tables present slightly different
ways of conceptualizing the operations that go to establish their maps.
Table 13.1 records the functional associations that connect each domain
with the next, taking the triplings of a sentence S_j, a cactus C_j, and
a proposition Q_j as basic data, and fixing the rest by recursion on these.
Table 13.2 records these associations in the form of equations, treating
sentences and graphs as alternative kinds of signs, and generalizing the
spiny bracket operator to indicate the proposition that either denotes.
It should be clear at this point that either scheme of translation puts
the sentences, the graphs, and the propositions that it associates with
each other roughly in the roles of the signs, the interpretants, and the
objects, respectively, whose triples define an appropriate sign relation.
Indeed, the "roughly" can be made "exactly" as soon as the domains of
a suitable sign relation are specified precisely.

A good way to illustrate the action of the conjunction and surjunction
operators is to demonstate how they can be used to construct all of the
boolean functions on k variables, just now, let us say, for k = 0, 1, 2.

A boolean function on 0 variables is just a boolean constant F^0 in the
boolean domain %B% = {%0%, %1%}.  Table 14 shows several different ways
of referring to these elements, just for the sake of consistency using
the same format that will be used in subsequent Tables, no matter how
degenerate it tends to appears in the immediate case:

   Column 1 lists each boolean element or boolean function under its
   ordinary constant name or under a succinct nickname, respectively.

   Column 2 lists each boolean function in a style of function name "F^i_j"
   that is constructed as follows:  The superscript "i" gives the dimension
   of the functional domain, that is, the number of its functional variables,
   and the subscript "j" is a binary string that recapitulates the functional
   values, using the obvious translation of boolean values into binary values.

   Column 3 lists the functional values for each boolean function, or possibly
   a boolean element appearing in the guise of a function, for each combination
   of its domain values.

   Column 4 shows the usual expressions of these elements in the cactus language,
   conforming to the practice of omitting the strike-throughs in display formats.
   Here I illustrate also the useful convention of sending the expression "(())"
   as a visible stand-in for the expression of a constantly "true" truth value,
   one that would otherwise be represented by a blank expression, and tend to
   elude our giving it much notice in the context of more demonstrative texts.

Table 14.  Boolean Functions on Zero Variables
o----------o----------o-------------------------------------------o----------o
| Constant | Function |                    F()                    | Function |
o----------o----------o-------------------------------------------o----------o
|          |          |                                           |          |
| %0%      | F^0_0    |                    %0%                    |    ()    |
|          |          |                                           |          |
| %1%      | F^0_1    |                    %1%                    |   (())   |
|          |          |                                           |          |
o----------o----------o-------------------------------------------o----------o

Table 15 presents the boolean functions on one variable, F^1 : %B% -> %B%,
of which there are precisely four.  Here, Column 1 codes the contents of
Column 2 in a more concise form, compressing the lists of boolean values,
recorded as bits in the subscript string, into their decimal equivalents.
Naturally, the boolean constants reprise themselves in this new setting
as constant functions on one variable.  Thus, one has the synonymous
expressions for constant functions that are expressed in the next
two chains of equations:

   F^1_0  =  F^1_00  =  %0% : %B% -> %B%

   F^1_3  =  F^1_11  =  %1% : %B% -> %B%

As for the rest, the other two functions are easily recognized as corresponding
to the one-place logical connectives, or the monadic operators on %B%.  Thus,
the function F^1_1  =  F^1_01 is recognizable as the negation operation, and
the function F^1_2  =  F^1_10 is obviously the identity operation.

Table 15.  Boolean Functions on One Variable
o----------o----------o-------------------------------------------o----------o
| Function | Function |                   F(x)                    | Function |
o----------o----------o---------------------o---------------------o----------o
|          |          |       F(%0%)        |       F(%1%)        |          |
o----------o----------o---------------------o---------------------o----------o
|          |          |                     |                     |          |
| F^1_0    | F^1_00   |         %0%         |         %0%         |   ( )    |
|          |          |                     |                     |          |
| F^1_1    | F^1_01   |         %0%         |         %1%         |   (x)    |
|          |          |                     |                     |          |
| F^1_2    | F^1_10   |         %1%         |         %0%         |    x     |
|          |          |                     |                     |          |
| F^1_3    | F^1_11   |         %1%         |         %1%         |  (( ))   |
|          |          |                     |                     |          |
o----------o----------o---------------------o---------------------o----------o

Table 16 presents the boolean functions on two variables, F^2 : %B%^2 -> %B%,
of which there are precisely sixteen in number.  As before, all of the boolean
functions of fewer variables are subsumed in this Table, though under a set of
alternative names and possibly different interpretations.  Just to acknowledge
a few of the more notable pseudonyms:

   The constant function %0% : %B%^2 -> %B% appears under the name of F^2_00.

   The constant function %1% : %B%^2 -> %B% appears under the name of F^2_15.

   The negation and identity of the first variable are F^2_03 and F^2_12, resp.

   The negation and identity of the other variable are F^2_05 and F^2_10, resp.

   The logical conjunction is given by the function F^2_08 (x, y)  =  x · y.

   The logical disjunction is given by the function F^2_14 (x, y)  =  ((x)(y)).

Functions expressing the "conditionals", "implications",
or "if-then" statements are given in the following ways:

   [x => y]  =  F^2_11 (x, y)  =  (x (y))  =  [not x without y].

   [x <= y]  =  F^2_13 (x, y)  =  ((x) y)  =  [not y without x].

The function that corresponds to the "biconditional",
the "equivalence", or the "if and only" statement is
exhibited in the following fashion:

   [x <=> y]  =  [x = y]  =  F^2_09 (x, y)  =  ((x , y)).

Finally, there is a boolean function that is logically associated with
the "exclusive disjunction", "inequivalence", or "not equals" statement,
algebraically associated with the "binary sum" or "bitsum" operation,
and geometrically associated with the "symmetric difference" of sets.
This function is given by:

   [x =/= y]  =  [x + y]  =  F^2_06 (x, y)  =  (x , y).

Table 16.  Boolean Functions on Two Variables
o----------o----------o-------------------------------------------o----------o
| Function | Function |                  F(x, y)                  | Function |
o----------o----------o----------o----------o----------o----------o----------o
|          |          | %1%, %1% | %1%, %0% | %0%, %1% | %0%, %0% |          |
o----------o----------o----------o----------o----------o----------o----------o
|          |          |          |          |          |          |          |
| F^2_00   | F^2_0000 |   %0%    |   %0%    |   %0%    |   %0%    |    ()    |
|          |          |          |          |          |          |          |
| F^2_01   | F^2_0001 |   %0%    |   %0%    |   %0%    |   %1%    |  (x)(y)  |
|          |          |          |          |          |          |          |
| F^2_02   | F^2_0010 |   %0%    |   %0%    |   %1%    |   %0%    |  (x) y   |
|          |          |          |          |          |          |          |
| F^2_03   | F^2_0011 |   %0%    |   %0%    |   %1%    |   %1%    |  (x)     |
|          |          |          |          |          |          |          |
| F^2_04   | F^2_0100 |   %0%    |   %1%    |   %0%    |   %0%    |   x (y)  |
|          |          |          |          |          |          |          |
| F^2_05   | F^2_0101 |   %0%    |   %1%    |   %0%    |   %1%    |     (y)  |
|          |          |          |          |          |          |          |
| F^2_06   | F^2_0110 |   %0%    |   %1%    |   %1%    |   %0%    |  (x, y)  |
|          |          |          |          |          |          |          |
| F^2_07   | F^2_0111 |   %0%    |   %1%    |   %1%    |   %1%    |  (x  y)  |
|          |          |          |          |          |          |          |
| F^2_08   | F^2_1000 |   %1%    |   %0%    |   %0%    |   %0%    |   x  y   |
|          |          |          |          |          |          |          |
| F^2_09   | F^2_1001 |   %1%    |   %0%    |   %0%    |   %1%    | ((x, y)) |
|          |          |          |          |          |          |          |
| F^2_10   | F^2_1010 |   %1%    |   %0%    |   %1%    |   %0%    |      y   |
|          |          |          |          |          |          |          |
| F^2_11   | F^2_1011 |   %1%    |   %0%    |   %1%    |   %1%    |  (x (y)) |
|          |          |          |          |          |          |          |
| F^2_12   | F^2_1100 |   %1%    |   %1%    |   %0%    |   %0%    |   x      |
|          |          |          |          |          |          |          |
| F^2_13   | F^2_1101 |   %1%    |   %1%    |   %0%    |   %1%    | ((x) y)  |
|          |          |          |          |          |          |          |
| F^2_14   | F^2_1110 |   %1%    |   %1%    |   %1%    |   %0%    | ((x)(y)) |
|          |          |          |          |          |          |          |
| F^2_15   | F^2_1111 |   %1%    |   %1%    |   %1%    |   %1%    |   (())   |
|          |          |          |          |          |          |          |
o----------o----------o----------o----------o----------o----------o----------o

Let me now address one last question that may have occurred to some.
What has happened, in this suggested scheme of functional reasoning,
to the distinction that is quite pointedly made by careful logicians
between (1) the connectives called "conditionals" and symbolized by
the signs "->" and "<-", and (2) the assertions called "implications"
and symbolized by the signs "=>" and "<=", and, in a related question:
What has happened to the distinction that is equally insistently made
between (3) the connective called the "biconditional" and signified by
the sign "<->" and (4) the assertion that is called an "equivalence"
and signified by the sign "<=>"?  My answer is this:  For my part,
I am deliberately avoiding making these distinctions at the level
of syntax, preferring to treat them instead as distinctions in
the use of boolean functions, turning on whether the function
is mentioned directly and used to compute values on arguments,
or whether its inverse is being invoked to indicate the fibers
of truth or untruth under the propositional function in question.

1.3.11.6. Stretching Exercises

For ease of reference, I repeat here a couple of the
definitions that are needed again in this discussion.

   | A "boolean connection" of degree k, also known as a "boolean function"
   | on k variables, is a map of the form F : %B%^k -> %B%.  In other words,
   | a boolean connection of degree k is a proposition about things in the
   | universe of discourse X = %B%^k.
   |
   | An "imagination" of degree k on X is a k-tuple of propositions
   | about things in the universe X.  By way of displaying the kinds
   | of notation that are used to express this idea, the imagination
   | #f# = <f_1, ..., f_k> is can be given as a sequence of indicator
   | functions f_j : X -> %B%, for j = 1 to k.  All of these features
   | of the typical imagination #f# can be summed up in either one of
   | two ways:  either in the form of a membership statement, stating
   | words to the effect that #f# belongs to the space (X -> %B%)^k,
   | or in the form of the type declaration that #f# : (X -> %B%)^k,
   | though perhaps the latter specification is slightly more precise
   | than the former.

The definition of the "stretch" operation and the uses of the
various brands of denotational operators can be reviewed here:

   IDS 133.  http://stderr.org/pipermail/inquiry/2004-June/001578.html
   IDS 134.  http://stderr.org/pipermail/inquiry/2004-June/001579.html
   IDS 136.  http://stderr.org/pipermail/inquiry/2004-June/001581.html
   IDS 137.  http://stderr.org/pipermail/inquiry/2004-June/001582.html

Taking up the preceding arrays of particular connections, namely,
the boolean functions on two or less variables, it possible to
illustrate the use of the stretch operation in a variety of
concrete cases.

For example, suppose that F is a connection of the form F : %B%^2 -> %B%,
that is, any one of the sixteen possibilities in Table 16, while p and q
are propositions of the form p, q : X -> %B%, that is, propositions about
things in the universe X, or else the indicators of sets contained in X.

Then one has the imagination #f# = <f_1, f_2> = <p, q> : (X -> %B%)^2,
and the stretch of the connection F to #f# on X amounts to a proposition
F^$ <p, q> : X -> %B%, usually written as "F^$ (p, q)" and vocalized as
the "stretch of F to p and q".  If one is concerned with many different
propositions about things in X, or if one is abstractly indifferent to
the particular choices for p and q, then one can detach the operator
F^$ : (X -> %B%)^2 -> (X -> %B%), called the "stretch of F over X",
and consider it in isolation from any concrete application.

When the "cactus notation" is used to represent boolean functions,
a single "$" sign at the end of the expression is enough to remind
a reader that the connections are meant to be stretched to several
propositions on a universe X.

For instance, take the connection F : %B%^2 -> %B% such that:

   F(x, y)  =  F^2_06 (x, y)  =  -(x, y)-.

This connection is the boolean function on a couple of variables x, y
that yields a value of %1% if and only if just one of x, y is not %1%,
that is, if and only if just one of x, y is %1%.  There is clearly an
isomorphism between this connection, viewed as an operation on the
boolean domain %B% = {%0%, %1%}, and the dyadic operation on binary
values x, y in !B! = GF(2) that is otherwise known as "x + y".

The same connection F : %B%^2 -> %B% can also be read as a proposition
about things in the universe X = %B%^2.  If S is a sentence that denotes
the proposition F, then the corresponding assertion says exactly what one
otherwise states by uttering "x is not equal to y".  In such a case, one
has -[S]- = F, and all of the following expressions are ordinarily taken
as equivalent descriptions of the same set:

   [| -[S]- |]  =  [| F |]

                =  F^(-1)(%1%)

                =  {<x, y> in %B%^2  :  S}

                =  {<x, y> in %B%^2  :  F(x, y) = %1%}

                =  {<x, y> in %B%^2  :  F(x, y)}

                =  {<x, y> in %B%^2  :  -(x, y)- = %1%}

                =  {<x, y> in %B%^2  :  -(x, y)- }

                =  {<x, y> in %B%^2  :  x exclusive-or y}

                =  {<x, y> in %B%^2  :  just one true of x, y}

                =  {<x, y> in %B%^2  :  x not equal to y}

                =  {<x, y> in %B%^2  :  x <=/=> y}

                =  {<x, y> in %B%^2  :  x =/= y}

                =  {<x, y> in %B%^2  :  x + y}

Notice the slight distinction, that I continue to maintain at this point,
between the logical values {false, true} and the algebraic values {0, 1}.
This makes it legitimate to write a sentence directly into the right side
of the set-builder expression, for instance, weaving the sentence S or the
sentence "x is not equal to y" into the context "{<x, y> in %B%^2 : ... }",
thereby obtaining the corresponding expressions listed above, while the
proposition F(x, y) can also be asserted more directly without equating
it to %1%, since it already has a value in {false, true}, and thus can
be taken as tantamount to an actual sentence.

If the appropriate safeguards can be kept in mind, avoiding all danger of
confusing propositions with sentences and sentences with assertions, then
the marks of these distinctions need not be forced to clutter the account
of the more substantive indications, that is, the ones that really matter.
If this level of understanding can be achieved, then it may be possible
to relax these restrictions, along with the absolute dichotomy between
algebraic and logical values, which tends to inhibit the flexibility
of interpretation.

This covers the properties of the connection F(x, y) = -(x, y)-,
treated as a proposition about things in the universe X = %B%^2.
Staying with this same connection, it is time to demonstrate how
it can be "stretched" into an operator on arbitrary propositions.

To continue the exercise, let p and q be arbitrary propositions about
things in the universe X, that is, maps of the form p, q : X -> %B%,
and suppose that p, q are indicator functions of the sets P, Q c X,
respectively.  In other words, one has the following set of data:

     p     =        -{P}-        :   X -> %B%

     q     =        -{Q}-        :   X -> %B%

   <p, q>  =  < -{P}- , -{Q}- >  :  (X -> %B%)^2

Then one has an operator F^$, the stretch of the connection F over X,
and a proposition F^$ (p, q), the stretch of F to <p, q> on X, with
the following properties:

   F^$         =  -( , )-^$   :  (X -> %B%)^2 -> (X -> %B%)

   F^$ (p, q)  =  -(p, q)-^$  :   X -> %B%

As a result, the application of the proposition F^$ (p, q) to each x in X
yields a logical value in %B%, all in accord with the following equations:

   F^$ (p, q)(x)   =   -(p, q)-^$ (x)  in  %B%

    ^                         ^
    |                         |
    =                         =
    |                         |
    v                         v

   F(p(x), q(x))   =   -(p(x), q(x))-  in  %B%

For each choice of propositions p and q about things in X, the stretch of
F to p and q on X is just another proposition about things in X, a simple
proposition in its own right, no matter how complex its current expression
or its present construction as F^$ (p, q) = -(p, q)^$ makes it appear in
relation to p and q.  Like any other proposition about things in X, it
indicates a subset of X, namely, the fiber that is variously described
in the following ways:

   [| F^$ (p, q) |]  =  [| -(p, q)-^$ |]

                     =  (F^$ (p, q))^(-1)(%1%)

                     =  {x in X  :  F^$ (p, q)(x)}

                     =  {x in X  :  -(p, q)-^$ (x)}

                     =  {x in X  :  -(p(x), q(x))- }

                     =  {x in X  :  p(x) + q(x)}

                     =  {x in X  :  p(x) =/= q(x)}

                     =  {x in X  :  -{P}- (x) =/= -{Q}- (x)}

                     =  {x in X  :  x in P <=/=> x in Q}

                     =  {x in X  :  x in P-Q or x in Q-P}

                     =  {x in X  :  x in P-Q |_| Q-P}

                     =  {x in X  :  x in P + Q}

                     =  P + Q          c  X

                     =  [|p|] + [|q|]  c  X

Which was to be shown.

1.3.12. Syntactic Transformations

We have been examining several distinct but closely related notions of indication. To discuss the import of these ideas in greater depth, it serves to establish a number of logical relations and set-theoretic identities that can be found to hold among their roughly parallel arrays of conceptions and constructions. Facilitating this task, in turn, requires a number of auxiliary concepts and notations.

The diverse notions of indication presently under discussion are expressed in a variety of different notations, enumerated as follows:

1. The functional language of propositions
2. The logical language of sentences
3. The geometric language of sets

Correspondingly, one way to explain the relationships that exist among the various notions of indication is to describe the translations that they induce among the associated families of notation.

1.3.12.1. Syntactic Transformation Rules

A good way to summarize the necessary translations between different styles of indication, and along the way to organize their use in practice, is by means of the "rules of syntactic transformation" (ROSTs) that partially formalize the translations in question.

Rudimentary examples of ROSTs are readily mined from the raw materials that are already available in this area of discussion. To begin as near the beginning as possible, let the definition of an indicator function be recorded in the following form:

o-------------------------------------------------o
| Definition 1.  Indicator Function               |
o-------------------------------------------------o
|                                                 |
| If      Q   c X,                                |
|                                                 |
| then  -{Q}- : X -> %B%                          |
|                                                 |
| such that, for all x in X:                      |
|                                                 |
o-------------------------------------------------o
|                                                 |
| D1a.  -{Q}-(x)  <=>  x in Q.                    |
|                                                 |
o-------------------------------------------------o

In practice, a definition like this is commonly used to substitute one of two logically equivalent expressions or sentences for the other in a context where the conditions of using the definition in this way are satisfied and where the change is perceived as potentially advancing a proof. The employment of a definition in this way can be expressed in the form of a ROST that allows one to exchange two expressions of logically equivalent forms for one another in every context where their logical values are the only consideration. To be specific, the logical value of an expression is the value in the boolean domain %B% = {%0%, %1%} that the expression represents to its context or that it stands for in its context.

In the case of Definition 1, the corresponding ROST permits one to exchange a sentence of the form "x in Q" with an expression of the form "-{Q}-(x)" in any context that satisfies the conditions of its use, namely, the conditions of the definition that lead up to the stated equivalence. The relevant ROST is recorded in Rule 1. By way of convention, I list the items that fall under a rule in rough order of their ascending conceptual subtlety or their increasing syntactic complexity, without regard for the normal or the typical orders of their exchange, since this can vary from widely from case to case.

o-------------------------------------------------o
| Rule 1                                          |
o-------------------------------------------------o
|                                                 |
| If      Q   c X,                                |
|                                                 |
| then  -{Q}- : X -> %B%,                         |
|                                                 |
| and if  x  in X,                                |
|                                                 |
| then the following are equivalent:              |
|                                                 |
o-------------------------------------------------o
|                                                 |
| R1a.   x in Q.                                  |
|                                                 |
| R1b.  -{Q}-(x).                                 |
|                                                 |
o-------------------------------------------------o

Conversely, any rule of this sort, properly qualified by the conditions under which it applies, can be turned back into a summary statement of the logical equivalence that is involved in its application. This mode of conversion between a static principle and a transformational rule, in other words, between a statement of equivalence and an equivalence of statements, is so automatic that it is usually not necessary to make a separate note of the "horizontal" versus the "vertical" versions of what amounts to the same abstract principle.

As another example of a ROST, consider the following logical equivalence, that holds for any \(X \subseteq U\!\) and for all \(u \in U.\)

-{X}-(u) <=> -{X}-(u) = 1.

In practice, this logical equivalence is used to exchange an expression of the form "-{X}-(u)" with a sentence of the form "-{X}-(u) = 1" in any context where one has a relatively fixed X c U in mind and where one is conceiving u in U to vary over its whole domain, namely, the universe U. This leads to the ROST that is given in Rule 2.

o-------------------------------------------------o
| Rule 2                                          |
o-------------------------------------------------o
|                                                 |
| If f : U -> %B%                                 |
|                                                 |
| and u in U,                                     |
|                                                 |
| then the following are equivalent:              |
|                                                 |
o-------------------------------------------------o
|                                                 |
| R2a.  f(u).                                     |
|                                                 |
| R2b.  f(u) = 1.                                 |
|                                                 |
o-------------------------------------------------o

Rules like these can be chained together to establish extended rules, just so long as their antecedent conditions are compatible. For example, Rules 1 and 2 combine to give the equivalents that are listed in Rule 3. This follows from a recognition that the function -{X}- : U -> %B% that is introduced in Rule 1 is an instance of the function f : U -> %B% that is mentioned in Rule 2. By the time one arrives in the "consequence box" of either Rule, then, one has in mind a comparatively fixed X c U, a proposition f or -{X}- about things in U, and a variable argument u in U.

o-------------------------------------------------o---------o
| Rule 3                                          |         |
o-------------------------------------------------o---------o
|                                                 |         |
| If X c U                                        |         |
|                                                 |         |
| and u in U,                                     |         |
|                                                 |         |
| then the following are equivalent:              |         |
|                                                 |         |
o-------------------------------------------------o---------o
|                                                 |         |
| R3a.  u in X.                                   | : R1a   |
|                                                 |   ::    |
| R3b.  -{X}-(u).                                 | : R1b   |
|                                                 | : R2a   |
|                                                 |   ::    |
| R3c.  -{X}-(u) = 1.                             | : R2b   |
|                                                 |         |
o-------------------------------------------------o---------o

A large stock of rules can be derived in this way, by chaining together segments that are selected from a stock of previous rules, with perhaps the whole process of derivation leading back to an axial body or a core stock of rules, with all recurring to and relying on an axiomatic basis. In order to keep track of their derivations, as their pedigrees help to remember the reasons for trusting their use in the first place, derived rules can be annotated by citing the rules from which they are derived.

In the present discussion, I am using a particular style of annotation for rule derivations, one that is called "proof by grammatical paradigm" or "proof by syntactic analogy". The annotations in the right margin of the Rule box can be read as the "denominators" of the paradigm that is being employed, in other words, as the alternating terms of comparison in a sequence of analogies. This can be illustrated by considering the derivation Rule 3 in detail. Taking the steps marked in the box one at a time, one can interweave the applications in the central body of the box with the annotations in the right margin of the box, reading "is to" for the ":" sign and "as" for the "::" sign, in the following fashion:

R3a.  "u in X"  is to  R1a, namely, "u in X",

      as

R3b.  "{X}(u)"  is to  R1b, namely, "{X}(u)",

      and

	"{X}(u)"  is to  R2a, namely, "f(u)",

      as

R3c.  "{X}(u) = 1"  is to  R2b, namely, "f(u) = 1".

Notice how the sequence of analogies pivots on the item R3b, viewing it first under the aegis of R1b, as the second term of the first analogy, and then turning to view it again under the guise of R2a, as the first term of the second analogy.

By way of convention, rules that are tailored to a particular application, case, or subject, and rules that are adapted to a particular goal, object, or purpose, I frequently refer to as "Facts".

Besides linking rules together into extended sequences of equivalents, there is one other way that is commonly used to get new rules from old. Novel starting points for rules can be obtained by extracting pairs of equivalent expressions from a sequence that falls under an established rule, and then by stating their equality in the proper form of equation. For example, by extracting the equivalent expressions that are annotated as "R3a" and "R3c" in Rule 3 and by explictly stating their equivalence, one obtains the specialized result that is recorded in Corollary 1.

Corollary 1

If	X	c	U

and	u	C	U,

then the following statement is true:

C1a.	u C X  <=>  {X}(u) = 1.	 R3a=R3c

There are a number of issues, that arise especially in establishing the proper use of ROSTs, that are appropriate to discuss at this juncture. The notation "[S]" is intended to represent "the proposition denoted by the sentence S". There is only one problem with the use of this form. There is, in general, no such thing as "the" proposition denoted by S. Generally speaking, if a sentence is taken out of context and considered across a variety of different contexts, there is no unique proposition that it can be said to denote. But one is seldom ever speaking at the maximum level of generality, or even found to be thinking of it, and so this notation is usually meaningful and readily understandable whenever it is read in the proper frame of mind. Still, once the issue is raised, the question of how these meanings and understandings are possible has to be addressed, especially if one desires to express the regulations of their syntax in a partially computational form. This requires a closer examination of the very notion of "context", and it involves engaging in enough reflection on the "contextual evaluation" of sentences that the relevant principles of its successful operation can be discerned and rationalized in explicit terms.

A sentence that is written in a context where it represents a value of 1 or 0 as a function of things in the universe U, where it stands for a value of "true" or "false", depending on how the signs that constitute its proper syntactic arguments are interpreted as denoting objects in U, in other words, where it is bound to lead its interpreter to view its own truth or falsity as determined by a choice of objects in U, is a sentence that might as well be written in the context "[ ... ]", whether or not this frame is explicitly marked around it.

More often than not, the context of interpretation fixes the denotations of most of the signs that make up a sentence, and so it is safe to adopt the convention that only those signs whose objects are not already fixed are free to vary in their denotations. Thus, only the signs that remain in default of prior specification are subject to treatment as variables, with a decree of functional abstraction hanging over all of their heads.

[u C X] = Lambda (u, C, X).(u C X).

As it is presently stated, Rule 1 lists a couple of manifest sentences, and it authorizes one to make exchanges in either direction between the syntactic items that have these two forms. But a sentence is any sign that denotes a proposition, and thus there are a number of less obvious sentences that can be added to this list, extending the number of items that are licensed to be exchanged. Consider the sense of equivalence among sentences that is recorded in Rule 4.

Rule 4

If	X	c	U	is fixed

and	u	C	U	is varied,

then the following are equivalent:

R4a.	u C X.

R4b.	[u C X].

R4c.	[u C X](u).

R4d.	{X}(u).

R4e.	{X}(u) = 1.

The first and last items on this list, namely, the sentences "u C X" and "{X}(u) = 1" that are annotated as "R4a" and "R4e", respectively, are just the pair of sentences from Rule 3 whose equivalence for all u C U is usually taken to define the idea of an indicator function {X} : U -> B. At first sight, the inclusion of the other items appears to involve a category confusion, in other words, to mix the modes of interpretation and to create an array of mismatches between their own ostensible types and the ruling type of a sentence. On reflection, and taken in context, these problems are not as serious as they initially seem. For instance, the expression "[u C X]" ostensibly denotes a proposition, but if it does, then it evidently can be recognized, by virtue of this very fact, to be a genuine sentence. As a general rule, if one can see it on the page, then it cannot be a proposition, but can be, at best, a sign of one.

The use of the basic connectives can be expressed in the form of a ROST as follows:

Logical Translation Rule 0

If	Sj	is a sentence

		about things in the universe U

and	Pj	is a proposition

		about things in the universe U

such that:

L0a.	[Sj] = Pj, for all j C J,

then the following equations are true:

L0b.	[ConcJj Sj]  =  ConjJj [Sj]  =  ConjJj Pj.

L0c.	[SurcJj Sj]  =  SurjJj [Sj]  =  SurjJj Pj.

As a general rule, the application of a ROST involves the recognition of an antecedent condition and the facilitation of a consequent condition. The antecedent condition is a state whose initial expression presents a match, in a formal sense, to one of the sentences that are listed in the STR, and the consequent condition is achieved by taking its suggestions seriously, in other words, by following its sequence of equivalents and implicants to some other link in its chain.

Generally speaking, the application of a rule involves the recognition of an antecedent condition as a case that falls under a clause of the rule. This means that the antecedent condition is able to be captured in the form, conceived in the guise, expressed in the manner, grasped in the pattern, or recognized in the shape of one of the sentences in a list of equivalents or a chain of implicants.

A condition is "amenable" to a rule if any of its conceivable expressions formally match any of the expressions that are enumerated by the rule. Further, it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that needs to be checked on input for whether it fits the antecedent condition and there are several types of output that are generated as a consequence, only a few of which are usually needed at any given time.

Logical Translation Rule 1

If	S	is a sentence

		about things in the universe U

and	P	is a proposition : U -> B, such that:

L1a.	[S]  =  P,

then the following equations hold:

L1b00.	[False]	=	()	=	0	:	U->B.

L1b01.	[Not S]	=	([S])	=	(P)	:	U->B.

L1b10.	[S]	=	[S]	=	P	:	U->B.

L1b11.	[True]	=	(())	=	1	:	U->B.

Geometric Translation Rule 1

If	X	c	U

and	P	:	U -> B, such that:

G1a.	{X}  =  P,

then the following equations hold:

G1b00.	{{}}	=	()	=	0	:	U->B.

G1b10.	{~X}	=	({X})	=	(P)	:	U->B.

G1b01.	{X}	=	{X}	=	P	:	U->B.

G1b11.	{U}	=	(())	=	1	:	U->B.

Logical Translation Rule 2

If	S, T	are sentences

		about things in the universe U

and	P, Q	are propositions: U -> B, such that:

L2a.	[S] = P  and  [T] = Q,

then the following equations hold:

L2b00.	[False]	=	()	=	0 : U->B.

L2b01.	[Neither S nor T]	=	([S])([T])	=	(P)(Q).

L2b02.	[Not S, but T]	=	([S])[T]	=	(P) Q.

L2b03.	[Not S]	=	([S])	=	(P).

L2b04.	[S and not T]	=	[S]([T])	=	P (Q).

L2b05.	[Not T]	=	([T])	=	(Q).

L2b06.	[S or T, not both]	=	([S], [T])	=	(P, Q).

L2b07.	[Not both S and T]	=	([S].[T])	=	(P Q).

L2b08.	[S and T]	=	[S].[T]	=	P.Q.

L2b09.	[S <=> T]	=	(([S], [T]))	=	((P, Q)).

L2b10.	[T]	=	[T]	=	Q.

L2b11.	[S => T]	=	([S]([T]))	=	(P (Q)).

L2b12.	[S]	=	[S]	=	P.

L2b13.	[S <= T]	=	(([S]) [T])	=	((P) Q).

L2b14.	[S or T]	=	(([S])([T]))	=	((P)(Q)).

L2b15.	[True]	=	(())	=	1 : U->B.

Geometric Translation Rule 2

If	X, Y	c	U

and	P, Q	U -> B, such that:

G2a.	{X} = P  and  {Y} = Q,

then the following equations hold:

G2b00.	{{}}	=	()	=	0 : U->B.

G2b01.	{~X n ~Y}	=	({X})({Y})	=	(P)(Q).

G2b02.	{~X n Y}	=	({X}){Y}	=	(P) Q.

G2b03.	{~X}	=	({X})	=	(P).

G2b04.	{X n ~Y}	=	{X}({Y})	=	P (Q).

G2b05.	{~Y}	=	({Y})	=	(Q).

G2b06.	{X + Y}	=	({X}, {Y})	=	(P, Q).

G2b07.	{~(X n Y)}	=	({X}.{Y})	=	(P Q).

G2b08.	{X n Y}	=	{X}.{Y}	=	P.Q.

G2b09.	{~(X + Y)}	=	(({X}, {Y}))	=	((P, Q)).

G2b10.	{Y}	=	{Y}	=	Q.

G2b11.	{~(X n ~Y)}	=	({X}({Y}))	=	(P (Q)).

G2b12.	{X}	=	{X}	=	P.

G2b13.	{~(~X n Y)}	=	(({X}) {Y})	=	((P) Q).

G2b14.	{X u Y}	=	(({X})({Y}))	=	((P)(Q)).

G2b15.	{U}	=	(())	=	1 : U->B.

Value Rule 1

If	v, w	C	B

then	"v = w" is a sentence about <v, w> C B2,

	[v = w] is a proposition : B2 -> B,

and the following are identical values in B:

V1a.	[ v = w ](v, w)

V1b.	[ v <=> w ](v, w)

V1c.	((v , w))

Value Rule 1

If	v, w	C	B,

then the following are equivalent:

V1a.	v = w.

V1b.	v <=> w.

V1c.	(( v , w )).

A rule that allows one to turn equivalent sentences into identical propositions:

(S <=> T) <=> ([S] = [T])

Consider [ v = w ](v, w) and [ v(u) = w(u) ](u)

Value Rule 1

If	v, w	C	B,

then the following are identical values in B:

V1a.	[ v = w ]

V1b.	[ v <=> w ]

V1c.	(( v , w ))

Value Rule 1

If	f, g	:	U -> B,

and	u	C	U

then the following are identical values in B:

V1a.	[ f(u) = g(u) ]

V1b.	[ f(u) <=> g(u) ]

V1c.	(( f(u) , g(u) ))

Value Rule 1

If	f, g	:	U -> B,

then the following are identical propositions on U:

V1a.	[ f = g ]

V1b.	[ f <=> g ]

V1c.	(( f , g ))$

Evaluation Rule 1

If	f, g	:	U -> B

and	u	C	U,

then the following are equivalent:

E1a.	f(u) = g(u).	:V1a

				::

E1b.	f(u) <=> g(u).	:V1b

				::

E1c.	(( f(u) , g(u) )).	:V1c

				:$1a

				::

E1d.	(( f , g ))$(u).	:$1b

Evaluation Rule 1

If	S, T	are sentences

		about things in the universe U,

	f, g	are propositions: U -> B,

and	u	C	U,

then the following are equivalent:

E1a.	f(u) = g(u).	:V1a

				::

E1b.	f(u) <=> g(u).	:V1b

				::

E1c.	(( f(u) , g(u) )).	:V1c

				:$1a

				::

E1d.	(( f , g ))$(u).	:$1b

Definition 2

If	X, Y	c	U,

then the following are equivalent:

D2a.	X = Y.

D2b.	u C X  <=>  u C Y, for all u C U.

Definition 3

If	f, g	:	U -> V,

then the following are equivalent:

D3a.	f = g.

D3b.	f(u) = g(u), for all u C U.

Definition 4

If	X	c	U,

then the following are identical subsets of UxB:

D4a.	{X}

D4b.	{< u, v> C UxB : v = [u C X]}

Definition 5

If	X	c	U,

then the following are identical propositions:

D5a.	{X}.

D5b.	f	:	U -> B

:	f(u)	=	[u C X], for all u C U.

Given an indexed set of sentences, Sj for j C J, it is possible to consider the logical conjunction of the corresponding propositions. Various notations for this concept are be useful in various contexts, a sufficient sample of which are recorded in Definition 6.

Definition 6

If	Sj	is a sentence

		about things in the universe U,

		for all j C J,

then the following are equivalent:

D6a.	Sj, for all j C J.

D6b.	For all j C J, Sj.

D6c.	Conj(j C J) Sj.

D6d.	ConjJ,j Sj.

D6e.	ConjJj Sj.

Definition 7

If	S, T	are sentences

		about things in the universe U,

then the following are equivalent:

D7a.	S <=> T.

D7b.	[S] = [T].

Rule 5

If	X, Y	c	U,

then the following are equivalent:

R5a.	X = Y.	:D2a

		::

R5b.	u C X  <=>  u C Y, for all u C U.	:D2b

		:D7a

		::

R5c.	[u C X] = [u C Y], for all u C U.	:D7b

		:???

		::

R5d.	{< u, v> C UxB : v = [u C X]}

	=

	{< u, v> C UxB : v = [u C Y]}.	:???

		:D5b

		::

R5e.	{X} = {Y}.	:D5a

Rule 6

If	f, g	:	U -> V,

then the following are equivalent:

R6a.	f = g.	:D3a

		::

R6b.	f(u) = g(u), for all u C U.	:D3b

		:D6a

		::

R6c.	ConjUu (f(u) = g(u)).	:D6e

Rule 7

If	P, Q	:	U -> B,

then the following are equivalent:

R7a.	P = Q.	:R6a

		::

R7b.	P(u) = Q(u), for all u C U.	:R6b

		::

R7c.	ConjUu (P(u)  =  Q(u)).	:R6c

		:P1a

		::

R7d.	ConjUu (P(u) <=> Q(u)).	:P1b

		::

R7e.	ConjUu (( P(u) , Q(u) )).	:P1c

		:$1a

		::

R7f.	ConjUu (( P , Q ))$(u).	:$1b

Rule 8

If	S, T	are sentences

		about things in the universe U,

then the following are equivalent:

R8a.	S <=> T.	:D7a

		::

R8b.	[S] = [T].	:D7b

		:R7a

		::

R8c.	[S](u) = [T](u), for all u C U.	:R7b

		::

R8d.	ConjUu ( [S](u)  =  [T](u) ).	:R7c

		::

R8e.	ConjUu ( [S](u) <=> [T](u) ).	:R7d

		::

R8f.	ConjUu (( [S](u) , [T](u) )).	:R7e

		::

R8g.	ConjUu (( [S] , [T] ))$(u).	:R7f

For instance, the observation that expresses the equality of sets in terms of their indicator functions can be formalized according to the pattern in Rule 9, namely, at lines (a, b, c), and these components of Rule 9 can be cited in future uses as "R9a", "R9b", "R9c", respectively. Using Rule 7, annotated as "R7", to adduce a few properties of indicator functions to the account, it is possible to extend Rule 9 by another few steps, referenced as "R9d", "R9e", "R9f", "R9g".

Rule 9

If	X, Y	c	U,

then the following are equivalent:

R9a.	X = Y.	:R5a

		::

R9b.	{X} = {Y}.	:R5e

		:R7a

		::

R9c.	{X}(u) = {Y}(u), for all u C U.	:R7b

		::

R9d.	ConjUu ( {X}(u)  =  {Y}(u) ).	:R7c

		::

R9e.	ConjUu ( {X}(u) <=> {Y}(u) ).	:R7d

		::

R9f.	ConjUu (( {X}(u) , {Y}(u) )).	:R7e

		::

R9g.	ConjUu (( {X} , {Y} ))$(u).	:R7f

Rule 10

If	X, Y	c	U,

then the following are equivalent:

R10a.	X = Y.	:D2a

			::

R10b.	u C X  <=>  u C Y, for all u C U.	:D2b

			:R8a

			::

R10c.	[u C X] = [u C Y].	:R8b

			::

R10d.	For all u C U,

		[u C X](u) = [u C Y](u).	:R8c

			::

R10e.	ConjUu ( [u C X](u)  =  [u C Y](u) ).	:R8d

			::

R10f.	ConjUu ( [u C X](u) <=> [u C Y](u) ).	:R8e

			::

R10g.	ConjUu (( [u C X](u) , [u C Y](u) )).	:R8f

			::

R10h.	ConjUu (( [u C X] , [u C Y] ))$(u).	:R8g

Rule 11

If	X	c	U

then the following are equivalent:

R11a.	X	=	{u C U : S}.	:R5a

					::

R11b.	{X}	=	{ {u C U : S} }.	:R5e

					::

R11c.	{X}	c	UxB

:	{X}	=	{< u, v> C UxB : v = [S](u)}.	:R

					::

R11d.	{X}	:	U -> B

:	{X}(u)	=	[S](u),	for all u C U.	:R

					::

R11e.	{X}	=	[S].		:R

An application of Rule 11 involves the recognition of an antecedent condition as a case under the Rule, that is, as a condition that matches one of the sentences in the Rule's chain of equivalents, and it requires the relegation of the other expressions to the production of a result. Thus, there is the choice of an initial expression that has to be checked on input for whether it fits the antecedent condition, and there is the choice of three types of output that are generated as a consequence, only one of which is generally needed at any given time. More often than not, though, a rule is applied in only a few of its possible ways. The usual antecedent and the usual consequents for Rule 11 can be distinguished in form and specialized in practice as follows:

a. R11a marks the usual starting place for an application of the Rule, that is, the standard form of antecedent condition that is likely to lead to an invocation of the Rule.

b. R11b records the trivial consequence of applying the spiny braces to both sides of the initial equation.

c. R11c gives a version of the indicator function with {X} c UxB, called its "extensional form".

d. R11d gives a version of the indicator function with {X} : U->B, called its "functional form".

Applying Rule 9, Rule 8, and the Logical Rules to the special case where S <=> (X = Y), one obtains the following general fact.

Fact 1

If	X,Y	c	U,

then the following are equivalent:

F1a.	S	<=>	X = Y.	:R9a

				::

F1b.	S	<=>	{X} = {Y}.	:R9b

				::

F1c.	S	<=>	{X}(u) = {Y}(u), for all u C U.	:R9c

				::

F1d.	S	<=>	ConjUu ( {X}(u) = {Y}(u) ).	:R9d

				:R8a

				::

F1e.	[S]	=	[ ConjUu ( {X}(u) = {Y}(u) ) ].	:R8b

				:???

				::

F1f.	[S]	=	ConjUu [ {X}(u) = {Y}(u) ].	:???

				::

F1g.	[S]	=	ConjUu (( {X}(u) , {Y}(u) )).	:$1a

				::

F1h.	[S]	=	ConjUu (( {X} , {Y} ))$(u).	:$1b

///

	{u C U : (f, g)$(u)}

	=	{u C U : (f(u), g(u))}

	=	{u C 

///

1.3.12.2. Derived Equivalence Relations

One seeks a method of general application for approaching the individual sign relation, a way to select an aspect of its form, to analyze it with regard to its intrinsic structure, and to classify it in comparison with other sign relations. With respect to a particular sign relation, one approach that presents itself is to examine the relation between signs and interpretants that is given directly by its connotative component and to compare it with the various forms of derived, indirect, mediate, or peripheral relationships that can be found to exist among signs and interpretants by way of secondary considerations or subsequent studies. Of especial interest are the relationships among signs and interpretants that can be obtained by working through the collections of objects that they commonly or severally denote.

A classic way of showing that two sets are equal is to show that every element of the first belongs to the second and that every element of the second belongs to the first. The problem with this strategy is that one can exhaust a considerable amount of time trying to prove that two sets are equal before it occurs to one to look for a counterexample, that is, an element of the first that does not belong to the second or an element of the second that does not belong to the first, in cases where that is precisely what one ought to be seeking. It would be nice if there were a more balanced, impartial, neutral, or nonchalant way to go about this task, one that did not require such an undue commitment to either side, a technique that helps to pinpoint the counterexamples when they exist, and a method that keeps in mind the original relation of "proving that" and "showing that" to probing, testing, and seeing "whether".

A different way of seeing that two sets are equal, or of seeing whether two sets are equal, is based on the following observation:

	Two sets are equal as sets

<=>	the indicator functions of these sets are equal as functions

<=>	the values of these functions are equal on all domain elements.

It is important to notice the hidden quantifier, of a universal kind, that lurks in all three equivalent statements but is only revealed in the last.

In making the next set of definitions and in using the corresponding terminology it is taken for granted that all of the references of signs are relative to a particular sign relation R c OxSxI that either remains to be specified or is already understood. Further, I continue to assume that S = I, in which case this set is called the "syntactic domain" of R.

In the following definitions let R c OxSxI, let S = I, and let x, y C S.

Recall the definition of Con(R), the connotative component of R, in the following form:

Con(R) = RSI = {< s, i> C SxI : <o, s, i> C R for some o C O}.

Equivalent expressions for this concept are recorded in Definition 8.

Definition 8

If	R	c	OxSxI,

then the following are identical subsets of SxI:

D8a.	RSI

D8b.	ConR

D8c.	Con(R)

D8d.	PrSI(R)

D8e.	{< s, i> C SxI : <o, s, i> C R for some o C O}

The dyadic relation RIS that constitutes the converse of the connotative relation RSI can be defined directly in the following fashion:

Con(R)^ = RIS = {< i, s> C IxS : <o, s, i> C R for some o C O}.

A few of the many different expressions for this concept are recorded in Definition 9.

Definition 9

If	R	c	OxSxI,

then the following are identical subsets of IxS:

D9a.	RIS

D9b.	RSI^

D9c.	ConR^

D9d.	Con(R)^

D9e.	PrIS(R)

D9f.	Conv(Con(R))

D9g.	{< i, s> C IxS : <o, s, i> C R for some o C O}

Recall the definition of Den(R), the denotative component of R, in the following form:

Den(R) = ROS = {<o, s> C OxS : <o, s, i> C R for some i C I}.

Equivalent expressions for this concept are recorded in Definition 10.

Definition 10

If	R	c	OxSxI,

then the following are identical subsets of OxS:

D10a.	ROS

D10b.	DenR

D10c.	Den(R)

D10d.	PrOS(R)

D10e.	{<o, s> C OxS : <o, s, i> C R for some i C I}

The dyadic relation RSO that constitutes the converse of the denotative relation ROS can be defined directly in the following fashion:

Den(R)^ = RSO = {< s, o> C SxO : <o, s, i> C R for some i C I}.

A few of the many different expressions for this concept are recorded in Definition 11.

Definition 11

If	R	c	OxSxI,

then the following are identical subsets of SxO:

D11a.	RSO

D11b.	ROS^

D11c.	DenR^

D11d.	Den(R)^

D11e.	PrSO(R)

D11f.	Conv(Den(R))

D11g.	{< s, o> C SxO : <o, s, i> C R for some i C I}

The "denotation of x in R", written "Den(R, x)", is defined as follows:

Den(R, x) = {o C O : <o, x> C Den(R)}.

In other words:

Den(R, x) = {o C O : <o, x, i> C R for some i C I}.

Equivalent expressions for this concept are recorded in Definition 12.

Definition 12

If	R	c	OxSxI,

and	x	C	S,

then the following are identical subsets of O:

D12a.	ROS.x

D12b.	DenR.x

D12c.	DenR|x

D12d.	DenR(, x)

D12e.	Den(R, x)

D12f.	Den(R).x

D12g.	{o C O : <o, x> C Den(R)}

D12h.	{o C O : <o, x, i> C R for some i C I}

Signs are "equiferent" if they refer to all and only the same objects, that is, if they have exactly the same denotations. In other language for the same relation, signs are said to be "denotatively equivalent" or "referentially equivalent", but it is probably best to check whether the extension of this concept over the syntactic domain is really a genuine equivalence relation before jumping to the conclusions that are implied by these latter terms.

To define the "equiference" of signs in terms of their denotations, one says that "x is equiferent to y under R", and writes "x =R y", to mean that Den(R, x) = Den(R, y). Taken in extension, this notion of a relation between signs induces an "equiference relation" on the syntactic domain.

For each sign relation R, this yields a binary relation Der(R) c SxI that is defined as follows:

Der(R) = DerR = {<x, y> C SxI : Den(R, x) = Den(R, y)}.

These definitions and notations are recorded in the following display.

Definition 13

If	R	c	OxSxI,

then the following are identical subsets of SxI:

D13a.	DerR

D13b.	Der(R)

D13c.	{<x,y> C SxI : DenR|x = DenR|y}

D13d.	{<x,y> C SxI : Den(R, x) = Den(R, y)}

The relation Der(R) is defined and the notation "x =R y" is meaningful in every situation where Den(-,-) makes sense, but it remains to check whether this relation enjoys the properties of an equivalence relation.

1. Reflexive property. Is it true that x =R x for every x C S = I? By definition, x =R x if and only if Den(R, x) = Den(R, x). Thus, the reflexive property holds in any setting where the denotations Den(R, x) are defined for all signs x in the syntactic domain of R.
2. Symmetric property. Does x =R y => y =R x for all x, y C S? In effect, does Den(R, x) = Den(R, y) imply Den(R, y) = Den(R, x) for all signs x and y in the syntactic domain S? Yes, so long as the sets Den(R, x) and Den(R, y) are well-defined, a fact which is already being assumed.
3. Transitive property. Does x =R y & y =R z => x =R z for all x, y, z C S? To belabor the point, does Den(R, x) = Den(R, y) and Den(R, y) = Den(R, z) imply Den(R, x) = Den(R, z) for all x, y, z in S? Yes, again, under the stated conditions.

It should be clear at this point that any question about the equiference of signs reduces to a question about the equality of sets, specifically, the sets that are indexed by these signs. As a result, so long as these sets are well-defined, the issue of whether equiference relations induce equivalence relations on their syntactic domains is almost as trivial as it initially appears.

Taken in its set-theoretic extension, a relation of equiference induces a "denotative equivalence relation" (DER) on its syntactic domain S = I. This leads to the formation of "denotative equivalence classes" (DEC's), "denotative partitions" (DEP's), and "denotative equations" (DEQ's) on the syntactic domain. But what does it mean for signs to be equiferent?

Notice that this is not the same thing as being "semiotically equivalent", in the sense of belonging to a single "semiotic equivalence class" (SEC), falling into the same part of a "semiotic partition" (SEP), or having a "semiotic equation" (SEQ) between them. It is only when very felicitous conditions obtain, establishing a concord between the denotative and the connotative components of a sign relation, that these two ideas coalesce.

In general, there is no necessity that the equiference of signs, that is, their denotational equivalence or their referential equivalence, induces the same equivalence relation on the syntactic domain as that defined by their semiotic equivalence, even though this state of accord seems like an especially desirable situation. This makes it necessary to find a distinctive nomenclature for these structures, for which I adopt the term "denotative equivalence relations" (DER's). In their train they bring the allied structures of "denotative equivalence classes" (DEC's) and "denotative partitions" (DEP's), while the corresponding statements of "denotative equations" (DEQ's) are expressible in the form "x =R y".

The uses of the equal sign for denoting equations or equivalences are recalled and extended in the following ways:

1. If E is an arbitrary equivalence relation,

then the equation "x =E y" means that <x, y> C E.

2. If R is a sign relation such that RSI is a SER on S = I,

then the semiotic equation "x =R y" means that <x, y> C RSI.

3. If R is a sign relation such that F is its DER on S = I,

then the denotative equation "x =R y" means that <x, y> C F,

in other words, that Den(R, x) = Den(R, y).

The uses of square brackets for denoting equivalence classes are recalled and extended in the following ways:

1. If E is an arbitrary equivalence relation,

then "[x]E" denotes the equivalence class of x under E.

2. If R is a sign relation such that Con(R) is a SER on S = I,

then "[x]R" denotes the SEC of x under Con(R).

3. If R is a sign relation such that Der(R) is a DER on S = I,

then "[x]R" denotes the DEC of x under Der(R).

By applying the form of Fact 1 to the special case where X = Den(R, x) and Y = Den(R, y), one obtains the following facts.

Fact 2.1

If	R	c	OxSxI,

then the following are identical subsets of SxI:

F2.1a.	DerR		:D13a

					::

F2.1b.	Der(R)	:D13b

					::

F2.1c.	{<x, y> C SxI :

		Den(R, x) = Den(R, y)

	}				:D13c

					:R9a

					::

F2.1d.	{<x, y> C SxI :

		{Den(R, x)} = {Den(R, y)}

	}				:R9b

					::

F2.1e.	{<x, y> C SxI :

		for all o C O

			{Den(R, x)}(o) = {Den(R, y)}(o)

	}				:R9c

					::

F2.1f.	{<x, y> C SxI :	

		Conj(o C O)

			{Den(R, x)}(o) = {Den(R, y)}(o)

	}				:R9d

					::

F2.1g.	{<x, y> C SxI :

		Conj(o C O)

			(( {Den(R, x)}(o) , {Den(R, y)}(o) ))

	}				:R9e

					::

F2.1h.	{<x, y> C SxI :

		Conj(o C O)

			(( {Den(R, x)} , {Den(R, y)} ))$(o)

	}				:R9f

					:D12e

					::

F2.1i.	{<x, y> C SxI :

		Conj(o C O)

			(( {ROS.x} , {ROS.y} ))$(o)

	}				:D12a

Fact 2.2

If	R	c	OxSxI,

then the following are equivalent:

F2.2a.	DerR	=	{<x, y> C SxI :

				Conj(o C O)

					{Den(R, x)}(o) =

					{Den(R, y)}(o)

			}						:R11a
								::

F2.2b.	{DerR}	=	{	{<x, y> C SxI :

					Conj(o C O)

						{Den(R, x)}(o) =

						{Den(R, y)}(o)

				}

			}						:R11b

									::

F2.2c.	{DerR}	c	SxIxB

	:

	{DerR}	=	{<x, y, v> C SxIxB :

				v =

					[	Conj(o C O)

							{Den(R, x)}(o) =

							{Den(R, y)}(o)

					]

			}						:R11c

									::

F2.2d.	{DerR}	=	{<x, y, v> C SxIxB :

				v =

					Conj(o C O)

						[	{Den(R, x)}(o) =

							{Den(R, y)}(o)

						]

			}						:Log

F2.2e.	{DerR}	=	{<x, y, v> C SxIxB :

				v =

					Conj(o C O)

						((	{Den(R, x)}(o),

								{Den(R, y)}(o)

						))

			}						:Log

F2.2f.	{DerR}	=	{<x, y, v> C SxIxB :

				v =

					Conj(o C O)

						((	{Den(R, x)},

								{Den(R, y)}

						))$(o)

			}						:$

Fact 2.3

If	R	c	OxSxI,

then the following are equivalent:

F2.3a.	DerR	=	{<x, y> C SxI :

				Conj(o C O)

					{Den(R, x)}(o) =

					{Den(R, y)}(o)

			}				:R11a

							::

F2.3b.	{DerR}	:	SxI -> B

	:

	{DerR}(x, y)	=	[	Conj(o C O)

					{Den(R, x)}(o) =

					{Den(R, y)}(o)

			]				:R11d

							::

F2.3c.	{DerR}(x, y)	=	Conj(o C O)

				[	{Den(R, x)}(o) =

					{Den(R, y)}(o)

				]			:Log

							::

F2.3d.	{DerR}(x, y)	=	Conj(o C O)

				[	{DenR}(o, x) =

					{DenR}(o, y)

				]			:Def

							::

F2.3e.	{DerR}(x, y)	=	Conj(o C O)

				((	{DenR}(o, x),

						{DenR}(o, y)

				))		:Log

							:D10b

							::

F2.3f.	{DerR}(x, y)	=	Conj(o C O)

				((	{ROS}(o, x),

						{ROS}(o, y)

				))		:D10a

1.3.12.3. Digression on Derived Relations

A better understanding of derived equivalence relations (DER's) can be achieved by placing their constructions within a more general context, and thus comparing the associated type of derivation operation, namely, the one that takes a triadic relation R into a dyadic relation Der(R), with other types of operations on triadic relations. The proper setting would permit a comparative study of all their constructions from a basic set of projections and a full array of compositions on dyadic relations.

To that end, let the derivation Der(R) be expressed in the following way:

{DerR}(x, y) = Conj(o C O) (( {RSO}(x, o) , {ROS}(o, y) )).

From this abstract a form of composition, temporarily notated as "P#Q", where P c XxM and Q c MxY are otherwise arbitrary dyadic relations, and where P#Q c XxY is defined as follows:

{P#Q}(x, y) = Conj(m C M) (( {P}(x, m) , {Q}(m, y) )).

Compare this with the usual form of composition, typically notated as "P.Q" and defined as follows:

{P.Q}(x, y) = Disj(m C M) ( {P}(x, m) . {Q}(m, y) ).

1.4. Outlook of the Project : All Ways Lead to Inquiry

I am using the word inquiry in a way that is roughly synonymous with the term scientific method. Use of inquiry is more convenient, aside from being the shorter term, because of the following advantages:

1. It allows one to broaden the scope of investigation to include any form of proceeding toward knowledge that merely aims at such a method.
2. It allows one to finesse the issue, for the time being, of how much "method" there is in science.

This Subdivision and the next deal with opposite aspects of inquiry. In many ways it might have been better to interlace the opposing points of comparison, taking them up in a parallel fashion, but this plan was judged to be too distracting for a first approach. In other ways, the negative sides of each topic are prior in point of time to the positive sides of the issue, but sensible people like to see the light at the end of the tunnel before they trouble themselves with the obscurities of the intervening journey. Thus, this Subdivision of the text emphasizes the positive features of inquiry and the positive qualities of its objective, while the next Subdivision is reserved to examine the negative aspects of each question.

In the order of nature, the absence of a feature naturally precedes the full development of its presence. In the order of discussion, however, positive terms must be proposed if it is desired to say anything at all.

The discussion in this Subdivision is placed to serve a primer, declaring at least the names of enough positive concepts to propose addressing the negative conditions of knowledge in which inquiry necessarily starts.

In this Subdivision I stand back once again from the problem of inquiry and allow myself take a more distant view of the subject, settling into what I think is a comfortable and a natural account of inquiry, the best that I have at my command, and attending to the task of describing its positive features in a positive light. I present my personal view of inquiry as I currently understand it, without stopping to justify every concept in detail or to examine every objection that might be made to this view. In the next Subdivision I discuss a few of the more obvious problems that stand in the way of this view and I try to remove a few of the more tractable obscurities that appear ready to be cleared up. The fact that I treat them as my "personal insights" does not mean that all of these ideas about inquiry originate with me, but only that I have come to adopt them for my personal use. There will be many occasions, the next time that I go over this ground, to point out the sources of these ideas, so far as I know them.

The reader may take my apology for this style of presentation to be implicit in its dogmatic character. It is done this way in a first approach for the sake of avoiding an immense number of distractions, each of which is not being slighted but demands to be addressed in its own good time. I want to convey the general drift of my current model, however conjectural, naive, uncritical, and unreflective it may seem.

1.4.1. The Matrix of Inquiry

Thus when mothers have children suffering from sleeplessness, and want to lull them to rest, the treatment they apply is to give them, not quiet, but motion, for they rock them constantly in their arms; and instead of silence, they use a kind of crooning noise; and thus they literally cast a spell upon the children (like the victims of a Bacchic frenzy) by employing the combined movements of dance and song as a remedy.

(Plato, Laws, VII, 790D).

Try as I might, I do not see a way to develop a theory of inquiry from nothing: To take for granted nothing more than is already given, to set out from nothing but absolutely certain beginnings, or to move forward with nothing but absolutely certain means of proceeding. In particular, the present inquiry into inquiry, \(y_0 = y \cdot y,\) ought not to be misconstrued as a device for magically generating a theory of inquiry from nothing. Like any other inquiry, it requires an agent to invest in a conjecture, to make a guess about the relevant features of the subject of interest, and to choose the actions, the aspects, and the attitudes with regard to the subject that are critical to achieving the objectives of the study.

I can sum all this up by saying that an inquiry requires an inquirer to suggest a hypothesis about the subject of interest and then to put that particular model of the subject to the test. This in turn requires one to devote a modicum of personal effort to the task of testing the chosen hypothesis, to put a quantum of personal interest at stake for the sake of finding out whether the model fits the subject, and, overall, to take the risk of being wrong. Any model that is feasible is also defeasible, at least, where it concerns a contingent subject of inquiry.

The first step, then, of an inquiry into inquiry, is to put forth a tentative model of inquiry, to make a hypothesis about the features of inquiry that are essential to explaining its experienced characteristics, and thus, in a sense, to make a guess at the very definition of inquiry. This requirement seems both obvious and outrageous at the same time. One is perfectly justified in objecting that there is much that precedes this so-called "first step", namely, the body of experience that prepares one to see it and the mass of observation that prompts one to take it. I can deal with this objection by making a distinction between mundane experience and olympian theory, and then by saying that the making of a conjecture is really the first "theoretical" step, but this is a hedge that covers the tracks of theory in a deceptive way, hiding how early in the empirical process the "cloven hoof" of theory actually enters.

Leaving behind the mythical conditions of pure experience and naive observation, and at least by the time that one comes to give a name to the subject of investigation, one's trek through the data is already half-shod, half-fettered by the connotations of the name, and in turn by all of the concepts that it invokes in its train. The name, the concepts that it suggests, and the tacit but vague definition of the subject that this complex of associations is already beginning to constellate, attract certain experiences to the complex and filter out other observations from having any bearing on the subject matter. By this point, one is already busy translating one's empirical acquaintance with the subject into an arrangement of concepts that is intended to define its essential nature.

An array of concepts that is set up to capture the essence of a subject is a provisional definition of it, an implicit model of the subject that contains the makings of an explicit theory. It amounts to a selection from the phenomenal aspects of the subject, expresses a guess about its relevant features, and constitutes a hypothesis in explanation of its experienced characteristics. This incipient order of model or theory is tantamount to a definition because it sets bounds on the "stretches" and the "holds" of a term — its extension, intension, and intention — but this is not the kind of definition that has to be taken on faith, or that constitutes the first and the last word on the subject. In other words, it is an empirical definition, one that is subject to being falsified in reference to its intended subject, by failing to indicate the necessary, the pertinent, or the relevant features that account for the presence of its phenomena or the persistence of its process.

If I reflect on the conduct of inquiry, seeking to fix it in a fitting image and trying to cast it in a positive light, the best I can do is this:

Inquiry is a process that aims at achieving belief or knowledge.

But even this simple a description already plunges the discussion deep into a number of obscurities. Most prominently, there is the disjunction between belief and knowledge that cries out to be explained or resolved. Stirring beneath the surface, and not quite fading into the background, many of the other terms that are invoked in the description are capable of hiding the entire contents of the original ignorance that the image as a whole is aimed to dispel. And yet, there is nothing that I can do in this avowedly positive context but to mark these points down as topics for future discussion.

There is already a model of inquiry that is implicit, at least partially, in the text of the above description. Let me see if I can tease out a few of its tacit assumptions.

1.4.1.1. Inquiry as Conduct

First of all, inquiry is conceived to be a form of conduct. This invokes the technical term conduct, referring to the species of prototypically human action that is both dynamic and deliberate, or conceived to fall under a form of purposeful control, usually conscious but possibly not. For the sake of clarity, it helps to seek a more formal definition of conduct, one that expresses the concept in terms of abstract features rather than trying to suggest it by means of typical examples.

Conduct is action with respect to an object. The distinction between action and conduct, reduced to the level of the most abstract formal relations that are involved, can be described in the following manner.

Action is a matter of going from A to B, whereas conduct is matter of going from A to B in relation to C. In describing particular cases and types of conduct, the phrase "in relation to" can be filled out in more detail as "on account of", "in the cause of", "in order to bring about", "for the sake of", "in the interests of", or in many other ways. Thus, action by itself has a dyadic character, involving transitions through pairs of states, while conduct has a triadic character, involving the kinds of transactions between states that relate throughout to an object.

With regard to this distinction, notice that "action" is used inclusively, to name the genus of which "conduct" names a species, and thus depicts whatever has the aspect of action, even if it is actually more complex.

This creates the difficulty that the reputed "genus" is less than fully "generative", "generic", "genetic", or even "genuine" -- and so it is necessary to remain on guard against this source of misunderstanding.

What does this definition of conduct say about the temporal ordering of the object with respect to the states? The states are conceived to be ordered in time, but so far nothing has been said to pin down where in relation to these states the object must be conceived to fall in time. Nor does the definition make any particular specification necessary. This makes the question of relative time a secular parameter of the definition, allowing the consideration of the following options:

1. If the object is thought to precede the action of the conduct, then it tends to be regarded as a creative act, an initial intention, an original stimulus, a principal cause, or a prime mover.
2. If the object is thought to succeed the action of the conduct, then it tends to be regarded as an end, a goal, or a purpose, in other words, a state envisioned to be fulfilled.
3. If the object is thought to be concurrent, immanent, or transcendent throughout the action of the conduct, then it tends to be regarded as falling under one of the following possibilities: a prevailing value, a controlling parameter, a universal system of effective forces, a pervasive field of potentials, a ruling law, or a governing principle.

A prevailing value or a controlling parameter, which guides the temporal development of a system, is a term that fits into a law or a principle, which governs the system at a higher level. The existence of a value or a law that rules a system, and the information that an agent of the system has about its parameters and its principles, are two different matters. Indeed, a major task of development for an inquiring agent is to inform itself about the values and the laws that form its own system. Thus, one of the objects of the conduct of inquiry is a description in terms of laws and values of the rules that govern and guide inquiry.

The elaboration of an object in terms of this rich vocabulary — as a cause, end, field, force, goal, intention, law, parameter, principle, purpose, system, or value — adds colorful detail and concrete sensation to the account, and it helps to establish connections with the arrays of terminology that are widely used to discuss these issues. From a formal and relational point of view, however, all of these concepts are simply different ways of describing, at possibly different levels of generality, the object of a form of conduct. With that in mind, I find it useful to return to the simpler form of description as often as possible.

This account of conduct brings to the fore a number of issues, some of them new and some of them familiar, but each of them allowing itself to be approached from a fresh direction by treating it as an implication of a critical thesis just laid down. I next examine these issues in accord with the tenets from which they stem.

1. Inquiry is a form of conduct.

This makes inquiry into inquiry a special case of inquiry into conduct.

Certainly, it must be possible to reason about conduct in general, especially if forms of conduct need to be learned, examined, modified, and improved.

Placing the subject of inquiry within the subject of conduct and making the inquiry into inquiry a subordinate part of the inquiry into conduct does not automatically further the investigation, especially if it turns out that the general subject of conduct is more difficult to understand than the specialized subject of inquiry. But in those realms of inquiry where it is feasible to proceed hypothetically and recursively, stretching the appropriate sort of hypothesis over a wider subject area can act to prime the pump of mathematical induction all the more generously, and actually increase the power of the recursion. Of course, the use of a recursive strategy comes at the expense of having to establish a more extended result at the base.

2. The existence of an object that rules a form of conduct and the information that an agent of the conduct has about the object are two different matters.

This means that the exact specification of the object can demand an order of information that the agent does not have available, at least, not for use in reflective action, or even require an amount of information that the agent lacks the capacity to store. No matter how true it is that the actual course of the agent's conduct exactly reflects the influence of the object, and thus, in a sense, represents the object exactly, the question is whether the agent possesses the equivalent of this information in any kind of accessible, exploitable, reflective, surveyable, or usable form of representation, in effect, in any mode of information that the agent can use to forsee, to modify, or to temper its own temporal course.

This issue may seem familiar as a repetition of the "meta" question.

Once again, there is a distinction between (a) the properties of an action, agent, conduct, or system, as expressible by the agent that is engaged in the conduct, or as representable within the system that is undergoing the action, and (b) the properties of the same entities, as evident from an "external viewpoint", or as statable by the equivalent of an "outside observer".

3. Reflection is a part of inquiry. Reflection is a form of conduct.

The task of reflection on conduct is to pass from a purely interior view of one's own conduct to an outlook that is, effectively, an exterior view.

What is sought is a wider perspective, one that is able to incorporate the sort of information that might be available to an outside observer, that ought to be evident from an external vantage point, or that one reasonably imagines might be obvious from an independent viewpoint. I am tempted to refer to such a view as a "quasi-objective perspective", but only so long as it possible to keep in mind that there is no such thing as a "completely outside perspective", at least, not one that a finite and mortal agent can hope to achieve, nor one that a reasonably socialized member of a community can wish to take up as a permanent station in life.

With these qualifications, reflection is a form of conduct that can serve inquiry into conduct. Inquiry and its component reflection, applied to a form of conduct, are intended to provide information that can be used to develop the conduct in question. The "reflective development" that occurs depends on the nature of the case. It can be the continuation, the correction, or the complete cessation of the conduct in question.

If it is to have the properties that it is commonly thought to have, then reflection must be capable of running in parallel, and not interfering too severely, with the conduct on which it reflects. If this turns out to be an illusion of reflection that is not really possible in actuality, then reflection must be capable, at the very least, of reviewing the memory record of the conduct in question, in ways that appear concurrent with a replay of its action. But these are the abilities that reflection is "pre-reflectively" thought to have, that is, before the reflection on reflection can get under way. If reflection is truly a form of conduct, then it becomes conceivable as a project to reflect on reflection itself, and this reflection can even lead to the conclusion that reflection does not have all of the powers that it is commonly portrayed to have.

First of all, inquiry is conceived to be a form of conduct. This invokes the technical term "conduct", referring to the species of prototypically human action that is both dynamic and deliberate, or conceived to fall under a form of purposeful control, usually conscious but possibly not. For the sake of clarity, it helps to seek a more formal definition of conduct, one that expresses the concept in terms of abstract features rather than trying to suggest it by means of typical examples.

Conduct is action with respect to an object. The distinction between action and conduct, reduced to the level of the most abstract formal relations that are involved, can be described in the following manner. Action is a matter of going from A to B, whereas conduct is matter of going from A to B in relation to C. In describing particular cases and types of conduct, the phrase "in relation to" can be filled out in more detail as "on account of", "in the cause of", "in order to bring about", "for the sake of", "in the interests of", or in many other ways. Thus, action by itself has a dyadic character, involving transitions through pairs of states, while conduct has a triadic character, involving the kinds of transactions between states that relate throughout to an object.

With regard to this distinction, notice that "action" is used inclusively, to name the genus of which "conduct" names a species, and thus depicts whatever has the aspect of action, even if it is actually more complex. This creates the difficulty that the reputed "genus" is less than fully "generative", "generic", "genetic", or even "genuine" - and so it is necessary to remain on guard against this source of misunderstanding.

What does this definition of conduct say about the temporal ordering of the object with respect to the states? The states are conceived to be ordered in time, but so far nothing has been said to pin down where in relation to these states the object must be conceived to fall in time. Nor does the definition make any particular specification necessary. This makes the question of relative time a secular parameter of the definition, allowing the consideration of the following options:

1. If the object is thought to precede the action of the conduct, then it tends to be regarded as a creative act, an initial intention, an original stimulus, a principal cause, or a prime mover.
2. If the object is thought to succeed the action of the conduct, then it tends to be regarded as an end, a goal, or a purpose, in other words, a state envisioned to be fulfilled.
3. If the object is thought to be concurrent, immanent, or transcendent throughout the action of the conduct, then it tends to be regarded as falling under one of the following possibilities: a prevailing value, a controlling parameter, a universal system of effective forces, a pervasive field of potentials, a ruling law, or a governing principle.

A prevailing value or a controlling parameter, which guides the temporal development of a system, is a term that fits into a law or a principle, which governs the system at a higher level. The existence of a value or a law that rules a system, and the information that an agent of the system has about its parameters and its principles, are two different matters. Indeed, a major task of development for an inquiring agent is to inform itself about the values and the laws that form its own system. Thus, one of the objects of the conduct of inquiry is a description in terms of laws and values of the rules that govern and guide inquiry.

The elaboration of an object in terms of this rich vocabulary — as a cause, end, field, force, goal, intention, law, parameter, principle, purpose, system, or value — adds colorful detail and concrete sensation to the account, and it helps to establish connections with the arrays of terminology that are widely used to discuss these issues. From a formal and relational point of view, however, all of these concepts are simply different ways of describing, at possibly different levels of generality, the object of a form of conduct. With that in mind, I find it useful to return to the simpler form of description as often as possible.

This account of conduct brings to the fore a number of issues, some of them new and some of them familiar, but each of them allowing itself to be approached from a fresh direction by treating it as an implication of a critical thesis just laid down. I next examine these issues in accord with the tenets from which they stem.

1. Inquiry is a form of conduct.

This makes inquiry into inquiry a special case of inquiry into conduct. Certainly, it must be possible to reason about conduct in general, especially if forms of conduct need to be learned, examined, modified, and improved.

Placing the subject of inquiry within the subject of conduct and making the inquiry into inquiry a subordinate part of the inquiry into conduct does not automatically further the investigation, especially if it turns out that the general subject of conduct is more difficult to understand than the specialized subject of inquiry. But in those realms of inquiry where it is feasible to proceed hypothetically and recursively, stretching the appropriate sort of hypothesis over a wider subject area can act to prime the pump of mathematical induction all the more generously, and actually increase the power of the recursion. Of course, the use of a recursive strategy comes at the expense of having to establish a more extended result at the base.

2. The existence of an object that rules a form of conduct and the information that an agent of the conduct has about the object are two different matters.

This means that the exact specification of the object can require an order of information that the agent does not have available, at least, not for use in reflective action, or even an amount of information that the agent lacks the capacity to store. No matter how true it is that the actual course of the agent's conduct exactly reflects the influence of the object, and thus, in a sense, represents the object exactly, the question is whether the agent possesses the equivalent of this information in any kind of accessible, exploitable, reflective, surveyable, or usable form of representation, in effect, any mode of information that the agent can use to forsee, to modify, or to temper its own temporal course.

This issue may seem familiar as a repetition of the "meta" question. Once again, there is a distinction between (a) the properties of an action, agent, conduct, or system, as expressible by the agent that is engaged in the conduct, or as representable within the system that is undergoing the action, and (b) the properties of the same entities, as evident from an "external viewpoint", or as statable by the equivalent of an "outside observer".

3. Reflection is a part of inquiry. Reflection is a form of conduct.

The task of reflection on conduct is to pass from a purely interior view of one's own conduct to an outlook that is, effectively, an exterior view. What is sought is a wider perspective, one that is able to incorporate the sort of information that might be available to an outside observer, that ought to be evident from an external vantage point, or that one reasonably imagines might be obvious from an independent viewpoint. I am tempted to refer to such a view as a "quasi-objective perspective", but only so long as it possible to keep in mind that there is no such thing as a "completely outside perspective", at least, not one that a finite and mortal agent can hope to achieve, nor one that a reasonably socialized member of a community can wish to take up as a permanent station in life.

With these qualifications, reflection is a form of conduct that can serve inquiry into conduct. Inquiry and its component reflection, applied to a form of conduct, are intended to provide information that can be used to develop the conduct in question. The "reflective development" that occurs depends on the nature of the case. It can be the continuation, the correction, or the complete cessation of the conduct in question.

If it is to have the properties that it is commonly thought to have, then reflection must be capable of running in parallel, and not interfering too severely, with the conduct on which it reflects. If this turns out to be an illusion of reflection that is not really possible in actuality, then reflection must be capable, at the very least, of reviewing the memory record of the conduct in question, in ways that appear concurrent with a replay of its action. But these are the abilities that reflection is "pre-reflectively" thought to have, that is, before the reflection on reflection can get under way. If reflection is truly a form of conduct, then it becomes conceivable as a project to reflect on reflection itself, and this reflection can even lead to the conclusion that reflection does not have all of the powers that it is commonly portrayed to have.

1.4.1.2. Types of Conduct

The chief distinction that applies to different forms of conduct is whether the object is the same sort of thing as the states or whether it is something entirely different, a thing apart, of a wholly other order. Although I am using different words for objects and states, it is always possible that these words are indicative of different roles in a formal relation and not indicative of substantially different types of things. If objects and states are but formal points and naturally belong to the same domain, then it is conceivable that a temporal sequence of states can include the object in its succession, in other words, that a path through a state space can reach or pass through an object of conduct. But if a form of conduct has an object that is completely different from any one of its temporal states, then the role of the object in regard to the action cannot be like the end or goal of a temporal development.

What names can be given to these two orders of conduct?

1.4.1.3. Perils of Inquiry

Now suppose that making a hypothesis is a kind of action, no matter how covert, or that testing a hypothesis takes an action that is more overt. If entertaining a hypothesis in any serious way requires action, and if action is capable of altering the situation in which it acts, then what prevents this action from interfering with the subject of inquiry in a way that undermines, with positive or negative intentions, the very aim of inquiry, namely, to understand the situation as it is in itself?

That making a hypothesis is a type of action may seem like a hypothesis that is too far-fetched, but it appears to follow without exception from thinking that thinking is a form of conduct, in other words, an activity with a purpose or an action that wants an end. The justification of a hypothesis is not to be found in a rational pedigree, by searching back through a deductive genealogy, or determined by that which precedes it in the logical order, since a perfectly trivial tautology caps them all. Since a logical tautology, that conveys no empirical information, finds every proposition appearing to implicate it, in other words, since it is an ultimate implication of every proposition and a conceivable conclusion that is implicit in every piece of reasoning, it is obvious that seeking logical precedents is the wrong way to go for empirical content.

In making a hypothesis or choosing a model, one appears to select from a vaster number of conceivable possibilities than a finite agent could ever enumerate in complete detail or consider as an articulate totality. As the very nature of a contingent description and the very character of a discriminate action is to apply in some cases but not in others, there is no escaping the making of a risky hypothesis or a speculative interpretation, even in the realm of a purely mental action. Thus, all significant thought, even thinking to any purpose about thought itself, demands a guess at the subject or a grasp of the situation that is contingent, dubious, fallible, and uncertain.

If all this is true — if inquiry begins with doubt, if every significant hypothesis is itself a dubious proposition, if the making and the testing of a hypothesis are instances of equally doubtful actions, and if every action has the potential to alter the very situation and the very subject matter that are being addressed — then it leads to the critical question: How is the conduct of inquiry, that begins by making a hypothesis and that continues by testing this description in action, supposed to help with the situation of uncertainty that incites it in the first place and that is supposed to maintain its motivation until the end is reached? The danger is that the posing of a hypothesis may literally introduce an irreversible change in the situation or the subject matter in question. The fear is that this change might be one that too conveniently fulfills or too perversely subverts the very hypothesis that engenders it, that it may obstruct the hypothesis from ever being viewed with equanimity again, and thus prevent the order of reflection that is needed to amend or discard the hypothesis when the occasion to do so arises.

If one fears that merely contemplating a special hypothesis is enough to admit a spurious demonstration into the foundations of one's reasoning, even to allow a specious demon to subvert all one's hopes of a future rationality and to destroy all one's chances of a reasonable share of knowledge, then one is hardly in a state of mind that can tolerate the tensions of a full-fledged, genuine inquiry. If one is beset with such radical doubts, then all inquiry is no more comfort than pure enchoiry. Sometimes it seems like the best you can do is sing yourself a song that soothes your doubts. Perhaps it is even quite literally true that all inquiry comes back at last to a form of "enchoiry", the invocation of a nomos, a way of life, or a song and a dance. But even if this is the ultimate case, it does no harm and it does not seem like a bad idea to store up in this song one or two bits of useful lore, and to weave into its lyric a few suggestions of a practical character.

Let us now put aside these more radical doubts. This putting aside of doubts is itself a form of inquiry, that is, a way of allaying doubts. The fact that I appear to do this by fiat, and to beg for tacit assent, tends to make me suspect the validity of this particular tactic. Still, it is not too inanely dismissive, as its appeal is based on an argument, the argument that continuing to entertain this type of doubt leads to a paralysis of the reason, and that paralyzing the ability to think is not in the interests of the agent concerned. Thus, I adopt the hypothesis that the relationship between the world and the mind is not so perverse that merely making a hypothesis is enough to alter the nature of either. If, in future, I or anyone sees the need to reconsider this hypothesis, then I see nothing about making it that prevents anyone from doing so. Indeed, making it explicit only renders it more subject to reflection.

Of course, a finite person can only take up so many causes in a single lifetime, and so there is always the excuse of time for not chasing down every conceivable hypothesis that comes to mind.

1.4.1.4. Forms of Relations

The next distinguishing trait that I can draw out of this incipient treatise is its emphasis on the forms of relations. From a sufficiently formal and relational point of view, many of the complexities that arise from throwing intentions, objectives, and purposes into the mix of discussion are conceivably due to the greater arity of triadic relations over dyadic relations, and do not necessarily implicate any differences of essence inhering in the entities and the states invoked. As far as this question goes, whether a dynamic object is essentially different from a deliberate object, I intend to remain as neutral as possible, at least, until forced by some good reason to do otherwise. In the meantime, the factors that are traceable to formal differences among relations are ready to be investigated and useful to examine. With this in mind, it it useful to make the following definition:

A conduct relation is a triadic relation involving a domain of objects and two domains of states. When a shorter term is desired, I refer to a conduct relation as a conduit. A conduit is given in terms of its extension as a subset C c XxYxZ, where X is the object domain and where Y and Z are the state domains. Typically, Y = Z.

In general, a conduct relation serves as a model of conduct (MOC), not always the kind of model that is meant to be emulated, but the type of model that captures an aspect of structure in a form of conduct.

The question arises: What is the relationship between signs and states? On the assumption that signs and states are comparable in their levels of generality, consider the following possibilities:

1. Signs are special cases of states.
2. Signs and states are the same sorts of things.
3. States are special cases of signs.

Depending on how one answers this question, one is also choosing among the following options:

1. Sign relations are special cases of conduct relations.
2. Sign relations and conduct relations are the same sorts of things.
3. Conduct relations are special cases of sign relations.

I doubt if there is any hard and fast answer to this question, but think that it depends on particular interpreters and particular observers, to what extent each one interprets a state as a sign, and to what degree each one recognizes a sign as a component of a state.

1.4.1.5 Models of Inquiry

The value of a hypothesis, or the worth of a model, is not to be given a prior justification, as by a deductive proof, but has to be examined in practice, as by an empirical probation. It is not intended to be taken for granted or to go untested, but its meaning in practice has to be articulated before its usefulness can be judged. This means that the conceivable practical import of the hypothesis or the model has to be developed in terms of its predicted and its promised consequences, after which it is judged by the comparison of these speculative consequences with the actual results. But this is not the end of the matter, for it can be a useful piece of information to discover that a particular kind of conception fails a particular kind of comparison. Thus, the final justification for a hypothesis or a model is contained in the order of work that it leads one to do, and the value of this work is often the same whether or not its premiss is true. Indeed, the fruitfulness of a suggestion can lie in the work that proves it untrue.

My plan then has to be, rather than trying to derive a model of inquiry in a deductive fashion from a number of conditions like y0 = y.y, only to propose a plausible model, and then to test it under such conditions. Each of these tests is a "two-edged sword", and the result of applying a particular test to a proposed model can have either one of two effects. If one believes that a particular test is a hard and fast rule of inquiry, or a condition that any inquiry is required to satisfy, then the failure of a model to live up to its standard tends only to rule out that model. If one has reason to believe that a particular model of inquiry covers a significant number of genuine examples, then the failure of these models to follow the prescribed rule can reflect badly on the test itself.

In order to prime the pump, therefore, let me offer the following account of inquiry in general, the whole of which can be taken as a plausible hypothesis about the nature of inquiry in general.

My observations of inquiry in general, together with a few suggestions that seem apt to me, have led me to believe that inquiry begins with a "surprise" or a "problem". The way I understand these words, they refer to departures, differences, or discrepancies among various modalities of experience, in particular, among "observations", "expectations", and "intentions".

1. A "surprise" is a departure of an observation from an expectation, and thus it invokes a comparison between present experience and past experience, since expectations are based on the remembered disposition of past experience.

2. A "problem" is a departure of an observation from an intention, and thus it invokes a comparison between present experience and future experience, since intentions choose from the envisioned disposition of future experience.

With respect to these

With respect to this hypothetical

I now test this model of inquiry under the conditions of an inquiry into inquiry, asking whether it is consistent in its application to itself. This leaves others to test the models they like best under the same conditions, should they ever see the need to do so.

Does the inquiry into inquiry begin with a surprise or a problem concerning the process or the conduct of inquiry? In other words, does the inquiry into inquiry start with one of the following forms of departure: (1) a surprising difference between what is expected of inquiry and what is observed about it, or (2) a problematic difference between what is observed about inquiry and what is intended for it?

1.4.2 The Moment of Inquiry

Every young man - not to speak of old men - on hearing or seeing anything unusual and strange, is likely to avoid jumping to a hasty and impulsive solution of his doubts about it, and to stand still; just as a man who has come to a crossroads and is not quite sure of his way, if he be travelling alone, will question himself, or if travelling with others, will question them too about the matter in doubt, and refuse to proceed until he has made sure by investigation of the direction of his path.

(Plato, Laws, VII, 799C).

Observe the paradox of this precise ambiguity: That both the occasion and the impulse of inquiry are instances of a negative moment. But the immediate discussion is aimed at the positive aspects of inquiry, and so I convert this issue into its corresponding positive form.

The positive aim of inquiry is a state of belief, certainty, or knowledge. There are distinctions that can be made in the use of these words, but the question remains as to what kind of distinctions these are. In my opinion, the differences that arise in practice have more to do with the purely grammatical distinctions of "case", "mood", "number", "person", and "voice", and thus raise the issues of plurality and point of view, as opposed to indicating substantial differences in the relevant features of state, as actually experienced by the agent concerned.

It is often claimed that there are signficant differences between the conditions of belief and knowledge, but the way that I understand the distinction is as follows. One says that a person "knows" something when that person believes exactly the same thing that one believes. When one is none other than the person in question, then one says that one "knows" exactly what one believes. Differences arise between the invocations of "belief" and "knowledge" only when more than one person is involved in the issue. Thus, there is no occasion for a difference between belief and knowledge unless there is more than one person that is being consulted about the matter in question, or else a single person in a divided state of opinion, in any case, when there is more than one impulse, moment, or occasion that currently falls under consideration.

In any case, belief or knowledge is the feature of state that an agent of inquiry lacks at the moment of setting out. Inquiry begins in a state of impoverishment, need, or privation, a state that is absent the quality of certainty. It is due to this feature that the agent is motivated, and it is on account of its continuing absence that the agent keeps on striving to achieve it, at least, with respect to the subject in question, and, at any rate, in sufficient measure to make action possible.

1.4.3 The Modes of Inquiry

Let the strange fact be granted, we say, that our hymns are now made into "nomes" (laws), just as the men of old, it would seem, gave this name to harp-tunes, - so that they, too, perhaps, would not wholly disagree with our present suggestion, but one of them may have divined it vaguely, as in a dream by night or a waking vision: anyhow, let this be the decree on the matter:- In violation of public tunes and sacred songs and the whole choristry of the young, just as in violation of any other "nome" (law), no person shall utter a note or move a limb in the dance.

(Plato, Laws, VII, 799E-800A).

In the present section, I am concerned with the kinds of reasoning that might be involved in the choice of a method, that is, in discovering a way to go about inquiry, in constructing a way to carry it through, and in justifying the way that one chooses. If the choice of a method can be established on the basis of reasoning, if it can be rationalized or reconstructed on grounds that are commonly thought to be sensible, or if it is likely to be affected or influenced in any way by a rational argument, then there is reason to examine the kinds of reasoning that go into this choice. All of this requires a minimal discussion of different modes of reasoning.

In this work as a whole, each instance of inquiry is analyzed in accord with various modes of reasoning, the prospective "elements of inquiry", and its structure as an object of inquiry is articulated, rationalized, and reconstructed with respect to the corresponding "form of analysis", "form of synthesis", or "objective genre" (OG).

According to my current understanding, the elements of inquiry can be found to rest on three types of steps, called "abductive", "deductive", and "inductive" modes of inference. As a result of this opinion, I do not believe that I can do any better at present than to articulate the structure of each instance of learning or reasoning according to these three types of motions of the mind. But since this work as a whole is nowhere near complete, I cannot dictate these steps in a dogmatic style, nor will it do for me to to call the tune of this form of analysis in a purely ritual or a wholly routine fashion.

Since the complexity of reasoning about different modes of reasoning is enough of a complication to occupy my attention at the present stage of development in this work, it is proably best to restrain this discussion along the majority of its other dimensions. A convenient way to do this is to limit its scope to simple examples and concrete situations, just enough to illustrate the selected modes of reasoning.

With all of these considerations in mind, the best plan that I can find for addressing the tasks of the present section is to proceed as follows: I make it my primary aim to examine only a few of the simplest settings in which these different modes of reasoning are able to appear, and I try to plot my path through this domain by way of concrete examples. Along the way, I discuss a few of the problems that are associated with reasoning about different modes of reasoning. Given the present stage of development, the majority of these issues have to be put aside almost as quickly as they are taken up. If they are ever going to be subject to resolution, it is not within reach of the present moment of discussion. In the body of this section, I therefore return to the initial strategy: to examine a few of the simplest cases and situations that can serve to illustrate the distinctions among the chosen modes of reasoning.

In trying to initiate a general discussion of the different modes of reasoning that might be available, and thus to motivate a model of this subject matter that makes an initial kind of sense to me, I meet once again with all the old "difficulties at the beginning", the kinds of obstructions that always seem to arise on trying to open up any new subject for discussion or in trying to introduce any new model of an old subject area. Much of this gratuitous bedevilment is probably due to the inherent conservatism of the human mind. Everything familiar is taken for granted, but each new picture of the situation is immediately subjected to the severest suspicions.

Now, I cannot reason with necessary force that the mind must use these particular modes of reasoning, any more than I can say that it must use a given language in order to express itself. But I can argue, relative to a particular model of thinking that must be proposed hypothetically, that certain modes of reasoning are available to the mind and are likely to be evident in its operation, if one only takes the trouble to look.

Ultimately, the model of thinking that I plan to propose makes use of the proposition that all thinking takes place in signs, and thus that inquiry is the transformation of a sign relation. Relative to this hypothesis, it would be possible to discharge the current assumptions about the basic modes of reasoning, that is, to derive the elementary modes of inquiry from a sign relational model of inquiry, and then to compare them with the current suggestions. Until this work is done, however, the assumption that these really are the most basic modes of reasoning has to be treated as a still more tentative hypothesis.

When a subject matter is so familiar that the logical connections between its parts are known both forwards and backwards, then it is reasonable and convenient to organize its presentation in an axiomatic fashion. This would not be such a bad idea, if it did not make it so easy to forget the nature of the reorganization that goes into a representation, and it would not constitute such a deceptive conception of the subject, if it did not mean that the exposition of the subject matter is just as often the falsification of its actual development and the covering up of its real excavation. Indeed, the logical order of axioms and theorems may have little to do with the original order of discovery and invention. In practice, the deepest axioms are often the last to come to light.

Once again, the structure of a reflective context means that each mode of reasoning is able to appear in a double role, once as an object and once as an instrument of the same extended discussion. And once again, the discussion runs into an array of obstructions, whose structures are becoming, if not more clear, at least, more familiar with each encounter. In particular, a description of different modes of reasoning involves a classification, and a classification presupposes a basis of distinctive features that cannot be treated as categorical, or objectively neutral, but has to be regarded as hypothetical, or potentially biased. In other words, the language that I use to describe different modes of reasoning may already have a particular model of reasoning built into it, and this disposition to a particular conception of logic may be lodged in such a way that it makes it nearly impossible to reflect on the operations and the limitations of this model.

Inquiry begins when a law is violated. It marks a time when a certain peace of mind is breached, it reigns all the while that a common accord is broken, disturbed, forgotten, or lost, and it rules right up until the time when a former condition of harmony is restored or until the moment when a new state of accord is established. Of course, the word "law" is a highly equivocal choice, especially to convey the sense of a founding principle. It renders not just its own meaning irrevocably subject to interpretation, but delivers into a similar subjection all the forms of understanding that depend on it. But the letter must release its hold on the spirit, if the word "law" is meant to evoke the requisite variety of connotations, and yet to maintain a sensible degree of order among their concrete meanings. Only in this way can it rise above the many different kinds of law that come into play.

There are descriptive laws, that organize experiences into expectations. There are prescriptive laws, that organize performances into intentions.

Other names for descriptive laws are "declarative" or "empirical" laws. Other names for prescriptive laws are "procedural" or "normative" laws.

Implicit in a descriptive law is the connection to be found or made, discovered or created, between past experience and present expectation. What one knows about these connections is kept in a descrptive model.

Implicit in a prescriptive law is the connection to be found or made, discovered or created, between current conduct and future experience. What one knows about these connections is kept in a prescriptive model.

A violation of an expectation, the contravention of a descriptive law, occurs when a present experience departs from a predicted experience, which is what a past expectation or description projected to be present. This is a "surprise", a state of affairs that calls for an explanation. An explanation points to other descriptions that better predict the actual experience, and suggests an alteration to the descriptive model that generated the expectation from a past experience.

A violation of an intention, the contravention of a prescriptive law, occurs when a present experience departs from a desired experience, which is what a past intention or prescription projected to be present. This is a "problem", a state of affairs that calls for a plan of action. , A plan of action points to other actions that better achieve the desired experience, and suggests an alteration to the prescriptive model that generated the conduct toward a prospective experience.

In the rest of this section, I treat the different modes of reasoning according to the forms that Aristotle gave them, collectively referred to as the "syllogistic" model. The discussion is kept within the bounds of propositional reasoning by considering only those "figures of syllogism" that are "purely universal", that is, the forms of argument all of whose premisses, and therefore all of whose conclusions, involve nothing but universal quantifications.

If it were only a matter of doing propositional reasoning as efficiently as possible, I would simply use the cactus language and be done with it, but there are several other reasons for revisiting the syllogistic model. Treating the discipline that is commonly called "logic" as a cultural subject with a rich and varied history of development, and attending to the thread of tradition in which I currently find myself, I observe what looks like a critical transition that occurs between the classical and the modern ages. Aside from supplying the barest essentials of a historical approach to the subject, a consideration of this elder standard makes it easier to appreciate the nature and the character of this transformation. In addition, and surprisingly enough to warrant further attention, there appear to be a number of cryptic relationships that exist between the syllogistic patterns of reasoning and the ostensibly more advanced forms of analysis and synthesis that are involved in the logic of relations.

1.4.3.1 Deductive Reasoning

In this subsection, I present a trimmed-down version of deductive reasoning in Aristotle, limiting the account to universal syllogisms, in effect, keeping to the level of propositional reasoning. Within these constraints, there are three basic "figures" of the syllogism.

In order to understand Aristotle's description of these figures, it is necessary to explain a few items of his technical terminology. In each figure of the syllogism, there are three "terms". Each term can be read as denoting either (1) a class of entities or (2) all of the members of a class of entities, depending on which interpretation the reader prefers. These terms are ranked in two ways: With respect to the "magnitudes" that they have in relation to each other, there are "major", "middle", and "minor" terms. With respect to the "positions" that they take up within the figure, there are "first", "intermediate", and "last" terms. The figures are distinguished by how the magnitudes correlate with the positions. However, the names for these rankings are not always used or translated in a rigorously systematic manner, so the reader has to be on guard to guess which type of ranking is meant.

In addition to this terminology, it is convenient to make use of the following nomenclature:

1. The "Fact" is the proposition that applies the term in the first position to the term in the third or last position.

2. The "Case" is the proposition that applies the term in the second or intermediate position to the term in the third or last position.

3. The "Rule" is the proposition that applies the term in the first position to the term in the second or intermediate position.

Because the roles of Fact, Case, and Rule are defined with regard to positions rather than magnitudes they are insensitive to whether the proposition in question is being used as a premiss or is being drawn as a conclusion.

The "first figure" of the syllogism is explained as follows:

When three terms are so related to one another that the last is wholly contained in the middle and the middle is wholly contained in or excluded from the first, the extremes must admit of perfect syllogism. By "middle term" I mean that which both is contained in another and contains another in itself, and which is the middle by its position also; and by "extremes" (a) that which is contained in another, and (b) that in which another is contained. For if A is predicated of all B, and B of all C, A must necessarily be predicated of all C. ... I call this kind of figure the First.

(Aristotle, Prior Analytics, 1.4).

For example, suppose A is "animal", B is "bird", and C is "canary". Then there is a deductive conclusion to be drawn in the first figure.

There is the Case:

"All canaries are birds." (C => B)

There is the Rule:

"All birds are animals." (B => A)

One deduces the Fact:

"All canaries are animals." (C => A)

The propositional content of this deduction is summarized on the right. Taken at this level of detail, deductive reasoning is nothing more than an application of the transitive rule for logical implications.

The "second figure" of the syllogism is explained as follows:

When the same term applies to all of one subject and to none of the other, or to all or none of both, I call this kind of figure the Second; and in it by the middle term I mean that which is predicated of both subjects; by the extreme terms, the subjects of which the middle is predicated; by the major term, that which comes next to the middle; and by the minor that which is more distant from it. The middle is placed outside the extreme terms, and is first by position. (Aristotle, Prior Analytics, 1.5).

For example, suppose M is "mammal", N is "newt", and O is "opossum". Then there is a deductive conclusion to be drawn in the second figure.

There is the Fact:

"All opossums are mammals." (O => M)

There is the Rule:

"No newts are mammals." (N.M = 0)

One deduces the Case:

"No newts are opossums." (N.O = 0)

The propositional content of this deduction is summarized on the right. Expressed in terms of the corresponding classes, it says that if O c M and if N intersects M trivially, then N must also intersect O trivially. Here, I use a raised dot "." to indicate either the conjunction of two propositions or the intersection of two classes, and I use a zero "0" to indicate either the identically false proposition or the empty class, leaving the choice of interpretation to the option of the reader.

The "third figure" of the syllogism is explained as follows:

If one of the terms applies to all and the other to none of the same subject, or if both terms apply to all or none of it, I call this kind of figure the Third; and in it by the middle I mean that of which both the predications are made; by extremes the predicates; by the major term that which is [further from] the middle; and by the minor that which is nearer to it. The middle is placed outside the extremes, and is last by position. Aristotle, Prior Analytics, 1.6).

It appears that this passage is only meant to mark out the limiting cases of the type. From the examples that Aristotle gives it is clear that he includes many other kinds of logical situation under this figure. Perhaps the phrase "applies to all or none" is intended to specify that a term applies "affirmatively or negatively" to another term, but is not meant to require that it applies universally so.

For example, suppose P is "poem", R is "rhapsody", and S is "sonnet". Then there is deductive conclusion to be drawn in the third figure:

There is the Fact:

"All sonnets are poems." (S => P)

There is the Case:

"Some sonnets are rhapsodies." (S.R > 0)

One deduces the Rule:

"Some rhapsodies are poems." (R.P > 0)

The propositional content of this deduction is summarized on the right. Expressed in terms of the corresponding classes, it says that if S c P and if R intersects S non-trivially then R must intersect P non-trivially.

1.4.3.2 Inductive Reasoning

(Aristotle, Prior Analytics, 2.23).

1.4.3.3 Abductive Reasoning

A choice of method cannot be justified by deduction or by induction, at least, not wholly, but involves an element of hypothesis. In ancient times, this mode of inference to an explanatory hypothesis was described by the Greek word "apagoge", articulating an action or a process that "carries", "drives", or "leads" in a direction "away", "from", or "off". This was later translated into the Latin "abductio", and that is the source of what is today called "abduction" or "abductive reasoning". Another residue of this sense survives today in the terminology for "abductor muscles", those that "draw away (say, a limb or an eye) from a position near or parallel to the median axis of the body" (Webster's).

If an image is needed, one may think of Prometheus, arrogating for the sake of an earthly purpose the divine prerogative of the gods, and then drawing the fire of their heavenly ire for the presumption of this act. This seems to sum up pretty well, not only the necessity and the utility of hypotheses, but also the risks that one incurs in making conjectures. In other guises, abductive reasoning is the mode of inference that is used to diagnose a complex situation, one that originally presents itself under a bewildering array of signs and symptoms, and fixes it subject to the terms of a succinct "nomen" or a summary predicate. Finally, by way of offering a personal speculation, I think it is likely that this entire trio of terms, "abduction", "deduction", and "induction", have reference to a style of geometric diagrams that the Ancients originally used to illustrate their reasonings.

Abductive reasoning has also been called by other names. C.S. Peirce at times called it "presumption", perhaps because it puts a plausible assumption logically prior to the observed facts, and at other times referred to it as "retroduction", because it reasons backwards from the consequent to the antecedent of a logical implication.

In its simplest form, abductive reasoning proceeds from a "fact" that A is true, using a "rule" that B => A, to presume a "case" that B is true. Thus, if A is a surprising fact that one happens to observe, and B => A is a rule to the effect that if B is true then A necessarily follows, then guessing the case that B is true is an instance of abductive reasoning. This is a backward form of reasoning, and therefore extremely fallible, but when it works it has the effect of reducing the amount of surprise in the initial observation, and thus of partially explaining the fact.

In a slightly more complicated version, abduction proceeds from a fact that C => A, using a rule that B => A, to presume a case that C => B. This is an inessential complication, since the rule of modus ponens and the rule of transitivity are essentially equivalent in their logical force, but it is often convenient to imagine that C is the "common subject" or the "current situation" that is implicit throughout the argument, namely, the existing entity that substantiates or instantiates all of the other predicates that are invoked in its course.

Suppose I have occasion to reason as follows:

"It looks like a duck, so I guess it is a duck."

Or even more simply:

"It looks blue, therefore it is blue."

These are instances in which I am using abductive reasoning, according to the pattern of the following schema:

I observe a Fact:

"It looks like X." (X')

I have in the back of my mind a general Rule:

"If it is X, then it looks like X." (X => X')

I reason my way back from the observed Fact and the assumed Rule to assert what I guess to be the Case:

"It is X." (X)

The abduction is a hypothetical inference that results in a diagnostic conclusion, that is, a statement of opinion as to what is conjectured to be the case. In each case the operation of abductive reasoning starts from a complex configuration, involving a number of explicit observations in the foreground and a class of implicit assumptions in the background, and it offers a provisional statement about certain possibility, one that is typically less conspicuous, obvious, or prominent, but still potentially present in the situation, and hopefully serving to explain the surprising or the problematic aspects of the whole state of affairs.

What results from the abductive inference is a concept and possibly a term, for instance, "duck" or "blue". The concept attempts to grasp a vast complex of appearances within a unitary form, and the term that connotes the concept is used to put explicit bounds on what it conveys. Working in tandem, they express an approximation or a simplification, "a reduction of the manifold of phenomena to a unified conception". Finite minds cannot operate for very long with anything more than this.

The reader may have noticed some obvious distinctions between the two examples of abductive reasoning that I gave above, between the case of "looking like a duck" and the case of "looking blue". Just to mention the most glaring difference: Although a person is occasionally heard to reason out loud after the fashion of the former example, it is rare to hear anyone naturally reasoning along the lines of the latter example. Indeed, it is more likely that any appearance of doing so is always an artificial performance and a self-conscious reconstruction, if not a complete fabrication, and it is doubtful that the process of arriving at a perceptual judgment can follow this rule in just so literal a fashion.

This is true and important, but it is beside the point of the immediate discussion, which is only to identify the logical form of the inference, that is, to specify up to informational equivalence the class of conduct that is involved in each example. Thus, considering the inference as an information process, I do not care at this point whether the process is implemented by a literal-minded variety of rule-following procedure, so long as it "follows", "obeys", or "respects" these rules in the form of what it does. One can say that an information process "obeys" a set of rules in a "figurative" and a "formal" sense if the transformation that occurs in the state of information between the beginning and the end of the process has the form of a relation that can be achieved by literally following these rules with respect to the prospective class of materials.

The general drift of the strategy that is being mapped out here, the "abstract", the "formal", or the "functional" approach, is now evident. Conceptually, one partitions the space of processes into "effective", "informational", or "pragmatic" equivalence classes and then adopts the inditement of a sequence of rules as a symbolic "nomen" for the class of processes that all achieve the same class of effects. At this level of functional abstraction, the conception of a process is indifferent to the particulars of its implemenation, so long as it lives within the means of the indicated constraints. Moreover, unless there is a way to detect the nature of the "actual" process without interfering too severely with it, that is, a path-sensitive but still unobtrusive measure that can sort out a finer structure from these equivalence classes, then it is not possible to inquire any further into the supposedly "actual" details.

Similar remarks apply to every case where one attributes "law-abiding" or "rule-governed" behavior to oneself, to another person, or even to a physical process. Across this diverse spectrum of cases, it ranges from likely but not certain to unlikely but still conceivable that the action in question depends on the agent "knowing" the laws that abide or the rules that are effectively being obeyed. With this in mind, I can draw this digression on appearances to a conclusion: When I say that agents are acting according to a particular pattern of rules, it only means that it "looks like" they are. In other words, they are acting "as if" they are consciously following these rules, or they are acting just like I act when I conscientiously follow such rules. A concise way to sum all of this up is to say that a pattern of rules constitutes a model of conduct, one that I can deliberately emulate, or one that I can attribute to others by way of explaining their conduct. In attributing this model to others, or even in using it to account for my own less deliberate behavior, I am making an abductive inference.

One way to appreciate the pertinence of this point is to notice that this entire digression, concerned with explaining the similarities between "looking like a duck" and "looking blue", is itself a form of argument, making a case of abductive inference to a case of abductive inference. In short, I am reasoning according to the following pattern:

It appears to be the making of an abductive inference,

so I guess it is the making of an abductive inference.

Anyone who thinks that this style of reasoning is too chancy to be tolerated ought to observe that it is only the pattern of inference that one follows in attributing minds to others, solely on the evidence that they exhibit roughly the same array of external behaviors in reaction to various external conditions as one employs to express one's experience of roughly the same conditions.

It goes without saying that abductive reasoning is extremely fallible. The fact that it looks like a duck does not necessarily mean that it is a duck - it might be a decoy. Moreover, in most cases of actual practice the implicit rule that serves to catalyze the abductive inference is not an absolute rule or a necessary truth in its own right but may be only a contingent rule or a probable premiss. For instance, not every case of being blue presents the fact of looking blue - the conditions of observation may be trickier than that. This brings to the fore another mark that distinguishes the two examples, highlighting a potentially important difference between "looking like a duck" and "looking blue". This is the amount of oversight, or awareness and control, that an agent has with regard to an inference, in other words, the extent to which an inference really does "go without saying".

The abductive inference from "it looks blue" to "it is blue" and the abductive inference from "it looks like a duck" to "it is a duck" differ in the degrees to which they exhibit a complex of correlated properties. These variations are summed up in one sense by saying that the first, more perceptual inference is more automatic, compulsive, habitual, incorrigible, and inveterate. The correlations are summed up in the opposite sense by saying that the second, more conceptual inference is more aware, controllable, correctable, critical, deliberate, guarded, and reflective. From a fully pragmatic standpoint, these differences are naturally of critical importance. But from a purely logical standpoint, they have to be regarded as incidental aspects or secondary features of the underlying forms of inference.

There is one thing yet missing from this description of abductive reasoning, and that is its creative aspect. The description so far is likely to leave the impression that the posing of a hypothesis always takes place against a narrowly circumscribed background of established terms that are available for describing cases, and thus that it amounts to nothing more original than picking out the right label for the case. Of course, the forming of a hypothesis may be bound by the generative potential of the language that is ultimately in force, but that is a far cry from a prescriptively finite list of more or less obvious choices.

How does all of this bear on the choice of a method? In order to make a start toward answering that question, I need to consider the part that abductive reasoning plays in the inquiry into method, which is, after all, just another name for the inquiry into inquiry.

There are times when choosing a method looks more like discovering or inventing a method, a purely spontaneous creation of a novel way to proceed, but normally the choice of a path picks its way through a landscape of familiar options and mapped out opportunities, and this presupposes a description of previously observed forms of conduct and a classification of different paths from which to choose. Hence the etymology of the word "method", indicating a review of means or a study of ways.

I would now like to examine several types situations where a choice of method is involved, paying special attention to the way that abductive reasoning enters into the consideration.

Example 1.

Suppose I have occasion to reason along the following lines:

This situation looks like one in which this method will work, therefore I will proceed on the hypothesis that it will work.

The current situation (C) looks amenable (A') to this method, so I guess it really is amenable (A) to this method.

In this type of situation, my observations of the situation are reduced to a form of description that portrays it in the light of a given method, amounting to an estimate of whether the situation is a case to which the method applies. The form of the entire argument hinges on the question of whether the assurance of this application is apparent or actual.

I express my observations of the situation as a Fact:

"The current situation looks amenable." (C => A')

I have in the back of my mind a general Rule:

"If it is amenable, then it looks amenable." (A => A')

I reason my way back from the observed Fact and the assumed Rule to assert what I guess to be the Case:

"The current situation is amenable." (C => A)

As far as it goes, this style of reasoning follows the basic pattern of abductive inference. Its obvious facticity is due to the fact that the situation is being described solely in the light of a pre-selected method. That is a relatively specious way to go about describing a situation, in spite of the fact that it may be inevitable in many of the most ultimate and limiting cases. The overall effect is noticeably strained, perhaps because it results from dictating an artificial setting, attempting to reduce a situation to the patterns that one is prepared to observe, and trying to fit what is there to see into a precut frame. A more natural way to describe a situation is in terms of the freely chosen perceptual features that inform a language of affects, impressions, and sensations. But here a situation is forced to be described in terms of the prevailing operational features that constitute a language of actions, forcing the description to be limited by the actions that are available within a prescribed framework of methods.

Instead of describing a situation solely in terms of its reactive bearing, that is, wholly in terms of how it reacts to the application of a method, one can try to describe it in terms that appear to be more its own, its independent, natural, observational, perceptual, or "proper" features. What the "proper" or "object-oriented" features are and whether they can be distinguished in the end from "reactive" or "method-oriented" features are questions that cannot be answered in the early phases of an investigation.

Example 2.

Suppose I find myself reasoning as follows:

If the current world (C) is a blessed world (B),

then it is a world in which my method works (A).

Here, I call to mind an independent property of being, B, that a world or a situation can have, and I use it as a middle term to reason along the lines of the following scheme:

I express my inquiry by questioning the possibility of a certain Fact, that is, by interrogating the following statement:

"The current world is amenable." (C =?> A)

I have in the back of my mind a general Rule:

"What is blessed, is amenable." (B => A)

I reason my way back from the interrogated Fact and the assumed Rule to guess that I ought to contemplate the chances of the following Case:

"The current world is blessed." (C =?> B)

Altogether, the argument that underlies the current question of method falls into line with the following example of abductive reasoning:

I hope that C is A, so I guess I hope that C is B.

To proceed with the application of a given method on the basis of such a piece of reasoning is tantamount to the faith, the hope, or the wish that there is already the right kind of justice in the world that would make the prejudices of one's favorite method turn out to be right, that one is just lucky enough to be playing in accord with a pre-established harmony. If such a confidence is all that allows one to go on inquiring, then there is no harm in assuming it, so long as one reserves the right to question every particular of its grant, should the occasion arise.

If one abstracts from the specific content of this example and examines its underlying structure, it reveals itself as the pattern of abductive reasoning that occurs in relating complex questions to simpler questions or in reducing difficult problems to easier problems. Furthermore, the iteration of this basic kind of step motivates a downward recursion from questions of fact to questions of cases, in a hopeful search for a level of cases where most of the answers are already known.

The previous examples of inquiry into method are not very satisfactory. Indeed, their schematic forms have an absurdly sketchy character about them, and they fail to convey the realistic sorts of problems that are usually involved in reasoning about the choice of a method. The first example characterizes a situation wholly in terms of a selected method. The second example characterizes a situation in terms of a property that is nominally independent of the method chosen, but the ad hoc character of this property remains obvious. In order to reason "properly" about the choice of method, it is necessary to contemplate properties of the methods themselves, and not just the situations in which they are used.

Example 3.

If I reason that scientific method is wise because wise people use it, then I am making the hypothesis that they use it because they are wise. Here, my reasoning can be explained according to the following pattern:

I observe a fact:

"A certain conduct is done by wise people." (C => X)

I have in mind a rule:

"If a wise act, then done by wise people." (A => X)

I abduce the case:

"A certain conduct is a wise act." (C => A)

Example 4.

If I reason that scientific method is a good method on account of the fact that it works for now, then I am guessing that it works for now precisely because it is good.

I observe a fact:

"Scientific method works for now." (C => X)

I have in mind a rule:

"What is good, works for now." (A => X)

I abduce the case:

"Scientific method is good." (C => A)

As always, the abductive argument is extremely fallible. The fact that scientific method works for now can be one of its accidental features, and not due to any essential goodness that it might be thought to have.

Finally, it is useful to consider an important variation on this style of argument, one that exhibits its close relation to reasoning by analogy or inference from example. Suppose that the above argument is presented in the following manner:

Scientific method (C) has many of the features that a good method needs to have, for instance, it works for now (X), so I reason that it has all of the features of a good method, in short, that it is a good method (A).

So far, the underlying argument is exactly the same. In particular, it is important to notice that the abductive argument does not depend on the prior establishment of any known cases of good methods. As of yet, the phrase "good method" is a purely hypothetical description, a term that could easily turn out to be vacuous. One has in mind a number of properties that one thinks a good method ought to have, but who knows if there is any thing that would satisfy all of these requirements? There may be some sort of subtle contradiction that is involved in the very juxtaposition of the terms "good" and "method". In sum, it can happen that scientific method is the very first method that is being considered for membership in the class of good methods, and so it is still unknown whether the class labeled "good methods" is empty or not.

But what if an example of a good method is already known to exist, one that has all of the commonly accepted properties that appear to define what a good method ought to be? In this case, the abductive argument acquires the additional strength of an argument from analogy.

1.4.3.4 Analogical Reasoning

The classical treatment of analogical reasoning by Aristotle explains it as a combination of induction and deduction. More recently, C.S. Peirce gave two different ways of viewing the use of analogy, analyzing it into complex patterns of reasoning that involve all three types of inference. In the appropriate place, it will be useful to consider these alternative accounts of analogy in detail. At the present point, it is more useful to illustrate the different versions of analogical reasoning as they bear on the topic of choosing a method.

The next example, ostensibly concerned with reasoning about a choice of method, is still too artificial to be taken seriously for this purpose, but it does serve to illustrate Aristotle's analysis of analogical reasoning as a mixed mode of inference, involving inductive and deductive phases.

Example 5.

Suppose I reason as follows. I think I can establish it as a fact that scientific method is a good method by taking it as a case of a method that always works and by using a rule that what always works is good. I think I can establish this rule, in turn, by pointing to one or more examples of methods that share the criterial property of always working and that are already acknowledged to be good. In form, this pattern of reasoning works by noticing examples of good methods, by identifying a reason why they are good, in other words, by finding a property of the examples that seems sufficient to prove them good, and by noticing that the method in question is similar to these examples precisely in the sense that it has in common this cause, criterion, property, or reason.

In this situation, I am said to be reasoning by way of analogy, example, or paradigm. That is, I am drawing a conclusion about the main subject of discussion by way of its likeness to similar examples. These cases are like the main subject in the possession of a certain property, and the relation of this critical feature to the consequential feature of interest is assumed to be conclusive. The examples that exhibit the criterial property are sometimes known as "analogues" or "paradigms". For many purposes, one can imagine that the whole weight of evidence present in a body of examples is represented by a single example of the type, an exemplary or typical case, in short, an archetype or epitome. With this in mind, the overall argument can be presented as follows:

Suppose that there is an exemplary method (E) that I already know to be a good method (A). Then it pays to examine the other properties of the exemplary method, in hopes of finding a property (B) that explains why it is good. If scientific method (C) shares this property, then it can serve to establish that scientific method is good.

The first part of the argument is the induction of a rule:

I notice the case:

"The exemplary method always works." (E => B)

I observe the fact:

"The exemplary method is a good method." (E => A)

I induce the rule:

"What always works, is good." (B => A)

The second part of the argument is the deduction of a fact:

I notice the case:

"Scientific method always works." (C => B)

I recall the rule:

"What always works, is good." (B => A)

I deduce the fact:

"Scientific method is good." (C => A)

Example 6.

Example 7.

Suppose that several examples (S1, S2, S3) of a good method are already known to exist, ones that have a number of the commonly accepted properties (P1, P2, P3) that appear to define what a good method is. Then the abductive argument acquires the additional strength of an argument from analogy.

The first part of the argument is the abduction of a case:

I observe a set of facts:

"Scientific method is P1, P2, P3." (C => P)

I recall a set of rules:

"Bona fide inquiry is P1, P2, P3." (B => P)

I abduce the case:

"Scientific method is bona fide inquiry." (C => B)

The second part of the argument is the induction of a rule:

I notice a set of cases:

"S1, S2, S3 exemplify bona fide inquiry." (S => B)

I observe a set of facts:

"S1, S2, S3 exemplify good method." (S => A)

I induce the rule:

"Bona fide inquiry is good method." (B => A)

The third part of the argument is the deduction of a fact:

I recall the case:

"Scientific method is bona fide inquiry." (C => B)

I recall the rule:

"Bona fide inquiry is good method." (B => A)

I deduce the fact:

"Scientfic method is good method." (C => A)

Now, logically and rationally in the purest sense, the argument by analogy to an example has no more force than the abductive argument, but, empirically and existentially, the example serves, not only as a model of the method to be emulated, but as an object of experimental variation and a source of further experience.

It is time to ask the question: Why do these examples continue to maintain their unrealistic character, their comical and even ridiculous appearance, in spite of all my continuing attempts to reform them in a sensible way? It is not merely their simplicity. A simple example can be telling, if it grasps the essence of the problem, that is, so long as it captures even a single essential feature or highlights even a single critical property of the thing that one seeks to understand. It is more likely due to the circumstance that I am describing agents, methods, and situations all in one piece, that is, without any analysis, articulation, or definition of what exactly constitutes the self, the scientific method, or the world in question. It is not completely useless to consider cases of this type, since many forms of automatic, customary, and unreflective practice are underlain by arguments that are not much better that this. Of course, on reflection, their "commedius" character becomes apparent, and all deny or laugh off the suggestion that they ever think this way, but that is just the way of reflection.

In order to improve the character of the discussion on this score ...

References

Aristotle, "On The Soul", in 'Aristotle, Volume 8',
W.S. Hett (trans.), Heinemann, London, UK, 1936, 1986.

Charniak, E. & McDermott, D.V.,
'Introduction to Artificial Intelligence',
Addison-Wesley, Reading, MA, 1985.

2.  Charniak, E., Riesbeck, C.K., & McDermott, D.V.  Artificial Intelligence Programming.  Lawrence Erlbaum Associates, Hillsdale, NJ, 1980.

3.  Holland, J.H., Holyoak, K.J., Nisbett, R.E., & Thagard, P.R.  Induction:  Processes of Inference, Learning, and Discovery.  MIT Press, Cambridge, MA, 1986.

4.  O'Rorke, P.  Review of AAAI 1990 Spring Symposium on Automated Abduction.  SIGART Bulletin, Vol. 1, No. 3.  ACM Press, October 1990, p. 12-17.

5.  Pearl, J.  Probabilistic Reasoning in Intelligent Systems:  Networks of Plausible Inference.  Revised 2nd printing.  Morgan Kaufmann, San Mateo, CA, 1991.

6.  Peng, Y. & Reggia, J.A.  Abductive Inference Models for Diagnostic Problem-Solving.  Springer-Verlag, New York, NY, 1990.

7.  Sowa, J.F.  Conceptual Structures:  Information Processing in Mind and Machine.  Addison-Wesley, Reading, MA, 1984.

8.  Sowa, J.F. (ed.)  Principles of Semantic Networks:  Explorations in the Representation of Knowledge.  Morgan Kaufmann, San Mateo, CA, 1991.

Dewey, J. (1991).  How We Think.  Buffalo, NY: Prometheus Books.  Originally published 1910.

Shakespeare, Wm.  (1988).  William Shakespeare:  The Complete Works.  Compact Edition.  S. Wells & G. Taylor (eds.).  Oxford University Press, Oxford, UK.

Notes

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IDS.  Additional Notes

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

CFR.  Note 78

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

MW = Matthew West:

MW: Do you have a Cactus Manual all in one place please?

the documentation for my 'theme one' program
that I wrote up for my quant psy master's
contains the last thing like an official
manual that I wrote, also an expository
introduction to the cactus language and
its application to prop calc examples.
may still have an ancient ascii version,
or else the medieval 'word' doc, or i can
send the mac belle version by snail express
if you can vouchsafe me your postal address.

in the mean time, i append a few of the expositions that
i have outlined here/elsewhere over the last year on-line.

pre-scanning this whole mess'o'messages for you,
I find one that looks to me shortest & sweetest:

http://suo.ieee.org/email/msg05694.html

since this particular synopsis is mercifully short, i will copy it out here
and use it to explain surcatenation, along with a few other thing that i am
guessing might be puzzling at first sight about what in hey's going on here.

o~~~~~~~~~o~~~~~~~~~o~ARCHIVE~o~~~~~~~~~o~~~~~~~~~o

Reflective Extension of Logical Graphs (Ref Log)

Here is a formal introduction to the RefLog Syntax.

Formally speaking, we have the following set-up:

Set out the "alphabet of punctuation marks" $M$ = {" ", ",", "(", ")"}.
The elements of $M$ are vocalized as "blank, "comma", "links", "right".

1.  There is a parametric family of formal languages of character strings
    such that, for each set $X$ of variable names $X$ = {"x_1", ..., "x_k"},
    there is a formal language L($X$) over the alphabet A($X$) = $M$ |_| $X$.
    The grammar can be given in gory detail, but most folks know it already.

| Examples.  If $X$ = {"x", "y"}, then these are typical strings in L($X$):
|
| " ", "( )", "x", "y", "(x)", "(y)", "x y", "(x y)", "(x, y)", "((x)(y))", "((x, y))", ...

2.  There is a parallel family of formal languages of graphical structures,
    generically known as "painted and rooted cacti" (PARC's), that exist in
    a one-to-one correspondence with these string expressions, being more or
    less roughly, at a suitable level of abstraction, their parse graphs as
    data structures in the computer.  The PARC's for the above formulas are:

| Examples.
|                                                                 x   y       x   y
|                                                                 o   o       o---o
|                         x      y             x y     x   y       \ /         \ /
|        o                o      o              o      o---o        o           o
|        |     x    y     |      |     x y      |       \ /         |           |
|  @     @     @    @     @      @      @       @        @          @           @       ...
|
| " ", "( )", "x", "y", "(x)", "(y)", "x y", "(x y)", "(x, y)", "((x)(y))", "((x, y))", ...

Together, these two families of formal languages constitute a system
that is called the "reflective extension of logical graphs" (Ref Log).

Strictly speaking, Ref Log is an abstract or "uninterpreted" formal system,
but its expressions enjoy, as a rule, two dual interpretations that assign
them the meanings of propositions or sentences in "zeroth order logic" (ZOL),
to wit, what Peirce called the "alpha level" of his systems of logical graphs.

For example, the string expression "(x (y))" parses into the following graph:

|       x   y
|       o---o
|       |
|       @

You can "deparse" the string off the graph by traversing
it like so, reading off the marks and varnames as you go.

|  o---x->(--y---o
|  ^             |
|  |   x  (  y   |
|  |   o-----o   v
|  |   |  )      )
|  (  (|)        )
|  ^   |         |
|  |   @         v

In the "existential" interpretation of RefLog,
in which I do my own thinking most of the time,
concatenation of expressions has the meaning of
logical conjunction, while "(x)" has the meaning
of "not x", and so the above string and graph have
a meaning of "x => y", "x implies y", "if x then y",
"not x without y", or anything else that's equivalent.
The blank expression is assigned the value of "true".
Hence, the expression "()" takes the value of "false".
The bracket expression "(x_1, x_2, ..., x_k)" is given
the meaning "Exactly one of the x_j is false, j=1..k".
Therefore, "((x_1),(x_2), ...,(x_k))" partitions the
universe of discourse, saying "Just one x_j is true".

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CFR.  Note 83

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

| Tantum ergo sacramentum
|   veneremur cernui,
| et antiquum documentum
|   novo cedat ritui,
| praestet fides supplementum
|   sensuum defectui.
|
| So great therefore a sacrifice
|   let us humbly adore
| and let the old law yield
|   to the new rite;
| let faith supplement
|   the shortcoming of the senses.
|
| Lyric by Thomas Aquinas,
| Music by Amadeus Mozart, KV 142 & 197.

The increasing ossification of asciification
is heaping up way too many old bones to bear.
So I am going to shift my anklage a bit, and
try out a new set of conventions for a while,
to see if I can lighten the overloading obit.

Let us try to reserve script and singly-underscored fake-fonts or formats
for the names of sets, as in the notations !O!, !S!, !I! that I will now
set aside and use from now on for the Object, Sign, Interpretant domains,
respectively, of an arbitrary sign relation !L! c !O! x !S! x !I!.

Among other benefits, this will serve to liberate the plain faced characters
for employment as the non-terminal symbols of our formal grammars, rendering
our formal grammatical productions far less $Capitalistic$, !Exclamatory!,
and overbearingly prescriptive than they be otherwise hell-bent to become.

So let me try out this new rite to see how it works out,
And I will not pause to rewrite the old law in its font,
But advise you solely of its transformed instantiations,
And fix my faith on imagination to sense the supplement.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

CFR.  Note 92

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

I need to try and say some things at his point about
why formal language theory is interesting and useful,
but all I have at the moment are random remembrances
and reflections that enter my mind from time to time.

In many ways, the study of formal languages and grammars
is a paradigm, more, a paragon, of the situation that we
face whenever we inquire into a complex reality, that is,
all of the ever-renewed sources of puzzling phenomena or
pressing problems that we call a world.

The archtypical place of formal language theory is well
understood in many quarters, and has been from the very
outset of its constellation as an independent viewpoint.

In this paradigmatic (analogical or exemplary) way of
understanding it, a formal language is the "data" and
a formal grammar is the "theory", and the question is,
as always, whether a theory accounts for and explains
the data, a "fitting" relationship that may be viewed
in many ways, for one, the way that a theory might be
said to "generate" the data, or perhaps better stated,
not just to "cook" in a precociously specious fashion
but more like to "regenerate" the form after the fact.

That's all that I can manage to express at the moment,
but maybe it will supply a grub-stake of motivational
victuals for the grueling labors of exploration ahead.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

IDS.  Incitatory Note 1

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

| Each ground-principle must be proved entirely
| by that same kind of inference which it supports.
|
| But we cannot arrive at any conclusion
| by mere deduction except about symbols.
|
| We cannot arrive at any conclusion
| by mere induction except about things.
|
| And we cannot arrive at any conclusion
| by mere hypothesis except about forms.
|
| C.S. Peirce, CE 1, page 290.
|
| Charles Sanders Peirce, "On the Logic of Science",
| Harvard University Lectures (1865), pages 161-302 in:
|'Writings of Charles S. Peirce:  A Chronological Edition',
|'Volume 1, 1857-1866', Peirce Edition Project,
| Indiana University Press, Bloomington, IN, 1982.

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

IDS.  Meditative Note 1

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

I would like to start from a "common sense practical" (CSP) point of view,
and, indeed, never to lose sight of what appears evident from that station,
no matter how many levels of abstract remove and abstruse mention it might
become necessary to interpose along the way.

So let's examine this initial caltrop
"descriptive/normative/prescriptive"
from the CSP POV, if you will.

Reading "Descriptive" to mean "What it is",
while "Normative" means "What it oughta be",
and "Prescriptive" says "Make it so, or else",
I will have very little to say about the last,
and only be able to focus on the distinctions
that may exist among the first two dimensions.

From the beginning, from this point of view, difficult words,
like "inquiry", "logic", "truth", and so on, must be taken
as initially indexical, inchoately succeeding at little
more than pointing to a realm of experience that may
or may not be common to the e-mitter and re-mitter.

I suspect that this stanza is likely to be controversial,
so I'll pause at this point for the countrapunctal verse.

Or for a rest ...

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IDS.  Meditative Note 2

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So I may begin with an object and a sign in a tenuous relation,
with the subject matter indexed under the topic name "inquiry",
where the sign originates from a "just noticeable differential"
of information about the object, and not a single "figit" more.
Few would call this a foundation -- I only call it a beginning.

Yet another of many ...

But it does provide us with a clue to a signficant difference,
however much this difference is bound by this origin to raise
itself from egg, germ, seed, spore, or whatever it is that is
infinitesimal in its initial condition.  In this disjointness
of an archetype where what begins, what leads, and what rules
are not so trivially identical to one another, one encounters
the brand of beginning that begins in the middle of the story,
and has no need of any other foundation but the medium itself.

["sign-ficant" [stet]]

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IDS.  Obligatory Note 1

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While I remain compelled to remain silent on the status of the absolute fiat,
the irrelative notion of the unmotivated motion and the disinterested stance,
let me then turn to the other axes of description, descriptive vs. normative.
Axes of description, indeed, you can almost hear one branch of the recursion
already beginning to wind up its whine to the verge of a howl, but toss it a
sop and try to persevere in the quest.

In this view, I regard the very idea of a norm as invoking its due pragma --
aim, business, concern, desire, end, function, goal, intention, interest,
objective, purpose, its names are legion -- and the good sense of the
norm is simply to suggest what one ought to do, contingent, of course,
on one's motive to achieve that pragma.

If we keep in mind the kinds of "applied research task" (ART) that your
everyday artist, designer, engineer, mathematician, scientist, or other
type of technical worker has to carry out on an everyday basis, we note
how these axes of description can be used to frame their activities and
to depict their forms of conduct, without mistaking either the frame or
the picture for the object of the picture so framed.  Nor does any body
imagine that the observer must flatten out into a single plane or align
with a single axis, in order to make a vantage of the frame so pictured.

Common sense practical wit tells us that effective action toward the
achievement of a desirable result will naturally depend on acquiring
good descriptions of the lay of the land in which we hope to advance.

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IDS.  Projective Note 1

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Good morning.  Thanks.  I had a bad night.
I blame Bernard Morand, who wrote me this:

BM: But this looks as some God's view.
    What about us, finite humans, occupied
    in counting the instants of our lives?
    And thus condemned to try to improve
    the fate of our successors?

When you think of this in the future, and of course you may never,
you may blame him too, for in writing this he has "erged" me on
to return to my deserted dissertation work, into which I have
poured my life for lo! these too many years to count, truly,
if you stop to contemplate the fact that time is relative.

In that time I have come to the view that we really need
a good "theory of inquiry" (TOI), for all sorts of very
practical and crucial reasons, also, that we cannot get
a good TOI without its being, at one and the same time,
a good "theory of information" (TOI too), and also that
an integral constituent of TOI 1 and TOI 2 would have to
be a good "theory of representation and semiosis" (TORAS) --
"Bull!?", you say, well, so be it.

Further, I think that it is abundantly evident by now that
we will get no such good theories of signs or science from
the "establishment philosophy of science" (EPOS?) -- which
has managed to mince and to trash the best available tries
at such theories for over a hundred years now.  But Hey! --
don't take my word for it -- waste a century of your own.

We just got our regular email back,
so I think that I can now get going --
Yes, I have lost the ability to think
if not literally writing 'to' somebody.

When it begins, it begins like this:

Why am I asking this question?

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IDS.  Projective Note 2

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So we may rest assured that we do have a "subject matter", an empirical domain,
or a realm of experience that is indexed, however dimly, generally, or vaguely,
by the word "inquiry", and only the question how best to describe it remains
in doubt at this stage of the play.  If we wanted to cast our net as widely
as possible, at the risk of anticipating a bounding hypothesis, we could
think of all the world's creatures bright and beautiful and of how they
conduct themselves when faced with some moment of uncertainty, where
their aim is to cope with a surprising phenomenon or to deal with
a problematic situation that meets them in the course of their
ever-ongoing struggles to live, to revive, and to thrive.

Now, neither the fact that we begin with a descriptive task,
nor the fact that it remains of interest for its own sake,
necessarily means that we must end there, for it is also
the means to a further end, of learning how to better
our own skill at inquiry, which means in our time
the building of tools that help with the task.

I hope I have made this sound as truly and
as trivially obvious as it ought to be.

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IDS.  Reflective Note 1

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In reflecting on what in the world a "Theory of Inquiry" (TOI) might be,
it occurs to me that there are many different things that one might mean
by such a theory.  It could just be any number of things that one asserts
or has a mind to assert about the ostensible subject matter.  But it has
been my experience that one can assert pretty much whatever one chooses,
and others will choose to heed it or ignore it on many different grounds,
the grounds themselves being a matter of choice, conditioning, or custom.

But I am looking for theories that work, that is to say, theories that
are subject to probation through proof, probability, and programming.

Astute readers will have noticed that I've already attempted to finesse
a very important, and most likely "infinessible" issue, to wit, that of
the scruples dividing descriptive, normative, and prescriptive theories.

I will think about that, and get back to you.

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IDS.  Reflective Note 2

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| 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 that 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?
| Accordingly, the questions of methodological self-application
| and self-referential consistency will be found at the center
| of this research.

These lines image in compact form the crux of the problem,
the crucible of the method, and the character that marks
relation between the two, if indeed they really are two,
in a form whose extended development will wind its way
through many a later page of the present exposition.

But let me just point out at this point some of
the reasons why I have found the prerequisite
of an inquiry into inquiry to be inescapable.

Let us entertain the idea, for the sake of getting the inquiry started,
if nothing else, that it is admissible to use a word like "inquiry" as
an initially indefinite indicator of an ostensible object of inquiry.
If we ever again find ourselves being puzzled how our reasoning can
chastize its own entailments this way, we may remind ourselves of
that fine old line between our "logica docens' (logic as taught)
and our "logica utens" (logic as used).  With this distinction
in mind, we can dispell the initial puzzlement by saying that
we are using a capacity for inquiry that we do not know how
to formalize yet in order to examine the forms of inquiry
that various thinkers have been able, at least partially,
to formalize.

The dilemma that we face has the following structure:

If we recommend to all a method of inquiry that
we ourselves do not use in a pinch, precisely
in a pinch where we need to study an issue
as important as the nature of inquiry,
then who would take our advice?

So it seems that there is no choice
but to study inquiry, the pragma,
by way of inquiry, the praxis,
that is to say, recursively.

Incidentally, many variations on this theme are
thoroughly developed in Peirce's "Lectures" of
1865 and 1866 and recapitulated in his early
study "On a New List of Categories" (1867).

http://members.door.net/arisbe/menu/library/bycsp/newlist/nl-main.htm

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IDS.  Work Area

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From this point of view, inquiry is form of conduct,
an applied research task, like many others that we
have to carry out, and that can be done either
better or worse.

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Document History

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| Document History
|
| Subject:  Inquiry Driven Systems:  An Inquiry Into Inquiry
| Contact:  Jon Awbrey <jawbrey@oakland.edu>
| Version:  Draft 10.00
| Created:  23 Jun 1996
| Revised:  02 Mar 2003
| Advisor:  M.A. Zohdy
| Setting:  Oakland University, Rochester, Michigan, USA

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Inquiry List

• http://stderr.org/pipermail/inquiry/2004-May/thread.html#1434
• http://stderr.org/pipermail/inquiry/2004-May/thread.html#1564
• http://stderr.org/pipermail/inquiry/2004-June/thread.html#1574