Inquiry Driven Systems:
An Inquiry Into Inquiry
Jon Awbrey
Oakland University
1. Research Proposal
1.1 Outline of the Project: Inquiry Driven Systems
1.2 Onus of the Project: No Way But Inquiry
1.3 Option of the Project: A Way Up To Inquiry
1.3.1 Initial Analysis of Inquiry
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Allegro Aperto
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1.3.2 Discussion of Discussion
1.3.3 Discussion of Formalization: General Topics
1.3.4 Discussion of Formalization: Concrete Examples
1.3.4.1 Formal Models: A Sketch 1.3.4.2 Sign Relations: A Primer To the extent that their structures and functions can be discussed at all, it is likely that all of the formal
entities destined to develop in this approach to inquiry will be instances of a class of three-place relations
called "sign relations". At any rate, all of the formal structures that I have examined so far in this area have
turned out to be, if not manifestly sign relations themselves, erither easily convertible to, or else ultimately
grounded in, some variation on sign relations. This class of triadic relations constitutes the main study
of the "pragmatic theory of signs", a branch of logical philosophy that is devoted to understanding
all manners of symbolic representation and all types of significant communication.
There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry.
In fact, the correspondence between the two studies exhibits so many parallels and coincidences that it is
often best to treat them as integral parts of one and the same subject. In a very real sense, inquiry is the
process by which sign relations come to be established and continue to evolve. In other words, inquiry,
"thinking" in its best sense, "is a term denoting the various ways in which things acquire significance"
(Dewey). Thus, there is an active and intricate form of cooperation that needs to be appreciated and
maintained between these converging modes of investigation. Its proper character is best understood
by realizing that the theory of inquiry is adapted to study the developmental aspects of sign relations,
a subject which the theory of signs is specialized to treat from structural and comparative points of view.
Because the examples in this section have been artificially constructed to be a simple as possible,
their detailed elaboration can run the risk of trivializing the whole theory of sign relations. Still,
these examples have subtleties of their own, and their careful treatment will serve to illustrate
important issues in the general theory of signs.
Imagine a discussion between two people, Ann and Bob, and attend only to that aspect of their
expressive and interpretive practice that involves the use of the following nouns and pronouns:
"Ann", "Bob", "I", "You".
The "object domain" of this discussion fragment is the set consiting of two people {Ann, Bob}.
The "syntactic domain" or the "sign system" of their discussion is limited to the set of four signs
{"Ann", "Bob", "I", "You"}.
In their discussion, Ann and Bob are not only the passive objects of nominative and accusative
references but also the active interpreters of the language they use. The system of interpretation (SOI)
associated with each language user can be represented in the form of an individual three-place relation
called the "sign relation" of that interpreter.
Understood in terms of its set-theoretic extension, a sign relation R is a subset of a cartesian product OxSxI.
Here, O, S, and I are three sets called the "object domain", the "sign domain", and the "interpretant domain",
respectively, of the sign relation R c OxSxI. In general, the three domains of a sign relation can be any
sets whatsoever, but the kinds of sign relation contemplated in a computational framework are usually
constrained to having I c S. In this case, interpretants are just a special type of signs, and this makes
it convenient to lump signs and interpretants together into a "syntactic domain". In the forthcoming
examples, S and I are identical as sets, so the very same elements appear in two distinct roles of the
pertinent sign relations. When it is necessary to refer to the whole set of objects and signs in the
union of the domains O, S, and I for a given sign relation R, I will call this the "world of R" and
write W = W(R) = O U S U I.
To facilitate an interest in the abstract structures of sign relations, and to keep the notations as brief
as possible when the examples get more complicated, I introduce the following abbreviations:
O = object domain;
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S = sign domain;
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I = interpretant domain.
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O
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=
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{ Ann, Bob }
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=
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{ A, B }.
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S
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=
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{"Ann", "Bob", "I", "You"}
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=
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{"A", "B", "i", "u"}.
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In the present examples, S = I = syntactic domain.
Tables 1 and 2 give the sign relations associated with the interpreters A and B, respectively, putting
them in the form of relational databases. Thus, the rows of each Table list the ordered triples <o, s, i>
that make up the corresponding sign relations: A, B c OxSxI. The issue of using the same names for
objects and for relations involving these objects will be taken up later, after the less problematic features
of these relations have been treated.
These Tables codify a rudimentary level of interpretive practice for the agents A and B, and provide
a basis for formalizing the initial semantics appropriate to their common syntactic domain. Each row
of a Table names an object and two co-referent signs, making up an ordered triple <o, s, i> called an
"elementary relation", that is, one element of the relation's set-theoretic extension.
Already in this elementary context, there are several different meanings that might attach to the project
of a "formal semantics". In the process of discussing these alternatives, I will introduce a few terms that
are occasionally used in the philosophy of language to point out the needed distinctions.
Table 1. Sign Relation of Interpreter A
Table 2. Sign Relation of Interpreter B
One aspect of semantics is concerned with the reference that a sign has to its object, which is often
called its "denotation". For signs in the most general type of situation, neither the existence nor the
uniqueness of a denotation is guaranteed. Thus, the denotation of a sign can refer to a plural, to a
singular, or to a vacuous number of objects. In the pragmatic theory of signs, these references of
signs to their objects are formalized as certain types of dyadic sub-relations that are found embedded
in the triadic sign relations. When it comes to dealing with the degenerate cases of signs that do not
denote, it is necessary to introduce what is, strictly speaking, a slightly more general concept than
a sign relation proper, namely, what is called a "sign-relational complex". But that is the subject
of a much later discussion. For now, I shall keep to signs which are known to have one or more
objects among their denotations.
The dyadic relation that constitutes the "denotative component" of a sign relation R is denoted by
"Den (R)". Information about the denotative component of semantics can be derived from R by taking
its "dyadic projection" on the object and sign domains, indicated by any one of the equivalent forms
"ProjOS(R)", "ROS", or "R12", and defined as:
Den (R) = ProjOS(R) = ROS = R12 = {<o, s> C OxS : <o, s, i> C R for some i C I}.
Looking to the denotative aspects of the present example, various rows of the Tables specify that
A uses "i" to denote A and "u" to denote B, whereas B uses "i" to denote B and "u" to denote A.
It is utterly amazing that even these impoverished remnants of natural language use have properties
that quickly bring the usual prospects of formal semantics to a screeching halt.
The other dyadic aspects of semantics that might be considered concern the reference that a sign has to its
interpretant and the reference that an interpretant has to its object. As before, either type of reference can
be multiple, unique, or empty in its collection of terminal points, and both can be formalized as different
kinds of dyadic sub-relations that can be found embedded in the triadic sign relations.
The connection that a sign makes to an interpretant is called its "connotation". In the general theory of
sign relations, this aspect of semantics includes the references that a sign has to ideas, concepts, affects,
intentions, and to the whole realm of an agent's mental states and allied activities, broadly encompassing
intellectual associations, emotional impressions, and motivational impulses. This complex ecosystem of
references is unlikely ever to be mapped in much detail, much less completely formalized, but the tangible
warp of its accumulated mass is commonly alluded to as the "connotative" import of language. Given a
particular sign relation R, the dyadic relation that constitutes the "connotative component" of R is denoted
by "Con (R)".
The bearing that an interpretant has toward a common object of its sign and itself has no standard name.
If an interpretant is considered to be a sign in its own right, then its independent reference to an object
can be taken as belonging to another moment of denotation, but this omits the mediational character
of the whole transaction.
Given the service that interpretants supply in furnishing a locus for critical, reflective, and explanatory
glosses on objective scenes and their descriptive texts, it is easy to regard them as "annotations" of both
objects and signs, but this function points in the opposite direction to what is needed in this connection.
What does one call the inverse of the annotation function? More generally asked, what is the converse
of the annotation relation?
In light of these considerations, I find myself still experimenting with terms to suit this last-mentioned
dimension of semantics. On a trial basis, I will refer to it as the "ideational", "intentional", or "canonical"
component of the sign relation, and I will try calling the reference of an interpretant sign to an object its
"ideation", "intention", or "conation". Given a particular sign relation R, the dyadic relation that constitutes
the "intentional component" of R is denoted by "Int (R)".
A full consideration of the connotative and intentional aspects of semantics would force a return to
difficult questions about the true nature of the interpretant sign in the general theory of sign relations.
It is best to defer these issues to a later discussion. Fortunately, omission of this material does not interfere
with understanding the purely formal aspects of the present example.
Formally, these new aspects of semantics present no additional problem. The connotative component
of a sign relation R can be formalized as its dyadic projection on the sign and interpretant domains,
defined as:
Con (R) = ProjSI(R) = RSI = R23 = {<s, i> C SxI : <o, s, i> C R for some o C O}.
The intentional component of semantics in a sign relation R, or the "second moment of denotation",
is captured by its dyadic projection on the object and interpretant domains, defined as:
Int (R) = ProjOI(R) = ROI = R13 = {<o, i> C OxI : <o, s, i> C R for some s C S}.
Indeed, the sign relations A and B in the present example are fully symmetric with respect to exchanging
signs and interpretants, so all the structure of AOS and BOS is merely echoed in AOI and BOI, respectively.
The concern of this project is not with every conceivable sign relation but only with those that are capable
of supporting inquiry processes. In these, the relationship between the denotational and connotational
aspects of meaning is not wholly arbitrary. Instead, this relationship must be naturally constrained or
deliberately designed in such a way that it (1) supports the achievement of particular purposes that
have intentional value for the agent and (2) represents the embodiment of significant properties that
have objective reality in the agent's domain. Therefore, my attention is directed toward understanding
the forms of correlation, coordination, and cooperation among the various components of sign relations
that form the necessary conditions for carrying out coherent inquiries.
1.3.4.3 Semiotic Equivalence Relations A nice property possessed by the sign relations A and B is that their connotative components ASI and BSI
constitute a pair of equivalence relations on their common syntactic domain S = I. It is convenient to refer
to such structures as "semiotic equivalence relations" (SER's) since they equate signs that mean the same
thing to somebody. Each of these semiotic equivalence relations ASI, BSI c SxI = SxS partitions the whole
collection of signs into "semiotic equivalence classes" (SEC's). This constitution makes for a strong form of
representation in that the structure of the participants' common object domain is reflected or reconstructed,
part for part, in the structure of each of their "semiotic partitions" (SEP's) of the syntactic domain.
The main trouble with this notion of semantics in the present situation is that the two semiotic partitions
for A and B are not the same, indeed, they are orthogonal to each other. This makes it difficult to interpret
either one of the partitions or equivalence relations on the syntactic domain as corresponding to any sort of
objective structure or invariant reality, independent of the individual interpreter's "point of view" (POV).
Information about the different forms of semiotic equivalence induced by the interpreters A and B
is summarized in Tables 3 and 4. The form of these Tables should suffice to explain what is meant
by saying that the SEP's for A and B are orthogonal to each other.
Table 3. Semiotic Partition of Interpreter A
Table 4. Semiotic Partition of Interpreter B
To discuss this situation further, I introduce the square bracket notation "[x]E" for "the equivalence class
of the element x under the equivalence relation E". A statement that the elements x and y are equivalent
under E is called an "equation". When the particular equivalence relation that qualifies an equation needs
to be made explicit, or cannot otherwise be taken for granted, as being implicitly understood, the equation
can be written in either one of two ways, as "[x]E = [y]E" or as "x =E y".
In the application to sign relations I extend this notation in the following ways. When R is a sign relation
whose "syntactic projection" or connotative component RSI is an equivalence relation on S, I write "[s]R"
for "the equivalence class of s under RSI". A statement that the signs x and y are synonymous under
a semiotic equivalence relation RSI is called a "semiotic equation" (SEQ), and can be written in either
of the forms: "[x]R = [y]R" or "x =R y".
In many situations there is one further adaptation of the square bracket notation that can be useful.
Namely, when there is known to exist a particular triple <o, s, i> C R, it is permissible to use "[o]R"
to mean the same thing as "[s]R". These modifications are designed to make the notation for semiotic
equivalence classes harmonize as well as possible with the frequent use of similar devices for the
denotations of signs and expressions.
In these terms, the SER for interpreter A yields the semiotic equations:
["A"]A
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=
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["i"]A
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,
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["B"]A
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=
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["u"]A
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,
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"A"
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=A
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"i"
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,
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"B"
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=A
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"u"
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,
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and the semiotic partition: {{"A", "i"}, {"B", "u"}}.
In contrast, the SER for interpreter B yields the semiotic equations:
["A"]B
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=
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["u"]B
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,
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["B"]B
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=
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["i"]B
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,
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"A"
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=B
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"u"
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,
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"B"
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=B
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"i"
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,
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and the semiotic partition: {{"A", "u"}, {"B", "i"}}.
1.3.4.4 Graphical Representations The dyadic components of sign relations can be given graph-theoretic representations, as "digraphs"
(or "directed graphs"), that provide concise pictures of their structural and potential dynamic properties.
By way of terminology, a directed edge <x, y> is called an "arc" from point x to point y, and a self-loop
<x, x> is called a "sling" at x.
The denotative components Den (A) and Den (B) can be represented as directed graphs on the six points
of their common world set W = O U S U I = {A, B, "A", "B", "i", "u"}. The arcs of the corresponding
digraphs are given as follows:
1.
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Den (A) has an arc from each point of {"A", "i"} to A and from each point of {"B", "u"} to B.
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2.
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Den (B) has an arc from each point of {"A", "u"} to A and from each point of {"B", "i"} to B.
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Den (A) and Den (B) can be interpreted as "transition digraphs" that chart the succession of steps or
the connection of states in a computational process. Read this way, the denotational arcs summarize
the "upshots" of the computations that are involved when the interpreters A and B evaluate the signs
in S according to their own lights, that is to say, in line with their own respective frames of reference.
The connotative components Con (A) and Con (B) can be represented as digraphs on the four points
of their common syntactic domain S = I = {"A", "B", "i", "u"}. Since Con (A) and Con (B) are SER's,
their digraphs conform to the generic pattern that is manifested by all digraphs of equivalence relations.
In general, a digraph of an equivalence relation falls into connected components that correspond to the
parts of the associated partition, with a "complete digraph" on the points of each part, and no other arcs.
By way of definition, a "complete digraph" is one that has all of the possible arcs on a given point set.
In the present case, the arcs of the digraphs for Con (A) and Con (B) are given as follows:
1.
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Con (A) has the structure of a SER on S, with a sling on each of the points in S,
two-way arcs between the points of the syntactic subset {"A", "i"}, and
two-way arcs between the points of the syntactic subset {"B", "u"}.
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2.
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Con (B) has the structure of a SER on S, with a sling on each of the points in S,
two-way arcs between the points of the syntactic subset {"A", "u"}, and
two-way arcs between the points of the syntactic subset {"B", "i"}.
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Taken as transition digraphs, Con (A) and Con (B) highlight the associations that are permitted
between equivalent signs, as this equivalence is judged by the interpreters A and B, respectively.
The theme running through the last two subsections, that associates different interpreters and
different aspects of interpretation with different kinds of relational structures on the same set
of points, heralds a topic that will be developed extensively in the sequel.
Document History:
Contact: <jawbrey@oakland.edu>
Version: Draft 8.2
Created: 23 Jun 1996
Revised: 30 Jun 2000
Advisor: M.A. Zohdy
Setting: Oakland University, Rochester, Michigan, USA
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