ONT Re: Sign Relations
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Sign Relation SIG,
Let us return to the "Story of A and B",
to see what fresh insights we might see.
Here is the subsection of my dissertation where I first introduce
the A & B example. It should serve to outline many of the basic
concepts that arise in this approach to pragmatic semiotics.
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| Document History:
|
| Subject: Inquiry Driven Systems: An Inquiry Into Inquiry
| Contact: Jon Awbrey <jawbrey@oakland.edu>
| Version: Draft 8.61
| Created: 23-Jun-1996
| Revised: 04-Sep-2001
| Advisor: M.A. Zohdy
| Setting: Oakland University, Rochester, Michigan, USA
| Excerpt: Subsection 1.3.4.2 (Sign Relations: A Primer)
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 easily converted to
or ultimately grounded in sign relations. This class of triadic relations
constitutes the main study of the "pragmatic theory of signs", a branch
of logical philosophy devoted to understanding all types of symbolic
representation and communication.
There is a close relationship between the pragmatic theory of signs and the
pragmatic theory of inquiry. In fact, the correspondence between the two
studies exhibits so many parallels and coincidences that it is often best
to treat them as integral parts of one and the same subject. In a very
real sense, inquiry is the process by which sign relations come to be
established and continue to evolve. In other words, inquiry, "thinking"
in its best sense, "is a term denoting the various ways in which things
acquire significance" (Dewey). Thus, there is an active and intricate
form of cooperation that needs to be appreciated and maintained between
these converging modes of investigation. Its proper character is best
understood by realizing that the theory of inquiry is adapted to study
the developmental aspects of sign relations, a subject which the theory
of signs is specialized to treat from structural and comparative points
of view.
Because the examples in this section have been artificially constructed to be
as simple as possible, their detailed elaboration can run the risk of trivializing
the whole theory of sign relations. Still, these examples have subtleties of their
own, and their careful treatment will serve to illustrate important issues in the
general theory of signs.
Imagine a discussion between two people, Ann and Bob, and attend only
to that aspect of their interpretive practice that involves the use of
the following nouns and pronouns: "Ann", "Bob", "I", "you".
The "object domain" of this discussion fragment is the set of two people {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 OxSxI. Here, O, S, and 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 OxSxI. In general, the three domains of a sign relation
can be any sets whatsoever, but the kinds of sign relations that are contemplated
in a computational framework are usually constrained to having I 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 "syntactic domain". In the forthcoming
examples, S and I are identical as sets, so the very same elements manifest themselves
in two distinct 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 the "world of L" and write W = W(L) = 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 become
more complicated, I introduce the following abbreviations:
| O = Object Domain.
| S = Sign Domain.
| I = Interpretant Domain.
|
| O = {Ann, Bob} = {A, B}.
|
| S = {"Ann", "Bob", "I", "You"} = {"A", "B", "i", "u"}.
|
| I = {"Ann", "Bob", "I", "You"} = {"A", "B", "i", "u"}.
|
| In the present examples, S = I = Syntactic Domain.
Tables 1 and 2 give the sign relations associated with the interpreters A and B,
respectively, putting them in the form of relational databases. Thus, the rows
of each Table list the ordered triples of the form <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
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.
Table 1. Sign Relation of Interpreter A
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| Object | Sign | Interpretant |
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| A | "A" | "A" |
| A | "A" | "i" |
| A | "i" | "A" |
| A | "i" | "i" |
| B | "B" | "B" |
| B | "B" | "u" |
| B | "u" | "B" |
| B | "u" | "u" |
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Table 2. Sign Relation of Interpreter B
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| Object | Sign | Interpretant |
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| A | "A" | "A" |
| A | "A" | "u" |
| A | "u" | "A" |
| A | "u" | "u" |
| B | "B" | "B" |
| B | "B" | "i" |
| B | "i" | "B" |
| B | "i" | "i" |
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One aspect of semantics is concerned with the reference that a sign has to its object,
which is called its "denotation". For signs in general, neither the existence nor the
uniqueness of a denotation is guaranteed. Thus, the denotation of a sign can refer to
a plural, a singular, or a vacuous number of objects. In the pragmatic theory of signs,
these references are formalized as certain types of dyadic relations that are obtained
by projection from the triadic sign relations.
The dyadic relation that constitutes the "denotative component" of a sign relation L
is denoted by "Den(L)". Information about the denotative component of semantics can
be derived from L by taking its "dyadic projection" on the plane that is generated
by the object and the sign domains, indicated by any one of the equivalent forms,
"Proj_OS(L)", "L_OS", or "L_12", and defined as:
Den(L) = Proj_OS(L) = L_OS = {<o, s> in OxS : <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 semantics that might be considered concern
the reference that a sign has to its interpretant and the reference that
an interpretant has to its object. As before, either type of reference
can be multiple, unique, or empty in its collection of terminal points,
and both can be formalized as different types of dyadic relations that
are obtained as planar projections of the triadic sign relations.
The connection that a sign makes to an interpretant is called its "connotation".
In the general theory of sign relations, this aspect of semantics includes the
references that a sign has to affects, concepts, impressions, intentions,
mental ideas, and to the whole realm of an agent's mental states and
allied activities, broadly encompassing intellectual associations,
emotional impressions, motivational impulses, and real conduct.
This complex ecosystem of references is unlikely ever to be
mapped in much detail, much less completely formalized, but
the tangible warp of its accumulated mass is commonly alluded
to as the "connotative" import of language. Given a particular
sign relation L, the dyadic relation that constitutes the
"connotative component" of L is denoted by "Con(L)".
The bearing that an interpretant has toward a common object of its sign and itself
has no standard name. If an interpretant is considered to be a sign in its own right,
then its independent reference to an object can be taken as belonging to another moment
of denotation, but this omits the mediational character of the whole transaction. Given
the service that interpretants supply in furnishing a locus for critical, reflective, and
explanatory glosses on objective scenes and their descriptive texts, it is easy to regard
them as "annotations" 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 L, the dyadic relation
that constitutes the "intentional component" of L is denoted by "Int(L)".
A full consideration of the connotative and intentional aspects of semantics
would force a return to difficult questions about the true nature of the
interpretant sign in the general theory of sign relations. It is best
to defer these issues to a later discussion. Fortunately, omission
of this material does not interfere with understanding the purely
formal aspects of the present example.
Formally, these new aspects of semantics present no additional problem:
The connotative component of a sign relation L can be formalized as its
dyadic projection on the sign and interpretant domains, defined as follows:
Con(L) = Proj_SI(L) = L_SI = {<s, i> in SxI : <o, s, i> in L for some o in O}.
The intentional component of semantics for a sign relation L, or
its "second moment of denotation", is adequately captured by its
dyadic projection on the object and interpretant domains, defined
as follows:
Int(L) = Proj_OI(L) = L_OI = {<o, i> in OxI : <o, s, i> in L for some s in S}.
As it happens, the sign relations A and B in the present example are
fully symmetric with respect to exchanging signs and interpretants, so
all of the structure of A_OS and B_OS is merely echoed in A_OI and B_OI,
respectively.
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,
the relationship between the connotational and the denotational aspects of meaning
is not wholly arbitrary. Instead, this relationship must be naturally constrained
or deliberately designed in such a way that it:
1. Represents the embodiment of significant properties
that have objective reality in the agent's domain.
2. Supports the achievement of particular purposes
that have intentional value for the agent.
Therefore, my attention is directed mainly 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.
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