Directory talk:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1

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1.3.4.2. Sign Relations : A Primer

To the extent that their structures and functions can be discussed at all, it is likely that all of the formal entities that are 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 \(\{ \text{Ann}, \text{Bob} \}\).

• The syntactic domain or the sign system of their discussion is limited to the set of four signs \(\{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}\).

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 \(O \times S \times I\). 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 \(L \subseteq O \times S \times I\).

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 \(I \subseteq S\). 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 very same elements manifest themselves in two different roles of the sign relations in question. When it is necessary to refer to the whole set of objects and signs in the union of the domains \(O\!\), \(S\!\), \(I\!\) for a given sign relation \(L\!\), one may refer to this set as the World of \(L\!\) and write \(W = W_L = O \cup S \cup I\).

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, it serves to introduce the following general notations:

\(\begin{array}{ccl} O & = & \text{Object Domain} \\[6pt] S & = & \text{Sign Domain} \\[6pt] I & = & \text{Interpretant Domain} \end{array}\)

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

\(\begin{array}{cclcl} O & = & \{ \text{Ann}, \text{Bob} \} & = & \{ \text{A}, \text{B} \} \\[6pt] S & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \\[6pt] I & = & \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \} & = & \{ {}^{\backprime\backprime} \text{A} {}^{\prime\prime}, {}^{\backprime\backprime} \text{B} {}^{\prime\prime}, {}^{\backprime\backprime} \text{i} {}^{\prime\prime}, {}^{\backprime\backprime} \text{u} {}^{\prime\prime} \} \end{array}\)

In the present example, \(S = I = \text{Syntactic Domain}\).

The sign relation associated with a given interpreter J is denoted L_J  or L(J). 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  ⊆ O × S × I. It is often tempting to use the same names for objects and for relations involving these objects, but it is best to avoid this in a first approach, taking up the issues that this practice raises after the less problematic features of these relations have been treated.

Table 1. Sign Relation of Interpreter A

Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"i"
A	"i"	"A"
A	"i"	"i"
B	"B"	"B"
B	"B"	"u"
B	"u"	"B"
B	"u"	"u"

Table 2. Sign Relation of Interpreter B

Object	Sign	Interpretant
A	"A"	"A"
A	"A"	"u"
A	"u"	"A"
A	"u"	"u"
B	"B"	"B"
B	"B"	"i"
B	"i"	"B"
B	"i"	"i"

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 semiotics, or a formal theory of meaning for signs. In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions.

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 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 domain and the sign domain, indicated by any one of the equivalent forms, Proj_OS L, L_OS , or L₁₂ , and defined as follows:

Den(L) = Proj_OS L = L_OS = {‹o, s› ∈ O × S : ‹o, s, i› ∈ L for some i ∈ 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 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 both of objects and of signs, but this function points in the opposite direction to what is needed in this connection. What does one call the inverse of the annotation function? More generally asked, what is the converse of the annotation relation?

In light of these considerations, I find myself still experimenting with terms to suit this last-mentioned dimension of semantics. On a trial basis, I refer to it as the ideational, the intentional, or the canonical component of the sign relation, and I provisionally refer to the reference of an interpretant sign to its object as its ideation, its intention, or its conation. Given a particular sign relation L, the dyadic relation that constitutes the intentional component of L is denoted 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 plane generated by the sign domain and the interpretant domain, defined as follows:

Con(L) = Proj_SI L = L_SI = {‹s, i› ∈ S × I : ‹o, s, i› ∈ L for some o ∈ 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 plane generated by the object domain and interpretant domain, defined as follows:

Int(L) = Proj_OI L = L_OI = {‹o, i› ∈ O × I : ‹o, s, i› ∈ L for some s ∈ 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 structure of (L_A)_OS  and (L_B)_OS  is merely echoed in (L_A)_OI  and (L_B)_OI , respectively.

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

JOS	=	Den(LJ )	=	ProjOS LJ	=	(LJ )OS	=	L(J)OS
JSI	=	Con(LJ )	=	ProjSI LJ	=	(LJ )SI	=	L(J)SI
JOI	=	Int(LJ )	=	ProjOI LJ	=	(LJ )OI	=	L(J)OI

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.