Inquiry Into Inquiry

One aspect of a sign’s complete meaning concerns the reference a sign has to its objects, which objects are collectively known as the denotation of the sign.  In the pragmatic theory of sign relations, denotative references fall within the projection of the sign relation on the plane spanned by its object domain and its sign domain.

The dyadic relation making up the denotative, referent, or semantic aspect of a sign relation is notated as   Information about the denotative aspect of meaning is obtained from by taking its projection on the object‑sign plane.  We may visualize this as the “shadow” casts on the 2‑dimensional space whose axes are the object domain and the sign domain   The denotative component of a sign relation variously written in any of the forms,       and is defined as follows.

Tables 3a and 3b show the denotative components of the sign relations associated with the interpreters and respectively.  The rows of each Table list the ordered pairs in the corresponding projections,

Looking to the denotative aspects of and various rows of the Tables specify, for example, that uses to denote and to denote while uses to denote and to denote

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For an arbitrary triadic relation whether it happens to be a sign relation or not, there are six dyadic relations obtained by projecting on one of the planes of the -space   The six dyadic projections of a triadic relation are defined and notated as shown in Table 2.

By way of unpacking the set‑theoretic notation, here is what the first definition says in ordinary language.

The dyadic relation resulting from the projection of on the -plane is written briefly as or written more fully as and is defined as the set of all ordered pairs in the cartesian product for which there exists an ordered triple in for some element in the set

In the case where is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of can be recognized as formalizing aspects of sign meaning which have received their share of attention from students of signs over the centuries, and thus they can be associated with traditional concepts and terminology.

Of course, traditions vary with respect to the precise formation and usage of such concepts and terms.  Other aspects of meaning have not received their fair share of attention and thus remain innominate in current anatomies of sign relations.

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Soon after I made my third foray into grad school, this time in Systems Engineering, I was trying to explain sign relations to my advisor and he — being the very model of a modern systems engineer — asked me to give a concrete example of a sign relation, as simple as possible without being trivial.  After much cudgeling of the grey matter I came up with a pair of examples which had the added benefit of bearing instructive relationships to each other.  Despite their simplicity, the examples to follow have subtleties of their own and their careful treatment serves to illustrate important issues in the general theory of signs.

Imagine a discussion between two people, Ann and Bob, and attend only to the aspects of their interpretive practice involving the use of the following nouns and pronouns.

“Ann”,   “Bob”,   “I”,   “you”.

• The object domain of their discussion is the set of two people
• The sign domain of their discussion is the set of four signs

Ann and Bob are not only the passive objects of linguistic 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 known as the sign relation of that interpreter.

In terms of its set-theoretic extension, a sign relation is a subset of a cartesian product   The three sets are known as the object domain, the sign domain, and the interpretant domain, respectively, of the sign relation

Broadly speaking, the three domains of a sign relation may be any sets at all but the types of sign relations contemplated in formal settings are usually constrained to having   In those cases it becomes convenient to lump signs and interpretants together in a single class called a sign system or syntactic domain.  In the forthcoming examples and are identical as sets, so the same elements manifest themselves in two different roles of the sign relations in question.

When it becomes necessary to refer to the whole set of objects and signs in the union of the domains for a given sign relation we will call this set the World of and write

To facilitate an interest in the formal structures of sign relations and to keep notations as simple as possible as the examples become more complicated, it serves to introduce the following general notations.

Introducing a few abbreviations for use in this Example, we have the following data.

In the present example,

Tables 1a and 1b show the sign relations associated with the interpreters and respectively.  In this arrangement the rows of each Table list the ordered triples of the form belonging to the corresponding sign relations,

The Tables codify a rudimentary level of interpretive practice for the agents and and provide a basis for formalizing the initial semantics appropriate to their common syntactic domain.  Each row of a Table lists an object and two co-referent signs, together forming an ordered triple called an elementary sign relation, in other words, one element of the relation’s set-theoretic extension.

Already in this elementary context, there are several meanings which might attach to the project of a formal semiotics, or a formal theory of meaning for signs.  In the process of discussing the alternatives, it is useful to introduce a few terms occasionally used in the philosophy of language to point out the needed distinctions.  That is the task we’ll turn to next.

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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 congruences and parallels 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, 38).

Tracing the passage of inquiry through the medium of signs calls for an active, intricate form of cooperation between our converging modes of investigation.  Its proper character is best understood by realizing the theory of inquiry is adapted to study the developmental aspects of sign relations, whose evolution the theory of signs is specialized to treat from comparative and structural points of view.

References

• Dewey, J. (1910), How We Think, D.C. Heath, Boston, MA.  Reprinted (1991), Prometheus Books, Buffalo, NY.  Online.
• Awbrey, J.L., and Awbrey, S.M. (1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  Archive.  Journal.  Online (doc) (pdf).

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One of Peirce’s clearest and most complete definitions of a sign is one he gives in the context of providing a definition for logic, and so it is informative to view it in that setting.

Logic will here be defined as formal semiotic.  A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time.  Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C.  It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic.  I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has virtually been quite generally held, though not generally recognized.  (C.S. Peirce, NEM 4, 20–21).

In the general discussion of diverse theories of signs, the question frequently arises whether signhood is an absolute, essential, indelible, or ontological property of a thing, or whether it is a relational, interpretive, and mutable role a thing can be said to have only within a particular context of relationships.

Peirce’s definition of a sign defines it in relation to its object and its interpretant sign, and thus defines signhood in relative terms, by means of a predicate with three places.  In this definition, signhood is a role in a triadic relation, a role a thing bears or plays in a given context of relationships — it is not an absolute, non‑relative property of a thing‑in‑itself, a status it maintains independently of all relationships to other things.

Some of the terms Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader.

• Correspondence.  From the way Peirce uses this term throughout his work it is clear he means what he elsewhere calls a “triple correspondence”, in short, just another way of referring to the whole triadic sign relation itself.  In particular, his use of this term should not be taken to imply a dyadic correspondence, as in the varieties of “mirror image” correspondence between realities and representations bandied about in contemporary controversies about “correspondence theories of truth”.

• Determination.  Peirce’s concept of determination is broader in several ways than the sense of the word referring to strictly deterministic causal‑temporal processes.  First, and especially in this context of defining logic, he uses a more general concept of determination, what is known as formal or informational determination, as we use in geometry when we say “two points determine a line”, rather than the more special cases of causal or temporal determinisms.  Second, he characteristically allows for the broader concept of determination in measure, that is, an order of determinism admitting a full spectrum of more and less determined relationships.

• Non‑Psychological.  Peirce’s “non‑psychological conception of logic” must be distinguished from any variety of anti‑psychologism.  He was quite interested in matters of psychology and had much of import to say about them.  But logic and psychology operate on different planes of study even when they happen to view the same data, as logic is a normative science where psychology is a descriptive science.  Thus they have distinct aims, methods, and rationales.

Reference

• Peirce, C.S. (1902), “Parts of Carnegie Application” (L 75), in Carolyn Eisele (ed., 1976), The New Elements of Mathematics by Charles S. Peirce, vol. 4, 13–73.  Online.

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Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun.

— C.S. Peirce, Collected Papers, CP 2.274

In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or semiosis, Peirce uses the technical term representamen for his concept of a sign, but the shorter word is precise enough, so long as one recognizes its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign.

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