## Sign Relations • Definition

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, 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

• Charles S. Peirce (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.

### Document History

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.

cc: CyberneticsOntolog • Peirce List (1) (2)Structural ModelingSystems Science

## Triadic Relations • Discussion 2

To everything there is a season,
A time for every purpose under heaven:
A time for building castles in the stratosphere,
A time to mind the anti-gravs that keep us here.

RY:
The aspiration to a form of knowledge ‘wisdom’ resonated with me, and, not withstanding the ‘university’ context (connotative?) the article was couched in, every time I read the word ‘university’, I mentally substituted it with ‘systems movement’ and the resonance was there.

Dear Rob,

Thanks for your comments and questions.  They took me back to the decade before the turn of the millennium when there was a general trend of thought to embrace chaos and complexity, seeking the order and simplicity on the other side.  (Oliver Wendell Holmes, but it appears in doubt whether Sr. or Jr.)

One thing I’ve learned in the mean time is just how poorly grounded and maintained are many of the abstract concepts and theories we need for grappling with the complexities of communication, computation, experimental information, and scientific inquiry.  So I’ve been doing what I can to reinforce the concrete bases and stabilize the working platforms of what otherwise tend to become empty à priori haunts.

I’ll have to be getting back to that.  For now I’ll just link to a few readings your remarks brought to mind.  The “Conceptual Barriers” paper from 2001 is the journal upgrade of a conference presentation from 1999, “Organizations of Learning or Learning Organizations : The Challenge of Creating Integrative Universities for the Next Century”.

Your reflex of jumping from universities to systems in general is very much on the mark.  Our work was motivated in large part by the movement toward Learning Organizations, that is, organizations able to apply organizational research to their own organizational development.  To put a fine point on it, all we are saying is, “Shouldn’t a University, as an Organization of Learning, also be a Learning Organization?”

Well, I had a lot more to relate at this point, but our dishwasher just went on the fritz, so I’ll leave it for now with a few links to Susan’s earlier work along these lines and try to get back to it later …

• Scott, David K., and Awbrey, Susan M. (1993), “Transforming Scholarship”, Change : The Magazine of Higher Learning, 25(4), 38–43. Online (1) (2).
• Papers by Susan Awbrey and David Scott • University of Massachusetts, Amherst.

### Reference

• Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”, Organization : The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, pp. 269–284. Abstract. Online.

## Sign Relations • Discussion 1

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

I’m working at reviewing and revising some pieces I’ve rewritten two score times over the last … lost count of years … and that bit from Peirce is one of my favorite epigraphs for the work ahead.  But I take it as an allegorical figure whose purpose is to illustrate a certain form of relation, and not to be taken too literally.  So I’m sympathetic to the reactions of several readers who find it clangs a bit if taken at face value.  I think there are clues in the passage, the hypothetical subjunctive construction, the unnatural qualification, “without further condition”, etc., telling us Peirce did not intend it as a truth of botany.  But taken rightly it does point to the shape of a proper definition to come.  So I’ll be getting to that …

### Resources

cc: CyberneticsOntolog • Peirce List (1) (2)Structural ModelingSystems Science

## Sign Relations • Anthesis

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.

### Resources

cc: CyberneticsOntolog • Peirce List (1) (2)Structural ModelingSystems Science

## Triadic Relations • Discussion 1

Loet Leydesdorff mentioned making extensive use of triads in a new paper.

• Leydesdorff, Loet, and Ivanova, Inga (2020), “The Measurement of ‘Interdisciplinarity’ and ‘Synergy’ in Scientific and Extra-Scientific Collaborations”. Online (1) (2).

Just off-hand this looks like the right ballpark for my long run interests but it will take me a few more posts just dusting off home plate and clearing the base lines.

Here’s a paper Susan Awbrey and I wrote a while back giving some hint of the Big Game in play here, the “scholarship of integration” needed to bring the harvests of information locked away in so many isolated silos to bear on our world of common problems.

• Awbrey, S.M., and Awbrey, J.L. (2001), “Conceptual Barriers to Creating Integrative Universities”, Organization : The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, pp. 269–284. Abstract. Online.

### Examples from Semiotics

The study of signs — the full variety of significant forms of expression — in relation to all the affairs signs are significant of, and in relation to all the beings signs are significant to, is known as semiotics or the theory of signs.  As described, semiotics treats of a 3-place relation among signs, their objects, and their interpreters.

The term semiosis refers to any activity or process involving signs.  Studies of semiosis focusing on its abstract form are not concerned with every concrete detail of the entities acting as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among those three roles.  In particular, the formal theory of signs does not consider all the properties of the interpretive agent but only the more striking features of the impressions signs make on a representative interpreter.  From a formal point of view this impact or influence may be treated as just another sign, called the interpretant sign, or the interpretant for short.  A triadic relation of this type, among objects, signs, and interpretants, is called a sign relation.

For example, consider the aspects of sign use involved when two people, say Ann and Bob, use their own proper names, “Ann” and “Bob”, along with the pronouns, “I” and “you”, to refer to themselves and each other.  For brevity, these four signs may be abbreviated to the set $\{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.$  The abstract consideration of how $\mathrm{A}$ and $\mathrm{B}$ use this set of signs leads to the contemplation of a pair of triadic relations, the sign relations $L_\mathrm{A}$ and $L_\mathrm{B},$ reflecting the differential use of these signs by $\mathrm{A}$ and $\mathrm{B},$ respectively.

Each of the sign relations $L_\mathrm{A}$ and $L_\mathrm{B}$ consists of eight triples of the form $(x, y, z),$ where the object $x$ belongs to the object domain $O = \{ \mathrm{A}, \mathrm{B} \},$ the sign $y$ belongs to the sign domain $S,$ the interpretant sign $z$ belongs to the interpretant domain $I,$ and where it happens in this case that $S = I = \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.$  The union $S \cup I$ is often referred to as the syntactic domain, but in this case $S = I = S \cup I.$

The set-up so far is summarized as follows:

$\begin{array}{ccc} L_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I \\[8pt] O & = & \{ \mathrm{A}, \mathrm{B} \} \\[8pt] S & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\[8pt] I & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \end{array}$

The relation $L_\mathrm{A}$ is the following set of eight triples in $O \times S \times I.$

$\begin{array}{cccccc} \{ & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & \\ & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}) & \}. \end{array}$

The triples in $L_\mathrm{A}$ represent the way interpreter $\mathrm{A}$ uses signs.  For example, the presence of $( \mathrm{B}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} )$ in $L_\mathrm{A}$ says $\mathrm{A}$ uses ${}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}$ to mean the same thing $\mathrm{A}$ uses ${}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}$ to mean, namely, $\mathrm{B}.$

The relation $L_\mathrm{B}$ is the following set of eight triples in $O \times S \times I.$

$\begin{array}{cccccc} \{ & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & \\ & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}) & \}. \end{array}$

The triples in $L_\mathrm{B}$ represent the way interpreter $\mathrm{B}$ uses signs.  For example, the presence of $( \mathrm{B}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} )$ in $L_\mathrm{B}$ says $\mathrm{B}$ uses ${}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}$ to mean the same thing $\mathrm{B}$ uses ${}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}$ to mean, namely, $\mathrm{B}.$

The triples in the relations $L_\mathrm{A}$ and $L_\mathrm{B}$ are conveniently arranged in the form of relational data tables, as shown below.

### Document History

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.

### Examples from Mathematics

For the sake of topics to be taken up later, it is useful to examine a pair of triadic relations in tandem.  We will construct two triadic relations, $L_0$ and $L_1,$ each of which is a subset of the same cartesian product $X \times Y \times Z.$  The structures of $L_0$ and $L_1$ can be described in the following way.

Each space $X, Y, Z$ is isomorphic to the boolean domain $\mathbb{B} = \{ 0, 1 \}$ so $L_0$ and $L_1$ are subsets of the cartesian power $\mathbb{B} \times \mathbb{B} \times \mathbb{B}$ or the boolean cube $\mathbb{B}^3.$

The operation of boolean addition, $+ : \mathbb{B} \times \mathbb{B} \to \mathbb{B},$ is equivalent to addition modulo 2, where $0$ acts in the usual manner but $1 + 1 = 0.$  In its logical interpretation, the plus sign can be used to indicate either the boolean operation of exclusive disjunction or the boolean relation of logical inequality.

The relation $L_0$ is defined by the following formula.

$L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.$

The relation $L_0$ is the following set of four triples in $\mathbb{B}^3.$

$L_0 ~=~ \{ ~ (0, 0, 0), ~ (0, 1, 1), ~ (1, 0, 1), ~ (1, 1, 0) ~ \}.$

The relation $L_1$ is defined by the following formula.

$L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.$

The relation $L_1$ is the following set of four triples in $\mathbb{B}^3.$

$L_1 ~=~ \{ ~ (0, 0, 1), ~ (0, 1, 0), ~ (1, 0, 0), ~ (1, 1, 1) ~ \}.$

The triples in the relations $L_0$ and $L_1$ are conveniently arranged in the form of relational data tables, as shown below.

### Document History

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.