Sign Relations • Ennotation

Posted on December 29, 2025 by Jon Awbrey

A third aspect of a sign’s complete meaning concerns the relation between its objects and its interpretants, which has no standard name in semiotics.  It would be called an induced relation in graph theory or the result of relational composition in relation theory.  If an interpretant is recognized as 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 neglects the mediational character of the whole transaction in which this occurs.  Denotation and connotation have to do with dyadic relations in which the sign plays an active role but here we are dealing with a dyadic relation between objects and interpretants mediated by the sign from an off‑stage position, as it were.

As a relation between objects and interpretants mediated by a sign, this third aspect of meaning may be referred to as the ennotation of a sign and the dyadic relation making up the ennotative aspect of a sign relation L may be notated as \mathrm{Enn}(L).  Information about the ennotative aspect of meaning is obtained from L by taking its projection on the object‑interpretant plane and visualized as the “shadow” L casts on the 2‑dimensional space whose axes are the object domain O and the interpretant domain I.  The ennotative component of a sign relation L, variously written as \mathrm{proj}_{OI} L,  L_{OI},  \mathrm{proj}_{13} L,  or L_{13}, is defined as follows.

Display 5

As it happens, the sign relations L_\mathrm{A} and L_\mathrm{B} are fully symmetric with respect to exchanging signs and interpretants, so all the data of \mathrm{proj}_{OS} L_\mathrm{A} is echoed unchanged in \mathrm{proj}_{OI} L_\mathrm{A} and all the data of \mathrm{proj}_{OS} L_\mathrm{B} is echoed unchanged in \mathrm{proj}_{OI} L_\mathrm{B}.

Tables 5a and 5b show the ennotative components of the sign relations associated with the interpreters \mathrm{A} and \mathrm{B}, respectively.  The rows of each Table list the ordered pairs (o, i) in the corresponding projections, \mathrm{Enn}(L_\mathrm{A}), \mathrm{Enn}(L_\mathrm{B}) \subseteq O \times I.

Ennotative Components Enn(L_A) and Enn(L_B)

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Sign Relations • Connotation

Posted on December 28, 2025 by Jon Awbrey

Another aspect of a sign’s complete meaning concerns the reference a sign has to its interpretants, which interpretants are collectively known as the connotation of the sign.  In the pragmatic theory of sign relations, connotative references fall within the projection of the sign relation on the plane spanned by its sign domain and its interpretant domain.

In the full theory of sign relations the connotative aspect of meaning includes the links a sign has to affects, concepts, ideas, impressions, intentions, and the whole realm of an interpretive agent’s mental states and allied activities, broadly encompassing intellectual associations, emotional impressions, motivational impulses, and real conduct.  Taken at the full, in the natural setting of semiotic phenomena, this complex system of references is unlikely ever to find itself 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.

Formally speaking, however, the connotative aspect of meaning presents no additional difficulty.  The dyadic relation making up the connotative aspect of a sign relation L is notated as \mathrm{Con}(L).  Information about the connotative aspect of meaning is obtained from L by taking its projection on the sign‑interpretant plane and visualized as the “shadow” L casts on the 2‑dimensional space whose axes are the sign domain S and the interpretant domain I.  The connotative component of a sign relation L, variously written as \mathrm{proj}_{SI} L,  L_{SI},  \mathrm{proj}_{23} L,  or L_{23}, is defined as follows.

Display 4

Tables 4a and 4b show the connotative components of the sign relations associated with the interpreters \mathrm{A} and \mathrm{B}, respectively.  The rows of each Table list the ordered pairs (s, i) in the corresponding projections, \mathrm{Con}(L_\mathrm{A}), \mathrm{Con}(L_\mathrm{B}) \subseteq S \times I.

Connotative Components Con(L_A) and Con(L_B)

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Sign Relations • Denotation

Posted on December 27, 2025 by Jon Awbrey

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 L is notated as \mathrm{Den}(L).  Information about the denotative aspect of meaning is obtained from L by taking its projection on the object‑sign plane.  The result may be visualized as the “shadow” L casts on the 2‑dimensional space whose axes are the object domain O and the sign domain S.  The denotative component of a sign relation L, variously written as \mathrm{proj}_{OS} L,  L_{OS},  \mathrm{proj}_{12} L,  or L_{12}, is defined as follows.

Display 3

Tables 3a and 3b show the denotative components of the sign relations associated with the interpreters \mathrm{A} and \mathrm{B}, respectively.  The rows of each Table list the ordered pairs (o, s) in the corresponding projections, \mathrm{Den}(L_\mathrm{A}), \mathrm{Den}(L_\mathrm{B}) \subseteq O \times S.

Denotative Components Den(L_A) and Den(L_B)

Looking to the denotative aspects of L_\mathrm{A} and L_\mathrm{B}, various rows of the Tables specify, for example, that \mathrm{A} uses ``\text{i}" to denote \mathrm{A} and ``\text{u}" to denote \mathrm{B}, while \mathrm{B} uses ``\text{i}" to denote \mathrm{B} and ``\text{u}" to denote \mathrm{A}.

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Sign Relations • Dyadic Aspects

Posted on December 25, 2025 by Jon Awbrey

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

\text{Table 2. Dyadic Aspects of Triadic Relations}

Dyadic Aspects of Triadic Relations

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 L on the OS‑plane O \times S is written briefly as L_{OS} or written more fully as \mathrm{proj}_{OS}(L) and is defined as the set of all ordered pairs (o, s) in the cartesian product O \times S for which there exists an ordered triple (o, s, i) in L for some element i in the set I.

In the case where L is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of L 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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Sign Relations • Examples

Posted on December 18, 2025 by Jon Awbrey

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.

\{ ``\text{Ann}", ``\text{Bob}", ``\text{I}", ``\text{you}" \}

  • The object domain of their discussion is the set of two people \{ \text{Ann}, \text{Bob} \}.
  • The sign domain of their discussion is the set of four signs \{ ``\text{Ann}", ``\text{Bob}", ``\text{I}", ``\text{you}" \}.

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 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 L is a subset of a cartesian product O \times S \times I.  The three sets O, S, I 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.

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 I \subseteq S.  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 S and I 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 O, S, I for a given sign relation L, we will call this set the World of L and write W = W_L = O \cup S \cup I.

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.

Display 1

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

Display 2

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

Tables 1a and 1b show the sign relations associated with the interpreters \mathrm{A} and \mathrm{B}, respectively.  In this arrangement the rows of each Table list the ordered triples of the form (o, s, i) belonging to the corresponding sign relations, L_\mathrm{A}, L_\mathrm{B} \subseteq O \times S \times I.

Sign Relation Twin Tables LA & LB

The Tables codify a rudimentary level of interpretive practice for the agents \mathrm{A} and \mathrm{B} 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 (o, s, i) 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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Sign Relations • Signs and Inquiry

Posted on December 16, 2025 by Jon Awbrey

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 the 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, a subject 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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Sign Relations • Definition

Posted on December 14, 2025 by Jon Awbrey

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, New Elements of Mathematics, vol. 4, 20–21

In the general discussion of diverse theories of signs, the question 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 may be said to have only within a particular context of relationships.

Peirce’s definition of a sign defines it in relation to its objects and its interpretant signs, and thus defines signhood in relative terms, by means of a predicate with three places.  In that definition, signhood is a role in a triadic relation, a role a thing bears or plays in a determinate context of relationships — it is not an absolute or non‑relative property of a thing‑in‑itself, one it possesses 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 the term throughout his work, it is clear he means what he elsewhere calls a “triple correspondence”, and thus this is just another way of referring to the whole triadic sign relation itself.  In particular, his use of the term should not be taken to imply a dyadic correspondence, like the kinds 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 directions than the sense of the word referring to strictly deterministic causal‑temporal processes.  First, and especially in this context, he is invoking a more general concept of determination, what is called a formal or informational determination, as in saying “two points determine a line”, rather than the more special cases of causal and temporal determinisms.  Second, he characteristically allows for what is called 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 have occasion to view the same data, as logic is a normative science where psychology is a descriptive science, and so they have very different 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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Sign Relations • Anthesis

Posted on December 12, 2025 by Jon Awbrey

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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Cactus Language • Semantics 8

Posted on November 1, 2025 by Jon Awbrey

The 16 boolean functions on two variables F^{(2)} : \mathbb{B}^2 \to \mathbb{B} are shown in the following Table.

\text{Boolean Functions on Two Variables}
Boolean Functions on Two Variables

As before, all boolean functions on proper subsets of the current variables are subsumed in the Table at hand.  In particular, we have the following inclusions.

  • The constant function 0 ~:~ \mathbb{B}^2 \to \mathbb{B} appears under the name F_{0}^{(2)}.
  • The constant function 1 ~:~ \mathbb{B}^2 \to \mathbb{B} appears under the name F_{15}^{(2)}.
  • The function expressing the assertion of the first variable is F_{12}^{(2)}.
  • The function expressing the negation of the first variable is F_{3}^{(2)}.
  • The function expressing the assertion of the second variable is F_{10}^{(2)}.
  • The function expressing the negation of the second variable is F_{5}^{(2)}.

Next come the functions on two variables whose output values change depending on changes in both input variables.  Notable among them are the following examples.

  • The logical conjunction is given by the function F_{8}^{(2)} (x, y) ~=~ x \cdot y.
  • The logical disjunction is given by the function F_{14}^{(2)} (x, y) ~=~ \texttt{((} ~x~ \texttt{)(} ~y~ \texttt{))}.

Functions expressing the conditionals, implications, or if‑then statements appear as follows.

  • [x \Rightarrow y] ~=~ F_{11}^{(2)} (x, y) ~=~ \texttt{(} ~x~ \texttt{(} ~y~ \texttt{))} ~=~ [\mathrm{not}~ x ~\mathrm{without}~ y].
  • [x \Leftarrow y] ~=~ F_{13}^{(2)} (x, y) ~=~ \texttt{((} ~x~ \texttt{)} ~y~ \texttt{)} ~=~ [\mathrm{not}~ y ~\mathrm{without}~ x].

The function expressing the biconditional, equivalence, or if‑and‑only‑if statement appears in the following form.

  • [x \Leftrightarrow y] ~=~ [x = y] ~=~ F_{9}^{(2)} (x, y) ~=~ \texttt{((} ~x~ \texttt{,} ~y~ \texttt{))}.

Finally, the boolean function expressing the exclusive disjunction, inequivalence, or not equals statement, algebraically associated with the binary sum operation, and geometrically associated with the symmetric difference of sets, appears as follows.

  • [x \neq y] ~=~ [x + y] ~=~ F_{6}^{(2)} (x, y) ~=~ \texttt{(} ~x~ \texttt{,} ~y~ \texttt{)}.

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Cactus Language • Semantics 7

Posted on October 30, 2025 by Jon Awbrey

A good way to illustrate the action of the conjunction and surjunction operators is to show how they can be used to construct the boolean functions on any finite number of variables.  Though it’s not much to look at let’s start with the case of zero variables, boolean constants by any other word, partly for completeness and partly to supply an anchor for the cases in its train.

A boolean function F^{(0)} on zero variables is just an element of the boolean domain \mathbb{B} = \{ 0, 1 \}.  The following Table shows several ways of referring to those elements, for the sake of consistency using the same format we’ll use in subsequent Tables, however degenerate it appears in this case.

\text{Boolean Functions on Zero Variables}
Boolean Functions on Zero Variables

  • Column 1 lists each boolean element or boolean function under its ordinary constant name or under a succinct nickname, respectively.
  • Column 2 lists each boolean function by means of a function name F_j^{(k)} of the following form.  The superscript (k) gives the dimension of the functional domain, in effect, the number of variables, and the subscript j is a binary string formed from the functional values, using the obvious coding of boolean values into binary values.
  • Column 3 lists the values each function takes for each combination of its domain values.
  • Column 4 lists the ordinary cactus expressions for each boolean function.  Here, as usual, the expression ``\texttt{(( ))}" renders the blank expression for logical truth more visible in context.

The next Table shows the four boolean functions on one variable, F^{(1)} : \mathbb{B} \to \mathbb{B}.

\text{Boolean Functions on One Variable}
Boolean Functions on One Variable

  • Column 1 lists the contents of Column 2 in a more concise form, converting the lists of boolean values in the subscript strings to their decimal equivalents.  Naturally, the boolean constants reprise themselves in this new setting as constant functions on one variable.  The constant functions are thus expressible in the following equivalent ways.

\begin{matrix} F_0^{(1)} & = & F_{00}^{(1)} & = & 0 ~:~ \mathbb{B} \to \mathbb{B}. \\[4pt] F_3^{(1)} & = & F_{11}^{(1)} & = & 1 ~:~ \mathbb{B} \to \mathbb{B}. \end{matrix}

  • The other two functions in the Table are easily recognized as the one‑place logical connectives or the monadic operators on \mathbb{B}.  Thus the function F_1^{(1)} = F_{01}^{(1)} is recognizable as the negation operation and the function F_2^{(1)} = F_{10}^{(1)} is obviously the identity operation.

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