Jon Awbrey 4:36 pm on April 21, 2026
Tags: Arithmetization ( 6 ), C.S. Peirce ( 49 ), Gödel Numbers ( 6 ), Higher Order Sign Relations ( 6 ), Inquiry Driven Systems ( 33 ), Inquiry Into Inquiry ( 9 ), logic ( 51 ), Mathematics ( 50 ), Quotation ( 6 ), Recursion ( 7 ), reflection ( 7 ), Reflective Interpretive Frameworks ( 6 ), Semiotics, Sign Relations ( 24 ), Triadic Relations ( 23 ), Use and Mention ( 6 ), Visualization ( 36 )

Reflection On Recursion • Discussion 1

Re: Reflection On Recursion • 1
Re: Laws of Form • John Mingers

JM:: This is a very important and interesting topic. I think you should consider the relationship to self‑reference, indeed are they really the same thing?
Also the work of Maturana and Varela on autopoiesis and the neurophysiology of cognition which also has recursion at its heart.

Thanks, John. Yes, we certainly find the whole array of self concepts coming into play here — selfhood, autopoiesis or self creation, self reference and self transformation, just to name a few. But one thing I need to emphasize from the start is how radically different such concepts appear when viewed in the x‑ray vision of Peirce’s pragmatic semiotics.

I forget where I first heard it, but it’s fairly common observation that the persistence of a recurring problem is a symptom of how unlikely it is to be solved in the paradigm where it keeps occurring.

After a while, it simply becomes time to change the paradigm …

Just by way of a first example, take the very idea of “self‑reference”. The moment we place it in the medium of triadic sign relations we realize signs do not refer to anything at all except insofar as an interpreter refers them.

And when we ask, “What is this, that we call an interpreter?”, the pragmatic theory of signs tells us we cannot tell when we turn out the light but under the x‑ray of the pragmatic maxim the sum of its effects is effectively modeled by an extended triadic sign relation.

Everything I’ll be working at here will be done within a framework like that.

Regards,
Jon

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Jon Awbrey 10:10 pm on April 18, 2026
Tags: Arithmetization ( 6 ), C.S. Peirce ( 49 ), Gödel Numbers ( 6 ), Higher Order Sign Relations ( 6 ), Inquiry Driven Systems ( 33 ), Inquiry Into Inquiry ( 9 ), logic ( 51 ), Mathematics ( 50 ), Quotation ( 6 ), Recursion ( 7 ), reflection ( 7 ), Reflective Interpretive Frameworks ( 6 ), Semiotics, Sign Relations ( 24 ), Triadic Relations ( 23 ), Use and Mention ( 6 ), Visualization ( 36 )

Reflection On Recursion • 4

A feature of special note in the recursion diagram is the function traversing the square from one triadic node to the other. It preserves an image of the object $n$ all the while its precedent $p(n)$ is being retrieved and processed — thus it injects a measure of parallel process and a modicum of extra memory over and above that afforded by the serial composition of functions.

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Jon Awbrey 8:30 am on April 14, 2026
Tags: Arithmetization ( 6 ), C.S. Peirce ( 49 ), Gödel Numbers ( 6 ), Higher Order Sign Relations ( 6 ), Inquiry Driven Systems ( 33 ), Inquiry Into Inquiry ( 9 ), logic ( 51 ), Mathematics ( 50 ), Quotation ( 6 ), Recursion ( 7 ), reflection ( 7 ), Reflective Interpretive Frameworks ( 6 ), Semiotics, Sign Relations ( 24 ), Triadic Relations ( 23 ), Use and Mention ( 6 ), Visualization ( 36 )

Reflection On Recursion • 3

One other feature of syntactic recursion deserves to be brought into higher relief. Evidence of it can be found in the recursion diagram by examining the places where three paths meet. On the descending side there is the point where three paths diverge. On the ascending side there is the point where the middlemost of the three divergent paths joins the upshot arrow in medias res.

The arrows of the diagram represent functions, a species of dyadic relations, but nodes of degree three signify aspects of triadic relations somewhere in the mix.

The three arrows from the initial node represent a function $F : \mathbb{N} \to \mathbb{N} \times \mathbb{N} \times \mathbb{N}$ such that $F(n) = ( p(n), n, f(n) ).$
The three arrows at the penultimate node represent a function $m : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ such that $m(j, k) = jk.$

For the sake of a first approach, many questions about triadic relations which might arise at this point can be safely left to later discussions, since the current level of generality is comprehensible enough in functional terms.

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Jon Awbrey 2:40 pm on April 9, 2026
Tags: Arithmetization ( 6 ), C.S. Peirce ( 49 ), Gödel Numbers ( 6 ), Higher Order Sign Relations ( 6 ), Inquiry Driven Systems ( 33 ), Inquiry Into Inquiry ( 9 ), logic ( 51 ), Mathematics ( 50 ), Quotation ( 6 ), Recursion ( 7 ), reflection ( 7 ), Reflective Interpretive Frameworks ( 6 ), Semiotics, Sign Relations ( 24 ), Triadic Relations ( 23 ), Use and Mention ( 6 ), Visualization ( 36 )

Reflection On Recursion • 2

Turning to the form of a simple recursive function $f(n) = m(n, f(p(n))),$ the clause we used to define it earns the title of “syntactic recursion” due to the way the function name $``f"$ occurring in the defined phrase $``f(n)"$ re‑occurs in the defining phrase $``m(n, f(p(n)))".$

It needs to be clear there is no circle in the definition — each instance of the type $f$ is defined in terms of an instance one step simpler until the base case is reached and fixed by fiat. Instead of a circle then we have two gyres, the gyre down via the precedent function $p$ and the gyre up via the modifier function $m.$

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Jon Awbrey 5:30 pm on April 6, 2026
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Reflection On Recursion • 1

Ongoing conversations with Dan Everett on Facebook have me backtracking to recurring questions about the relationship between formal language theory (as I once learned it) and the properties of natural languages as they are found occurring in the field. A point of particular interest is the role of recursion in formal and natural languages, along with collateral questions about its role in the cognitive sciences at large.

It has taken me quite a while to bring my reflections up to the threshold of minimal coherence — and the inquiry remains ongoing — but it may catalyze the thinking process if I simply share what I’ve thought so far …

Comment 1

Recursion is where you find it — so, myself not being a natural language researcher, when someone who is says they don’t find it in a given corpus I just take them at their word …

Comment 2

The question to which I keep returning has to do with the relationship between two ways we find recursion occurring.

One way I’d call pragmatic recursion — if I wanted to be precise and cover its full scope — since so many of its operations occur without conscious direction, but for now I’ll defer to more familiar language, calling it cognitive or conceptual recursion.

Comment 3

If we discard from the idea of recursion what is not of its essence, we find recursion occurs when our understanding of one situation has recourse to our understanding of other situations.

Very typically, the object situation presents itself as complex, difficult, or unfamiliar while the resource situations are regarded as being better understood.

It must be appreciated, however, that any ranking of situations by level of understanding is contingent on the circumstances in view and may vary radically in alternate settings.

Comment 4

Recursion occurs more markedly in syntactic recursion, where the recursive process shows its character as such in the symbols of its syntactic expression.

A sense of the difference can be gained by looking at a case of ostensible syntactic recursion. (How much substance backs the ostentation is a subject we’ll take up, maybe at length, but later …)

Consider the following diagram for the computation of a simple recursive function.

For example, the factorial function $f(n) = n!$ has a definition in terms of the predecessor function $p(n) = n-1$ and the multiplier function $m(j, k) = j \cdot k.$

Comment 5

Recursion is rife in mathematics and computation, typically sporting its recursive character on its sleeve in the fashion of syntax sketched above. But mathematics and computation are overlearned subjects and practices, enjoying long histories of being gone over with an eye to articulating every last detail of any way they might be conceived and conducted. So it’s fair to ask whether all that artifice truly tutors nature or only creates a rationalized reconstruction of it. Then again, even if that’s all it does, is there anything of use to be learned from it?

Comment 6

The prevalence of recursion in mathematics arises from the architecture of mathematical systems.

Mathematical systems grow from a fourfold root.

Primitives are taken as initial terms.
Definitions expound ever more complex terms in relation to the primitives.
Axioms are taken as initial truths.
Theorems follow from the axioms by way of inference rules.

Recursive definitions of mathematical objects and inductive proofs of the corresponding theorems follow closely parallel patterns. And again, in computation, recursive programs follow the same patterns in action.

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Jon Awbrey 8:36 am on March 28, 2026
Tags: Arithmetization ( 6 ), C.S. Peirce ( 49 ), Gödel Numbers ( 6 ), Higher Order Sign Relations ( 6 ), Inquiry Driven Systems ( 33 ), Inquiry Into Inquiry ( 9 ), logic ( 51 ), Mathematics ( 50 ), Quotation ( 6 ), Recursion ( 7 ), reflection ( 7 ), Reflective Interpretive Frameworks ( 6 ), Semiotics, Sign Relations ( 24 ), Triadic Relations ( 23 ), Use and Mention ( 6 ), Visualization ( 36 )

Reflective Interpretive Frameworks • Incident 1

Re: William Waites • The Agent That Doesn’t Know Itself

WW: ❝Why Has Nobody Done This?❞

People who study C.S. Peirce would say reflective reasoning requires triadic relations at core and there is work being done on that. One of the challenges is clarifying the role of triadic relations in category theory and raising them into higher relief as fundamental operations.

Note. I was looking for a word to describe a random encounter with something that jogs one’s memory of a recurring theme — incident plays into the reflection theme and looked worth trying for now.

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Jon Awbrey 8:36 am on January 16, 2026
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Sign Relations • Graphical Representations

The dyadic components of sign relations have graph‑theoretic representations, as digraphs (or directed graphs), which provide concise pictures of their structural and potential dynamic properties.

By way of terminology, a directed edge $(x, y)$ is called an arc from point $x$ to point $y,$ and a self‑loop $(x, x)$ is called a sling at $x.$

The denotative components $\mathrm{Den}(L_\mathrm{A})$ and $\mathrm{Den}(L_\mathrm{B})$ can be represented as digraphs on the six points of their common world set $W = O \cup S \cup I =$ $\{ \mathrm{A}, \mathrm{B}, ``\text{A}", ``\text{B}", ``\text{i}", ``\text{u}" \}.$ The arcs are given as follows.

Denotative Component $\mathrm{Den}(L_\mathrm{A})$: $\mathrm{Den}(L_\mathrm{A})$ has an arc from each point of $\{ ``\text{A}", ``\text{i}" \}$ to $\mathrm{A}.$
$\mathrm{Den}(L_\mathrm{A})$ has an arc from each point of $\{ ``\text{B}", ``\text{u}" \}$ to $\mathrm{B}.$
Denotative Component $\mathrm{Den}(L_\mathrm{B})$: $\mathrm{Den}(L_\mathrm{B})$ has an arc from each point of $\{ ``\text{A}", ``\text{u}" \}$ to $\mathrm{A}.$
$\mathrm{Den}(L_\mathrm{B})$ has an arc from each point of $\{ ``\text{B}", ``\text{i}" \}$ to $\mathrm{B}.$

$\mathrm{Den}(L_\mathrm{A})$ and $\mathrm{Den}(L_\mathrm{B})$ can be interpreted as transition digraphs which chart the succession of steps or the connection of states in a computational process. If the graphs are read in that way, the denotational arcs summarize the upshots of the computations involved when the interpreters $\mathrm{A}$ and $\mathrm{B}$ evaluate the signs in $S$ according to their own frames of reference.

The connotative components $\mathrm{Con}(L_\mathrm{A})$ and $\mathrm{Con}(L_\mathrm{B})$ can be represented as digraphs on the four points of their common syntactic domain $S = I =$ $\{ ``\text{A}", ``\text{B}", ``\text{i}", ``\text{u}" \}.$ Since $\mathrm{Con}(L_\mathrm{A})$ and $\mathrm{Con}(L_\mathrm{B})$ are semiotic equivalence relations, their digraphs conform to the pattern manifested by all digraphs of equivalence relations. In general, a digraph of an equivalence relation falls into connected components which correspond to the parts of the associated partition, with a complete digraph on the points of each part, and no other arcs. In the present case, the arcs are given as follows.

Connotative Component $\mathrm{Con}(L_\mathrm{A})$: $\mathrm{Con}(L_\mathrm{A})$ has the structure of a semiotic equivalence relation on $S.$
There is a sling at each point of $S,$ arcs in both directions between the points of $\{ ``\text{A}", ``\text{i}" \},$ and arcs in both directions between the points of $\{ ``\text{B}", ``\text{u}" \}.$
Connotative Component $\mathrm{Con}(L_\mathrm{B})$: $\mathrm{Con}(L_\mathrm{B})$ has the structure of a semiotic equivalence relation on $S.$
There is a sling at each point of $S,$ arcs in both directions between the points of $\{ ``\text{A}", ``\text{u}" \},$ and arcs in both directions between the points of $\{ ``\text{B}", ``\text{i}" \}.$

Taken as transition digraphs, $\mathrm{Con}(L_\mathrm{A})$ and $\mathrm{Con}(L_\mathrm{B})$ highlight the associations permitted between equivalent signs, as the equivalence is judged by the respective interpreters $\mathrm{A}$ and $\mathrm{B}.$

Resources

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Jon Awbrey 5:28 pm on January 5, 2026
Tags: Algebra ( 2 ), Arithmetic, Combinatorics ( 2 ), Computation, Graph Theory ( 26 ), Group Theory ( 3 ), logic ( 51 ), Mathematics ( 50 ), Number Theory, Recursion ( 7 ), Representation, Riffs and Rotes, Semiotics, Visualization ( 36 )

Riffs and Rotes • Happy New Year 2026

$\text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}.$

$\begin{array}{llcl} \text{Then} & 2026 & = & 2 \cdot 1013 \\ && = & p_1 p_{170} \\ && = & p_1 p_{2 \cdot 5 \cdot 17} \\ && = & p_1 p_{p_1 p_3 p_7} \\ && = & p_1 p_{p_1 p_{p_2} p_{p_4}} \\ && = & p_1 p_{p_1 p_{p_{p_1}} p_{p_{{p_1}^{p_1}}}} \end{array}$

No information is lost by dropping the terminal 1s. Thus we may write the following form.

$2026 = p p_{p p_{p_p} p_{p_{p^p}}}$

The article linked below tells how forms of that order correspond to a family of digraphs called riffs and a family of graphs called rotes. The riff and rote for $2026$ are shown in the next two Figures.

Riff 2026

Rote 2026

Reference

Riffs and Rotes

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Jon Awbrey 10:10 am on January 1, 2026
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Sign Relations • Semiotic Equivalence Relations 2

A few items of notation are useful in discussing equivalence relations in general and semiotic equivalence relations in particular.

In general, if $E$ is an equivalence relation on a set $X$ then every element $x$ of $X$ belongs to a unique equivalence class under $E$ called the equivalence class of $x$ under $E$ . Convention provides the square bracket notation for denoting such equivalence classes, in either the form $[x]_E$ or the simpler form $[x]$ when the subscript $E$ is understood. A statement that the elements $x$ and $y$ are equivalent under $E$ is called an equation or an equivalence and may be expressed in any of the following ways.

Thus we have the following definitions.

In the application to sign relations it is useful to extend the square bracket notation in the following ways. If $L$ is a sign relation whose connotative component $L_{SI}$ is an equivalence relation on $S = I,$ let $[s]_L$ be the equivalence class of $s$ under $L_{SI}.$ In short, $[s]_L = [s]_{L_{SI}}.$ A statement that the signs $x$ and $y$ belong to the same equivalence class under a semiotic equivalence relation $L_{SI}$ is called a semiotic equation (SEQ) and may be written in either of the following forms.

In many situations there is one further adaptation of the square bracket notation for semiotic equivalence classes which can be useful. Namely, when there is known to exist a particular triple $(o, s, i)$ in a sign relation $L,$ it is permissible to let $[o]_L$ be defined as $[s]_L.$ This lets the notation for semiotic equivalence classes harmonize more smoothly with the frequent use of similar devices for the denotations of signs and expressions.

Applying the array of equivalence notations to the sign relations for $A$ and $B$ will serve to illustrate their use and utility.

The semiotic equivalence relation for interpreter $\mathrm{A}$ yields the following semiotic equations.

In this way the SER for $\mathrm{A}$ induces the following semiotic partition.

The semiotic equivalence relation for interpreter $\mathrm{B}$ yields the following semiotic equations.

In this way the SER for $\mathrm{B}$ induces the following semiotic partition.

Taken all together we have the following picture.

Resources

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Jon Awbrey 12:08 pm on December 30, 2025
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Sign Relations • Semiotic Equivalence Relations 1

A semiotic equivalence relation (SER) is a special type of equivalence relation arising in the analysis of sign relations. Generally speaking, any equivalence relation induces a partition of the underlying set of elements, known as the domain or space of the relation, into a family of equivalence classes. In the case of a SER the equivalence classes are called semiotic equivalence classes (SECs) and the partition is called a semiotic partition (SEP).

The sign relations $L_\mathrm{A}$ and $L_\mathrm{B}$ have many interesting properties over and above those possessed by sign relations in general. Some of those properties have to do with the relation between signs and their interpretant signs, as reflected in the projections of $L_\mathrm{A}$ and $L_\mathrm{B}$ on the $SI$ ‑plane, notated as $\mathrm{proj}_{SI} L_\mathrm{A}$ and $\mathrm{proj}_{SI} L_\mathrm{B},$ respectively. The dyadic relations on $S \times I$ induced by those projections are also referred to as the connotative components of the corresponding sign relations, notated as $\mathrm{Con}(L_\mathrm{A})$ and $\mathrm{Con}(L_\mathrm{B}),$ respectively. Tables 6a and 6b show the corresponding connotative components.

A nice property of the sign relations $L_\mathrm{A}$ and $L_\mathrm{B}$ is that their connotative components $\mathrm{Con}(L_\mathrm{A})$ and $\mathrm{Con}(L_\mathrm{B})$ form a pair of equivalence relations on their common syntactic domain $S = I.$ This type of equivalence relation is called a semiotic equivalence relation (SER) because it equates signs having the same meaning to some interpreter.

Each of the semiotic equivalence relations, $\mathrm{Con}(L_\mathrm{A}), \mathrm{Con}(L_\mathrm{B}) \subseteq S \times I \cong S \times S$ partitions the collection of signs into semiotic equivalence classes. This constitutes a strong form of representation in that the structure of the interpreters’ common object domain $\{ \mathrm{A}, \mathrm{B} \}$ is reflected or reconstructed, part for part, in the structure of each one’s semiotic partition of the syntactic domain $\{ ``\text{A}", ``\text{B}", ``\text{i}", ``\text{u}" \}.$

It’s important to observe the semiotic partitions for interpreters $\mathrm{A}$ and $\mathrm{B}$ are not identical, indeed, they are orthogonal to each other. Thus we may regard the form of the partitions as corresponding to an objective structure or invariant reality, but not the literal sets of signs themselves, independent of the individual interpreter’s point of view.

Information about the contrasting patterns of semiotic equivalence corresponding to the interpreters $\mathrm{A}$ and $\mathrm{B}$ is summarized in Tables 7a and 7b. The form of the Tables serves to explain what is meant by saying the SEPs for $\mathrm{A}$ and $\mathrm{B}$ are orthogonal to each other.

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Jon Awbrey 10:15 am on December 29, 2025
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Sign Relations • Ennotation

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.

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, $\mathrm{Enn}(L_\mathrm{A}), \mathrm{Enn}(L_\mathrm{B}) \subseteq O \times I.$

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Jon Awbrey 11:45 am on December 28, 2025
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Sign Relations • Connotation

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.

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, $\mathrm{Con}(L_\mathrm{A}), \mathrm{Con}(L_\mathrm{B}) \subseteq S \times I.$

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Jon Awbrey 10:17 am on December 27, 2025
Tags: C.S. Peirce ( 49 ), Connotation ( 11 ), Denotation ( 11 ), inquiry ( 18 ), logic ( 51 ), Logic of Relatives ( 15 ), Mathematics ( 50 ), Relation Theory ( 16 ), Semiosis ( 14 ), Semiotic Equivalence Relations ( 11 ), Semiotics, Sign Relations ( 24 ), Triadic Relations ( 23 )

Sign Relations • Denotation

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.

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, $\mathrm{Den}(L_\mathrm{A}), \mathrm{Den}(L_\mathrm{B}) \subseteq O \times S.$

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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Jon Awbrey 11:24 am on December 25, 2025
Tags: C.S. Peirce ( 49 ), Connotation ( 11 ), Denotation ( 11 ), inquiry ( 18 ), logic ( 51 ), Logic of Relatives ( 15 ), Mathematics ( 50 ), Relation Theory ( 16 ), Semiosis ( 14 ), Semiotic Equivalence Relations ( 11 ), Semiotics, Sign Relations ( 24 ), Triadic Relations ( 23 )

Sign Relations • Dyadic Aspects

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.

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

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.

Resources

Sign Relation • OEIS • MyWikiBiz • Wikiversity
Survey of Semiotics, Semiosis, Sign Relations

cc: Academia.edu • Laws of Form • Research Gate • Syscoi
cc: Cybernetics • Structural Modeling • Systems Science

Jon Awbrey 6:00 pm on December 19, 2025
Tags: C.S. Peirce ( 49 ), Connotation ( 11 ), Denotation ( 11 ), inquiry ( 18 ), logic ( 51 ), Logic of Relatives ( 15 ), Mathematics ( 50 ), Relation Theory ( 16 ), Semiosis ( 14 ), Semiotic Equivalence Relations ( 11 ), Semiotics, Sign Relations ( 24 ), Triadic Relations ( 23 )

Sign Relations • Examples

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.

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

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, $L_\mathrm{A}, L_\mathrm{B} \subseteq O \times S \times I.$

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.