Sign Relations, Triadic Relations, Relation Theory • Discussion 12

Posted on February 24, 2024 by Jon Awbrey

Re: Sign Relations, Triadic Relations, Relation Theory • 1

A note from a longtime correspondent points out a search of the available texts turns up no use of the plural form “semiotics” by Peirce and just one place where he uses the plural form “Semeiotics”.  That prompts me to make the following excuse for my use or abuse of Peirce’s terms, as the case may be.

Peirce has always been one of my chief resources in the quest to understand how logic and math and science work.  There is much to be gained by getting his distinctive ideas across to active practitioners in those fields.  In doing that I find it better to tweak the words a bit, if that’s what it takes to preserve the idea, than to hallow the words at the risk of losing the idea.

As far as semiotics by any name goes, what seems to work best without too much clanging in modern ears is parsing semiotics in line with words like mathematics and cybernetics, plus we can now use the singular form as the adjective semiotic.

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Sign Relations, Triadic Relations, Relation Theory • 4

Posted on February 22, 2024 by Jon Awbrey

For ease of reference, here are two variants of Peirce’s 1902 definition of a sign, which he gives in the process of defining logic.

Selections from C.S. Peirce, “Carnegie Application” (1902)

No. 12.  On the Definition of Logic

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.  (NEM 4, 20–21).

No. 12.  On the Definition of Logic [Earlier Draft]

Logic is formal semiotic.  A sign is something, A, which brings something, B, its interpretant sign, determined or created by it, into the same sort of correspondence (or a lower implied sort) with something, C, its object, as that in which itself stands to C.  This definition no more involves any reference to human thought than does the definition of a line as the place within which a particle lies during a lapse of time.  It is from this definition that I deduce the principles of logic by mathematical reasoning, and by mathematical reasoning that, I aver, will support criticism of Weierstrassian severity, and that is perfectly evident.  The word “formal” in the definition is also defined.  (NEM 4, 54).

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, Triadic Relations, Relation Theory • 3

Posted on February 20, 2024 by Jon Awbrey

The middle ground between relations in general and the sign relations informing logic, inquiry, and communication is occupied by triadic relations, also called ternary or 3‑place relations.

Triadic relations are some of the most pervasive in mathematics, over and above the importance of sign relations for logic.

For a primer on triadic relations, with examples from mathematics and semiotics, see the article linked below.

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Sign Relations, Triadic Relations, Relation Theory • 2

Posted on February 19, 2024 by Jon Awbrey

I always have trouble deciding whether to start with the genus and drive down to the species or begin with concrete examples and accompany Sisyphus up Mt. Abstraction.

To start at the wide end of the funnel, the following article takes up relations in general, focusing on the discrete mathematical varieties we find most useful in applications, for example, as background for empirical data sets and relational data bases.

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Sign Relations, Triadic Relations, Relation Theory • 1

Posted on February 18, 2024 by Jon Awbrey

To understand how signs work in Peirce’s theory of triadic sign relations, or “semiotics”, we have to understand, in order of increasing generality, sign relations, triadic relations, and relations in general, each as conceived in Peirce’s logic of relative terms and the corresponding mathematics of relations.

Toward that understanding, here are the current versions of articles I long ago contributed to Wikipedia and Wikiversity and continue to develop at a number of other places.

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Tone Token Type • Discussion 1

Posted on February 17, 2024 by Jon Awbrey

Re: FB | Daniel Everett

DE:
People who believe that Peirce’s terms firstness, secondness, and thirdness are complicated might have overlooked the fact that they almost certainly already use two of the three terms via Peirce’s other terms type (thirdness) and token (secondness).  What is missing is only Peirce’s other term tone, which refers to firstness.

These distinctions are crucial.  Take linguistic fieldwork.  When the fieldworker first hears something or sees something but has no idea about it other than it is “strange” or unexpected, that is a tone/firstness.  When the linguist proposes the phones of a language, the list are tokens/secondnesses.  When the linguist proposes phonemes, those are types/thirdnesses.  (And underlying form would be a thirdness/type and the surface form a secondness/token.)

Daniel,

The way Peirce shades the matter of signs along the lines of a Tone‑Token‑Type spectrum is a topic of recurring discussion.  There’s a selection of Peirce quotes and a few comments from me on the following page.

Re: C.S. Peirce • Note 1

CSP:
For a “possible” Sign I have no better designation than a Tone,
though I am considering replacing this by “Mark”.

I’ve seen some readers be confused by Peirce’s sometime alternative of Mark for Tone, thinking he meant something like a scratch‑mark on paper, but he is using Mark in the sense of Character(istic), Distinctive Feature, or Quality.  I don’t know whether he had it in mind but that particular use was also common among 19th Century mathematicians in the early years of the subject known as the Representation Theory of Groups.

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Sign Relations • Semiotic Equivalence Relations 2

Posted on February 16, 2024 by Jon Awbrey

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.

\begin{array}{clc} (x, y) & \in & E \\[4pt] x & \in & [y]_E \\[4pt] y & \in & [x]_E \\[4pt] [x]_E & = & [y]_E \\[4pt] x & =_E & y \end{array}

Thus we have the following definitions.

\begin{array}{ccc} [x]_E & = & \{ y \in X : (x, y) \in E \} \\[6pt] x =_E y & \Leftrightarrow & (x, y) \in E \end{array}

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.

\begin{array}{clc} [x]_L & = & [y]_L \\[6pt] x & =_L & y \end{array}

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.

Connotative Components Con(L_A) and Con(L_B)

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

\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{A}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{A}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{A}} \end{matrix}

or

\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{A}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \end{matrix}

In this way it induces the following semiotic partition.

\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \} \}.

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

\begin{matrix} [ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} ]_{L_\mathrm{B}} \\[6pt] [ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} ]_{L_\mathrm{B}} & = & [ {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} ]_{L_\mathrm{B}} \end{matrix}

or

\begin{matrix} {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \\[6pt] {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} & =_{L_\mathrm{B}} & {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \end{matrix}

In this way it induces the following semiotic partition.

\{ \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}, \{ {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} \} \}.

Semiotic Partitions for Interpreters A and B

Resources

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Sign Relations • Semiotic Equivalence Relations 1

Posted on February 15, 2024 by Jon Awbrey

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.

Connotative Components Con(L_A) and Con(L_B)

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 \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.

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.

Semiotic Partitions for Interpreters A and B

Resources

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Sign Relations • Ennotation

Posted on February 14, 2024 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.  We may visualize this 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 in any of the forms, \mathrm{proj}_{OI} L,  L_{OI},  \mathrm{proj}_{13} L,  and L_{13}, is defined as follows.

\begin{matrix} \mathrm{Enn}(L) & = & \mathrm{proj}_{OI} L & = & \{ (o, i) \in O \times I ~:~ (o, s, i) \in L ~\text{for some}~ s \in S \}. \end{matrix}

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)

Resources

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Sign Relations • Connotation

Posted on February 12, 2024 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.  We may visualize this 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 in any of the forms, \mathrm{proj}_{SI} L,  L_{SI},  \mathrm{proj}_{23} L,  and L_{23}, is defined as follows.

\begin{matrix} \mathrm{Con}(L) & = & \mathrm{proj}_{SI} L & = & \{ (s, i) \in S \times I ~:~ (o, s, i) \in L ~\text{for some}~ o \in O \}. \end{matrix}

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)

Resources

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