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.  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, alternatively written in any of 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)

References

  • 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.
  • 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.  ArchiveJournalOnline.

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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.  We may visualize this 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, alternatively written in any of forms, \mathrm{proj}_{OS} L,  L_{OS},  \mathrm{proj}_{12} L,  and L_{12}, is defined as follows.

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

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 {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} to denote \mathrm{A} and {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} to denote \mathrm{B}, while \mathrm{B} uses {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime} to denote \mathrm{B} and {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} to denote \mathrm{A}.

References

  • 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.
  • 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.  ArchiveJournalOnline.

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Sign Relations • Discussion 6

Re: Sign RelationsDefinitionExamples
Re: Ontolog Forum • Alex Shkotin (1) (2) (3)

Dear Alex,

We all love natural languages, our native tongues, but each one has a mind of its own and a habit of saying both more and less and something other than the meanings we intend at the moment of utterance.  So maybe it’s a love-hate relationship, or at least a Liebeskampf.

Whether we are endowed with an inborn faculty for language, even a genetic blueprint for selected species of languages on a par with our naturally evolved motor and sense organs, or whether we acquire our initial languages from scratch, every natural language worth its salt preserves a rich heritage of biological and cultural meanings its users will assimilate, consciously or otherwise.  I would not say “resistance is futile” but habits of thought built into our first and second natures demand persistent habits of critical reflection to break.

We do use natural language paraphrases to “express the meaning of [a logical formula] using different words, especially to achieve greater clarity” and up to a point they serve that end.  But there’s a catch.  If a natural language paraphrase could express the precise meaning of a logical formula with greater clarity, what would be the use of the formula?

Well, that’s the beginning of a post I started on the spectrum of formality from form to formal object to formula to paraphrase.  But I decided to let it simmer for another day.  Now that we have a workbench stocked with concrete examples of triadic relations and sign relations we might as well use them to illustrate the abstractions while keeping our feet on more solid ground.

I’ll turn to that task next.

References

  • 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.  AbstractOnline.
  • 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 • Dyadic Aspects

For an arbitrary triadic relation L \subseteq O \times S \times I, whether it is 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.} ~~ \text{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 interpretant i in the interpretant domain 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 may vary as 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 anonymous on the contemporary scene of sign studies.

References

  • 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.
  • 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.  ArchiveJournalOnline.

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Animated Logical Graphs • 33

A reader’s request for more examples of animated logical graphs prompted me to look again at the User Guide for my Theme One Program, whose exposition develops a series of logical graphs increasing in complexity from extremely simple to more substantial than any I’ve posted so far.

I’m thinking now it may be worthwhile to look at those examples again and see if they’re suitable for recycling as a series of blog posts.

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Differential Logic • Discussion 3

Re: R.J. LiptonP<NP

Instead of boolean circuit complexity I would look at logical graph complexity, where those logical graphs are constructed from minimal negation operators.

Physics once had a frame problem (complexity of dynamic updating) long before AI did but physics learned to reduce complexity through the use of differential equations and group symmetries (combined in Lie groups).  One of the promising features of minimal negation operators is their relationship to differential operators.  So I’ve been looking into that.  Here’s a link, a bit in medias res, but what I’ve got for now.

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Sign Relations • Examples

Because the examples to follow have been artificially constructed to be as simple as possible, their detailed elaboration can run the risk of trivializing the whole theory of sign relations.  Despite their simplicity, however, these examples have subtleties of their own, and their careful treatment will serve to illustrate many important issues in the general theory of signs.

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

“Ann”,   “Bob”,   “I”,   “you”.

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

Ann and Bob are not only the passive objects of linguistic references but also the active interpreters of the language they use.  The system of interpretation (SOI) associated with each language user can be represented in the form of an individual three-place relation known as the sign relation of that interpreter.

In terms of its set-theoretic extension, a sign relation 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 situations it becomes convenient to lump signs and interpretants together in a single class called the sign system or the 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:

\begin{array}{ccl}  O & = & \text{Object Domain}  \\[6pt]  S & = & \text{Sign Domain}  \\[6pt]  I & = & \text{Interpretant Domain}  \end{array}

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

\begin{array}{cclcl}  O  & = &  \{ \text{Ann}, \text{Bob} \} & = & \{ \mathrm{A}, \mathrm{B} \}  \\[6pt]  S  & = &  \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}  & = &  \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}  \\[6pt]  I  & = &  \{ {}^{\backprime\backprime} \text{Ann} {}^{\prime\prime}, {}^{\backprime\backprime} \text{Bob} {}^{\prime\prime}, {}^{\backprime\backprime} \text{I} {}^{\prime\prime}, {}^{\backprime\backprime} \text{you} {}^{\prime\prime} \}  & = &  \{ {}^{\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}

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

These 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, making up an ordered triple of the form (o, s, i) called an elementary relation, that is, one element of the relation’s set-theoretic extension.

Already in this elementary context, there are several different meanings that might attach to the project of a formal semiotics, or a formal theory of meaning for signs.  In the process of discussing these 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.

References

  • 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.
  • 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.  ArchiveJournalOnline.

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Sign Relations • Discussion 5

Re: Sign Relations • Discussion 4
Re: Peirce ListGary FuhrmanJon Alan Schmidt

The transformative idea in Peirce’s case of the French interpreter is not the convertibility of term logic, propositional logic, and monadic predicate logic — a commonplace of logic from the time of Aristotle, if not in those words, obscured only by the false subtleties of the Frege-Russell tradition, though even Quine was woke enough in time to write a nice essay on it — but rather the transformation from interpreter models to interpretant models of semiosis.  The latter models are what Peirce and all in his train require for constructing abstract formal theories neutral on psychologism, materialism, biologism, and various other all too stolid -isms.

There’s more discussion of Peirce’s passage to the interpretant at the following locations.

References

  • 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.
  • Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Cited as (CE volume, page).

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Sign Relations • Discussion 4

Re: Peirce ListEdwina Taborsky

A note on a couple of recurring themes may be useful at this point.

  1. Peirce’s “metaphorical argument” for transforming discussion of interpretive agents, whether individuals or communities, to discussion of interpretant signs is as follows.

I think we need to reflect upon the circumstance that every word implies some proposition or, what is the same thing, every word, concept, symbol has an equivalent term — or one which has become identified with it, — in short, has an interpretant.

Consider, what a word or symbol is;  it is a sort of representation.  Now a representation is something which stands for something.  …   A thing cannot stand for something without standing to something for that something.  Now, what is this that a word stands to?  Is it a person?

We usually say that the word homme stands to a Frenchman for man.  It would be a little more precise to say that it stands to the Frenchman’s mind — to his memory.  It is still more accurate to say that it addresses a particular remembrance or image in that memory.  And what image, what remembrance?  Plainly, the one which is the mental equivalent of the word homme — in short, its interpretant.  Whatever a word addresses then or stands to, is its interpretant or identified symbol.  …

The interpretant of a term, then, and that which it stands to are identical.  Hence, since it is of the very essence of a symbol that it should stand to something, every symbol — every word and every conception — must have an interpretant — or what is the same thing, must have information or implication.  (Peirce, CE 1, 466–467).

There’s additional discussion of this passage at the following locations.

  1. When we employ mathematical models to describe any domain of phenomena, we are always proceeding hypothetically and tentatively, and the modality of all mathematics, in its own right, is the possible.  That is because mathematical existence is existence in the modest sense of “whatever’s not inconsistent”.  In the idiom, “It’s would-be’s all the way down.”  In effect the ordinary scales of modality are flattened down to one mode, to wit, Be ♭.  It is not until we take the risk of acting on our abduced model that we encounter genuine brute force Secondness.

References

  • 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.
  • Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Cited as (CE volume, page).

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Animated Logical Graphs • 32

Re: R.J. Lipton and K.W. ReganProof Checking

Dear Dick/Ken,

Here’s a place where I explore different shapes of proofs in a propositional calculus deriving from the graphical systems of Charles S. Peirce and G. Spencer Brown.

I don’t know whether that helps any with \mathrm{P} \overset{\underset{?}{}}{=} \mathrm{NP} but it does supply a lot of nice pictures to contemplate.

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