Sign Relations • Discussion 10

Re: CyberneticsKlaus KrippendorffBernard Scott
Re: OntologMihai NadinJohn SowaAlex Shkotin
Re: Peirce ListHelmut RaulienEdwina Taborsky

While engaged in a number of real and imaginary dialogues with people I continue to owe full replies, I thought it might be a good time to stand back and take in the view from this vantage point.  I summed up the desired outlook a few days ago in the following way.

The important thing now is to extend our perspective beyond one sign at a time and one object, sign, interpretant at a time to comprehending a sign relation as a specified set of object, sign, interpretant triples embedded in the set of all possible triples in a specified context.

If we now comprehend each sign relation L as an extended collection of triples (o, s, i), where each object o belongs to a set O of objects, each sign s belongs to a set S of signs, each interpretant i belongs to a set I of interpretants, and the whole sign relation L is embedded as a subset in the product space O \times S \times I, then our level of description ascends to the point where we take whole sign relations of this sort as the principal subjects of classification and structural analysis.

Once we adopt a whole systems perspective on sign relations we begin to see many commonplace topics in a fresh light.

Agency

That Peirce remodels his theory of semiosis from speaking of interpretive agents to speaking of interpretant signs is a familiar theme by now.  By way of reminder, we discussed this transformation recently in Discussion 4 and Discussion 5 of this series.

But we have to wonder:  Why does Peirce make this shift, this change of basis from interpreters to interpretants?  He does this because the idea of an interpreter stands in need of clarification and his method for clarifying ideas is to apply the pragmatic maxim.  The result is an operational definition of an interpreter in terms of its effects on signs in relation to their objects.

It would seem we have replaced an interpreter with a sign relation.  To be more precise, we are taking a sign relation as our effective model for the interpreter in question.  But we must not take this the wrong way.  There is no suggestion of reducing the hypostatic agent to a sign relation.  It falls within our capacity merely to clarify our concept of the agent to a moderate degree, to construct a model or a representation capturing aspects of the agent’s activity bearing on a particular application.

With that I’ve run out of time for today.  The topic for next time will be Context …

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.

Resources

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Sign Relations • Discussion 9

Re: Sign Relations • Ennotation
Re: Peirce ListHelmut Raulien

Dear Helmut,

Thanks for your comments.  They prompt me to say a little more about the mathematical character of the sign relational models I’m using.

Peirce without mathematics is like science without mathematics.  In every direction of research he pioneered or prospected, information, inquiry, logic, semiotics, we trace his advances only so far, barely scratch the surface before we need to bring in mathematical models adequate to the complexity of the phenomena under investigation.

In recent years there has been a tendency in certain quarters to ignore the mathematical substrate of Peirce’s pragmatic thought, even a refusal to use the mathematical tools he crafted to the task of sharpening our understanding.  I do not recall that attitude being prevalent when I began my studies of Peirce’s work some fifty years ago.  The issue in the “reception of Peirce” over most of that time has largely been the tendency of people imbued in the traditions of “analytic philosophy” to dismiss Peirce out of hand.  But that school of thought had no problem with using mathematics, aside from the short-sighted attempts to reduce mathematics to logic and all relations to dyadic ones.

Maybe this late resistance to Peirce’s mathematical groundwork has come about through an overly selective viewing of his entire spectrum of work or maybe it’s just a matter of taste.  Whatever the case, it’s critical for people who are looking for adequate models of the complex phenomena involved in belief systems, communication, intelligent systems, knowledge representation, scientific inquiry, and so on to recognize that all the resources we need for working with relations in general as sets of ordered tuples and sign relations in particular as sets of ordered triples are already available in Peirce’s technical works from 1870 on.

Okay, it looks like I’ve used up my morning again with more preliminary matters but it seemed important to clear up a few things about the overall mathematical approach.

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.

Resources

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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.

\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}.  That is, let [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 that 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.  These modifications are designed to make the notation for semiotic equivalence classes harmonize as well as possible 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}

Thus it induces the 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}

Thus it induces the 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

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.

Resources

Document History

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

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Sign Relations • Discussion 8

Re: Sign Relations • Ennotation
Re: Peirce ListHelmut Raulien

Dear Helmut,

The important thing now is to extend our perspective beyond one sign at a time and one object, sign, interpretant at a time to comprehending a sign relation as a specified set of object, sign, interpretant triples embedded in the set of all possible triples in a specified context.

In my mind’s eye, no doubt influenced by my early interest in Gestalt Psychology, I always picture a sign relation as a gestalt composed of figure and ground.  The triples in the sign relation form a figure set in relief against the background of all possible triples and the triples left over form the ground of the gestalt.

From a mathematical point of view, the set of possible triples is a cartesian product of the following form.

O \times S \times I = \{ (o, s, i) : o \in O \land s \in S \land i \in I \}.

Here, O is the object domain, the set of objects under discussion, S is the sign domain, the specified set of signs, and I is the interpretant domain, the specified set of interpretants.

On this canvass, in this frame, any number of sign relations might be set as figures and each of them would be delimited as a salient subset of the cartesian product in view.  Letting L be any such sign relation, mathematical convention provides the following description of its relation to the set of possible triples.

L \subseteq O \times S \times I.

It’s important to note at this point that the specified cartesian product and the specified subset of it are equally critical parts of the sign relation’s definition.

Well, it took a lot longer to set the scene than I thought it would when I got up this morning, so I’ll break here and get back to your specific comments when I next get a chance.

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.

Resources

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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 these 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 these 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 makes for 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} \}.  But it needs to be observed that the semiotic partitions for interpreters \mathrm{A} and \mathrm{B} are not identical, indeed, they are orthogonal to each other.  This allows us to regard the form of these 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 these 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

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.

Resources

Document History

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

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Sign Relations • Ennotation

A third aspect of a sign’s complete meaning concerns the reference its objects have to 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, alternatively written in any of 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)

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.

Resources

Document History

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

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Posted in C.S. Peirce, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Philosophy, Pragmatic Semiotic Information, Pragmatism, Relation Theory, Semiotics, Sign Relations, Thirdness, Triadic Relations, Triadicity | Tagged , , , , , , , , , , , , , , | 3 Comments

Sign Relations • Discussion 7

Re: Sign Relations • Definition
Re: Ontolog ForumAlex Shkotin

Dear Alex,

Please forgive the long and winding dissertation.  I’ve been through many discussions of Peirce’s definition of “logic as formal semiotic” but I keep discovering new ways of reading what I once regarded as a straightforward proposition.  That’s all useful information but it makes me anxious to avoid any missteps of interpretation I might have made in the past.  At any rate, I think I’ve set enough background and context to begin addressing your comments now.

For ease of reference here is Peirce’s twofold definition again.

Logic will here be defined as formal semiotic.  A definition of a sign will be given which no more refers to human thought than does the definition of a line as the place which a particle occupies, part by part, during a lapse of time.  Namely, a sign is something, A, which brings something, B, its interpretant sign determined or created by it, into the same sort of correspondence with something, C, its object, as that in which itself stands to C.  It is from this definition, together with a definition of “formal”, that I deduce mathematically the principles of logic.  I also make a historical review of all the definitions and conceptions of logic, and show, not merely that my definition is no novelty, but that my non-psychological conception of logic has virtually been quite generally held, though not generally recognized.  (C.S. Peirce, NEM 4, 20–21).

Turning to your first comment

A Sign is unusually active in Peirce’s definition:

A (a sign) brings B (interpretant sign) into correspondence with C (object of sign).

Moreover, A determines B or even creates B.

It would be nice to get an example of such an active sign, its interpretant sign, and an object.  My point is to make the Peirce definition as clear as to be formalized.

Several issues stand out.  There are questions about paraphrases, the active character of signs, and the nature of what is being defined.

  • The problem of paraphrases arises at this point because it affects how literally we ought to take the words in a natural language proxy for a logical or mathematical formula.

    For example, a conventional idiom in describing a mathematical function f : X \to Y is to say f “maps” or “sends” an element of X to an element of Y.  A concrete verb may quicken the intuition but the downside is its power to evoke excess meanings beyond the abstract intention.  It is only as we become more familiar with the formal subject matter of sign relations that we can decide what kind of “bringing” and “creating” and “determining” is really going on in all that sign, object, interpretant relating, whether at the abstract level or in a given application.

  • There is the question of a sign’s active character.  Where’s the dynamic function in all that static structure?  Klaus Krippendorff raised the same question in regard to the Parable of the Sunflower back at the beginning of this discussion.

    [Peirce’s] triadic explanations do not cover the dynamics of the sunflower’s behavior.  It favors static descriptions which cybernetics is fundamentally opposed to, moreover including the cybernetician as enactor of his or her conceptual system.

    I have not forgotten this question.  Indeed, it’s the question at the heart of my work on Inquiry Driven Systems, which led me back to grad school in Systems Engineering “to develop mutual applications of systems theory and artificial intelligence to each other”.

    Anything approaching an adequate answer to that question is going to be one of those things requiring more background and context, all in good time, but there are a few hints we can take from Peirce’s text about the way forward.

    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.

    My reading of that tells me about a division of labor across three levels of abstraction.  There is a level of psychological experience and social activity, a level of dynamic process and temporal pattern, and a level of mathematical form.

To be continued …

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.

Resources

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Survey of Inquiry Driven Systems • 2

This is a Survey of previous blog and wiki posts on Inquiry Driven Systems, material I plan to refine toward a more compact and systematic treatment of the subject.

An inquiry driven system is a system having among its state variables some representing its state of information with respect to various topics of interest, for example, its own state and the state of any potential object systems.  Thus it has a component of state tracing a trajectory though an information state space.

Elements

Developments

Applications

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Peirce’s Categories • 21

Re: Peirce ListRobert Marty
Re: Peirce ListRobert Marty

Dear Robert,

Let’s go back to a point where paths diverged in the yarrow wood and a lot of synchronicity was lost …

Variant understandings of words like axiom, definition, predicate, proposition, proof, relation, theory, and the like make mutual understanding difficult.  For example, when I mention Peirce’s definition of a sign, many people will bring to mind a long list of short statements Peirce made in describing the properties of signs, and when I refer to Peirce’s theory of signs, many people will bring to mind the entire corpus of Peirce’s writings on signs, so far as they know it, augmented perhaps with reliable reports of statements he may have made about signs.

There are fields of study where such expansive understandings of definitions and theories are the prevailing ones, perhaps the only feasible ones.  One example would be scriptural hermeneutics, where the full sense of a word’s meaning is determined by its use in every context where it occurs.  Thus the use of concordances to bring the diversity of contextual meanings into harmony.  We plow this field in a hermeneutic circle, according each bit of authoritative text equal priority, none privileged above the other, as if equidistant from a central point radiating a pervasive message.  It’s all you can do when there’s nothing but the text in view.

Curiously enough, the branch of mathematical logic known as model theory sets out with an equally expansive view, taking a maximally inclusive definition of theory as its initial point of departure and defining a theory as an arbitrary set of sentences from a formal language.  Naturally, logical and mathematical attention almost immediately shifts to more focused spheres of theory.

A set \Gamma of sentences is called a theory.  A theory is said to be closed iff every consequence of \Gamma belongs to \Gamma.  A set \Delta of sentences is said to be a set of axioms for a theory \Gamma iff \Gamma and \Delta have the same consequences.  A theory is called finitely axiomatizable iff it has a finite set of axioms.  Since we may form the conjunction of a finite set of axioms, a finitely axiomatizable theory actually always has a single axiom.  The set \bar\Gamma of all consequences of \Gamma is the unique closed theory which has \Gamma as a set of axioms.  (Chang and Keisler, p. 12).

That’s all well and good as far as esoteric technical usage goes but outside those cloisters I would recommend using the word corpus when we want to talk about an arbitrary set of sentences or texts and reserving the word theory for those corpora possessing more differentiated and substantial anatomies than a mere hermeneutic blastula.

Reference

  • Chang, C.C., and Keisler, H.J. (1973), Model Theory, North-Holland, Amsterdam.

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Posted in Abstraction, Aristotle, C.S. Peirce, Category Theory, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Phenomenology, Pragmatic Maxim, Relation Theory, Semiotics, Triadic Relations, Triadicity, Type Theory | Tagged , , , , , , , , , , , , , , , | Leave a comment

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

Posted in C.S. Peirce, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Philosophy, Pragmatic Semiotic Information, Pragmatism, Relation Theory, Semiotics, Sign Relations, Thirdness, Triadic Relations, Triadicity | Tagged , , , , , , , , , , , , , , | 1 Comment