Inquiry Into Inquiry

Re: Cybernetics • Cliff Joslyn

CJ:

For a given arbitrary triadic relation (let’s say that and are all finite, non‑empty sets), I’m interested to understand what additional axioms you’re saying are necessary and sufficient to make a sign relation.  I checked Sign Relations • Definition, but it wasn’t obvious, or at least, not formalized.

Dear Cliff,

Peirce claims a definition of logic as formal semiotic and goes on to define a sign in terms of its relation to its interpretant sign and its object.

For ease of reference, here’s the cited paragraph 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).

Let me cut to the chase and say what I see in that passage.  Peirce draws our attention to a category of mathematical structures of use in understanding various domains of complex phenomena by capturing aspects of objective structure immanent in those domains.

The domains of complex phenomena of interest to logic in its broadest sense encompass all that appears on the discourse side of any universe of discourse we happen to discuss.  That’s a big enough sky for anyone to live under, but for the moment I am focusing on the ways we transform signs in activities like communication, computation, inquiry, learning, proof, and reasoning in general.  I’m especially focused on the ways we do now and may yet use computation to advance the other pursuits on that list.

To be continued …

Reference

• Charles S. Peirce (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.

Sources

• C.S. Peirce • On the Definition of Logic
• C.S. Peirce • Logic as Semiotic
• C.S. Peirce • Objective Logic

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

In general, if is an equivalence relation on a set then every element of belongs to a unique equivalence class under called the equivalence class of under .  Convention provides the square bracket notation for denoting such equivalence classes, in either the form or the simpler form when the subscript is understood.  A statement that the elements and are equivalent under 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 is a sign relation whose connotative component is an equivalence relation on let be the equivalence class of under   In short,   A statement that the signs and belong to the same equivalence class under a semiotic equivalence relation 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 in a sign relation it is permissible to let be defined as   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 and will serve to illustrate their use and utility.

The semiotic equivalence relation for interpreter yields the following semiotic equations.

or

In this way it induces the following semiotic partition.

The semiotic equivalence relation for interpreter yields the following semiotic equations.

or

In this way it induces the following semiotic partition.

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.  Archive.  Journal.  Online (doc) (pdf).

Resources

• Inquiry
• Semeiotic
• Logic Syllabus
• Sign Relations
• Triadic Relations
• Relation Theory

Document History

See OEIS Wiki • Sign Relation • Document History.

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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 and 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 and on the ‑plane, notated as and respectively.  The dyadic relations on induced by those projections are also referred to as the connotative components of the corresponding sign relations, notated as and respectively.  Tables 6a and 6b show the corresponding connotative components.

A nice property of the sign relations and is that their connotative components and form a pair of equivalence relations on their common syntactic domain   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, 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 is reflected or reconstructed, part for part, in the structure of each one’s semiotic partition of the syntactic domain

It’s important to observe the semiotic partitions for interpreters and 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 and is summarized in Tables 7a and 7b.  The form of the Tables serves to explain what is meant by saying the SEPs for and are orthogonal to each other.

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.  Archive.  Journal.  Online (doc) (pdf).

Resources

• Inquiry
• Semeiotic
• Logic Syllabus
• Sign Relations
• Triadic Relations
• Relation Theory

Document History

See OEIS Wiki • Sign Relation • Document History.

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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 may be notated as   Information about the ennotative aspect of meaning is obtained from by taking its projection on the object‑interpretant plane.  We may visualize this as the “shadow” casts on the 2‑dimensional space whose axes are the object domain and the interpretant domain   The ennotative component of a sign relation variously written in any of the forms,       and is defined as follows.

As it happens, the sign relations and are fully symmetric with respect to exchanging signs and interpretants, so all the data of is echoed unchanged in and all the data of is echoed unchanged in

Tables 5a and 5b show the ennotative components of the sign relations associated with the interpreters and respectively.  The rows of each Table list the ordered pairs in the corresponding projections,

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.  Archive.  Journal.  Online (doc) (pdf).

Resources

• Inquiry
• Semeiotic
• Logic Syllabus
• Sign Relations
• Triadic Relations
• Relation Theory

Document History

See OEIS Wiki • Sign Relation • Document History.

cc: FB | Semeiotics • Cybernetics • Laws of Form • Ontolog Forum
cc: Conceptual Graphs • Structural Modeling • Systems Science

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 is notated as   Information about the connotative aspect of meaning is obtained from by taking its projection on the sign-interpretant plane.  We may visualize this as the “shadow” casts on the 2-dimensional space whose axes are the sign domain and the interpretant domain   The connotative component of a sign relation variously written in any of the forms,       and is defined as follows.

Tables 4a and 4b show the connotative components of the sign relations associated with the interpreters and respectively.  The rows of each Table list the ordered pairs in the corresponding projections,

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.  Archive.  Journal.  Online (doc) (pdf).

Resources

• Inquiry
• Semeiotic
• Logic Syllabus
• Sign Relations
• Triadic Relations
• Relation Theory

Document History

See OEIS Wiki • Sign Relation • Document History.

cc: FB | Semeiotics • Cybernetics • Laws of Form • Ontolog Forum
cc: Conceptual Graphs • Structural Modeling • Systems Science

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 is notated as   Information about the denotative aspect of meaning is obtained from by taking its projection on the object-sign plane.  We may visualize this as the “shadow” casts on the 2-dimensional space whose axes are the object domain and the sign domain   The denotative component of a sign relation variously written in any of the forms,       and is defined as follows.

Tables 3a and 3b show the denotative components of the sign relations associated with the interpreters and respectively.  The rows of each Table list the ordered pairs in the corresponding projections,

Looking to the denotative aspects of and various rows of the Tables specify, for example, that uses to denote and to denote while uses to denote and to denote

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.  Archive.  Journal.  Online (doc) (pdf).

Resources

• Inquiry
• Semeiotic
• Logic Syllabus
• Sign Relations
• Triadic Relations
• Relation Theory

Document History

See OEIS Wiki • Sign Relation • Document History

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

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 on the -plane is written briefly as or written more fully as and is defined as the set of all ordered pairs in the cartesian product for which there exists an ordered triple in for some element in the set

In the case where is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of 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.  Archive.  Journal.  Online (doc) (pdf).

Resources

• Inquiry
• Semeiotic
• Logic Syllabus
• Sign Relations
• Triadic Relations
• Relation Theory

Document History

See OEIS Wiki • Sign Relation • Document History.

cc: FB | Semeiotics • Cybernetics • Laws of Form • Ontolog Forum
cc: Conceptual Graphs • Structural Modeling • Systems Science

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.

“Ann”,   “Bob”,   “I”,   “you”.

• The object domain of their discussion is the set of two people
• The sign domain of their discussion is the set of four signs

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 is a subset of a cartesian product   The three sets are known as the object domain, the sign domain, and the interpretant domain, respectively, of the sign relation

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   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 and 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 for a given sign relation we will call this set the World of and write

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 this Example, we have the following data.

In the present example,

Tables 1a and 1b show the sign relations associated with the interpreters and respectively.  In this arrangement the rows of each Table list the ordered triples of the form belonging to the corresponding sign relations,

The Tables codify a rudimentary level of interpretive practice for the agents and 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 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 that 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.

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.  Archive.  Journal.  Online (doc) (pdf).

Resources

• Inquiry
• Semeiotic
• Logic Syllabus
• Sign Relations
• Triadic Relations
• Relation Theory

Document History

See OEIS Wiki • Sign Relation • Document History.

cc: FB | Semeiotics • Cybernetics • Laws of Form • Ontolog Forum
cc: Conceptual Graphs • Structural Modeling • Systems Science