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☞ This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].

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A sign relation is the basic construct in the theory of signs, also known as [[semeiotic]] or [[semiotics]], as developed by Charles Sanders Peirce.

==Anthesis==

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Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun.

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Thus, if a sunflower, in turning towards the sun, becomes by that very act fully capable, without further condition, of reproducing a sunflower which turns in precisely corresponding ways toward the sun, and of doing so with the same reproductive power, the sunflower would become a Representamen of the sun.&~~nbsp~~; (C.S. Peirce~~, “Syllabus” (c. 1902)~~, Collected Papers, CP 2.274).

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— C.S. Peirce, Collected Papers, CP 2.274

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In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or [[semiosis]], Peirce uses the technical term representamen for his concept of a sign, but the shorter word is precise enough, so long as one recognizes ~~that~~ its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign.

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In his picturesque illustration of a sign relation, along with his tracing of a corresponding sign process, or semiosis, Peirce uses the technical term representamen for his concept of a sign, but the shorter word is precise enough, so long as one recognizes its meaning in a particular theory of signs is given by a specific definition of what it means to be a sign.

==Definition==

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One of Peirce's clearest and most complete definitions of a sign is one he gives in the context of providing a definition for logic, and so it is informative to view it in that setting.

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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.

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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.&~~nbsp~~; (C.S. Peirce, ~~NEM~~ 4, 20–21).

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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.

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— C.S. Peirce, New Elements of Mathematics, vol. 4, 20–21

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In the general discussion of diverse theories of signs, the question ~~frequently~~ arises whether signhood is an absolute, essential, indelible, or ontological property of a thing, or whether it is a relational, interpretive, and mutable role a thing ~~can~~ be said to have only within a particular context of relationships.

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In the general discussion of diverse theories of signs, the question arises whether signhood is an absolute, essential, indelible, or ontological property of a thing, or whether it is a relational, interpretive, and mutable role a thing may be said to have only within a particular context of relationships.

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Peirce's definition of a sign defines it in relation to its ~~object~~ and its interpretant ~~sign~~, and thus it defines signhood in ~~[[logic of relatives|~~relative terms]], by means of a predicate with three places.  In ~~this~~ definition, signhood is a role in a [[triadic relation]], a role a thing bears or plays in a ~~given~~ context of relationships — it is not an absolute, non‑relative property of a thing‑in‑itself, one it possesses independently of all relationships to other things.

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Peirce's definition of a sign defines it in relation to its objects and its interpretant signs, and thus defines signhood in relative terms, by means of a predicate with three places.  In that definition, signhood is a role in a triadic relation, a role a thing bears or plays in a determinate context of relationships — it is not an absolute or non‑relative property of a thing‑in‑itself, one it possesses independently of all relationships to other things.

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Some of the terms ~~that~~ Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader.

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Some of the terms Peirce uses in his definition of a sign may need to be elaborated for the contemporary reader.

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<ul><li>Correspondence.  From the way Peirce uses the term throughout his work, it is clear he means what he elsewhere calls a “triple correspondence”, and thus this is just another way of referring to the whole triadic sign relation itself.  In particular, his use of the term should not be taken to imply a dyadic correspondence, like the kinds of “mirror image” correspondence between realities and representations bandied about in contemporary controversies about “correspondence theories of truth”.</li></ul>

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<li>Correspondence.  From the way ~~that~~ Peirce uses ~~this~~ term throughout his work, it is clear ~~that~~ he means what he elsewhere calls a “triple correspondence”, and thus this is just another way of referring to the whole triadic sign relation itself.  In particular, his use of ~~this~~ term should not be taken to imply a dyadic correspondence, like the kinds of “mirror image” correspondence between realities and representations ~~that are~~ bandied about in contemporary controversies about “correspondence theories of truth”.</li>

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<li>Determination.  Peirce's concept of determination is broader in several directions than the sense of the word ~~that refers~~ to strictly deterministic causal-temporal processes.  First, and especially in this context, he is invoking a more general concept of determination, what is called a formal or informational determination, as in saying “two points determine a line”, rather than the more special cases of causal and temporal determinisms.  Second, he characteristically allows for what is called determination in measure, that is, an order of determinism ~~that admits~~ a full spectrum of more and less determined relationships.</li>

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<ul><li>Determination.  Peirce's concept of determination is broader in several directions than the sense of the word referring to strictly deterministic causal‑temporal processes.  First, and especially in this context, he is invoking a more general concept of determination, what is called a formal or informational determination, as in saying “two points determine a line”, rather than the more special cases of causal and temporal determinisms.  Second, he characteristically allows for what is called determination in measure, that is, an order of determinism admitting a full spectrum of more and less determined relationships.</li></ul>

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<li>Non-psychological.  Peirce's “non-psychological conception of logic” must be distinguished from any variety of anti-psychologism.  He was quite interested in matters of psychology and had much of import to say about them.  But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a [[normative science]] where psychology is a [[descriptive science]], and so they have very different aims, methods, and rationales.</li>

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<ul><li>Non‑psychological.  Peirce's “non‑psychological conception of logic” must be distinguished from any variety of anti‑psychologism.  He was quite interested in matters of psychology and had much of import to say about them.  But logic and psychology operate on different planes of study even when they have occasion to view the same data, as logic is a normative science where psychology is a descriptive science, and so they have very different aims, methods, and rationales.</li></ul>

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==Signs and inquiry==

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: Main article : [[Inquiry]]

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<ul><li>Main article : [[Inquiry]]</li></ul>

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There is a close relationship between the pragmatic theory of signs and the pragmatic theory of [[inquiry]].  In fact, the correspondence between the two studies exhibits so many congruences and parallels ~~that~~ it is often best to treat them as integral parts of one and the same subject.  In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve.  In other words, inquiry, “thinking” in its best sense, “is a term denoting the various ways in which things acquire significance” (~~John~~ Dewey).~~  Thus, there is~~ an active ~~and~~ intricate form of cooperation ~~that needs to be appreciated and maintained~~ between ~~these~~ converging modes of investigation.  Its proper character is best understood by realizing ~~that~~ the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject ~~which~~ the theory of signs is specialized to treat from structural ~~and comparative~~ points of view.

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There is a close relationship between the pragmatic theory of signs and the pragmatic theory of inquiry.  In fact, the correspondence between the two studies exhibits so many congruences and parallels it is often best to treat them as integral parts of one and the same subject.  In a very real sense, inquiry is the process by which sign relations come to be established and continue to evolve.  In other words, inquiry, “thinking” in its best sense, “is a term denoting the various ways in which things acquire significance” (Dewey, 38).

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Tracing the passage of inquiry through the medium of signs calls for an active, intricate form of cooperation between the converging modes of investigation.  Its proper character is best understood by realizing the theory of inquiry is adapted to study the developmental aspects of sign relations, a subject the theory of signs is specialized to treat from comparative and structural points of view.

==Examples of sign relations==

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~~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.

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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 that aspect of their interpretive practice that involves the use of the following nouns and pronouns:  “Ann”, “Bob”, “I”, “you”.

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==Dyadic aspects of sign relations==

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For an arbitrary triadic relation <math>L \subseteq O \times S \times I,</math> whether it is a sign relation or not, there are six dyadic relations ~~that can be~~ obtained by projecting <math>L</math> on one of the planes of the <math>OSI</math>-space <math>O \times S \times I.</math>  The six dyadic projections of a triadic relation <math>L</math> are defined and notated as ~~follows:~~

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For an arbitrary triadic relation <math>L \subseteq O \times S \times I,</math> whether it happens to be a sign relation or not, there are six dyadic relations obtained by projecting <math>L</math> on one of the planes of the <math>OSI</math>‑space <math>O \times S \times I.</math>  The six dyadic projections of a triadic relation <math>L</math> are defined and notated as shown in Table 2.

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By way of unpacking the set-theoretic notation, here is what the first definition says in ordinary language.

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By way of unpacking the set‑theoretic notation, here is what the first definition says in ordinary language.

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The dyadic relation ~~that results~~ from the projection of <math>L</math> on the <math>OS</math>-plane <math>O \times S</math> is written briefly as <math>L_{OS}</math> or written more fully as <math>\mathrm{proj}_{OS}(L),</math> and it is defined as the set of all ordered pairs <math>(o, s)</math> in the cartesian product <math>O \times S</math> for which there exists an ordered triple <math>(o, s, i)</math> in <math>L</math> for some ~~interpretant~~ <math>i</math> in the ~~interpretant domain~~ <math>I.</math>

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The dyadic relation resulting from the projection of <math>L</math> on the <math>OS</math>‑plane <math>O \times S</math> is written briefly as <math>L_{OS}</math> or written more fully as <math>\mathrm{proj}_{OS}(L)</math> and is defined as the set of all ordered pairs <math>(o, s)</math> in the cartesian product <math>O \times S</math> for which there exists an ordered triple <math>(o, s, i)</math> in <math>L</math> for some element <math>i</math> in the set <math>I.</math>

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In the case where <math>L</math> is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of <math>L</math> can be recognized as formalizing aspects of sign meaning ~~that~~ 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~~.

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In the case where <math>L</math> is a sign relation, which it becomes by satisfying one of the definitions of a sign relation, some of the dyadic aspects of <math>L</math> 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.

===Denotation===

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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.

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The dyadic relation making up the denotative, referent, or semantic aspect of a sign relation <math>L</math> is notated as <math>\mathrm{Den}(L).</math>  Information about the denotative aspect of meaning is obtained from <math>L</math> by taking its projection on the object-sign plane.  We may ~~visualize this~~ as the “shadow” <math>L</math> casts on the 2-dimensional space whose axes are the object domain <math>O</math> and the sign domain <math>S.</math>  The denotative component of a sign relation <math>L,</math> ~~alternatively~~ written ~~in any of forms,~~ <math>\mathrm{proj}_{OS} L,</math>  <math>L_{OS},</math>  <math>\mathrm{proj}_{12} L,</math>  ~~and~~ <math>L_{12},</math> is defined as follows.

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The dyadic relation making up the denotative, referent, or semantic aspect of a sign relation <math>L</math> is notated as <math>\mathrm{Den}(L).</math>  Information about the denotative aspect of meaning is obtained from <math>L</math> by taking its projection on the object‑sign plane.  The result may be visualized as the “shadow” <math>L</math> casts on the 2‑dimensional space whose axes are the object domain <math>O</math> and the sign domain <math>S.</math>  The denotative component of a sign relation <math>L,</math> variously written as <math>\mathrm{proj}_{OS} L,</math>  <math>L_{OS},</math>  <math>\mathrm{proj}_{12} L,</math>  or <math>L_{12},</math> is defined as follows.

[[File:Sign Relation Display 3.png|550px]]

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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.

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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.

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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.

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Formally speaking, however, the connotative aspect of meaning presents no additional difficulty.  The dyadic relation making up the connotative aspect of a sign relation <math>L</math> is notated as <math>\mathrm{Con}(L).</math>  Information about the connotative aspect of meaning is obtained from <math>L</math> by taking its projection on the sign-interpretant plane~~.  We may visualize this~~ as the “shadow” <math>L</math> casts on the 2-dimensional space whose axes are the sign domain <math>S</math> and the interpretant domain <math>I.</math>  The connotative component of a sign relation <math>L,</math> ~~alternatively~~ written ~~in any of forms,~~ <math>\mathrm{proj}_{SI} L,</math>  <math>L_{SI},</math>  <math>\mathrm{proj}_{23} L,</math>  ~~and~~ <math>L_{23},</math> is defined as follows.

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Formally speaking, however, the connotative aspect of meaning presents no additional difficulty.  The dyadic relation making up the connotative aspect of a sign relation <math>L</math> is notated as <math>\mathrm{Con}(L).</math>  Information about the connotative aspect of meaning is obtained from <math>L</math> by taking its projection on the sign‑interpretant plane and visualized as the “shadow” <math>L</math> casts on the 2‑dimensional space whose axes are the sign domain <math>S</math> and the interpretant domain <math>I.</math>  The connotative component of a sign relation <math>L,</math> variously written as <math>\mathrm{proj}_{SI} L,</math>  <math>L_{SI},</math>  <math>\mathrm{proj}_{23} L,</math>  or <math>L_{23},</math> is defined as follows.

[[File:Sign Relation Display 4.png|550px]]

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===Ennotation===

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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.

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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.

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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 <math>L</math> may be notated as <math>\mathrm{Enn}(L).</math>  Information about the ennotative aspect of meaning is obtained from <math>L</math> by taking its projection on the object-interpretant plane~~.  We may visualize this~~ as the “shadow” <math>L</math> casts on the 2-dimensional space whose axes are the object domain <math>O</math> and the interpretant domain <math>I.</math>  The ennotative component of a sign relation <math>L,</math> ~~alternatively~~ written ~~in any of forms,~~ <math>\mathrm{proj}_{OI} L,</math>  <math>L_{OI},</math>  <math>\mathrm{proj}_{13} L,</math>  ~~and~~ <math>L_{13},</math> is defined as follows.

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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 <math>L</math> may be notated as <math>\mathrm{Enn}(L).</math>  Information about the ennotative aspect of meaning is obtained from <math>L</math> by taking its projection on the object‑interpretant plane and visualized as the “shadow” <math>L</math> casts on the 2‑dimensional space whose axes are the object domain <math>O</math> and the interpretant domain <math>I.</math>  The ennotative component of a sign relation <math>L,</math> variously written as <math>\mathrm{proj}_{OI} L,</math>  <math>L_{OI},</math>  <math>\mathrm{proj}_{13} L,</math>  or <math>L_{13},</math> is defined as follows.

[[File:Sign Relation Display 5.png|550px]]

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==Graphical representations==

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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.

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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.

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By way of terminology, a directed edge <math>(x, y)</math> is called an arc from point <math>x</math> to point <math>y,</math> and a self-loop <math>(x, x)</math> is called a sling at <math>x.</math>

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

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

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

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<dt>Denotative Component <math>\mathrm{Den}(L_\mathrm{A})</math></dt>

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<math>\mathrm{Den}(L_\mathrm{A})</math> has an arc from each point of <math>\{ \text{“A”}, \text{“i”} \}</math> to <math>\mathrm{A}</math> ~~and~~ an arc from each point of <math>\{ \text{“B”}, \text{“u”} \}</math> to <math>\mathrm{B}.</math>

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<dd><math>\mathrm{Den}(L_\mathrm{A})</math> has an arc from each point of <math>\{ \text{“A”}, \text{“i”} \}</math> to <math>\mathrm{A}.</math> <math>\mathrm{Den}(L_\mathrm{A})</math> has an arc from each point of <math>\{ \text{“B”}, \text{“u”} \}</math> to <math>\mathrm{B}.</math></dd>

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<dt>Denotative Component <math>\mathrm{Den}(L_\mathrm{B})</math></dt>

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<math>\mathrm{Den}(L_\mathrm{B})</math> has an arc from each point of <math>\{ \text{“A”}, \text{“u”} \}</math> to <math>\mathrm{A}</math> ~~and~~ an arc from each point of <math>\{ \text{“B”}, \text{“i”} \}</math> to <math>\mathrm{B}.</math>

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<dd><math>\mathrm{Den}(L_\mathrm{B})</math> has an arc from each point of <math>\{ \text{“A”}, \text{“u”} \}</math> to <math>\mathrm{A}.</math> <math>\mathrm{Den}(L_\mathrm{B})</math> has an arc from each point of <math>\{ \text{“B”}, \text{“i”} \}</math> to <math>\mathrm{B}.</math></dd>

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</dl>

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<math>\mathrm{Den}(L_\mathrm{A})</math> and <math>\mathrm{Den}(L_\mathrm{B})</math> 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 <math>\mathrm{A}</math> and <math>\mathrm{B}</math> evaluate the signs in <math>S</math> according to their own frames of reference.

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<math>\mathrm{~~Den~~}(L_\mathrm{A})</math> and <math>\mathrm{~~Den~~}(L_\mathrm{B})</math> 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 this way, the denotational arcs summarize the~~ ~~upshots~~ ~~of the computations involved when the interpreters~~ <math>\mathrm{A}</math> and <math>\mathrm{B}</math> ~~evaluate~~ the ~~signs in <math>S</math> according~~ to ~~their own frames~~ of ~~reference~~.

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The connotative components <math>\mathrm{Con}(L_\mathrm{A})</math> and <math>\mathrm{Con}(L_\mathrm{B})</math> can be represented as digraphs on the four points of their common syntactic domain <math>S = I =</math> <math>\{ \text{“A”}, \text{“B”}, \text{“i”}, \text{“u”} \}.</math>  Since <math>\mathrm{Con}(L_\mathrm{A})</math> and <math>\mathrm{Con}(L_\mathrm{B})</math> 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.

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~~The connotative components~~ <math>\mathrm{Con}(L_\mathrm{A})</math> ~~and~~ <math>\mathrm{Con}(L_\mathrm{B})</math> ~~can be represented as digraphs~~ on ~~the four points~~ of ~~their common syntactic domain~~ <math>S ~~= I =~~</math> <math>\{ \text{“A”}, \text{~~“B”~~}, \~~text{“i”~~}, ~~\text{“u”} \}.~~</math>~~  Since~~ <math>\~~mathrm~~{~~Con}(L_~~\~~mathrm~~{A}~~)</math> and <math>~~\~~mathrm~~{~~Con~~}~~(L_~~\~~mathrm{B~~})</math> 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 that 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:

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<dt>Connotative Component <math>\mathrm{Con}(L_\mathrm{A})</math></dt>

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<dd><math>\mathrm{Con}(L_\mathrm{A})</math> has the structure of a semiotic equivalence relation on <math>S.</math> There is a sling at each point of <math>S,</math> arcs in both directions between the points of <math>\{ \text{“A”}, \text{“i”} \},</math> and arcs in both directions between the points of <math>\{ \text{“B”}, \text{“u”} \}.</math></dd>

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<dt>Connotative Component <math>\mathrm{Con}(L_\mathrm{B})</math></dt>

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<dd><math>\mathrm{Con}(L_\mathrm{B})</math> has the structure of a semiotic equivalence relation on <math>S.</math> There is a sling at each point of <math>S,</math> arcs in both directions between the points of <math>\{ \text{“A”}, \text{“u”} \},</math> and arcs in both directions between the points of <math>\{ \text{“B”}, \text{“i”} \}.</math></dd>

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<math>\mathrm{Con}(L_\mathrm{A})</math> has the structure of a semiotic equivalence relation on <math>S,</math> with a sling at each point of <math>S,</math> arcs in both directions between the points of <math>\{ \text{“A”}, \text{“i”} \},</math> and arcs in both directions between the points of <math>\{ \text{“B”}, \text{“u”} \}.</~~math></p~~>

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</dl>

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<math>\mathrm{Con}(L_\mathrm{B})</math> has the structure of a semiotic equivalence relation on <math>S,</math> ~~with~~ a sling at each point of <math>S,</math> arcs in both directions between the points of <math>\{ \text{“A”}, \text{“u”} \},</math> and arcs in both directions between the points of <math>\{ \text{“B”}, \text{“i”} \}.</math>

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Taken as transition digraphs, <math>\mathrm{Con}(L_\mathrm{A})</math> and <math>\mathrm{Con}(L_\mathrm{B})</math> highlight the associations permitted between equivalent signs, as ~~this~~ equivalence is judged by the interpreters <math>\mathrm{A}</math> and <math>\mathrm{B},</math> ~~respectively.~~

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

==Six ways of looking at a sign relation==

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==References==

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* Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  [https://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive].  [https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052 Journal].  [https://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry ~~Online~~].

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* Awbrey, J.L., and Awbrey, S.M. (Autumn 1995), “Interpretation as Action : The Risk of Inquiry”, Inquiry : Critical Thinking Across the Disciplines 15(1), pp. 40–52.  [https://web.archive.org/web/19970626071826/http://chss.montclair.edu/inquiry/fall95/awbrey.html Archive].  [https://www.pdcnet.org/inquiryct/content/inquiryct_1995_0015_0001_0040_0052 Journal].  Online [https://independent.academia.edu/JonAwbrey/Papers/1302117/Interpretation_as_Action_The_Risk_of_Inquiry (doc)] [https://www.academia.edu/57812482/Interpretation_as_Action_The_Risk_of_Inquiry (pdf)].

* Deledalle, Gérard (2000), C.S. Peirce's Philosophy of Signs, Indiana University Press, Bloomington, IN.

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* Dewey, John. (1910), How We Think, D.C. Heath, Boston, MA. Reprinted (1991), Prometheus Books, Buffalo, NY. [https://www.gutenberg.org/files/37423/37423-h/37423-h.htm Online].

* Eisele, Carolyn (1979), in Studies in the Scientific and Mathematical Philosophy of C.S. Peirce, Richard Milton Martin (ed.), Mouton, The Hague.

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* Murphey, M. (1961), The Development of Peirce's Thought.  Reprinted, Hackett, Indianapolis, IN, 1993.

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* 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.  [https://cspeirce.com/menu/library/bycsp/l75/l75.htm Online].

* Percy, Walker (2000), pp. 271–291 in Signposts in a Strange Land, P. Samway (ed.), Saint Martin's Press.

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* [https://en.wikipedia.org/w/index.php?title=Sign_relation&oldid=161631069 Sign Relation], [https://en.wikipedia.org/ Wikipedia]

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Sign relation (view source)

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