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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.

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==~~{{anchor|Examples}}~~Examples of sign relations==

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==Examples of sign relations==

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.

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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 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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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.

{| align="center" cellpadding="0" cellspacing="0" style="text-align:center"

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{| align="center" cellpadding="6" width="90%"

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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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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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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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~~{| align~~="~~center" cellspacing="6" width="90%~~"

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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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~~{| align="center" cellspacing="6" width="90%"~~

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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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[[Category:Cognitive science]]

[[Category:Computer science]]

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[[Category:Differential logic]]

[[Category:Graph theory]]

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[[Category:Hermeneutics]]

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[[Category:Information systems]]

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[[Category:Information theory]]

[[Category:Inquiry]]

[[Category:Intelligent systems]]

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[[Category:Knowledge representation]]

[[Category:Logic]]

[[Category:Logical graphs]]

Jon Awbrey

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Sign relation (view source)

Revision as of 14:08, 16 January 2026