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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.

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.

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

==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.

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.

Already in this elementary context, there are several different meanings that might attach to the project of a <i>formal semiotics</i>, or a formal theory of meaning for signs.&nbsp; In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions.

Already in this elementary context, there are several different meanings that might attach to the project of a <i>formal semiotics</i>, or a formal theory of meaning for signs.&nbsp; In the process of discussing these alternatives, it is useful to introduce a few terms that are occasionally used in the philosophy of language to point out the needed distinctions.

=={{anchor|Dyadic Aspects}}Dyadic aspects of sign relations==

==Dyadic aspects of sign relations==

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 <i>projecting</i> <math>L</math> on one of the planes of the <math>OSI</math>&#8209;space <math>O \times S \times I.</math>&nbsp; The six dyadic projections of a triadic relation <math>L</math> are defined and notated as shown in Table&nbsp;2.

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 <i>projecting</i> <math>L</math> on one of the planes of the <math>OSI</math>&#8209;space <math>O \times S \times I.</math>&nbsp; The six dyadic projections of a triadic relation <math>L</math> are defined and notated as shown in Table&nbsp;2.

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=={{anchor|Semiotic Equivalence Relations 1}}Semiotic equivalence relations==

==Semiotic equivalence relations==

A <i>semiotic equivalence relation</i> (SER) is a special type of equivalence relation arising in the analysis of sign relations.&nbsp; As a general rule, any equivalence relation is closely associated with a family of equivalence classes which partition the underlying set of elements, frequently called the <i>domain</i> or <i>space</i> of the relation.&nbsp; In the case of a SER, the equivalence classes are called <i>semiotic equivalence classes</i> (SECs) and the partition is called a <i>semiotic partition</i> (SEP).

A <i>semiotic equivalence relation</i> (SER) is a special type of equivalence relation arising in the analysis of sign relations.&nbsp; As a general rule, any equivalence relation is closely associated with a family of equivalence classes which partition the underlying set of elements, frequently called the <i>domain</i> or <i>space</i> of the relation.&nbsp; In the case of a SER, the equivalence classes are called <i>semiotic equivalence classes</i> (SECs) and the partition is called a <i>semiotic partition</i> (SEP).

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

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

==Graphical representations==

==Graphical representations==

The dyadic components of sign relations have graph-theoretic representations, as <i>digraphs</i> (or <i>directed graphs</i>), which provide concise pictures of their structural and potential dynamic properties.

The dyadic components of sign relations have graph&#8209;theoretic representations, as <i>digraphs</i> (or <i>directed graphs</i>), which provide concise pictures of their structural and potential dynamic properties.

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

By way of terminology, a directed edge <math>(x, y)</math> is called an <i>arc</i> from point <math>x</math> to point <math>y,</math> and a self&#8209;loop <math>(x, x)</math> is called a <i>sling</i> at <math>x.</math>

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>&nbsp; The arcs are given as follows:

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>&nbsp; The arcs are given as follows.

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

<dl style="margin-left:28px">

<p><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></p>

<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> <br> <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>

 

<p><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></p>

<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> <br> <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>

 

<math>\mathrm{Den}(L_\mathrm{A})</math> and <math>\mathrm{Den}(L_\mathrm{B})</math> can be interpreted as <i>transition digraphs</i> which chart the succession of steps or the connection of states in a computational process.&nbsp; If the graphs are read in that way, the denotational arcs summarize the <i>upshots</i> 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.

<math>\mathrm{Den}(L_\mathrm{A})</math> and <math>\mathrm{Den}(L_\mathrm{B})</math> can be interpreted as <i>transition digraphs</i> which chart the succession of steps or the connection of states in a computational process.&nbsp; If the graphs are read this way, the denotational arcs summarize the <i>upshots</i> 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.

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>&nbsp; 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.&nbsp; 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.&nbsp; In the present case, the arcs are given as follows.

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>&nbsp; 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.&nbsp; 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.&nbsp; In the present case, the arcs are given as follows:

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

<dd><math>\mathrm{Con}(L_\mathrm{A})</math> has the structure of a semiotic equivalence relation on <math>S.</math> <br> 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>

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

<dd><math>\mathrm{Con}(L_\mathrm{B})</math> has the structure of a semiotic equivalence relation on <math>S.</math> <br> 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>

<p><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>

<p><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></p>

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.

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==

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