Difference between revisions of "Triadic relation"

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<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
 
<font size="3">&#9758;</font> This page belongs to resource collections on [[Logic Live|Logic]] and [[Inquiry Live|Inquiry]].
  
−
In [[logic]], [[mathematics]], and [[semiotics]], a '''triadic relation''' is an important special case of a [[relation (mathematics)|polyadic or finitary relation]], one in which the number of places in the relation is three.  In other language that is often used, a triadic relation is called a '''ternary relation'''.  One may also see the adjectives ''3-adic'', ''3-ary'', ''3-dimensional'', or ''3-place'' being used to describe these relations.
+
In logic, mathematics, and semiotics, a '''triadic relation''' is an important special case of a [[relation (mathematics)|polyadic or finitary relation]], one in which the number of places in the relation is three.  In other language that is often used, a triadic relation is called a '''ternary relation'''.  One may also see the adjectives ''3-adic'', ''3-ary'', ''3-dimensional'', or ''3-place'' being used to describe these relations.
  
 
Mathematics is positively rife with examples of 3-adic relations, and a [[sign relation]], the arch-idea of the whole field of semiotics, is a special case of a 3-adic relation.  Therefore it will be useful to consider a few concrete examples from each of these two realms.
 
Mathematics is positively rife with examples of 3-adic relations, and a [[sign relation]], the arch-idea of the whole field of semiotics, is a special case of a 3-adic relation.  Therefore it will be useful to consider a few concrete examples from each of these two realms.
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The first order of business is to define the space in which the relations <math>L_0\!</math> and <math>L_1\!</math> take up residence.  This space is constructed as a 3-fold [[cartesian power]] in the following way.
 
The first order of business is to define the space in which the relations <math>L_0\!</math> and <math>L_1\!</math> take up residence.  This space is constructed as a 3-fold [[cartesian power]] in the following way.
  
−
The ''[[boolean domain]]'' is the set <math>\mathbb{B} = \{ 0, 1 \}.</math>
+
The ''[[boolean domain]]'' is the set <math>\mathbb{B} = \{ 0, 1 \}.\!</math>
  
−
The ''plus sign'' <math>^{\backprime\backprime} + ^{\prime\prime},</math> used in the context of the boolean domain <math>\mathbb{B},</math> denotes addition modulo 2.  Interpreted for logic, the plus sign can be used to indicate either the boolean operation of ''[[exclusive disjunction]]'', <math>\operatorname{XOR} : \mathbb{B} \times \mathbb{B} \to \mathbb{B},</math> or the boolean relation of ''logical inequality'', <math>\operatorname{NEQ} \subseteq \mathbb{B} \times \mathbb{B}.</math>
+
The ''plus sign'' <math>{}^{\backprime\backprime} + {}^{\prime\prime},\!</math> used in the context of the boolean domain <math>\mathbb{B},\!</math> denotes addition modulo 2.  Interpreted for logic, the plus sign can be used to indicate either the boolean operation of ''[[exclusive disjunction]]'', <math>\mathrm{XOR} : \mathbb{B} \times \mathbb{B} \to \mathbb{B},\!</math> or the boolean relation of ''logical inequality'', <math>\mathrm{NEQ} \subseteq \mathbb{B} \times \mathbb{B}.\!</math>
  
−
The third cartesian power of <math>\mathbb{B}</math> is the set <math>\mathbb{B}^3 = \mathbb{B} \times \mathbb{B} \times \mathbb{B} = \{ (x_1, x_2, x_3) : x_j \in \mathbb{B} ~\text{for}~ j = 1, 2, 3 \}.</math>
+
The third cartesian power of <math>\mathbb{B}\!</math> is the set <math>\mathbb{B}^3 = \mathbb{B} \times \mathbb{B} \times \mathbb{B} = \{ (x_1, x_2, x_3) : x_j \in \mathbb{B} ~\text{for}~ j = 1, 2, 3 \}.\!</math>
  
−
In what follows, the space <math>X \times Y \times Z</math> is isomorphic to <math>\mathbb{B} \times \mathbb{B} \times \mathbb{B} ~=~ \mathbb{B}^3.</math>
+
In what follows, the space <math>X \times Y \times Z\!</math> is isomorphic to <math>\mathbb{B} \times \mathbb{B} \times \mathbb{B} ~=~ \mathbb{B}^3.\!</math>
  
 
The relation <math>L_0\!</math> is defined as follows:
 
The relation <math>L_0\!</math> is defined as follows:
  
−
: <math>L_0 = \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.</math>
+
{| align="center" cellpadding="6" width="90%"
 +
| <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.~\!</math>
 +
|}
  
 
The relation <math>L_0\!</math> is the set of four triples enumerated here:
 
The relation <math>L_0\!</math> is the set of four triples enumerated here:
  
−
: <math>L_0 = \{ (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) \}.\!</math>
+
{| align="center" cellpadding="6" width="90%"
 +
| <math>L_0 ~=~ \{ (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) \}.\!</math>
 +
|}
  
 
The relation <math>L_1\!</math> is defined as follows:
 
The relation <math>L_1\!</math> is defined as follows:
  
−
: <math>L_1 = \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.</math>
+
{| align="center" cellpadding="6" width="90%"
 +
| <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.~\!</math>
 +
|}
  
 
The relation <math>L_1\!</math> is the set of four triples enumerated here:
 
The relation <math>L_1\!</math> is the set of four triples enumerated here:
  
−
: <math>L_1 = \{ (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1) \}.\!</math>
+
{| align="center" cellpadding="6" width="90%"
 +
| <math>L_1 ~=~ \{ (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1) \}.\!</math>
 +
|}
  
−
The triples that make up the relations <math>L_0\!</math> and <math>L_1\!</math> are conveniently arranged in the form of ''[[relational database|relational data tables]]'', as follows:
+
The triples that make up the relations <math>L_0\!</math> and <math>L_1\!</math> are conveniently arranged in the form of ''relational data tables'', as follows:
  
 
<br>
 
<br>
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:60%"
+
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
−
|+ <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}</math>
+
|+ style="height:30px" | <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!</math>
−
|- style="background:#e6e6ff"
+
|- style="height:40px; background:ghostwhite"
−
! <math>X\!</math> !! <math>Y\!</math> !! <math>Z\!</math>
+
| <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math>
 
|-
 
|-
 
| <math>0\!</math> || <math>0\!</math> || <math>0\!</math>
 
| <math>0\!</math> || <math>0\!</math> || <math>0\!</math>
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<br>
 
<br>
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:60%"
+
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
−
|+ <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}</math>
+
|+ style="height:30px" | <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!</math>
−
|- style="background:#e6e6ff"
+
|- style="height:40px; background:ghostwhite"
−
! <math>X\!</math> !! <math>Y\!</math> !! <math>Z\!</math>
+
| <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math>
 
|-
 
|-
 
| <math>0\!</math> || <math>0\!</math> || <math>1\!</math>
 
| <math>0\!</math> || <math>0\!</math> || <math>1\!</math>
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The term ''[[semiosis]]'' refers to any activity or process that involves signs.  Studies of semiosis that deal with its more abstract form are not concerned with every concrete detail of the entities that act as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among these three roles.  In particular, the formal theory of signs does not consider all of the properties of the interpretive agent but only the more striking features of the impressions that signs make on a representative interpreter.  In its formal aspects, that impact or influence may be treated as just another sign, called the ''interpretant sign'', or the ''interpretant'' for short.  Such a 3-adic relation, among objects, signs, and interpretants, is called a ''[[sign relation]]''.
 
The term ''[[semiosis]]'' refers to any activity or process that involves signs.  Studies of semiosis that deal with its more abstract form are not concerned with every concrete detail of the entities that act as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among these three roles.  In particular, the formal theory of signs does not consider all of the properties of the interpretive agent but only the more striking features of the impressions that signs make on a representative interpreter.  In its formal aspects, that impact or influence may be treated as just another sign, called the ''interpretant sign'', or the ''interpretant'' for short.  Such a 3-adic relation, among objects, signs, and interpretants, is called a ''[[sign relation]]''.
  
−
For example, consider the aspects of sign use that concern two people &mdash; let us say <math>\operatorname{Ann}</math> and <math>\operatorname{Bob}\!</math> &mdash; in using their own proper names, <math>^{\backprime\backprime} \operatorname{Ann} ^{\prime\prime}</math> and <math>^{\backprime\backprime} \operatorname{Bob} ^{\prime\prime},</math> together with the pronouns, <math>^{\backprime\backprime} \operatorname{I} ^{\prime\prime}</math> and <math>^{\backprime\backprime} \operatorname{you} ^{\prime\prime}.</math>  For brevity, these four signs may be abbreviated to the set <math>\{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \}.</math>  The abstract consideration of how <math>\operatorname{A}</math> and <math>\operatorname{B}</math> use this set of signs to refer to themselves and each other leads to the contemplation of a pair of 3-adic relations, the sign relations <math>L_\operatorname{A}</math> and <math>L_\operatorname{B},</math> that reflect the differential use of these signs by <math>\operatorname{A}</math> and <math>\operatorname{B},</math> respectively.
+
For example, consider the aspects of sign use that concern two people &mdash; let us say <math>\mathrm{Ann}\!</math> and <math>\mathrm{Bob}\!</math> &mdash; in using their own proper names, <math>{}^{\backprime\backprime} \mathrm{Ann} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathrm{Bob} {}^{\prime\prime},\!</math> together with the pronouns, <math>{}^{\backprime\backprime} \mathrm{I} {}^{\prime\prime}\!</math> and <math>{}^{\backprime\backprime} \mathrm{you} {}^{\prime\prime}.\!</math>  For brevity, these four signs may be abbreviated to the set <math>\{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\!</math>  The abstract consideration of how <math>\mathrm{A}\!</math> and <math>\mathrm{B}\!</math> use this set of signs to refer to themselves and each other leads to the contemplation of a pair of 3-adic relations, the sign relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> that reflect the differential use of these signs by <math>\mathrm{A}\!</math> and <math>\mathrm{B},\!</math> respectively.
  
−
Each of the sign relations, <math>L_\operatorname{A}</math> and <math>L_\operatorname{B},</math> consists of eight triples of the form <math>(x, y, z),\!</math> where the ''object'' <math>x\!</math> is an element of the ''object domain'' <math>O = \{ \operatorname{A}, \operatorname{B} \},</math> where the ''sign'' <math>y\!</math> is an element of the ''sign domain'' <math>S\!,</math> where the ''interpretant sign'' <math>z\!</math> is an element of the interpretant domain <math>I,\!</math> and where it happens in this case that <math>S = I = \{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \}.</math>  In general, it is convenient to refer to the union <math>S \cup I</math> as the ''syntactic domain'', but in this case <math>S ~=~ I ~=~ S \cup I.</math>
+
Each of the sign relations, <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B},\!</math> consists of eight triples of the form <math>(x, y, z),\!</math> where the ''object'' <math>x\!</math> is an element of the ''object domain'' <math>O = \{ \mathrm{A}, \mathrm{B} \},\!</math> where the ''sign'' <math>y\!</math> is an element of the ''sign domain'' <math>S\!,</math> where the ''interpretant sign'' <math>z\!</math> is an element of the interpretant domain <math>I,\!</math> and where it happens in this case that <math>S = I = \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\!</math>  In general, it is convenient to refer to the union <math>S \cup I\!</math> as the ''syntactic domain'', but in this case <math>S ~=~ I ~=~ S \cup I.\!</math>
  
 
The set-up so far is summarized as follows:
 
The set-up so far is summarized as follows:
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|
 
|
 
<math>\begin{array}{ccc}
 
<math>\begin{array}{ccc}
−
L_\operatorname{A}, L_\operatorname{B} & \subseteq & O \times S \times I \\
+
L_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I \\
 
\\
 
\\
−
O & = & \{ \operatorname{A}, \operatorname{B} \} \\
+
O & = & \{ \mathrm{A}, \mathrm{B} \} \\
 
\\
 
\\
−
S & = & \{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \} \\
+
S & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\
 
\\
 
\\
−
I & = & \{ \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime} \, \} \\
+
I & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\
 
\\
 
\\
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
  
−
The relation <math>L_\operatorname{A}</math> is the set of eight triples enumerated here:
+
The relation <math>L_\mathrm{A}\!</math> is the set of eight triples enumerated here:
  
 
{| align="center" cellpadding="8" width="90%"
 
{| align="center" cellpadding="8" width="90%"
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<math>\begin{array}{cccccc}
 
<math>\begin{array}{cccccc}
 
\{ &
 
\{ &
−
(\operatorname{A}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}), &
+
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &
−
(\operatorname{A}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}), &
+
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), &
−
(\operatorname{A}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}), &
+
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &
−
(\operatorname{A}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}), &
+
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), &
 
\\
 
\\
 
&
 
&
−
(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}), &
+
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &
−
(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}), &
+
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), &
−
(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}), &
+
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &
−
(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}) &
+
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}) &
 
\}.
 
\}.
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
  
−
The triples in <math>L_\operatorname{A}</math> represent the way that interpreter <math>\operatorname{A}</math> uses signs.  For example, the listing of the triple <math>(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime})</math> in <math>L_\operatorname{A}</math> represents the fact that <math>\operatorname{A}</math> uses <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math> to mean the same thing that <math>\operatorname{A}</math> uses <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math> to mean, namely, <math>\operatorname{B}.</math>
+
The triples in <math>L_\mathrm{A}\!</math> represent the way that interpreter <math>\mathrm{A}\!</math> uses signs.  For example, the listing of the triple <math>(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\!</math> in <math>L_\mathrm{A}\!</math> represents the fact that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> to mean the same thing that <math>\mathrm{A}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math> to mean, namely, <math>\mathrm{B}.\!</math>
  
−
The relation <math>L_\operatorname{B}</math> is the set of eight triples enumerated here:
+
The relation <math>L_\mathrm{B}\!</math> is the set of eight triples enumerated here:
  
 
{| align="center" cellpadding="8" width="90%"
 
{| align="center" cellpadding="8" width="90%"
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<math>\begin{array}{cccccc}
 
<math>\begin{array}{cccccc}
 
\{ &
 
\{ &
−
(\operatorname{A}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}), &
+
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &
−
(\operatorname{A}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}), &
+
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), &
−
(\operatorname{A}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{A} ^{\prime\prime}), &
+
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), &
−
(\operatorname{A}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{u} ^{\prime\prime}), &
+
(\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), &
 
\\
 
\\
 
&
 
&
−
(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}), &
+
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &
−
(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}), &
+
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), &
−
(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime}), &
+
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), &
−
(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}) &
+
(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}) &
 
\}.
 
\}.
 
\end{array}</math>
 
\end{array}</math>
 
|}
 
|}
  
−
The triples in <math>L_\operatorname{B}</math> represent the way that interpreter <math>\operatorname{B}</math> uses signs.  For example, the listing of the triple <math>(\operatorname{B}, \, ^{\backprime\backprime} \operatorname{i} ^{\prime\prime}, \, ^{\backprime\backprime} \operatorname{B} ^{\prime\prime})</math> in <math>L_\operatorname{B}</math> represents the fact that <math>\operatorname{B}</math> uses <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math> to mean the same thing that <math>\operatorname{B}</math> uses <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math> to mean, namely, <math>\operatorname{B}.</math>
+
The triples in <math>L_\mathrm{B}\!</math> represent the way that interpreter <math>\mathrm{B}\!</math> uses signs.  For example, the listing of the triple <math>(\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\!</math> in <math>L_\mathrm{B}\!</math> represents the fact that <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math> to mean the same thing that <math>\mathrm{B}\!</math> uses <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math> to mean, namely, <math>\mathrm{B}.\!</math>
  
−
The triples that make up the relations <math>L_\operatorname{A}</math> and <math>L_\operatorname{B}</math> are conveniently arranged in the form of ''[[relational database|relational data tables]]'', as follows:
+
The triples that make up the relations <math>L_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are conveniently arranged in the form of ''relational data tables'', as follows:
  
 
<br>
 
<br>
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:60%"
+
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
−
|+ <math>L_\operatorname{A} ~=~ \operatorname{Sign~Relation~of~Interpreter~A}</math>
+
|+ style="height:30px" | <math>L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\!</math>
−
|- style="background:#e6e6ff"
+
|- style="height:40px; background:ghostwhite"
−
! style="width:33%" | <math>\operatorname{Object}</math>
+
| style="width:33%" | <math>\text{Object}\!</math>
−
! style="width:33%" | <math>\operatorname{Sign}</math>
+
| style="width:33%" | <math>\text{Sign}\!</math>
−
! style="width:33%" | <math>\operatorname{Interpretant}</math>
+
| style="width:33%" | <math>\text{Interpretant}\!</math>
 
|-
 
|-
−
| <math>\operatorname{A}</math>
+
| <math>\mathrm{A}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
 
|-
 
|-
−
| <math>\operatorname{A}</math>
+
| <math>\mathrm{A}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
 
|-
 
|-
−
| <math>\operatorname{A}</math>
+
| <math>\mathrm{A}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
 
|-
 
|-
−
| <math>\operatorname{A}</math>
+
| <math>\mathrm{A}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
 
|-
 
|-
−
| <math>\operatorname{B}</math>
+
| <math>\mathrm{B}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
 
|-
 
|-
−
| <math>\operatorname{B}</math>
+
| <math>\mathrm{B}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
 
|-
 
|-
−
| <math>\operatorname{B}</math>
+
| <math>\mathrm{B}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
 
|-
 
|-
−
| <math>\operatorname{B}</math>
+
| <math>\mathrm{B}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
 
|}
 
|}
  
 
<br>
 
<br>
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:60%"
+
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
−
|+ <math>L_\operatorname{B} ~=~ \operatorname{Sign~Relation~of~Interpreter~B}</math>
+
|+ style="height:30px" | <math>L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\!</math>
−
|- style="background:#e6e6ff"
+
|- style="height:40px; background:ghostwhite"
−
! style="width:33%" | <math>\operatorname{Object}</math>
+
| style="width:33%" | <math>\text{Object}\!</math>
−
! style="width:33%" | <math>\operatorname{Sign}</math>
+
| style="width:33%" | <math>\text{Sign}\!</math>
−
! style="width:33%" | <math>\operatorname{Interpretant}</math>
+
| style="width:33%" | <math>\text{Interpretant}\!</math>
 
|-
 
|-
−
| <math>\operatorname{A}</math>
+
| <math>\mathrm{A}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
 
|-
 
|-
−
| <math>\operatorname{A}</math>
+
| <math>\mathrm{A}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
 
|-
 
|-
−
|<math>\operatorname{A}</math>
+
|<math>\mathrm{A}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{A} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
 
|-
 
|-
−
| <math>\operatorname{A}</math>
+
| <math>\mathrm{A}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{u} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
 
|-
 
|-
−
| <math>\operatorname{B}</math>
+
| <math>\mathrm{B}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
 
|-
 
|-
−
| <math>\operatorname{B}</math>
+
| <math>\mathrm{B}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
 
|-
 
|-
−
| <math>\operatorname{B}</math>
+
| <math>\mathrm{B}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{B} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
 
|-
 
|-
−
| <math>\operatorname{B}</math>
+
| <math>\mathrm{B}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
−
| <math>^{\backprime\backprime} \operatorname{i} ^{\prime\prime}</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
 
|}
 
|}
  
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===Focal nodes===
 
===Focal nodes===
  
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* [[Inquiry Live]]
 
* [[Inquiry Live]]
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* [[Logic Live]]
 
* [[Logic Live]]
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===Peer nodes===
 
===Peer nodes===
  
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{{col-begin}}
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* [http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation Triadic Relation @ InterSciWiki]
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* [http://mywikibiz.com/Triadic_relation Triadic Relation @ MyWikiBiz]
 
* [http://mywikibiz.com/Triadic_relation Triadic Relation @ MyWikiBiz]
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* [http://mathweb.org/wiki/Triadic_relation Triadic Relation @ MathWeb Wiki]
 
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* [http://netknowledge.org/wiki/Triadic_relation Triadic Relation @ NetKnowledge]
 
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* [http://wiki.oercommons.org/mediawiki/index.php/Triadic_relation Triadic Relation @ OER Commons]
 
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* [http://p2pfoundation.net/Triadic_Relation Triadic Relation @ P2P Foundation]
 
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* [http://semanticweb.org/wiki/Triadic_relation Triadic Relation @ SemanticWeb]
 
 
* [http://ref.subwiki.org/wiki/Triadic_relation Triadic Relation @ Subject Wikis]
 
* [http://ref.subwiki.org/wiki/Triadic_relation Triadic Relation @ Subject Wikis]
 +
* [http://en.wikiversity.org/wiki/Triadic_relation Triadic Relation @ Wikiversity]
 
* [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation @ Wikiversity Beta]
 
* [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation @ Wikiversity Beta]
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===Logical operators===
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===Related articles===
 
===Related articles===
  
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Semiotic_Information Jon Awbrey, &ldquo;Semiotic Information&rdquo;]
+
{{col-begin}}
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Introduction_to_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Introduction To Inquiry Driven Systems&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Cactus_Language Cactus Language]
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* [http://intersci.ss.uci.edu/wiki/index.php/Futures_Of_Logical_Graphs Futures Of Logical Graphs]
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Essays/Prospects_For_Inquiry_Driven_Systems Jon Awbrey, &ldquo;Prospects For Inquiry Driven Systems&rdquo;]
+
* [http://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Inquiry_Driven_Systems Jon Awbrey, &ldquo;Inquiry Driven Systems : Inquiry Into Inquiry&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_:_Introduction Differential Logic : Introduction]
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+
* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Propositional_Calculus Differential Propositional Calculus]
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems Jon Awbrey, &ldquo;Propositional Equation Reasoning Systems&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
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+
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_:_Introduction Jon Awbrey, &ldquo;Differential Logic : Introduction&rdquo;]
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* [http://intersci.ss.uci.edu/wiki/index.php/Introduction_to_Inquiry_Driven_Systems Introduction to Inquiry Driven Systems]
−
 
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* [http://intersci.ss.uci.edu/wiki/index.php/Prospects_for_Inquiry_Driven_Systems Prospects for Inquiry Driven Systems]
−
* [http://planetmath.org/encyclopedia/DifferentialPropositionalCalculus.html Jon Awbrey, &ldquo;Differential Propositional Calculus&rdquo;]
+
* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
−
 
+
{{col-end}}
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* [http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0 Jon Awbrey, &ldquo;Differential Logic and Dynamic Systems&rdquo;]
 
  
 
==Document history==
 
==Document history==
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Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.
 
Portions of the above article were adapted from the following sources under the [[GNU Free Documentation License]], under other applicable licenses, or by permission of the copyright holders.
  
−
{{col-begin}}
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* [http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation Triadic Relation], [http://intersci.ss.uci.edu/ InterSciWiki]
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{{col-break}}
 
 
* [http://mywikibiz.com/Triadic_relation Triadic Relation], [http://mywikibiz.com/ MyWikiBiz]
 
* [http://mywikibiz.com/Triadic_relation Triadic Relation], [http://mywikibiz.com/ MyWikiBiz]
−
* [http://mathweb.org/wiki/Triadic_relation Triadic Relation], [http://mathweb.org/ MathWeb Wiki]
+
* [http://planetmath.org/TriadicRelation Triadic Relation], [http://planetmath.org/ PlanetMath]
−
* [http://netknowledge.org/wiki/Triadic_relation Triadic Relation], [http://netknowledge.org/ NetKnowledge]
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* [http://en.wikiversity.org/wiki/Triadic_relation Triadic Relation], [http://en.wikiversity.org/ Wikiversity]
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* [http://wiki.oercommons.org/mediawiki/index.php/Triadic_relation Triadic Relation], [http://wiki.oercommons.org/ OER Commons]
 
−
* [http://p2pfoundation.net/Triadic_Relation Triadic Relation], [http://p2pfoundation.net/ P2P Foundation]
 
−
* [http://semanticweb.org/wiki/Triadic_relation Triadic Relation], [http://semanticweb.org/ SemanticWeb]
 
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{{col-break}}
 
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* [http://planetmath.org/encyclopedia/TriadicRelation.html Triadic Relation], [http://planetmath.org/ PlanetMath]
 
 
* [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation], [http://beta.wikiversity.org/ Wikiversity Beta]
 
* [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation], [http://beta.wikiversity.org/ Wikiversity Beta]
−
* [http://getwiki.net/-Triadic_Relation Triadic Relation], [http://getwiki.net/ GetWiki]
 
−
* [http://wikinfo.org/index.php/Triadic_relation Triadic Relation], [http://wikinfo.org/ Wikinfo]
 
−
* [http://textop.org/wiki/index.php?title=Triadic_relation Triadic Relation], [http://textop.org/wiki/ Textop Wiki]
 
 
* [http://en.wikipedia.org/w/index.php?title=Triadic_relation&oldid=108548758 Triadic Relation], [http://en.wikipedia.org/ Wikipedia]
 
* [http://en.wikipedia.org/w/index.php?title=Triadic_relation&oldid=108548758 Triadic Relation], [http://en.wikipedia.org/ Wikipedia]
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[[Category:Syntax]]

Revision as of 22:00, 15 November 2015

☞ This page belongs to resource collections on Logic and Inquiry.

In logic, mathematics, and semiotics, a triadic relation is an important special case of a polyadic or finitary relation, one in which the number of places in the relation is three. In other language that is often used, a triadic relation is called a ternary relation. One may also see the adjectives 3-adic, 3-ary, 3-dimensional, or 3-place being used to describe these relations.

Mathematics is positively rife with examples of 3-adic relations, and a sign relation, the arch-idea of the whole field of semiotics, is a special case of a 3-adic relation. Therefore it will be useful to consider a few concrete examples from each of these two realms.

Examples from mathematics

For the sake of topics to be taken up later, it is useful to examine a pair of 3-adic relations in tandem, \(L_0\!\) and \(L_1,\!\) that can be described in the following manner.

The first order of business is to define the space in which the relations \(L_0\!\) and \(L_1\!\) take up residence. This space is constructed as a 3-fold cartesian power in the following way.

The boolean domain is the set \(\mathbb{B} = \{ 0, 1 \}.\!\)

The plus sign \({}^{\backprime\backprime} + {}^{\prime\prime},\!\) used in the context of the boolean domain \(\mathbb{B},\!\) denotes addition modulo 2. Interpreted for logic, the plus sign can be used to indicate either the boolean operation of exclusive disjunction, \(\mathrm{XOR} : \mathbb{B} \times \mathbb{B} \to \mathbb{B},\!\) or the boolean relation of logical inequality, \(\mathrm{NEQ} \subseteq \mathbb{B} \times \mathbb{B}.\!\)

The third cartesian power of \(\mathbb{B}\!\) is the set \(\mathbb{B}^3 = \mathbb{B} \times \mathbb{B} \times \mathbb{B} = \{ (x_1, x_2, x_3) : x_j \in \mathbb{B} ~\text{for}~ j = 1, 2, 3 \}.\!\)

In what follows, the space \(X \times Y \times Z\!\) is isomorphic to \(\mathbb{B} \times \mathbb{B} \times \mathbb{B} ~=~ \mathbb{B}^3.\!\)

The relation \(L_0\!\) is defined as follows:

\(L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.~\!\)

The relation \(L_0\!\) is the set of four triples enumerated here:

\(L_0 ~=~ \{ (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) \}.\!\)

The relation \(L_1\!\) is defined as follows:

\(L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.~\!\)

The relation \(L_1\!\) is the set of four triples enumerated here:

\(L_1 ~=~ \{ (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1) \}.\!\)

The triples that make up the relations \(L_0\!\) and \(L_1\!\) are conveniently arranged in the form of relational data tables, as follows:


\(L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!\)
\(X\!\) \(Y\!\) \(Z\!\)
\(0\!\) \(0\!\) \(0\!\)
\(0\!\) \(1\!\) \(1\!\)
\(1\!\) \(0\!\) \(1\!\)
\(1\!\) \(1\!\) \(0\!\)


\(L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!\)
\(X\!\) \(Y\!\) \(Z\!\)
\(0\!\) \(0\!\) \(1\!\)
\(0\!\) \(1\!\) \(0\!\)
\(1\!\) \(0\!\) \(0\!\)
\(1\!\) \(1\!\) \(1\!\)


Examples from semiotics

The study of signs — the full variety of significant forms of expression — in relation to the things that signs are significant of, and in relation to the beings that signs are significant to, is known as semiotics or the theory of signs. As just described, semiotics treats of a 3-place relation among signs, their objects, and their interpreters.

The term semiosis refers to any activity or process that involves signs. Studies of semiosis that deal with its more abstract form are not concerned with every concrete detail of the entities that act as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among these three roles. In particular, the formal theory of signs does not consider all of the properties of the interpretive agent but only the more striking features of the impressions that signs make on a representative interpreter. In its formal aspects, that impact or influence may be treated as just another sign, called the interpretant sign, or the interpretant for short. Such a 3-adic relation, among objects, signs, and interpretants, is called a sign relation.

For example, consider the aspects of sign use that concern two people — let us say \(\mathrm{Ann}\!\) and \(\mathrm{Bob}\!\) — in using their own proper names, \({}^{\backprime\backprime} \mathrm{Ann} {}^{\prime\prime}\!\) and \({}^{\backprime\backprime} \mathrm{Bob} {}^{\prime\prime},\!\) together with the pronouns, \({}^{\backprime\backprime} \mathrm{I} {}^{\prime\prime}\!\) and \({}^{\backprime\backprime} \mathrm{you} {}^{\prime\prime}.\!\) For brevity, these four signs may be abbreviated to the set \(\{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\!\) The abstract consideration of how \(\mathrm{A}\!\) and \(\mathrm{B}\!\) use this set of signs to refer to themselves and each other leads to the contemplation of a pair of 3-adic relations, the sign relations \(L_\mathrm{A}\!\) and \(L_\mathrm{B},\!\) that reflect the differential use of these signs by \(\mathrm{A}\!\) and \(\mathrm{B},\!\) respectively.

Each of the sign relations, \(L_\mathrm{A}\!\) and \(L_\mathrm{B},\!\) consists of eight triples of the form \((x, y, z),\!\) where the object \(x\!\) is an element of the object domain \(O = \{ \mathrm{A}, \mathrm{B} \},\!\) where the sign \(y\!\) is an element of the sign domain \(S\!,\) where the interpretant sign \(z\!\) is an element of the interpretant domain \(I,\!\) and where it happens in this case that \(S = I = \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \}.\!\) In general, it is convenient to refer to the union \(S \cup I\!\) as the syntactic domain, but in this case \(S ~=~ I ~=~ S \cup I.\!\)

The set-up so far is summarized as follows:

\(\begin{array}{ccc} L_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I \\ \\ O & = & \{ \mathrm{A}, \mathrm{B} \} \\ \\ S & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\ \\ I & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\ \\ \end{array}\)

The relation \(L_\mathrm{A}\!\) is the set of eight triples enumerated here:

\(\begin{array}{cccccc} \{ & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & \\ & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}) & \}. \end{array}\)

The triples in \(L_\mathrm{A}\!\) represent the way that interpreter \(\mathrm{A}\!\) uses signs. For example, the listing of the triple \((\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\!\) in \(L_\mathrm{A}\!\) represents the fact that \(\mathrm{A}\!\) uses \({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!\) to mean the same thing that \(\mathrm{A}\!\) uses \({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!\) to mean, namely, \(\mathrm{B}.\!\)

The relation \(L_\mathrm{B}\!\) is the set of eight triples enumerated here:

\(\begin{array}{cccccc} \{ & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}), & (\mathrm{A}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}), & \\ & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}), & (\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}) & \}. \end{array}\)

The triples in \(L_\mathrm{B}\!\) represent the way that interpreter \(\mathrm{B}\!\) uses signs. For example, the listing of the triple \((\mathrm{B}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime})\!\) in \(L_\mathrm{B}\!\) represents the fact that \(\mathrm{B}\!\) uses \({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!\) to mean the same thing that \(\mathrm{B}\!\) uses \({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!\) to mean, namely, \(\mathrm{B}.\!\)

The triples that make up the relations \(L_\mathrm{A}\!\) and \(L_\mathrm{B}\!\) are conveniently arranged in the form of relational data tables, as follows:


\(L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\!\)
\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)
\(\mathrm{A}\!\) \({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!\)
\(\mathrm{A}\!\) \({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!\)
\(\mathrm{A}\!\) \({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!\)
\(\mathrm{A}\!\) \({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!\)
\(\mathrm{B}\!\) \({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!\)
\(\mathrm{B}\!\) \({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!\)
\(\mathrm{B}\!\) \({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!\)
\(\mathrm{B}\!\) \({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!\)


\(L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\!\)
\(\text{Object}\!\) \(\text{Sign}\!\) \(\text{Interpretant}\!\)
\(\mathrm{A}\!\) \({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!\)
\(\mathrm{A}\!\) \({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!\)
\(\mathrm{A}\!\) \({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!\)
\(\mathrm{A}\!\) \({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!\)
\(\mathrm{B}\!\) \({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!\)
\(\mathrm{B}\!\) \({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!\)
\(\mathrm{B}\!\) \({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!\)
\(\mathrm{B}\!\) \({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!\) \({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!\)


Syllabus

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

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