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]].
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<p style="margin-left:40%; margin-bottom:0;">Of triadic Being the multitude of forms is so terrific that I have usually shrunk from the task of enumerating them;&nbsp; and for the present purpose such an enumeration would be worse than superfluous:&nbsp; it would be a great inconvenience.</p>
  
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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.
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<p align="right" style="margin-top:0;">&mdash; [https://inquiryintoinquiry.com/2012/06/14/c-s-peirce-%e2%80%a2-of-triadic-being/ C.S.&nbsp;Peirce, <i>Collected Papers</i>, CP&nbsp;6.347]</p>
  
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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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A '''triadic relation''' (or '''ternary relation''') is a special case of a [[relation|polyadic or finitary relation]], one in which the number of places in the relation is three.&nbsp; 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 triadic relations and the field of [[semeiotic|semiotics]] is rich in its harvest of [[sign relation]]s, which are special cases of triadic relations.&nbsp; In either subject, as Peirce observes, the multitude of forms is truly terrific, so it's best to begin with concrete examples just complex enough to illustrate the distinctive features of each type.&nbsp; The discussion to follow takes up a pair of simple but instructive examples from each of the realms of mathematics and semiotics.
  
 
==Examples from mathematics==
 
==Examples from mathematics==
  
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For the sake of topics to be taken up later, it is useful to examine a pair of 3-adic relations in tandem, <math>L_0\!</math> and <math>L_1,\!</math> that can be described in the following manner.
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For the sake of topics to be taken up later, it is useful to examine a pair of triadic relations in tandem.&nbsp; In what follows we construct two triadic relations, <math>L_0</math> and <math>L_1,</math> each of which is a subset of the same cartesian product <math>X \times Y \times Z.</math>&nbsp; The structures of <math>L_0</math> and <math>L_1</math> can be described in the following way.
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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.
 
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The ''[[boolean domain]]'' is the set <math>\mathbb{B} = \{ 0, 1 \}.\!</math>
 
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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>
 
  
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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>
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Each space <math>X, Y, Z</math> is isomorphic to the <i>[[boolean domain]]</i> <math>\mathbb{B} = \{ 0, 1 \}</math> so <math>L_0</math> and <math>L_1</math> are subsets of the cartesian power <math>\mathbb{B} \times \mathbb{B} \times \mathbb{B}</math> or the <i>boolean cube</i> <math>\mathbb{B}^3.</math>
  
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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>
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The operation of <i>boolean addition</i>, <math>+ : \mathbb{B} \times \mathbb{B} \to \mathbb{B},</math> is equivalent to addition modulo 2, where <math>0</math> acts in the usual manner but <math>1 + 1 = 0.</math>&nbsp; In its logical interpretation, the plus sign can be used to indicate either the boolean operation of <i>[[exclusive disjunction]]</i> or the boolean relation of <i>logical inequality</i>.
  
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The relation <math>L_0\!</math> is defined as follows:
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The relation <math>L_0</math> is defined by the following formula.
  
 
{| align="center" cellpadding="6" width="90%"
 
{| align="center" cellpadding="6" width="90%"
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| <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.~\!</math>
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| <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.</math>
 
|}
 
|}
  
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The relation <math>L_0\!</math> is the set of four triples enumerated here:
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The relation <math>L_0</math> is the following set of four triples in <math>\mathbb{B}^3.</math>
  
 
{| align="center" cellpadding="6" width="90%"
 
{| align="center" cellpadding="6" width="90%"
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| <math>L_0 ~=~ \{ (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) \}.\!</math>
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| <math>L_0 ~=~ \{ ~ (0, 0, 0), ~ (0, 1, 1), ~ (1, 0, 1), ~ (1, 1, 0) ~ \}.</math>
 
|}
 
|}
  
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The relation <math>L_1\!</math> is defined as follows:
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The relation <math>L_1</math> is defined by the following formula.
  
 
{| align="center" cellpadding="6" width="90%"
 
{| align="center" cellpadding="6" width="90%"
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| <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.~\!</math>
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| <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.</math>
 
|}
 
|}
  
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The relation <math>L_1\!</math> is the set of four triples enumerated here:
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The relation <math>L_1</math> is the following set of four triples in <math>\mathbb{B}^3.</math>
  
 
{| align="center" cellpadding="6" width="90%"
 
{| align="center" cellpadding="6" width="90%"
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| <math>L_1 ~=~ \{ (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1) \}.\!</math>
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| <math>L_1 ~=~ \{ ~ (0, 0, 1), ~ (0, 1, 0), ~ (1, 0, 0), ~ (1, 1, 1) ~ \}.</math>
 
|}
 
|}
  
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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:
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The triples in the relations <math>L_0</math> and <math>L_1</math> are conveniently arranged in the form of ''relational data tables'', as shown below.
  
 
<br>
 
<br>
  
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{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
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{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; font-size:larger; text-align:center; width:40%"
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|+ style="height:30px" | <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}\!</math>
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|+ style="height:30px" | <math>L_0 ~=~ \{ (x, y, z) : x + y + z = 0 \}</math>
 
|- style="height:40px; background:ghostwhite"
 
|- style="height:40px; background:ghostwhite"
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| <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math>
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| style="border-bottom:1px solid black" | <math>x</math>
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| style="border-bottom:1px solid black" | <math>y</math>
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| style="border-bottom:1px solid black" | <math>z</math>
 
|-
 
|-
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| <math>0\!</math> || <math>0\!</math> || <math>0\!</math>
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| <math>0</math> || <math>0</math> || <math>0</math>
 
|-
 
|-
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| <math>0\!</math> || <math>1\!</math> || <math>1\!</math>
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| <math>0</math> || <math>1</math> || <math>1</math>
 
|-
 
|-
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| <math>1\!</math> || <math>0\!</math> || <math>1\!</math>
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| <math>1</math> || <math>0</math> || <math>1</math>
 
|-
 
|-
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| <math>1\!</math> || <math>1\!</math> || <math>0\!</math>
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| <math>1</math> || <math>1</math> || <math>0</math>
 
|}
 
|}
  
 
<br>
 
<br>
  
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{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
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{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; font-size:larger; text-align:center; width:40%"
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|+ style="height:30px" | <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}\!</math>
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|+ style="height:30px" | <math>L_1 ~=~ \{ (x, y, z) : x + y + z = 1 \}</math>
 
|- style="height:40px; background:ghostwhite"
 
|- style="height:40px; background:ghostwhite"
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| <math>X\!</math> || <math>Y\!</math> || <math>Z\!</math>
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| style="border-bottom:1px solid black" | <math>x</math>
 +
| style="border-bottom:1px solid black" | <math>y</math>
 +
| style="border-bottom:1px solid black" | <math>z</math>
 
|-
 
|-
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| <math>0\!</math> || <math>0\!</math> || <math>1\!</math>
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| <math>0</math> || <math>0</math> || <math>1</math>
 
|-
 
|-
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| <math>0\!</math> || <math>1\!</math> || <math>0\!</math>
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| <math>0</math> || <math>1</math> || <math>0</math>
 
|-
 
|-
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| <math>1\!</math> || <math>0\!</math> || <math>0\!</math>
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| <math>1</math> || <math>0</math> || <math>0</math>
 
|-
 
|-
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| <math>1\!</math> || <math>1\!</math> || <math>1\!</math>
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| <math>1</math> || <math>1</math> || <math>1</math>
 
|}
 
|}
  
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==Examples from semiotics==
 
==Examples from semiotics==
  
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The study of signs &mdash; the full variety of significant forms of expression &mdash; 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''.
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The study of signs &mdash; the full variety of significant forms of expression &mdash; in relation to all the affairs signs are significant ''of'', and in relation to all the beings signs are significant ''to'', is known as ''[[semiotics]]'' or the ''theory of signs''.&nbsp; As described, semiotics treats of a 3-place relation among ''signs'', their ''objects'', and their ''interpreters''.
  
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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]]''.
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The term ''semiosis'' refers to any activity or process involving signs.&nbsp; Studies of semiosis focusing on its abstract form are not concerned with every concrete detail of the entities acting as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among those three roles.&nbsp; In particular, the formal theory of signs does not consider all the properties of the interpretive agent but only the more striking features of the impressions signs make on a representative interpreter.&nbsp; From a formal point of view this impact or influence may be treated as just another sign, called the ''interpretant sign'', or the ''interpretant'' for short.&nbsp; A triadic relation of this type, among objects, signs, and interpretants, is called a ''[[sign relation]]''.
  
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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.
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For example, consider the aspects of sign use involved when two people, say <math>\mathrm{Ann}</math> and <math>\mathrm{Bob},</math> use their own proper names, <math>{}^{\backprime\backprime} \mathrm{Ann} {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \mathrm{Bob} {}^{\prime\prime},</math> along with the pronouns, <math>{}^{\backprime\backprime} \mathrm{I} {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \mathrm{you} {}^{\prime\prime},</math> to refer to themselves and each other.&nbsp; 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>&nbsp; The abstract consideration of how <math>\mathrm{A}</math> and <math>\mathrm{B}</math> use this set of signs leads to the contemplation of a pair of triadic relations, the sign relations <math>L_\mathrm{A}</math> and <math>L_\mathrm{B},</math> reflecting the differential use of these signs by <math>\mathrm{A}</math> and <math>\mathrm{B},</math> respectively.
  
−
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>
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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> the ''sign'' <math>y</math> is an element of the ''sign domain'' <math>S,</math> 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>&nbsp; The union <math>S \cup I</math> is often referred to 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_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I \\
+
L_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I
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\\
+
\\[5pt]
−
O & = & \{ \mathrm{A}, \mathrm{B} \} \\
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O & = & \{ \mathrm{A}, \mathrm{B} \}
−
\\
+
\\[5pt]
−
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} \, \} \\
+
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} \, \}
−
\\
+
\\[5pt]
−
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} \, \} \\
+
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>
 
|}
 
|}
  
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The relation <math>L_\mathrm{A}\!</math> is the set of eight triples enumerated here:
+
The relation <math>L_\mathrm{A}</math> is the following set of eight triples in <math>O \times S \times I.</math>
  
 
{| align="center" cellpadding="8" width="90%"
 
{| align="center" cellpadding="8" width="90%"
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|}
 
|}
  
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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 triples in <math>L_\mathrm{A}</math> represent the way interpreter <math>\mathrm{A}</math> uses signs.&nbsp; For example, the presence of <math>( \mathrm{B}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} )</math> in <math>L_\mathrm{A}</math> tells us <math>\mathrm{A}</math> uses <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math> to mean the same thing <math>\mathrm{A}</math> uses <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math> to mean, namely, <math>\mathrm{B}.</math>
  
−
The relation <math>L_\mathrm{B}\!</math> is the set of eight triples enumerated here:
+
The relation <math>L_\mathrm{B}</math> is the following set of eight triples in <math>O \times S \times I.</math>
  
 
{| align="center" cellpadding="8" width="90%"
 
{| align="center" cellpadding="8" width="90%"
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|}
 
|}
  
−
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 in <math>L_\mathrm{B}</math> represent the way interpreter <math>\mathrm{B}</math> uses signs.&nbsp; For example, the presence of <math>( \mathrm{B}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} )</math> in <math>L_\mathrm{B}</math> tells us <math>\mathrm{B}</math> uses <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math> to mean the same thing <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_\mathrm{A}\!</math> and <math>L_\mathrm{B}\!</math> are conveniently arranged in the form of ''relational data tables'', as follows:
+
The triples in the relations <math>L_\mathrm{A}</math> and <math>L_\mathrm{B}</math> are conveniently arranged in the form of ''relational data tables'', as shown below.
  
 
<br>
 
<br>
  
−
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
+
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; font-size:larger; text-align:center; width:40%"
−
|+ style="height:30px" | <math>L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}\!</math>
+
|+ style="height:30px" | <math>L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}</math>
 
|- style="height:40px; background:ghostwhite"
 
|- style="height:40px; background:ghostwhite"
−
| style="width:33%" | <math>\text{Object}\!</math>
+
| style="border-bottom:1px solid black; width:33%" | <math>\text{Object}</math>
−
| style="width:33%" | <math>\text{Sign}\!</math>
+
| style="border-bottom:1px solid black; width:33%" | <math>\text{Sign}</math>
−
| style="width:33%" | <math>\text{Interpretant}\!</math>
+
| style="border-bottom:1px solid black; width:33%" | <math>\text{Interpretant}</math>
 
|-
 
|-
−
| <math>\mathrm{A}\!</math>
+
| <math>\mathrm{A}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
 
|-
 
|-
−
| <math>\mathrm{A}\!</math>
+
| <math>\mathrm{A}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
 
|-
 
|-
−
| <math>\mathrm{A}\!</math>
+
| <math>\mathrm{A}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
 
|-
 
|-
−
| <math>\mathrm{A}\!</math>
+
| <math>\mathrm{A}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
 
|-
 
|-
−
| <math>\mathrm{B}\!</math>
+
| style="border-top:1px solid black" | <math>\mathrm{B}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
+
| style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
+
| style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
 
|-
 
|-
−
| <math>\mathrm{B}\!</math>
+
| <math>\mathrm{B}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
 
|-
 
|-
−
| <math>\mathrm{B}\!</math>
+
| <math>\mathrm{B}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
 
|-
 
|-
−
| <math>\mathrm{B}\!</math>
+
| <math>\mathrm{B}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
 
|}
 
|}
  
 
<br>
 
<br>
  
−
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:center; width:60%"
+
{| align="center" cellpadding="6" cellspacing="0" style="border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; font-size:larger; text-align:center; width:40%"
−
|+ style="height:30px" | <math>L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}\!</math>
+
|+ style="height:30px" | <math>L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}</math>
 
|- style="height:40px; background:ghostwhite"
 
|- style="height:40px; background:ghostwhite"
−
| style="width:33%" | <math>\text{Object}\!</math>
+
| style="border-bottom:1px solid black; width:33%" | <math>\text{Object}</math>
−
| style="width:33%" | <math>\text{Sign}\!</math>
+
| style="border-bottom:1px solid black; width:33%" | <math>\text{Sign}</math>
−
| style="width:33%" | <math>\text{Interpretant}\!</math>
+
| style="border-bottom:1px solid black; width:33%" | <math>\text{Interpretant}</math>
 
|-
 
|-
−
| <math>\mathrm{A}\!</math>
+
| <math>\mathrm{A}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
 
|-
 
|-
−
| <math>\mathrm{A}\!</math>
+
| <math>\mathrm{A}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
 
|-
 
|-
−
|<math>\mathrm{A}\!</math>
+
| <math>\mathrm{A}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
 
|-
 
|-
−
| <math>\mathrm{A}\!</math>
+
| <math>\mathrm{A}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
 
|-
 
|-
−
| <math>\mathrm{B}\!</math>
+
| style="border-top:1px solid black" | <math>\mathrm{B}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
+
| style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
+
| style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
 
|-
 
|-
−
| <math>\mathrm{B}\!</math>
+
| <math>\mathrm{B}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
 
|-
 
|-
−
| <math>\mathrm{B}\!</math>
+
| <math>\mathrm{B}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
 
|-
 
|-
−
| <math>\mathrm{B}\!</math>
+
| <math>\mathrm{B}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
−
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\!</math>
+
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
 
|}
 
|}
  
 
<br>
 
<br>
  
−
==Syllabus==
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==Resources==
  
−
===Focal nodes===
+
* [[Logic Syllabus]]
  
−
* [[Inquiry Live]]
+
==Document history==
−
* [[Logic Live]]
 
  
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===Peer nodes===
+
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.
  
−
* [http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation Triadic Relation @ InterSciWiki]
+
* [http://web.archive.org/web/20190328161600/http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation Triadic Relation], [http://web.archive.org/web/20191124081501/http://intersci.ss.uci.edu/wiki/index.php/User:Jon_Awbrey InterSciWiki]
−
* [http://mywikibiz.com/Triadic_relation Triadic Relation @ MyWikiBiz]
+
* [http://en.wikiversity.org/wiki/Triadic_relation Triadic Relation], [http://en.wikiversity.org/ Wikiversity]
−
* [http://ref.subwiki.org/wiki/Triadic_relation Triadic Relation @ Subject Wikis]
+
* [http://en.wikipedia.org/w/index.php?title=Triadic_relation&oldid=108548758 Triadic Relation], [http://en.wikipedia.org/ Wikipedia]
−
* [http://en.wikiversity.org/wiki/Triadic_relation Triadic Relation @ Wikiversity]
 
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* [http://beta.wikiversity.org/wiki/Triadic_relation Triadic Relation @ Wikiversity Beta]
 
−
 
 
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===Logical operators===
 
−
 
 
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{{col-begin}}
 
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{{col-break}}
 
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* [[Exclusive disjunction]]
 
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* [[Logical conjunction]]
 
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* [[Logical disjunction]]
 
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* [[Logical equality]]
 
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{{col-break}}
 
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* [[Logical implication]]
 
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* [[Logical NAND]]
 
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* [[Logical NNOR]]
 
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* [[Logical negation|Negation]]
 
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{{col-end}}
 
−
 
 
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===Related topics===
 
−
 
 
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{{col-begin}}
 
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{{col-break}}
 
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* [[Ampheck]]
 
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* [[Boolean domain]]
 
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* [[Boolean function]]
 
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* [[Boolean-valued function]]
 
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* [[Differential logic]]
 
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{{col-break}}
 
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* [[Logical graph]]
 
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* [[Minimal negation operator]]
 
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* [[Multigrade operator]]
 
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* [[Parametric operator]]
 
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* [[Peirce's law]]
 
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{{col-break}}
 
−
* [[Propositional calculus]]
 
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* [[Sole sufficient operator]]
 
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* [[Truth table]]
 
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* [[Universe of discourse]]
 
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* [[Zeroth order logic]]
 
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{{col-end}}
 
−
 
 
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===Relational concepts===
 
−
 
 
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{{col-begin}}
 
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{{col-break}}
 
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* [[Continuous predicate]]
 
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* [[Hypostatic abstraction]]
 
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* [[Logic of relatives]]
 
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* [[Logical matrix]]
 
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{{col-break}}
 
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* [[Relation (mathematics)|Relation]]
 
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* [[Relation composition]]
 
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* [[Relation construction]]
 
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* [[Relation reduction]]
 
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{{col-break}}
 
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* [[Relation theory]]
 
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* [[Relative term]]
 
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* [[Sign relation]]
 
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* [[Triadic relation]]
 
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{{col-end}}
 
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===Information, Inquiry===
 
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{{col-begin}}
 
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{{col-break}}
 
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* [[Inquiry]]
 
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* [[Dynamics of inquiry]]
 
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{{col-break}}
 
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* [[Semeiotic]]
 
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* [[Logic of information]]
 
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{{col-break}}
 
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* [[Descriptive science]]
 
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* [[Normative science]]
 
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{{col-break}}
 
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* [[Pragmatic maxim]]
 
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* [[Truth theory]]
 
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{{col-end}}
 
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===Related articles===
 
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{{col-begin}}
 
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{{col-break}}
 
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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://intersci.ss.uci.edu/wiki/index.php/Propositional_Equation_Reasoning_Systems Propositional Equation Reasoning Systems]
 
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{{col-break}}
 
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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://intersci.ss.uci.edu/wiki/index.php/Differential_Logic_and_Dynamic_Systems_2.0 Differential Logic and Dynamic Systems]
 
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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]
 
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* [http://intersci.ss.uci.edu/wiki/index.php/Inquiry_Driven_Systems Inquiry Driven Systems : Inquiry Into Inquiry]
 
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{{col-end}}
 
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==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.
 
  
 
* [http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation Triadic Relation], [http://intersci.ss.uci.edu/ InterSciWiki]
 
* [http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation Triadic Relation], [http://intersci.ss.uci.edu/ InterSciWiki]
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* [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:Algebra]]
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[[Category:Syntax]]

Revision as of 17:36, 27 May 2020

Of triadic Being the multitude of forms is so terrific that I have usually shrunk from the task of enumerating them;  and for the present purpose such an enumeration would be worse than superfluous:  it would be a great inconvenience.

— C.S. Peirce, Collected Papers, CP 6.347

A triadic relation (or ternary relation) is a special case of a polyadic or finitary relation, one in which the number of places in the relation is three.  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 triadic relations and the field of semiotics is rich in its harvest of sign relations, which are special cases of triadic relations.  In either subject, as Peirce observes, the multitude of forms is truly terrific, so it's best to begin with concrete examples just complex enough to illustrate the distinctive features of each type.  The discussion to follow takes up a pair of simple but instructive examples from each of the realms of mathematics and semiotics.

Examples from mathematics

For the sake of topics to be taken up later, it is useful to examine a pair of triadic relations in tandem.  In what follows we construct two triadic relations, \(L_0\) and \(L_1,\) each of which is a subset of the same cartesian product \(X \times Y \times Z.\)  The structures of \(L_0\) and \(L_1\) can be described in the following way.

Each space \(X, Y, Z\) is isomorphic to the boolean domain \(\mathbb{B} = \{ 0, 1 \}\) so \(L_0\) and \(L_1\) are subsets of the cartesian power \(\mathbb{B} \times \mathbb{B} \times \mathbb{B}\) or the boolean cube \(\mathbb{B}^3.\)

The operation of boolean addition, \(+ : \mathbb{B} \times \mathbb{B} \to \mathbb{B},\) is equivalent to addition modulo 2, where \(0\) acts in the usual manner but \(1 + 1 = 0.\)  In its logical interpretation, the plus sign can be used to indicate either the boolean operation of exclusive disjunction or the boolean relation of logical inequality.

The relation \(L_0\) is defined by the following formula.

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

The relation \(L_0\) is the following set of four triples in \(\mathbb{B}^3.\)

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

The relation \(L_1\) is defined by the following formula.

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

The relation \(L_1\) is the following set of four triples in \(\mathbb{B}^3.\)

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

The triples in the relations \(L_0\) and \(L_1\) are conveniently arranged in the form of relational data tables, as shown below.


\(L_0 ~=~ \{ (x, y, z) : x + y + z = 0 \}\)
\(x\) \(y\) \(z\)
\(0\) \(0\) \(0\)
\(0\) \(1\) \(1\)
\(1\) \(0\) \(1\)
\(1\) \(1\) \(0\)


\(L_1 ~=~ \{ (x, y, z) : x + y + z = 1 \}\)
\(x\) \(y\) \(z\)
\(0\) \(0\) \(1\)
\(0\) \(1\) \(0\)
\(1\) \(0\) \(0\)
\(1\) \(1\) \(1\)


Examples from semiotics

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

The term semiosis refers to any activity or process involving signs.  Studies of semiosis focusing on its abstract form are not concerned with every concrete detail of the entities acting as signs, as objects, or as agents of semiosis, but only with the most salient patterns of relationship among those three roles.  In particular, the formal theory of signs does not consider all the properties of the interpretive agent but only the more striking features of the impressions signs make on a representative interpreter.  From a formal point of view this impact or influence may be treated as just another sign, called the interpretant sign, or the interpretant for short.  A triadic relation of this type, among objects, signs, and interpretants, is called a sign relation.

For example, consider the aspects of sign use involved when two people, say \(\mathrm{Ann}\) and \(\mathrm{Bob},\) use their own proper names, \({}^{\backprime\backprime} \mathrm{Ann} {}^{\prime\prime}\) and \({}^{\backprime\backprime} \mathrm{Bob} {}^{\prime\prime},\) along with the pronouns, \({}^{\backprime\backprime} \mathrm{I} {}^{\prime\prime}\) and \({}^{\backprime\backprime} \mathrm{you} {}^{\prime\prime},\) to refer to themselves and each other.  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 leads to the contemplation of a pair of triadic relations, the sign relations \(L_\mathrm{A}\) and \(L_\mathrm{B},\) reflecting 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 \(O = \{ \mathrm{A}, \mathrm{B} \},\) the sign \(y\) is an element of the sign domain \(S,\) the interpretant sign \(z\) is an element of the interpretant domain \(I,\) and where it happens in this case that \(S = I = \{ {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \}.\)  The union \(S \cup I\) is often referred to as the syntactic domain, but in this case \(S = I = S \cup I.\)

The set-up so far is summarized as follows:

\(\begin{array}{ccc} L_\mathrm{A}, L_\mathrm{B} & \subseteq & O \times S \times I \\[5pt] O & = & \{ \mathrm{A}, \mathrm{B} \} \\[5pt] S & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \\[5pt] I & = & \{ \, {}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, \, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime} \, \} \end{array}\)

The relation \(L_\mathrm{A}\) is the following set of eight triples in \(O \times S \times I.\)

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

The triples in \(L_\mathrm{A}\) represent the way interpreter \(\mathrm{A}\) uses signs.  For example, the presence of \(( \mathrm{B}, {}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} )\) in \(L_\mathrm{A}\) tells us \(\mathrm{A}\) uses \({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\) to mean the same thing \(\mathrm{A}\) uses \({}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}\) to mean, namely, \(\mathrm{B}.\)

The relation \(L_\mathrm{B}\) is the following set of eight triples in \(O \times S \times I.\)

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

The triples in \(L_\mathrm{B}\) represent the way interpreter \(\mathrm{B}\) uses signs.  For example, the presence of \(( \mathrm{B}, {}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime} )\) in \(L_\mathrm{B}\) tells us \(\mathrm{B}\) uses \({}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}\) to mean the same thing \(\mathrm{B}\) uses \({}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}\) to mean, namely, \(\mathrm{B}.\)

The triples in the relations \(L_\mathrm{A}\) and \(L_\mathrm{B}\) are conveniently arranged in the form of relational data tables, as shown below.


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


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


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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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