Difference between revisions of "Triadic relation"

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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.
+
<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>
  
−
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.
+
<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>
 +
 
 +
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==
  
−
For the sake of topics to be taken up later, it is useful to examine a pair of 3-adic relations in tandem, '''L'''<sub>0</sub> and '''L'''<sub>1</sub>, that can be described in the following manner.
+
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.
  
−
The first order of business is to define the space in which the relations '''L'''<sub>0</sub> and '''L'''<sub>1</sub> take up residence.  This space is constructed as a 3-fold [[cartesian power]] in the following way.
+
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>
  
−
The '''[[boolean domain]]''' is the set '''B''' = {0,&nbsp;1}.
+
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>.
−
The plus sign "+", used in the context of the boolean domain '''B''', denotes addition mod 2.  Interpreted for logic, this amounts to the same thing as the boolean operation of ''exclusive-or'' or ''not-equal-to''.
 
  
−
The third cartesian power of '''B''' is '''B'''<sup>3</sup> = {(''x''<sub>1</sub>, ''x''<sub>2</sub>, ''x''<sub>3</sub>) : ''x''<sub>''j''</sub> in '''B''' for ''j'' = 1, 2, 3}= '''B''' &times; '''B''' &times; '''B'''.
+
The relation <math>L_0</math> is defined by the following formula.
  
−
In what follows, the space '''X''' &times; '''Y''' &times; '''Z''' is isomorphic to '''B''' &times; '''B''' &times; '''B''' = '''B'''<sup>3</sup>.
+
{| align="center" cellpadding="6" width="90%"
 +
| <math>L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.</math>
 +
|}
  
−
The relation '''L'''<sub>0</sub> is defined as follows:
+
The relation <math>L_0</math> is the following set of four triples in <math>\mathbb{B}^3.</math>
  
−
: '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') in '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}.
+
{| align="center" cellpadding="6" width="90%"
 +
| <math>L_0 ~=~ \{ ~ (0, 0, 0), ~ (0, 1, 1), ~ (1, 0, 1), ~ (1, 1, 0) ~ \}.</math>
 +
|}
  
−
The relation '''L'''<sub>0</sub> is the set of four triples enumerated here:
+
The relation <math>L_1</math> is defined by the following formula.
  
−
: '''L'''<sub>0</sub> = {(0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0)}.
+
{| align="center" cellpadding="6" width="90%"
 +
| <math>L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.</math>
 +
|}
  
−
The relation '''L'''<sub>1</sub> is defined as follows:
+
The relation <math>L_1</math> is the following set of four triples in <math>\mathbb{B}^3.</math>
  
−
: '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') in '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}.
+
{| align="center" cellpadding="6" width="90%"
 +
| <math>L_1 ~=~ \{ ~ (0, 0, 1), ~ (0, 1, 0), ~ (1, 0, 0), ~ (1, 1, 1) ~ \}.</math>
 +
|}
  
−
The relation '''L'''<sub>1</sub> is the set of four triples enumerated here:
+
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.
  
−
: '''L'''<sub>1</sub> = {(0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1)}.
+
<br>
  
−
The triples that make up the relations '''L'''<sub>0</sub> and '''L'''<sub>1</sub> are conveniently arranged in the form of ''[[relational database|relational data tables]]'', as follows:
+
{| 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_0 ~=~ \{ (x, y, z) : x + y + z = 0 \}</math>
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"
+
|- style="height:40px; background:ghostwhite"
−
|+ '''L'''<sub>0</sub> = {(''x'', ''y'', ''z'') &isin; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 0}
+
| style="border-bottom:1px solid black" | <math>x</math>
−
|- style="background:paleturquoise"
+
| style="border-bottom:1px solid black" | <math>y</math>
−
! X !! Y !! Z
+
| style="border-bottom:1px solid black" | <math>z</math>
 
|-
 
|-
−
| '''0''' || '''0''' || '''0'''
+
| <math>0</math> || <math>0</math> || <math>0</math>
 
|-
 
|-
−
| '''0''' || '''1''' || '''1'''
+
| <math>0</math> || <math>1</math> || <math>1</math>
 
|-
 
|-
−
| '''1''' || '''0''' || '''1'''
+
| <math>1</math> || <math>0</math> || <math>1</math>
 
|-
 
|-
−
| '''1''' || '''1''' || '''0'''
+
| <math>1</math> || <math>1</math> || <math>0</math>
 
|}
 
|}
 +
 
<br>
 
<br>
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; 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%"
−
|+ '''L'''<sub>1</sub> = {(''x'', ''y'', ''z'') &isin; '''B'''<sup>3</sup> : ''x'' + ''y'' + ''z'' = 1}
+
|+ style="height:30px" | <math>L_1 ~=~ \{ (x, y, z) : x + y + z = 1 \}</math>
−
|- style="background:paleturquoise"
+
|- style="height:40px; background:ghostwhite"
−
! X !! Y !! Z
+
| 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>
 
|-
 
|-
−
| '''0''' || '''0''' || '''1'''
+
| <math>0</math> || <math>0</math> || <math>1</math>
 
|-
 
|-
−
| '''0''' || '''1''' || '''0'''
+
| <math>0</math> || <math>1</math> || <math>0</math>
 
|-
 
|-
−
| '''1''' || '''0''' || '''0'''
+
| <math>1</math> || <math>0</math> || <math>0</math>
 
|-
 
|-
−
| '''1''' || '''1''' || '''1'''
+
| <math>1</math> || <math>1</math> || <math>1</math>
 
|}
 
|}
 +
 
<br>
 
<br>
  
 
==Examples from semiotics==
 
==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 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''.
  
−
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 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]]''.
  
−
For example, consider the aspects of sign use that concern two people, say, Ann and Bob, in using their own proper names, "Ann" and "Bob", and in using the pronouns "I" and "you". For brevity, these four signs may be abbreviated to the set {"A",&nbsp;"B",&nbsp;"i",&nbsp;"u"}.  The abstract consideration of how A and B use this set of signs to refer to themselves and to each other leads to the contemplation of a pair of 3-adic relations, the sign relations '''L'''<sub>A</sub> and '''L'''<sub>B</sub>, that reflect the differential use of these signs by A and by B, respectively.
+
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, '''L'''<sub>A</sub> and '''L'''<sub>B</sub>, consists of eight triples of the form (''x'',&nbsp;''y'',&nbsp;''z''), where the object ''x'' belongs to the object domain '''O''' = {A,&nbsp;B}, where the sign ''y'' belongs to the sign domain '''S''', where the interpretant sign ''z'' belongs to the interpretant domain '''I''', and where it happens in this case that '''S''' = '''I''' = {"A",&nbsp;"B",&nbsp;"i",&nbsp;"u"}. In general, it is convenient to refer to the union '''S'''&nbsp;&cup;&nbsp;'''I''' as the "syntactic domain", but in this case '''S''' = '''I''' = '''S'''&nbsp;&cup;&nbsp;'''I'''.
+
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 to this point can be summarized as follows:
+
The set-up so far is summarized as follows:
  
−
: '''L'''<sub>A</sub>, '''L'''<sub>B</sub> &sube; '''O''' &times; '''S''' &times; '''I'''
+
{| align="center" cellpadding="8" width="90%"
 +
|
 +
<math>\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}</math>
 +
|}
  
−
: '''O''' = {A, B}
+
The relation <math>L_\mathrm{A}</math> is the following set of eight triples in <math>O \times S \times I.</math>
  
−
: '''S''' = {"A", "B", "i", "u"}
+
{| align="center" cellpadding="8" width="90%"
 +
|
 +
<math>\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}</math>
 +
|}
  
−
: '''I''' = {"A", "B", "i", "u"}
+
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 '''L'''<sub>A</sub> 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>
  
−
: {(A, "A", "A"), (A, "A", "i"), (A, "i", "A"), (A, "i", "i"),
+
{| align="center" cellpadding="8" width="90%"
−
: &nbsp;(B, "B", "B"), (B, "B", "u"), (B, "u", "B"), (B, "u", "u")}.
+
|
 +
<math>\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}</math>
 +
|}
  
−
The triples in '''L'''<sub>A</sub> represent the way that interpreter A uses signs. For example, the listing of the triple (B, "u", "B") in '''L'''<sub>A</sub> represents the fact that A uses "B" to mean the same thing that A uses "u" to mean, namely, B.
+
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 relation '''L'''<sub>B</sub> is the set of eight triples enumerated here:
+
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.
  
−
: {(A, "A", "A"), (A, "A", "u"), (A, "u", "A"), (A, "u", "u"),
+
<br>
−
: &nbsp;(B, "B", "B"), (B, "B", "i"), (B, "i", "B"), (B, "i", "i")}.
 
−
 
 
−
The triples in '''L'''<sub>B</sub> represent the way that interpreter B uses signs.  For example, the listing of the triple (B, "i", "B") in '''L'''<sub>B</sub> represents the fact that B uses "B" to mean the same thing that B uses "i" to mean, namely, B.
 
−
 
 
−
The triples that make up the relations '''L'''<sub>A</sub> and '''L'''<sub>B</sub> are conveniently arranged in the form of ''[[relational database|relational data tables]]'', as follows:
 
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; 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%"
−
|+ '''L'''<sub>A</sub> = Sign Relation of Interpreter A
+
|+ style="height:30px" | <math>L_\mathrm{A} ~=~ \text{Sign Relation of Interpreter A}</math>
−
|- style="background:paleturquoise"
+
|- style="height:40px; background:ghostwhite"
−
! style="width:20%" | Object
+
| style="border-bottom:1px solid black; width:33%" | <math>\text{Object}</math>
−
! style="width:20%" | Sign
+
| style="border-bottom:1px solid black; width:33%" | <math>\text{Sign}</math>
−
! style="width:20%" | Interpretant
+
| style="border-bottom:1px solid black; width:33%" | <math>\text{Interpretant}</math>
 
|-
 
|-
−
| '''A''' || '''"A"''' || '''"A"'''
+
| <math>\mathrm{A}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
 
|-
 
|-
−
| '''A''' || '''"A"''' || '''"i"'''
+
| <math>\mathrm{A}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
 
|-
 
|-
−
| '''A''' || '''"i"''' || '''"A"'''
+
| <math>\mathrm{A}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
 
|-
 
|-
−
| '''A''' || '''"i"''' || '''"i"'''
+
| <math>\mathrm{A}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
 
|-
 
|-
−
| '''B''' || '''"B"''' || '''"B"'''
+
| style="border-top:1px solid black" | <math>\mathrm{B}</math>
 +
| style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
 +
| style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
 
|-
 
|-
−
| '''B''' || '''"B"''' || '''"u"'''
+
| <math>\mathrm{B}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
 
|-
 
|-
−
| '''B''' || '''"u"''' || '''"B"'''
+
| <math>\mathrm{B}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
 
|-
 
|-
−
| '''B''' || '''"u"''' || '''"u"'''
+
| <math>\mathrm{B}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
 
|}
 
|}
 +
 
<br>
 
<br>
  
−
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; 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%"
−
|+ '''L'''<sub>B</sub> = Sign Relation of Interpreter B
+
|+ style="height:30px" | <math>L_\mathrm{B} ~=~ \text{Sign Relation of Interpreter B}</math>
−
|- style="background:paleturquoise"
+
|- style="height:40px; background:ghostwhite"
−
! style="width:20%" | Object
+
| style="border-bottom:1px solid black; width:33%" | <math>\text{Object}</math>
−
! style="width:20%" | Sign
+
| style="border-bottom:1px solid black; width:33%" | <math>\text{Sign}</math>
−
! style="width:20%" | Interpretant
+
| style="border-bottom:1px solid black; width:33%" | <math>\text{Interpretant}</math>
 
|-
 
|-
−
| '''A''' || '''"A"''' || '''"A"'''
+
| <math>\mathrm{A}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
 
|-
 
|-
−
| '''A''' || '''"A"''' || '''"u"'''
+
| <math>\mathrm{A}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
 
|-
 
|-
−
| '''A''' || '''"u"''' || '''"A"'''
+
| <math>\mathrm{A}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{A} {}^{\prime\prime}</math>
 
|-
 
|-
−
| '''A''' || '''"u"''' || '''"u"'''
+
| <math>\mathrm{A}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{u} {}^{\prime\prime}</math>
 
|-
 
|-
−
| '''B''' || '''"B"''' || '''"B"'''
+
| style="border-top:1px solid black" | <math>\mathrm{B}</math>
 +
| style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
 +
| style="border-top:1px solid black" | <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
 
|-
 
|-
−
| '''B''' || '''"B"''' || '''"i"'''
+
| <math>\mathrm{B}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
 
|-
 
|-
−
| '''B''' || '''"i"''' || '''"B"'''
+
| <math>\mathrm{B}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{B} {}^{\prime\prime}</math>
 
|-
 
|-
−
| '''B''' || '''"i"''' || '''"i"'''
+
| <math>\mathrm{B}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
 +
| <math>{}^{\backprime\backprime} \mathrm{i} {}^{\prime\prime}</math>
 
|}
 
|}
 +
 
<br>
 
<br>
  
−
==See also==
+
==Resources==
−
{|
+
 
−
| valign=top |
+
* [[Logic Syllabus]]
−
* [[Relation (mathematics)|Relation]]
+
 
−
* [[Relation composition]]
+
==Document history==
−
* [[Relation construction]]
+
 
−
* [[Relation reduction]]
+
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.
−
* [[Logic of relatives]]
+
 
−
| valign=top |
+
{{col-begin}}
−
* [[Logical matrix]]
+
{{col-break}}
−
* [[Semeiotic]]
+
* [https://oeis.org/wiki/Triadic_relation Triadic Relation], [https://oeis.org/wiki/ OEIS Wiki]
−
* [[Semiotic]]
+
* [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]
−
* [[Semiotic information theory|Semiotic information]]
+
* [http://mywikibiz.com/Triadic_relation Triadic Relation], [http://mywikibiz.com/ MyWikiBiz]
−
* [[Sign relation]]
+
{{col-break}}
−
|}
+
* [https://planetmath.org/TriadicRelation Triadic Relation], [https://planetmath.org/ PlanetMath]
 +
* [https://en.wikiversity.org/wiki/Triadic_relation Triadic Relation], [https://en.wikiversity.org/ Wikiversity]
 +
* [https://en.wikipedia.org/w/index.php?title=Triadic_relation&oldid=108548758 Triadic Relation], [https://en.wikipedia.org/ Wikipedia]
 +
{{col-end}}
 +
 
 +
[[Category:Algebra]]
 +
[[Category:Boolean algebra]]
 +
[[Category:Boolean functions]]
 +
[[Category:Category theory]]
 +
[[Category:Combinatorics]]
 +
[[Category:Computer science]]
 +
[[Category:Discrete mathematics]]
 +
[[Category:Graph theory]]
 +
[[Category:Group theory]]
 +
[[Category:Logic]]
 +
[[Category:Mathematics]]
 +
[[Category:Peirce, Charles Sanders]]
 +
[[Category:Pragmatics]]
 +
[[Category:Relation theory]]
 +
[[Category:Semantics]]
 +
[[Category:Semiotics]]
 +
[[Category:Syntax]]

Latest revision as of 18:02, 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.


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


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


Resources

Document history

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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