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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
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{{DISPLAYTITLE:Peirce's 1870 Logic Of Relatives}}
{{DISPLAYTITLE:Peirce's 1870 Logic Of Relatives}}
'''Note.''' The MathJax parser is not rendering this page properly.<br>Until it can be fixed please see the [http://intersci.ss.uci.edu/wiki/index.php/Peirce's_1870_Logic_Of_Relatives InterSciWiki version].
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
'''Author: [[User:Jon Awbrey|Jon Awbrey]]'''
Peirce's text employs lower case letters for logical terms of general reference and upper case letters for logical terms of individual reference. General terms fall into types, for example, absolute terms, dyadic relative terms, or higher adic relative terms, and Peirce employs different typefaces to distinguish these. The following Tables indicate the typefaces that are used in the text below for Peirce's examples of general terms.
Peirce's text employs lower case letters for logical terms of general reference and upper case letters for logical terms of individual reference. General terms fall into types — absolute terms, dyadic relative terms, higher adic relative terms — and Peirce employs different typefaces to distinguish these. The following Tables indicate the typefaces that are used in the text below for Peirce's examples of general terms.
<br>
<br>
The path to understanding Peirce's system and its wider implications for logic can be smoothed by paraphrasing his notations in a variety of contemporary mathematical formalisms, while preserving the semantics as much as possible. Remaining faithful to Peirce's orthography while adding parallel sets of stylistic conventions will, however, demand close attention to typography-in-context. Current style sheets for mathematical texts specify italics for mathematical variables, with upper case letters for sets and lower case letters for individuals. So we need to keep an eye out for the difference between the individual <math>\mathrm{X}\!</math> of the genus <math>\mathrm{x}\!</math> and the element <math>x\!</math> of the set <math>X\!</math> as we pass between the two styles of text.
The path to understanding Peirce's system and its wider implications for logic can be smoothed by paraphrasing his notations in a variety of contemporary mathematical formalisms, while preserving the semantics as much as possible. Remaining faithful to Peirce's orthography while adding parallel sets of stylistic conventions will, however, demand close attention to typography-in-context. Current style sheets for mathematical texts specify italics for mathematical variables, with upper case letters for sets and lower case letters for individuals. So we need to keep an eye out for the difference between the individual <math>\mathrm{X}\!</math> of the genus <math>\mathrm{x}\!</math> and the element <math>x\!</math> of the set <math>X\!</math> as we pass between the two styles of text.
__TOC__
==Selection 1==
==Selection 1==
<p>The third class embraces terms whose logical form involves the conception of bringing things into relation, and which require the addition of more than one term to complete the denotation. They discriminate not only with consciousness of discrimination, but with consciousness of its origin. They regard an object as medium or third between two others, that is as conjugative; as giver of —— to ——, or buyer of —— for —— from ——. These may be termed ''conjugative terms''.</p>
<p>The third class embraces terms whose logical form involves the conception of bringing things into relation, and which require the addition of more than one term to complete the denotation. They discriminate not only with consciousness of discrimination, but with consciousness of its origin. They regard an object as medium or third between two others, that is as conjugative; as giver of —— to ——, or buyer of —— for —— from ——. These may be termed ''conjugative terms''.</p>
<p>The conjugative term involves the conception of ''third'', the relative that of second or ''other'', the absolute term simply considers ''an'' object. No fourth class of terms exists involving the conception of ''fourth'', because when that of ''third'' is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship. Whether this ''reason'' for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory of not, the fact itself is made perfectly evident by the study of the logic of relatives.</p>
<p>The conjugative term involves the conception of ''third'', the relative that of second or ''other'', the absolute term simply considers ''an'' object. No fourth class of terms exists involving the conception of ''fourth'', because when that of ''third'' is introduced, since it involves the conception of bringing objects into relation, all higher numbers are given at once, inasmuch as the conception of bringing objects into relation is independent of the number of members of the relationship. Whether this ''reason'' for the fact that there is no fourth class of terms fundamentally different from the third is satisfactory or not, the fact itself is made perfectly evident by the study of the logic of relatives.</p>
<p>(Peirce, CP 3.63).</p>
<p>(Peirce, CP 3.63).</p>
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<math>\begin{array}{l}
<math>\begin{array}{l}
^{\backprime\backprime}\, \text{lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}
^{\backprime\backprime}\, \text{lover of}\, \underline{~~ ~~}\, ^{\prime\prime}
\\[6pt]
\\[6pt]
^{\backprime\backprime}\, \text{betrayer to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}
^{\backprime\backprime}\, \text{betrayer to}\, \underline{~~ ~~}\, \text{of}\, \underline{~~ ~~}\, ^{\prime\prime}
\\[6pt]
\\[6pt]
^{\backprime\backprime}\, \text{winner over of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{from}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}
^{\backprime\backprime}\, \text{winner over of}\, \underline{~~ ~~}\, \text{to}\, \underline{~~ ~~}\, \text{from}\, \underline{~~ ~~}\, ^{\prime\prime}
\end{array}</math>
\end{array}</math>
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{| align="center" cellspacing="6" width="90%"
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<p>The relative term <math>^{\backprime\backprime}\, \text{lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}</math></p>
<p>The relative term <math>^{\backprime\backprime}\, \text{lover of}\, \underline{~~ ~~}\, ^{\prime\prime}</math></p>
<p>can be reached by removing the absolute term <math>^{\backprime\backprime}\, \text{Emilia}\, ^{\prime\prime}</math></p>
<p>can be reached by removing the absolute term <math>^{\backprime\backprime}\, \text{Emilia}\, ^{\prime\prime}</math></p>
<p><math>\text{Iago}</math> is a lover of <math>\text{Emilia},</math> so the relate-correlate pair <math>\mathrm{I}:\mathrm{E}</math></p>
<p><math>\text{Iago}</math> is a lover of <math>\text{Emilia},</math> so the relate-correlate pair <math>\mathrm{I}:\mathrm{E}</math></p>
<p>lies in the 2-adic relation associated with the relative term <math>^{\backprime\backprime}\, \text{lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}.</math></p>
<p>lies in the 2-adic relation associated with the relative term <math>^{\backprime\backprime}\, \text{lover of}\, \underline{~~ ~~}\, ^{\prime\prime}.</math></p>
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<p>The relative term <math>^{\backprime\backprime}\, \text{betrayer to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}</math></p>
<p>The relative term <math>^{\backprime\backprime}\, \text{betrayer to}\, \underline{~~ ~~}\, \text{of}\, \underline{~~ ~~}\, ^{\prime\prime}</math></p>
<p>can be reached by removing the absolute terms <math>^{\backprime\backprime}\, \text{Othello}\, ^{\prime\prime}</math> and <math>^{\backprime\backprime}\, \text{Desdemona}\, ^{\prime\prime}</math></p>
<p>can be reached by removing the absolute terms <math>^{\backprime\backprime}\, \text{Othello}\, ^{\prime\prime}</math> and <math>^{\backprime\backprime}\, \text{Desdemona}\, ^{\prime\prime}</math></p>
<p><math>\text{Iago}</math> is a betrayer to <math>\text{Othello}</math> of <math>\text{Desdemona},</math> so the relate-correlate-correlate triple <math>\mathrm{I}:\mathrm{O}:\mathrm{D}</math></p>
<p><math>\text{Iago}</math> is a betrayer to <math>\text{Othello}</math> of <math>\text{Desdemona},</math> so the relate-correlate-correlate triple <math>\mathrm{I}:\mathrm{O}:\mathrm{D}</math></p>
<p>lies in the 3-adic relation assciated with the relative term <math>^{\backprime\backprime}\, \text{betrayer to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}.\!</math></p>
<p>lies in the 3-adic relation assciated with the relative term <math>^{\backprime\backprime}\, \text{betrayer to}\, \underline{~~ ~~}\, \text{of}\, \underline{~~ ~~}\, ^{\prime\prime}.\!</math></p>
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<p>The relative term <math>^{\backprime\backprime}\, \text{winner over of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{from}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}</math></p>
<p>The relative term <math>^{\backprime\backprime}\, \text{winner over of}\, \underline{~~ ~~}\, \text{to}\, \underline{~~ ~~}\, \text{from}\, \underline{~~ ~~}\, ^{\prime\prime}</math></p>
<p>can be reached by removing the absolute terms <math>^{\backprime\backprime}\, \text{Othello}\, ^{\prime\prime},</math> <math>^{\backprime\backprime}\, \text{Iago}\, ^{\prime\prime},</math> and <math>^{\backprime\backprime}\, \text{Cassio}\, ^{\prime\prime}</math></p>
<p>can be reached by removing the absolute terms <math>^{\backprime\backprime}\, \text{Othello}\, ^{\prime\prime},</math> <math>^{\backprime\backprime}\, \text{Iago}\, ^{\prime\prime},</math> and <math>^{\backprime\backprime}\, \text{Cassio}\, ^{\prime\prime}</math></p>
<p><math>\text{Iago}</math> is a winner over of <math>\text{Othello}</math> to <math>\text{Iago}</math> from <math>\text{Cassio},\!</math> so the elementary relative term <math>\mathrm{I}:\mathrm{O}:\mathrm{I}:\mathrm{C}</math></p>
<p><math>\text{Iago}</math> is a winner over of <math>\text{Othello}</math> to <math>\text{Iago}</math> from <math>\text{Cassio},\!</math> so the elementary relative term <math>\mathrm{I}:\mathrm{O}:\mathrm{I}:\mathrm{C}</math></p>
<p>lies in the 4-adic relation associated with the relative term <math>^{\backprime\backprime}\, \text{winner over of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{to}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{from}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}.</math></p>
<p>lies in the 4-adic relation associated with the relative term <math>^{\backprime\backprime}\, \text{winner over of}\, \underline{~~ ~~}\, \text{to}\, \underline{~~ ~~}\, \text{from}\, \underline{~~ ~~}\, ^{\prime\prime}.</math></p>
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Returning to the Othello example, let us take up the 2-adic relatives <math>^{\backprime\backprime}\, \text{lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}</math> and <math>^{\backprime\backprime}\, \text{servant of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}.</math>
Returning to the Othello example, let us take up the 2-adic relatives <math>^{\backprime\backprime}\, \text{lover of}\, \underline{~~ ~~}\, ^{\prime\prime}</math> and <math>^{\backprime\backprime}\, \text{servant of}\, \underline{~~ ~~}\, ^{\prime\prime}.</math>
Ignoring the many splendored nuances appurtenant to the idea of love, we may regard the relative term <math>\mathit{l}\!</math> for <math>^{\backprime\backprime}\, \text{lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}</math> to be given by the following equation:
Ignoring the many splendored nuances appurtenant to the idea of love, we may regard the relative term <math>\mathit{l}\!</math> for <math>^{\backprime\backprime}\, \text{lover of}\, \underline{~~ ~~}\, ^{\prime\prime}</math> to be given by the following equation:
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{| align="center" cellspacing="6" width="90%"
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If for no better reason than to make the example more interesting, let us put aside all distinctions of rank and fealty, collapsing the motley crews of attendant, servant, subordinate, and so on, under the heading of a single service, denoted by the relative term <math>\mathit{s}\!</math> for <math>^{\backprime\backprime}\, \text{servant of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, ^{\prime\prime}.</math> The terms of this service are:
If for no better reason than to make the example more interesting, let us put aside all distinctions of rank and fealty, collapsing the motley crews of attendant, servant, subordinate, and so on, under the heading of a single service, denoted by the relative term <math>\mathit{s}\!</math> for <math>^{\backprime\backprime}\, \text{servant of}\, \underline{~~ ~~}\, ^{\prime\prime}.</math> The terms of this service are:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
\mathit{l}\mathit{s}
\mathit{l}\mathit{s}
& = &
& = &
\text{lover of a servant of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
\text{lover of a servant of}\, \underline{~~ ~~}
\\[6pt]
\\[6pt]
& = &
& = &
\mathit{s}\mathit{l}
\mathit{s}\mathit{l}
& = &
& = &
\text{servant of a lover of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
\text{servant of a lover of}\, \underline{~~ ~~}
\\[6pt]
\\[6pt]
& = &
& = &
<math>\begin{array}{*{11}{c}}
<math>\begin{array}{*{11}{c}}
\mathrm{m,}
\mathrm{m,}
& = & \text{man that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{man that is}\, \underline{~~ ~~}
& = & \mathrm{C}:\mathrm{C}
& = & \mathrm{C}:\mathrm{C}
& +\!\!, & \mathrm{I}:\mathrm{I}
& +\!\!, & \mathrm{I}:\mathrm{I}
\\[6pt]
\\[6pt]
\mathrm{n,}
\mathrm{n,}
& = & \text{noble that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{noble that is}\, \underline{~~ ~~}
& = & \mathrm{C}:\mathrm{C}
& = & \mathrm{C}:\mathrm{C}
& +\!\!, & \mathrm{D}:\mathrm{D}
& +\!\!, & \mathrm{D}:\mathrm{D}
\\[6pt]
\\[6pt]
\mathrm{w,}
\mathrm{w,}
& = & \text{woman that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{woman that is}\, \underline{~~ ~~}
& = & \mathrm{B}:\mathrm{B}
& = & \mathrm{B}:\mathrm{B}
& +\!\!, & \mathrm{D}:\mathrm{D}
& +\!\!, & \mathrm{D}:\mathrm{D}
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\mathbf{1,}
\mathbf{1,}
& = & \text{anything that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{anything that is}\, \underline{~~ ~~}
\\[6pt]
\\[6pt]
& = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
& = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
\\[9pt]
\\[9pt]
\mathrm{m,}
\mathrm{m,}
& = & \text{man that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{man that is}\, \underline{~~ ~~}
\\[6pt]
\\[6pt]
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
\\[9pt]
\\[9pt]
\mathrm{n,}
\mathrm{n,}
& = & \text{noble that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{noble that is}\, \underline{~~ ~~}
\\[6pt]
\\[6pt]
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
\\[9pt]
\\[9pt]
\mathrm{w,}
\mathrm{w,}
& = & \text{woman that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{woman that is}\, \underline{~~ ~~}
\\[6pt]
\\[6pt]
& = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E}
& = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E}
<math>\begin{array}{lll}
<math>\begin{array}{lll}
\mathbf{1,}
\mathbf{1,}
& = & \text{anything that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{anything that is}\, \underline{~~ ~~}
\\[6pt]
\\[6pt]
& = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
& = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
\\[9pt]
\\[9pt]
\mathrm{m,}
\mathrm{m,}
& = & \text{man that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{man that is}\, \underline{~~ ~~}
\\[6pt]
\\[6pt]
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
\\[9pt]
\\[9pt]
\mathrm{n,}
\mathrm{n,}
& = & \text{noble that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{noble that is}\, \underline{~~ ~~}
\\[6pt]
\\[6pt]
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
& = & \mathrm{C}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}
\\[9pt]
\\[9pt]
\mathrm{w,}
\mathrm{w,}
& = & \text{woman that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}
& = & \text{woman that is}\, \underline{~~ ~~}
\\[6pt]
\\[6pt]
& = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E}
& = & \mathrm{B}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E}
\mathit{l}, ~=
\mathit{l}, ~=
\\[6pt]
\\[6pt]
\text{lover that is}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\, \text{of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)} ~=
\text{lover that is}\, \underline{~~ ~~}\, \text{of}\, \underline{~~ ~~} ~=
\\[6pt]
\\[6pt]
(\mathrm{B}\!:\!\mathrm{B}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{C}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}\!:\!\mathrm{D})
(\mathrm{B}\!:\!\mathrm{B}\!:\!\mathrm{C} ~+\!\!,~ \mathrm{C}\!:\!\mathrm{C}\!:\!\mathrm{B} ~+\!\!,~ \mathrm{D}\!:\!\mathrm{D}\!:\!\mathrm{O} ~+\!\!,~ \mathrm{E}\!:\!\mathrm{E}\!:\!\mathrm{I} ~+\!\!,~ \mathrm{I}\!:\!\mathrm{I}\!:\!\mathrm{E} ~+\!\!,~ \mathrm{J}\!:\!\mathrm{J}\!:\!\mathrm{D} ~+\!\!,~ \mathrm{O}\!:\!\mathrm{O}\!:\!\mathrm{D})
There's a critical transition point in sight of Peirce's 1870 Logic of Relatives and it's a point that turns on the teridentity relation.
There's a critical transition point in sight of Peirce's 1870 Logic of Relatives and it's a point that turns on the teridentity relation.
In taking up the next example of relational composition, let's substitute the relation <math>\mathit{t} = \text{taker of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\!</math> for Peirce's relation <math>\mathit{o} = \text{owner of}\, \underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)},\!</math> simply for the sake of avoiding conflicts in the symbols we use. In this way, Figure 17 is transformed into Figure 22.
In taking up the next example of relational composition, let's substitute the relation <math>\mathit{t} = \text{taker of}\, \underline{~~ ~~}\!</math> for Peirce's relation <math>\mathit{o} = \text{owner of}\, \underline{~~ ~~},\!</math> simply for the sake of avoiding conflicts in the symbols we use. In this way, Figure 17 is transformed into Figure 22.
{| align="center" cellpadding="10"
{| align="center" cellpadding="10"
|-
|-
| and
| and
| <math>\mathit{t} = \text{tooth of}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}.</math>
| <math>\mathit{t} = \text{tooth of}\,\underline{~~ ~~}.</math>
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Thus, in a universe of perfect human dentition, the number of the relative term <math>{}^{\backprime\backprime} \text{tooth of}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)} {}^{\prime\prime}\!</math> is equal to the number of teeth of humans divided by the number of humans, that is, <math>32.\!</math>
Thus, in a universe of perfect human dentition, the number of the relative term <math>{}^{\backprime\backprime} \text{tooth of}\,\underline{~~ ~~} {}^{\prime\prime}\!</math> is equal to the number of teeth of humans divided by the number of humans, that is, <math>32.\!</math>
The dyadic relative term <math>t\!</math> determines a dyadic relation <math>T \subseteq X \times Y,</math> where <math>X\!</math> contains all the teeth and <math>Y\!</math> contains all the people that happen to be under discussion.
The dyadic relative term <math>t\!</math> determines a dyadic relation <math>T \subseteq X \times Y,</math> where <math>X\!</math> contains all the teeth and <math>Y\!</math> contains all the people that happen to be under discussion.
|-
|-
| and
| and
| <math>\mathit{t} = \text{tooth of}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}.\!</math>
| <math>\mathit{t} = \text{tooth of}\,\underline{~~ ~~}.\!</math>
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That is to say, the number of the relative term <math>\text{tooth of}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}\!</math> is equal to the number of teeth of humans divided by the number of humans. In a universe of perfect human dentition this gives a quotient of <math>32.\!</math>
That is to say, the number of the relative term <math>\text{tooth of}\,\underline{~~ ~~}\!</math> is equal to the number of teeth of humans divided by the number of humans. In a universe of perfect human dentition this gives a quotient of <math>32.\!</math>
The dyadic relative term <math>t\!</math> determines a dyadic relation <math>T \subseteq X \times Y,</math> where <math>X\!</math> contains all the teeth and <math>Y\!</math> contains all the people that happen to be under discussion.
The dyadic relative term <math>t\!</math> determines a dyadic relation <math>T \subseteq X \times Y,</math> where <math>X\!</math> contains all the teeth and <math>Y\!</math> contains all the people that happen to be under discussion.
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In this picture the order of relational composition flows down the page. For convenience in composing relations, the absolute term <math>\mathrm{f} = \text{Frenchman}\!</math> is inflected by the comma functor to form the dyadic relative term <math>\mathrm{f,} = \text{Frenchman that is}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)},\!</math> which in turn determines the idempotent representation of Frenchmen as a subset of mankind, <math>F \subseteq Y \times Y.\!</math>
In this picture the order of relational composition flows down the page. For convenience in composing relations, the absolute term <math>\mathrm{f} = \text{Frenchman}\!</math> is inflected by the comma functor to form the dyadic relative term <math>\mathrm{f,} = \text{Frenchman that is}\,\underline{~~ ~~},\!</math> which in turn determines the idempotent representation of Frenchmen as a subset of mankind, <math>F \subseteq Y \times Y.\!</math>
By way of a legend for the figure, we have the following data:
By way of a legend for the figure, we have the following data:
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As before, it is convenient to represent the absolute term <math>\mathrm{b} = \text{black}\!</math> by means of the corresponding idempotent term <math>\mathrm{b,} = \text{black that is}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}.</math>
As before, it is convenient to represent the absolute term <math>\mathrm{b} = \text{black}\!</math> by means of the corresponding idempotent term <math>\mathrm{b,} = \text{black that is}\,\underline{~~ ~~}.</math>
Consider the bigraph for the composition:
Consider the bigraph for the composition:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>\mathrm{m,}\mathrm{b,} ~=~ \text{man that is black that is}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}.</math>
| <math>\mathrm{m,}\mathrm{b,} ~=~ \text{man that is black that is}\,\underline{~~ ~~}.</math>
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| height="40" | <math>W \subseteq X\!</math> is the 1-adic relation, or set, whose elements fall under the absolute term <math>\mathrm{w} = \text{woman}.\!</math> The elements of <math>W\!</math> are sometimes referred to as the ''denotation'' or the set-theoretic ''extension'' of the term <math>\mathrm{w}.\!</math>
| height="40" | <math>W \subseteq X\!</math> is the 1-adic relation, or set, whose elements fall under the absolute term <math>\mathrm{w} = \text{woman}.\!</math> The elements of <math>W\!</math> are sometimes referred to as the ''denotation'' or the set-theoretic ''extension'' of the term <math>\mathrm{w}.\!</math>
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| height="40" | <math>L \subseteq X \times X\!</math> is the 2-adic relation associated with the relative term <math>\mathit{l} = \text{lover of}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}.\!</math>
| height="40" | <math>L \subseteq X \times X\!</math> is the 2-adic relation associated with the relative term <math>\mathit{l} = \text{lover of}\,\underline{~~ ~~}.\!</math>
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| height="40" | <math>S \subseteq X \times X\!</math> is the 2-adic relation associated with the relative term <math>\mathit{s} = \text{servant of}\,\underline{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:08, 17 November 2015 (UTC)}.\!</math>
| height="40" | <math>S \subseteq X \times X\!</math> is the 2-adic relation associated with the relative term <math>\mathit{s} = \text{servant of}\,\underline{~~ ~~}.\!</math>
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