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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 02:36, 3 May 2009
, 02:36, 3 May 2009→Commentary Note 12.1
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_u ~=~ \prod_{x \in X} \mathfrak{L}_{ux}^{\mathfrak{W}_{x}}</math>
| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_u ~=~ \prod_{x \in X} \mathfrak{L}_{ux}^{\mathfrak{W}_x}</math>
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An abstract formula of this kind is more easily grasped with the aid of a freely chosen example and a picture of the relations involved.
An abstract formula of this kind is more easily grasped with the aid of a concrete example and a picture of the relations involved. The Figure below represents a universe of discourse <math>X\!</math> that is subject to the following data:
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{array}{*{14}{c}}
X & = & \{ & a, & b, & c, & d, & e, & f, & g, & h, & i & \}
\\[6pt]
W & = & \{ & d, & f & \}
\\[6pt]
L & = & \{ & b\!:\!a, & b\!:\!c, & c\!:\!b, & c\!:\!d, & e\!:\!d, & e\!:\!f, & g\!:\!f, & g\!:\!h, & h\!:\!g, & h\!:\!i & \}
\end{array}</math>
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{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|}
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To highlight the role of <math>W\!</math> more clearly, the Figure represents the absolute term <math>^{\backprime\backprime} \mathrm{w} ^{\prime\prime}</math> by means of the idempotent relative term <math>^{\backprime\backprime} \mathrm{w}, ^{\prime\prime}</math> that conveys the same information.
With the above picture in mind, we can visualize the computation of <math>(\mathfrak{L}^\mathfrak{W})_u = \textstyle\prod_{x \in X} \mathfrak{L}_{ux}^{\mathfrak{W}_x}</math> as follows:
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
|
| valign="top" | 1.
<math>\begin{array}{*{14}{c}}
| Pick a specific <math>u\!</math> in the bottom row of the Figure.
|-
| valign="top" | 2.
\{ & a, & b, & c, & d, & e, & f, & g, & h, & i & \}
| Pan across the elements <math>x\!</math> in the middle row of the Figure.
|-
| valign="top" | 3.
| If <math>u\!</math> links to <math>x\!</math> then <math>\mathfrak{L}_{ux} = 1,</math> otherwise <math>\mathfrak{L}_{ux} = 0.</math>
\{ & d, & f & \}
|-
\\[6pt]
| valign="top" | 4.
L
| If <math>x\!</math> in the middle row links to <math>x\!</math> in the top row then <math>\mathfrak{W}_x = 1,</math> otherwise <math>\mathfrak{W}_x = 0.</math>
|-
| valign="top" | 5.
| Compute the value <math>\mathfrak{L}_{ux}^{\mathfrak{W}_x} = (\mathfrak{L}_{ux}\!\Leftarrow\!\mathfrak{W}_x)</math> for each <math>x\!</math> in the middle row.
|-
| valign="top" | 6.
| If any of the values <math>\mathfrak{L}_{ux}^{\mathfrak{W}_x}</math> is <math>0\!</math> then the product <math>\textstyle\prod_{x \in X} \mathfrak{L}_{ux}^{\mathfrak{W}_x}</math> is <math>0,\!</math> otherwise it is <math>1.\!</math>
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===Commentary Note 12.2===
===Commentary Note 12.2===