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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 14:08, 3 May 2009
, 14:08, 3 May 2009→Commentary Note 12.2
| 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>
| 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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{| align="center" cellspacing="6" width="90%"
|
<pre>
a b c d e f g h i
o o o o o o o o o X
/ \ : | : |
/ \ 0 1 0 1 L
/ \ : | : |
o o o o + - + + o X
\ | / : : | |
\ | / 0 0 1 1 S
\|/ : : | |
o o o o o o o o o X
a b c d e f g h i
</pre>
|}
There is a "servant of every lover of" link between <math>u\!</math> and <math>v\!</math> if and only if <math>u \cdot S ~\supseteq~ L \cdot v.</math> But the vacuous inclusions will make this non-intuitive.
{| align="center" cellspacing="6" width="90%"
| <math>(\mathfrak{S}^\mathfrak{L})_{uv} ~=~ \prod_{x \in X} \mathfrak{S}_{ux}^{\mathfrak{L}_{xv}}</math>
|}
In other words, <math>(\mathfrak{S}^\mathfrak{L})_{uv} = 0</math> if and only if there exists an <math>x \in X</math> such that <math>\mathfrak{S}_{ux} = 0</math> and <math>\mathfrak{L}_{xv} = 1.</math>
==References==
==References==