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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 11:25, 4 May 2009
, 11:25, 4 May 2009→Example 6
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>\mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} L \cdot x</math>
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{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>(\mathfrak{L}^\mathfrak{W})_u ~=~ \prod_{x \in X} \mathfrak{L}_{ux}^{\mathfrak{W}_x}</math>
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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.
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 relative term <math>^{\backprime\backprime} \mathrm{w}, ^{\prime\prime}</math> that conveys the same information.
Computing the denotation of <math>\mathit{l}^\mathrm{w}\!</math> by way of the set-theoretic formula, we can show our work as follows:
{| align="center" cellspacing="6" width="90%"
| <math>\mathit{l}^\mathrm{w} ~=~ \bigcap_{x \in W} L \cdot x ~=~ L \cdot d ~\cap~ L \cdot f ~=~ \{ c, e \} \cap \{ e, g \} ~=~ \{ e \}</math>
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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:
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: