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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 02:00, 17 April 2009
, 02:00, 17 April 2009→Commentary Note 11.17
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>\mathrm{f} < \mathrm{m} ~\Rightarrow~ [\mathrm{f}] < [\mathrm{m}].</math>
|
<math>\begin{matrix}
\mathrm{f} < \mathrm{m} & \Rightarrow & [\mathrm{f}] < [\mathrm{m}].
\end{matrix}</math>
|}
|}
{| align="center" cellspacing="6" width="90%"
{| align="center" cellspacing="6" width="90%"
| <math>x <_1 y ~\Rightarrow~ F(x) <_2 F(y).</math>
|
<math>\begin{matrix}
x <_1 y & \Rightarrow & F(x) <_2 F(y).
\end{matrix}</math>
|}
|}
The "number of" map <math>v : (S, <_1) \to (\mathbb{R}, <_2)</math> has just this character, as exemplified by its application to the following case:
The "number of" map <math>v : (S, <_1) \to (\mathbb{R}, <_2)</math> has just this character, as exemplified in the case at hand:
{| align="center" cellspacing="6" width="90%"
|
<math>\begin{matrix}
\mathrm{f} & < & \mathrm{m} & \Rightarrow & [\mathrm{f}] & < & [\mathrm{m}]
\\[6pt]
\mathrm{f} & < & \mathrm{m} & \Rightarrow & v\mathrm{f} & < & v\mathrm{m}
\end{matrix}</math>
|}
Here, to be more exacting, we may interpret the "<" on the left as "proper subsumption", that is, excluding the equality case, while we read the "<" on the right as the usual "less than".
Here, to be more exacting, the <math>^{\backprime\backprime}\!\!<\!^{\prime\prime}</math> on the left is read as ''proper subsumption'', that is, the relation of being a subset but not being equal, while the <math>^{\backprime\backprime}\!\!<\!^{\prime\prime}</math> on the right is read as the usual ''less than'' relation.
===Commentary Note 11.18===
===Commentary Note 11.18===