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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 14:27, 26 November 2007
, 14:27, 26 November 2007→Commentary Note 11.6: markup
===Commentary Note 11.6===
===Commentary Note 11.6===
Let's continue to work our way through the rest of the first set of definitions, making up appropriate examples as we go.
Let's continue to work our way through the rest of the first
set of definitions, making up appropriate examples as we go.
<blockquote>
Let ''P'' ⊆ ''X'' × ''Y'' be an arbitrary 2-adic relation. The following properties of ''P'' can be defined:
:{| cellpadding="6"
| ''P'' is "total" at ''X''
|
| iff
| P is "total" at X iff P is (>=1)-regular at X.
| ''P'' is (≥1)-regular at ''X''.
|
|-
| P is "total" at Y iff P is (>=1)-regular at Y.
| ''P'' is "total" at ''Y''
|
| iff
| P is "tubular" at X iff P is (=<1)-regular at X.
| ''P'' is (≥1)-regular at ''Y''.
|
|-
| P is "tubular" at Y iff P is (=<1)-regular at Y.
| ''P'' is "tubular" at ''X''
| iff
| ''P'' is (≤1)-regular at ''X''.
|-
| ''P'' is "tubular" at ''Y''
| iff
| ''P'' is (≤1)-regular at ''Y''.
|}
</blockquote>
''E''<sub>1</sub> exemplifies the quality of "totality at ''X''".
<pre>
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
o o o o o o o o o o X
o o o o o o o o o o X
o o o o o o o o o o Y
o o o o o o o o o o Y
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
</pre>
''E''<sub>2</sub> exemplifies the quality of "totality at ''Y''".
<pre>
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
o o o o o o o o o o X
o o o o o o o o o o X
o o o o o o o o o o Y
o o o o o o o o o o Y
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
</pre>
''E''<sub>3</sub> exemplifies the quality of "tubularity at ''X''".
<pre>
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
o o o o o o o o o o X
o o o o o o o o o o X
o o o o o o o o o o Y
o o o o o o o o o o Y
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
</pre>
''E''<sub>4</sub> exemplifies the quality of "tubularity at ''Y''".
<pre>
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
o o o o o o o o o o X
o o o o o o o o o o X
o o o o o o o o o o Y
o o o o o o o o o o Y
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
</pre>
<blockquote>
<p>If ''P'' ⊆ ''X'' × ''Y'' is tubular at ''X'', then ''P'' is known as a "partial function" or a "pre-function" from ''X'' to ''Y'', frequently signalized by renaming ''P'' with an alternative lower case name, say "''p''", and writing ''p'' : ''X'' ~> ''Y''.</p>
<p>Just by way of formalizing the definition:</p>
</pre>
<p>''P'' is a "pre-function" ''P'' : ''X'' ~> ''Y'' iff ''P'' is tubular at ''X''.</p>
<p>''P'' is a "pre-function" ''P'' : ''X'' <~ ''Y'' iff ''P'' is tubular at ''Y''.</p>
</blockquote>
So, ''E''<sub>3</sub> is a pre-function ''e''<sub>3</sub> : ''X'' ~> ''Y'', and ''E''<sub>4</sub> is a pre-function ''e''<sub>4</sub> : ''X'' <~ ''Y''.
===Commentary Note 11.7===
===Commentary Note 11.7===