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Directory:Jon Awbrey/Papers/Peirce's 1870 Logic Of Relatives (view source)
Revision as of 21:10, 3 December 2007
, 21:10, 3 December 2007→Commentary Note 11.9: markup
===Commentary Note 11.9===
===Commentary Note 11.9===
Among the vast variety of conceivable regularities affecting 2-adic relations, we pay special attention to the ''c''-regularity conditions where ''c'' is equal to 1.
Among the vast variety of conceivable regularities affecting 2-adic relations,
we pay special attention to the c-regularity conditions where c is equal to 1.
<blockquote>
<p>Let ''P'' ⊆ ''X'' × ''Y'' be an arbitrary 2-adic relation. The following properties of P can be defined:</p>
| Let P c X x Y be an arbitrary 2-adic relation.
{| cellpadding="4"
| ''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>
We have already looked at 2-adic relations that
We have already looked at 2-adic relations that separately exemplify each of these regularities.
separately exemplify each of these regularities.
Also, we introduced a few bits of additional terminology and
Also, we introduced a few bits of additional terminology and special-purpose notations for working with tubular relations:
special-purpose notations for working with tubular relations:
| P is a "pre-function" P : X ~> Y iff P is tubular at X.
:{| cellpadding="4"
|
| ''P'' is a "pre-function" ''P'' : ''X'' ~> ''Y''
| P is a "pre-function" P : X <~ Y iff P is tubular at Y.
| iff
| ''P'' is tubular at ''X''.
|-
| ''P'' is a "pre-function" ''P'' : ''X'' <~ ''Y''
| iff
| ''P'' is tubular at ''Y''.
|}
Thus, we arrive by way of this winding stair at the very special stamps
Thus, we arrive by way of this winding stair at the very special stamps of 2-adic relations ''P'' ⊆ ''X'' × ''Y'' that are "total prefunctions" at ''X'' (or ''Y''), "total and tubular" at ''X'' (or ''Y''), or "1-regular" at ''X'' (or ''Y''), more often celebrated as "functions" at ''X'' (or ''Y'').
of 2-adic relations P c X x Y that are "total prefunctions" at X (or Y),
"total and tubular" at X (or Y), or "1-regular" at X (or Y), more often
celebrated as "functions" at X (or Y).
<pre>
| If P is a pre-function P : X ~> Y that happens to be total at X, then P
| If P is a pre-function P : X ~> Y that happens to be total at X, then P
| is known as a "function" from X to Y, typically indicated as P : X -> Y.
| is known as a "function" from X to Y, typically indicated as P : X -> Y.