Post : Peirce's 1870 “Logic Of Relatives” • Comment 11.9
http://inquiryintoinquiry.com/2014/05/07/peirces-1870-logic-of-relatives-%e2%80%a2-comment-11-9/
Posted : May 7, 2014 at 2:00 pm
Author : Jon Awbrey
Peircers,
Among the variety of regularities affecting dyadic relations we pay special attention to the
c-regularity conditions where c is equal to 1.
Let P ⊆ X × Y be an arbitrary dyadic relation. The following properties of P
can then be defined:
• P is total at X ⇔ P is (≥ 1)-regular at X.
• P is total at Y ⇔ P is (≥ 1)-regular at Y.
• P is tubular at X ⇔ P is (≤ 1)-regular at X.
• P is tubular at Y ⇔ P is (≤ 1)-regular at Y.
We previously examined dyadic relations that separately exemplified each of these regularity
conditions. And we introduced a few bits of terminology and special-purpose notations for working
with tubular relations:
• P is a pre-function P : X ⇀ Y ⇔ P is tubular at X.
• P is a pre-function P : X ↼ Y ⇔ P is tubular at Y.
We arrive by way of this winding stair at the special stamps of dyadic relations P ⊆ X × Y that are
variously described as ''1-regular'', ''total and tubular'', or ''total prefunctions'' on specified
domains, either X or Y or both, and that are more often celebrated as ''functions'' on those domains.
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.
To say that a relation P ⊆ X × Y is ''total and tubular'' at X is to say that P is 1-regular at X.
Thus, we may formalize the following definitions:
• P is a function P : X → Y ⇔ P is 1-regular at X.
• P is a function P : X ← Y ⇔ P is 1-regular at Y.
For example, let X = Y = {0, …, 9} and let F ⊆ X × Y be
the dyadic relation depicted in the bigraph below:
Figure 39. Dyadic Relation F ⊆ X × Y, Functional at Y
☞http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-39.jpg
We observe that F is a function at Y and we record this fact
in either of the manners F : X ← Y or F : Y → X.
Regards,
Jon
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