Post : Peirce's 1870 “Logic Of Relatives” • Comment 11.5
http://inquiryintoinquiry.com/2014/05/02/peirces-1870-logic-of-relatives-%e2%80%a2-comment-11-5/
Posted : May 2, 2014 at 5:00 pm
Author : Jon Awbrey
Peircers,
Everyone knows that the right sort of diagram can be a great aid in rendering complex matters
comprehensible, so let's extract the all too compressed bits of the Relation Theory article that it
takes to illuminate Peirce's 1870 “Logic of Relatives” and use them to fashion what icons we can
within the current frame of discussion.
Relation Theory
☞http://intersci.ss.uci.edu/wiki/index.php/Relation_Theory
For the immediate present, we may start with dyadic relations and describe the most frequently
encountered species of relations and functions in terms of their local and numerical incidence
properties.
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.
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 alternate lower case name, say “p”, and writing p
: X ⇀ Y.
Just by way of formalizing the definition:
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.
To illustrate these properties, let us fashion a generic example of a dyadic relation, P ⊆ X × Y,
where X = Y = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, and where the bigraph picture of E looks like this:
Figure 30. Bigraph Representation of the Dyadic Relation Example E
☞http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-30.jpg
If we scan along the X dimension from 0 to 9 we see that the incidence degrees of the X nodes with
the Y domain are 0, 1, 2, 3, 1, 1, 1, 2, 0, 0 in that order.
If we scan along the Y dimension from 0 to 9 we see that the incidence degrees of the Y nodes with
the X domain are 0, 0, 3, 2, 1, 1, 2, 1, 1, 0 in that order.
Thus, E is not total at either X or Y since there are nodes in both X and Y having incidence degrees
less than 1.
Also, E is not tubular at either X or Y since there are nodes in both X and Y having incidence
degrees greater than 1.
Clearly then the relation E cannot qualify as a pre-function, much less as a function on either of
its relational domains.
Regards,
Jon
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