[Inquiry] Re: Logic Of Relatives

[Jon Awbrey]

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LOR.  Note 47

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It always helps me to draw lots of pictures of stuff,
so let's extract the somewhat overly compressed bits
of the "Relations In General" thread that we'll need
right away for the applications to Peirce's 1870 LOR,
and draw what icons we can within the frame of Ascii.

For the immediate present, we may start with 2-adic relations
and describe the customary species of relations and functions
in terms of their local and numerical incidence properties.

Let P c X x Y be an arbitrary 2-adic relation.
The following properties of P can be defined:

P is "total" at X     iff   P is (>=1)-regular at X.

P is "total" 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.

To illustrate these properties, let us fashion
a "generic enough" example of a 2-adic relation,
E c X x Y, where X = Y = {0, 1, ..., 8, 9}, and
where the bigraph picture of E looks like this:

0   1   2   3   4   5   6   7   8   9
o   o   o   o   o   o   o   o   o   o   X
     \  |\ /|\   \   \  |   |\
      \ | / | \   \   \ |   | \         E
       \|/ \|  \   \   \|   |  \
o   o   o   o   o   o   o   o   o   o   Y
0   1   2   3   4   5   6   7   8   9

If we scan along the X dimension we see that the "Y incidence degrees"
of the X nodes 0 through 9 are 0, 1, 2, 3, 1, 1, 1, 2, 0, 0, in order.

If we scan along the Y dimension we see that the "X incidence degrees"
of the Y nodes 0 through 9 are 0, 0, 3, 2, 1, 1, 2, 1, 1, 0, in order.

Thus, E is not total at either X or Y,
since there are nodes in both X and Y
having incidence degrees that equal 0.

Also, E is not tubular at either X or Y,
since there exist nodes in both X and Y
having incidence degrees greater than 1.

Clearly, then, E cannot qualify as a pre-function
or a function on either of its relational domains.

Jon Awbrey

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