ONT Re: Logic Of Relatives
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LOR. Note 53
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The preceding exercises were intended to beef-up our
functional literacy skills to the point where we can
read our functional alphabets backwards and forwards
and to ferret out the local functionalites that may
be immanent in relative terms no matter where they
locate themselves within the domains of relations.
I am hopeful that these skills will serve us in
good stead as we work to build a catwalk from
Peirce's platform to contemporary scenes on
the logic of relatives, and back again.
By way of extending a few very tentative plancks,
let us experiment with the following definitions:
1. A relative term 'p' and the corresponding relation P c X x Y are both
called "functional on relates" if and only if P is a function at X,
in symbols, P : X -> Y.
2. A relative term 'p' and the corresponding relation P c X x Y are both
called "functional on correlates" if and only if P is function at Y,
in symbols, P : X <- Y.
When a relation happens to be a function, it may be excusable
to use the same name for it in both applications, writing out
explicit type markers like P : X x Y, P : X -> Y, P : X <- Y,
as the case may be, when and if it serves to clarify matters.
From this current, perhaps transient, perspective, it appears that
our next task is to examine how the known properties of relations
are modified when an aspect of functionality is spied in the mix.
Let us then return to our various ways of looking at relational composition,
and see what changes and what stays the same when the relations in question
happen to be functions of various different kinds at some of their domains.
Here is one generic picture of relational composition,
cast in a style that hews pretty close to the line of
potentials inherent in Peirce's syntax of this period.
o-----------------------------------------------------------o
| |
| P o Q |
| ____________@____________ |
| / \ |
| / P Q \ |
| / @ @ \ |
| / / \ / \ \ |
| / / \ / \ \ |
| o o o o o o |
| X X Y Y Z Z |
| 1,__# #'p'__$ $'q'__% %1 |
| o o o o o o |
| \ / \ / \ / |
| \ / \ / \ / |
| \ / \ / \ / |
| @ @ @ |
| !1! !1! !1! |
| |
o-----------------------------------------------------------o
Figure 16. Anything that is a 'p' of a 'q' of Anything
From this we extract the "hypergraph picture" of relational composition:
o-----------------------------------------------------------o
| |
| P P o Q Q |
| @ @ @ |
| / \ / \ / \ |
| / \ / \ / \ |
| o o o o o o |
| X Y X Z Y Z |
| o o o o o o |
| \ \ / \ / / |
| \ \ / \ / / |
| \ / \ / |
| \ / \ / \ / |
| \ / \___ ___/ \ / |
| @ @ @ |
| !1! !1! !1! |
| |
o-----------------------------------------------------------o
Figure 17. Relational Composition P o Q
All of the relevant information of these Figures can be compressed
into the form of a "spreadsheet", or constraint satisfaction table:
Table 18. Relational Composition P o Q
o---------o---------o---------o---------o
| # !1! | !1! | !1! |
o=========o=========o=========o=========o
| P # X | Y | |
o---------o---------o---------o---------o
| Q # | Y | Z |
o---------o---------o---------o---------o
| P o Q # X | | Z |
o---------o---------o---------o---------o
So the following presents itself as a reasonable plan of study:
Let's see how much easy mileage we can get in our exploration
of functions by adopting the above templates as a paradigm.
Jon Awbrey
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