Post : Peirce's 1870 “Logic Of Relatives” • Comment 11.11
http://inquiryintoinquiry.com/2014/05/09/peirces-1870-logic-of-relatives-%e2%80%a2-comment-11-11/
Posted : May 9, 2014 at 3:00 pm
Author : Jon Awbrey
Peircers,
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 recognize the local functionalities
that are immanent in relative terms no matter where they locate themselves within the domains of
relations. These skills will serve us in good stead as we work to build a catwalk from Peirce's
platform of 1870 to contemporary scenes on the logic of relatives, and back again.
By way of extending a few very tentative planks, let us experiment with the
following definitions:
• A relative term “p” and the corresponding relation P ⊆ X × Y are both called ''functional on
relates'' if and only if P is a function at X. We write this in symbols as P : X → Y.
• A relative term “p” and the corresponding relation P ⊆ X × Y are both called ''functional on
correlates'' if and only if P is a function at Y. We write this in symbols as 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 × 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 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.
Figure 44. Anything that is a ''p'' of a ''q'' of Anything
☞http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-44.jpg
From this we extract the ''hypergraph picture'' of relational composition:
Figure 45. Relational Composition P ∘ Q
☞http://inquiryintoinquiry.files.wordpress.com/2014/05/lor-1870-figure-45.jpg
All of the information contained in these Figures can be expressed in the form of a constraint
satisfaction table, or ''spreadsheet picture'' of relational composition:
Table 46. Relational Composition P ∘ Q
..... | 1 | 1 | 1 |
===================
P ... | X | Y | . |
Q ... | . | Y | Z |
P ◦ Q | X | . | Z |
The following plan of study then presents itself, to see what easy mileage we can get in our
exploration of functions by adopting the above templates as the primers of a paradigm.
Regards,
Jon
--
academia: http://independent.academia.edu/JonAwbrey
my word press blog: http://inquiryintoinquiry.com/
inquiry list: http://stderr.org/pipermail/inquiry/
isw: http://intersci.ss.uci.edu/wiki/index.php/JLA
oeiswiki: http://www.oeis.org/wiki/User:Jon_Awbrey
facebook page: https://www.facebook.com/JonnyCache
-----------------------------
PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L
to this message. PEIRCE-L posts should go to [email protected] . To
UNSUBSCRIBE, send a message not to PEIRCE-L but to [email protected] with the
line "UNSubscribe PEIRCE-L" in the BODY of the message. More at
http://www.cspeirce.com/peirce-l/peirce-l.htm .
