Helmut, List,
Looking over those articles with fresh eyes this morning
I see they are rather thick with abstract generalities at
the beginning and it would be better to skip down to the
concrete examples on a first run-through. I promise to
keep that in mind the next time I rewrite them. At any
rate, we can always go through the material in a more
leisurely fashion on the List.
Looking back over many previous discussions, I think one
of the main things keeping people from being on the same
page, or even being able to understand what others write
on their individual pages, is the question of what makes
a relation.
There's a big difference between a single ordered tuple, say,
(x_1, x_2, ..., x_k), and a whole set of ordered tuples that
it takes to make up a k-place relation. The language we use
to get a handle on the structure of relations goes like this:
Say the variable x_1 ranges over the set X_1,
and the variable x_2 ranges over the set X_2,
and ...
and the variable x_k ranges over the set X_k.
Then the set of all possible k-tuples (x_1, x_2, ..., x_k)
ranges over a set that is notated as X_1 × X_2 × ... × X_k,
called the “cartesian product” of the “domains” X_1 to X_k.
There are two different ways of defining
a k-place relation that are in common use:
1. Some define a relation L on the domains X_1 to X_k
as a subset of the cartesian product X_1 × ... × X_k,
in symbols, L ⊆ X_1 × ... × X_k.
2. Others like to make the domains of the relation
an explicit part of the definition, saying that
a relation L is a list of domains plus a subset
of their cartesian product.
Sounds like a mess but it's usually pretty easy to
translate between the two conventions, so long as
one remains aware of difference.
By way of a geometric image, we can picture the
cartesian product X_1 × ... × X_k as a space in
which many different relations reside, each one
cutting a different figure in that space.
To be continued ...
Jon
On 4/15/2017 12:00 AM, Jon Awbrey wrote:
Helmut, List,
I think it would be a good idea to continue reviewing basic concepts
and get better acquainted with the relational context that is needed
to ground all the higher order functions, properties, and structures
we might wish to think about. Once we understand what relations are
then we can narrow down to triadic relations and then sign relations
will fall more easily within our grasp.
I've written up intros to these topics many times before, and you
can find my latest editions, if still very much works in progress,
on the InterSciWiki site, though in this case it may be preferable
to take them up in order from special to general:
Sign Relations
http://intersci.ss.uci.edu/wiki/index.php/Sign_relation
Triadic Relations
http://intersci.ss.uci.edu/wiki/index.php/Triadic_relation
Relation Theory
http://intersci.ss.uci.edu/wiki/index.php/Relation_theory
I think most of the material you mentioned on Relational Reducibility,
Compositional and Projective, is summarized in the following article:
Relation Reduction
http://intersci.ss.uci.edu/wiki/index.php/Relation_reduction
Regards,
Jon
--
inquiry into inquiry: https://inquiryintoinquiry.com/
academia: https://independent.academia.edu/JonAwbrey
oeiswiki: https://www.oeis.org/wiki/User:Jon_Awbrey
isw: http://intersci.ss.uci.edu/wiki/index.php/JLA
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 peirce-L@list.iupui.edu . To
UNSUBSCRIBE, send a message not to PEIRCE-L but to l...@list.iupui.edu with the
line "UNSubscribe PEIRCE-L" in the BODY of the message. More at
http://www.cspeirce.com/peirce-l/peirce-l.htm .
