ONT Re: Identity & Teridentity
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
HC = Howard Callaway
JA = Jon Awbrey
KK = Kenneth Ketner
KK: Howard and Jon: All this nonreduction of triads stuff and the wrongness
of Quine's supposed reduction to a dyadic predicate is laid out with the
highest mathematical rigor in Robert Burch's A PEIRCEAN REDUCTION THESIS:
THE FOUNDATIONS OF TOPOLOGICAL LOGIC. Arisbebooks has some copies (search
via google). This volume is an absolute essential in any literature review
for persons working this area.
JA: Ergo, all notions of analysis, composition, reduction, synthesis, whatever,
contain a notion of triadic relations as a part of their very constitution.
HC: I find the related arguments very slippery. Certainly I am no fan of reductionism,
but it seems the the presuppositions involved in the proofs offered of the reduction
or non-reduction of "genuine" triadic relations (Correct me if I am wrong, but I think
Peirce uses the term "genuine" triadic relations, so that there is some distinction
between genuinely triadic relations and those which may just appear to be so, and
are open to some analysis) -- the various presuppositions -- seem not so obvious.
HC: So, please be aware that I do not have in mind to defend Quine's proof of
reduction to non-triadic relations. But that he gives a proof or apparent
proof seems a chief point of interest. On the other hand, Peirce's contrary
arguments to the effect that triadic relations cannot be reduced, seems to
involve presuppositions connected with his use of teridentity, in contrast
to the usual versions we see in logic books.
HC: I have my serious doubts that we actually need the concept of teridentity for
logical purposes generally. The cross-reference and possible cross-reference
of the variables seems to take the place of teridentity as things are usually
formulated.
HC: If you can clarify this matter, then I think you will
provide a benefit to readers of the list, myself included.
Howard, Ken, & All,
I apparently failed to receive this earlier message from Howard.
I will put Burch's book on my wish list, but the basic facts have
long been beyond question in the necks of the mathematical woods
that I was once accustomed to frequent. We had long battles over
this on SUO, and even when I got one of the reviewers who panned
Burch's book in print to concede off-list -- they always do it by
saying that it was always already trivial in the first place --
he/she never would fess up in public. So I have little hope.
There are at least three levels to the irreducibility half of it,
and I have seen all the relevant Peirce quotes already pass under
the bridge several times here with not so much cognizance of what
they say.
At level one, the argument is over before it starts. The very idea
of reducing 1 thing to 1 thing plus 1 other thing is a triadic idea.
At level two, there is ordinary relational composition, or as Peirce
called it "relative multiplication", and it defines the composition
of two 2-adic relations to be a 2-adic relation, which means that
you can never ever get a 3-adic relation as the composition of
two 2-adic relations. Peirce just takes this much as given
from the start, as it is already clear in his first papers.
At level three, which a player reaches in an orderly way
only by conceding the issue of irreducibility on the first
two scores, and this is where, as I currently understand it,
the whole business about "genuineness" comes into play, one is
given a single 3-adic relation, say, the one that corresponds to
logical conjunction, in terms of truth values 'and' : B x B -> B,
along with the whole stock of 2-adic relations previously granted.
Then one tries to see what other 3-adics can be generated from these
augmented resources. The ones that can be so produced are reducible
over this resource base; the ones that can't be so constructed are
then called "irreducible" in a stronger sense, the genuine 3-adics.
At any rate, this seems to fit the examples that Peirce provides.
I am putting a detailed "work in progress" on these issues here:
http://www.nexist.org/wiki/Doc16596Document
http://www.nexist.org/wiki/Doc16600Document
It can take a couple of minutes to load the pages, though.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o