ONT Re: Identity & Teridentity
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Howard, by way of translation, let us return to the orginal.
What Peirce says in the following quote, that Joe posted on
23 Nov 2002 08:36:18 -0600, covers the same facts that I am
indicating under my rubric of level 1 and level 2. This is
all still terating of relations as sets of tuples, before
we even get as far as mentioning sign relations, per se.
| The criticism which I make on [my] algebra of dyadic relations, with which I am by no means
| in love, though I think it is a pretty thing, is that the very triadic relations which it
| does not recognize, it does itself employ. For every combination of relatives to make a
| new relative is a triadic relation irreducible to dyadic relations. Its 'inadequacy' is
| shown in other ways, but in this way it is in a conflict with itself 'if it be regarded',
| as I never did regard it, 'as sufficient for the expression of all relations'.
|
| C.S. Peirce, 'Collected Papers', CP 8.331
JA: 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.
JA: 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.
JA: 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.
JA: 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.
JA: I am putting a detailed "work in progress" on these issues here:
JA: http://www.nexist.org/wiki/Doc16596Document
http://www.nexist.org/wiki/Doc16600Document
JA: It can take a couple of minutes to load the pages, though.
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