ONT Re: Identity & Teridentity
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I&T. Note 14
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Re: Quine's supposed "bisection of the triad".
I will make another attempt to explain what is going on here.
We had a long wrangle on the Standard/Ontology lists over this,
and it left me too burned out even to look up the links right now,
but if you keep pushing it ... consider that your fair warning.
Quine is just wrong here. And what he is commonly taken to have said
is even wronger still. As for Quine, the errors are so glaring that
I can only guess that he must have been operating under the influence
of a strong reductionist toxin, perhaps a mutant strain of behaviorism,
or something equally stupifying.
Up til now, I have mainly focused on what Peirce meant by saying that
triadic relations are irreducible, with special reference to the way
that he pictured the obviousness of it all in the Existential Graphs.
Now, if one grasps the morphism between relations and graphs, then
the basic fact about graphs was already proved by the one who is
commonly recognized as the first graph theorist, namely, Euler.
So the only wiggle room here is in denying the aptness of the
putative morphism h : Relations -> Graphs. But the facts
are clear enough in the source domain, at any rate.
As far as what Peirce actually claimed, it is a mathematical fact.
Though less familiar, it is literally a more elementary fact than
the facts that 2, 3, 5, 7, 11, 13 are a prime numbers, since these
facts would take a bit of proving from a suitable axiomatic basis,
while the fact that the set of 2-adic relations is closed under
ordinary relational composition is simply a matter of definition.
To be ignorant of that definition is a severe 'ignoratio elenchi'.
Down from this scene, is possible to define other sorts of algebraic
operations on relations or relative terms -- Peirce and his students,
especially Christine Ladd, later Franklin, were especially ubertous
in thinking up new ones -- but all of these involve the use of basic
logical operations like conjunction and disjunction in the mix, and
so they do not bear on the validity of the original question, since
"binary operations are ternary relations", as my very first abstract
algebra book once put it.
But if we put aside the mere technicality of what Peirce actually said,
you must try to comprehend what a total no-brainer this whole thing is.
The very notion of putting two things together
to produce a third involves a triadic relation!
Ergo, all notions of analysis, composition, reduction, synthesis, whatever,
contain a notion of triadic relations as a part of their very constitution.
Jon Awbrey
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