ONT Re: Theory Of Relations
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TOR. Note 3
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Thus we see the origin and meaning of the term "numerical identity".
For the account that I gave last time, enumerating the first three
generations of relations over the universe X = {i, j, k} will ever
after serve to remind us of one of the things that can give us the
most confidence that we have comprehended the numerical identities
of any types of entities, to wit, a putative ability to count them.
Let us make a few observations of general bearing that we can
already see exhibited in this early but blossoming universe X,
and also take the occasion to set down a few bits of notation.
Let me introduce a bit of language that comes up here.
Very roughly speaking -- for speaking this way ignores
a point of subtlety concerning the distinction between
extensions and intensions, and another point concerned
with the distinction between the "relative term" and
the "relation" proper -- Peirce called the elements
of a relation, its tuples, by the suggestive name
of "elementary relations". Let us do likewise.
For example, the elementary 2-adic relations that
serve as a basis for all of the 2-adic relations
over X are just the 3 x 3 = 9 ordered 2-tuples
that I list here:
i:i, i:j, i:k,
j:i, j:j, j:k,
k:i, k:j, k:k.
I hope that you will discover this form to be
a suggestive array and not a block to inquiry.
Now that we have an initial notion of what a relation is,
namely, an aggregate, class, collection, set, logical sum,
by whatever name of a similar sort you may wish to call it,
of ordered tuples, and now that we will forever after never
confuse a relation with one of its elemental tuples -- for,
yes, indeed, even a set that consists of a single element,
a "singleton" so-called, is counted as a different entity
from the element thereof -- we may begin to consider the
types of operations to which these relations are subject.
Jon Awbrey
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