ONT Theory Of Relations
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TOR. Note 1
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Let's see if we can build up a working theory of relations,
starting out as simply as we possibly can, forgetting most
of the finer subtleties of Peirce's distinctions, and yet
trying to build a system that will be roughly compatible
with the sorts of concepts that Peirce appeared to have
in mind, as it appears, that is, from reading what he
wrote, and from what we may know about the generic
mathematical background of his day.
If it were me, I would begin with a toy universe
like X = {i, j, k}, where the signs "i", "j", "k"
are taken to denote the distinct objects i, j, k,
repectively. It's not much, but it's enough for
a start.
Here are some relations that immediatedly,
if not exactly unmediatedly, come to mind:
The "2-identity relation" I_2 on X is the following set of ordered pairs:
I_2 = {(i, i), (j, j), (k, k)}
I will probably call it "I", not to be confused with me,
and bowing to convention call it the "identity relation".
For ease of expression, I will write relations in one
of the styles that Peirce was accustomed to write them,
in which the identity relation would be written like so:
I = i:i + j:j + k:k
He often called sets by the name of "aggregates" or "logical sums",
and so the plus sign here only signifies the aggregation of these
ordered pairs into a logical sum, or a "set" to us.
In this vein, the 3-identity relation over X would take the form:
I_3 = i:i:i + j:j:j + k:k:k
In general, a term of the form "x:y:z" denotes the
ordered triple that by any other name is (x, y, z).
To be continued ...
Jon Awbrey
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