[Inquiry] Re: Questions Involving Pure Symbols -- Discussion

[Jon Awbrey]

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QUIPS.  Discussion Note 56

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JW = Jim Willgoose

JW: I need to get a little clearer on terminology.  For instance, a set
    of k-tuples could have any cardinality N whatsoever regardless of the
    adicity of the relation.  But the replacement of k by its ordinal number
    w should identify the adicity.  I think of the extension of the collection
    as its "members".  I think of the components or "elements" of a tuple as
    positions marked, say, by a positive integer.  So, if I consider a set of
    10-adic relations, the number "10" is the ordinal w.  The reason I ask all
    this is simply to get clear as to whether Peirce builds a 10-adic relation
    out of members of a set of 3-adic relations by compounding a member and then
    adding 1 for the domain "s" or not.

Jim,

It is necessary to sort out the different levels or types of things that
we are talking about, otherwise we will fall into all sorts of so-called
"category errors".  There are, just at the outset, these three levels:

         o        Relation L c X = X_1 x ... x X_k
        / \
       /   \
      /     \
     o  ...  o    Elementary Relations x_i = <x_i_1, ..., x_i_k> in L
    /|\     /|\
   o...o   o...o  Components of Elementary Relations:
   x x x   x x x  ) x_i_j is the general form of a component.
   1 1 1   m m m  > x_i runs from x_1 to x_m, where m = number of k-tuples in L.
   1...k   1...k  ) x_i_j runs from x_i_1 to x_i_k, where k is the adicity of L.

Think of a practical/empirical situation, for instance, where you have to keep
a "relational database" (RDB) of some sort of relation, say, "spouse/spouse",
"employer/employee", or "seller of item_number to buyer".  The relation of
interest defines the adicity, the "columns" of the RDB table that you make,
while the number of k-tuples in the database, the number of "rows" in the
RDB table, is determined by the vicissitudes of the application setting,
or by sampling considerations in a more formal study.

In loose speech, we often say "relation" when we really mean
"elementary relation", that is, one k-tuple or one element of
the whole collection of k-tuples that constitute the relation.
But we will have to cure ourselves of that habit if we want to
get beyond the confusions of loose speech.

The terms "component" or "role" or sometimes "slot" are
convenient for the ordered places of a particular tuple.
Calling them "elements" will likely engender confusions.

I didn't understand the last part of your question.

Jon Awbrey

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