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

[Jon Awbrey]

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

QUIPS.  Discussion Note 53

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o

JW = Jim Willgoose

Re: QUIPS-DIS 50.  http://stderr.org/pipermail/inquiry/2005-May/002744.html
In: QUIPS-DIS.     http://stderr.org/pipermail/inquiry/2005-May/thread.html#2602

JW: Thanks for some of the clarifications.
    It forced me to study a little set theory again.

JW: You say:

JA: The gist of this is that setting the ground is
    really tantamount to defining the relation itself.
    This definition can be extensional or intensional, or
    both.  A k-adic relation is defined in extension as a
    subset of a cartesian product, say, L c X_1 x ... x X_k.
    Consequently, the information that determines the ground g
    is equivalent to the information that determines the relation L.
    Up to information equivalence, they are one and the same thing.

JW: In another words, the domain of the relation ("ground") may be
    identified with the term to be defined (ie. "sign").  Is that
    correct?  If so, I have the temptation to treat the relation
    as 4-adic.

Jim,

It's a common temptation,
This natural inclination:
Fourth from paradise and
Strait away to perdition.

By way of keeping track of the different levels of
entities that enter into the extensional definition
of a relation, it serves to make these distinctions:

         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.

The ground of a relative term, like "sign" or "teacher", is supposed
to pin down the sense in which the term applies in a given discussion.
Nothing short of a definition suffices to do that, so however we do it
amounts to a virtual definition of the term, as used in a given context.

In general, then, to specify the ground of a relation is just to specify
the relation itself, namely, the particular subset of a cartesian product
that is the extension of the relational term.  In particular, in the case
of those special sorts of 3-adic relations that we call "sign relations",
the "ground" is just the particular sign relation in view, L c O x S x I.

Have to break here ...

Jon Awbrey

o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
inquiry e-lab: http://stderr.org/pipermail/inquiry/
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o