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

[Jon Awbrey]

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QUIPS.  Discussion Note 54

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JW = Jim Willgoose

Re: QUIPS-DIS 53.  http://stderr.org/pipermail/inquiry/2005-May/002752.html
In: QUIPS-DIS.     http://stderr.org/pipermail/inquiry/2005-May/thread.html#2602

JW: You say:

JA: A triadic relation is a set of ordered triples.
    Thus the relation can have many, many elements.
    It is only the number of components in a triple
    that is limited to three.

JW: I had said "one triadic relation with a small number of elements".
    I should have said more accurately, "one interpretation of the
    relation".  I must have been thinking of the definition of a sign.

JW: You say:

JA: Properties like the ones that you mention are
    not defined at the level of individual tuples,
    but definable as properties of sets of tuples,
    since these are what constitutes the relation.

JW: Sure.  I am still thinking of the definition of sign rather than the language
    of set theory.  Outside of set theory, people talk about relational properties
    without defining them in terms of sets of tuples.  But I understand a little
    more clearly now that an n-tuple or a k-ordering is dependant on a cartesian
    product.  Defining a sign extensionally serves no immediate purpose for me.
    But using the definition to interpret a set of n-tuples seems to make sense.

Jim,

I've been emphasizing the extensional definition of relations,
in particular, sign relations, for several reasons.  One reason
is that extensional definitions get short shrift in philosophical
discussions, and as a consequence a whole lot of overgeneralization
from insufficient examples tends to keep everything up in the clouds.
The extension of a concept is "where the rubber meets the road", where
the concept makes contact with reality, and it is this sort of empirical
grounding in concrete examples that affords a check on runaway abstraction.
Another reason is that intensional definitions are actually a lot harder to
work with successfully than many people seem to think.  A good intensional
definition is not just a verbal description, the nuances of which can be
developed by blue sky intuition, but a definition that supports exact
forms of necessary reasoning.

JW: My next question is what to call the sets
    A, B, C in the cartesian product A x B x C.

The sets A, B, C in the cartesian product A x B x C
are called its "domains" or its "relational domains".

Jon Awbrey

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