SUO: Re: Mapping from one notation to another
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JA = Jon Awbrey
JS = John Sowa
JS: Yes, there is more to be said. But getting the point
across with just one instance is sufficient to dispell
many of the confusions about notation.
Perhaps, but the evidence is overwhelmingly against any notion
that single 3-tuple examples are sufficient to dispell any of
the confusion about what a relation is, much less about what
it means to reduce one relation to a set of simpler ones.
JA: And no matter how you syntacticize the description
or graphicize the depiction of that single triple,
which latter task CG's naturally do beautifully,
that triple is not the 3-adic relation "Gives",
but only a single instance of it.
JS: Actually there is no single "best" way to show a relation.
A table of all instances is only possible for a finite relation,
and even then, it only shows the extension, not the intension.
For most relations that one encounters in ordinary language,
the intension is far more important than the extension.
I thought we were talking about concrete and simple examples.
Finite extension examples are eminently concrete and simple,
and if they are sufficient to model the relevant properties,
then by all means I think that we should jump at the chance.
If you think that there is any chance of achieving a quick consensus on
what an "intension" is, much less what "the" intension of your average
ordinary language statement is, then I might be tempted to set foot
on that path. At any rate, it is neither concrete nor simple.
JS: In any case, any syntactic representation of the
issues for a single instance can be generalized
to a similar representation for any axioms that
define the intension.
That is most especially false when it comes to reducibility properties,
and this false notion has been the source of a great deal of confusion.
Jon Awbrey
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