ONT Re: Relations And Their Divisitudes
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
RATD. Note 18
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
TJ = Tom Johnston
In speaking of relations as relational database tables,
we need to be conscious of using a particular metaphor.
In order to keep clear about this, let's resort to the
old distinctions that Peirce adopted, or adapted, from
Aristotle, and who knows where he got it, that discern
from amidst the extension of a term and the intensions
of a thing, what we may call an "exemplary enumeration"
of the things that are denoted by a given term or that
share in all the properties of the things thus denoted.
Each such enumeration is merely a sample of the term's
full extension, and this is the sort of sample that we
actually find in all the realworld's working databases.
TJ: My own thoughts about the difference between a row of
a table (a tuple) and the table itself (a relation) is
that the former is part of the extension of the relation,
while the latter (more specifically, the set membership
conditions which define it) represents the intension
of the relation.
Yes, but. When we use database language to talk about formal concepts
and mathematical objects, that is, whole spaces of imaginary extent or
axiomatically ruled extensions of the corresponding models, and so on,
then we imagine ourselves blessed with sorts of samples that are total
with regard to the coverage of the terms and the theories in question.
Of course, in statistics, even our dreams are controlled experiments,
but the wishful bit of it is that many other practical nuisances of
sampling and data coding may be ignored, well, so long as we sleep.
As it falls out in real world studies, we often go about collecting data
in a somewhat "unintensional" (sic) fashion, only very gradually, indeed,
if ever, grasping the properties that are most apt to unify the manifold
of sensuous impressions, to borrow a phrase from an old systems engineer.
In Peirce's earliest mature work -- I figure that he was
mature enough for our age's sake at about the age of 26 --
all of these sorts of pragmatic considerations that I've
just remarked were quite clearly marked in his writings.
In sum, for the moment, the k-tuple is a singular sample
of one enumeration of the complete extension of whatever
relation we may have in front of us at a particular time.
Jon Awbrey
TJ: Next, that one cannot always infer the intensional rules from
the extensional instances because, at any given moment, the set
of all those instances may not define the boundary conditions of
all those rules.
TJ: To take a simple example, if one column of the table is a status-code
column, whose domain is the letters A to X inclusive, it might be that
the status-code value of P is not instantiated in any tuple.
TJ: Or: a column might be defined as nullable, although
all of the current rows have a value in that column.
TJ: In short: that intensional rules might be inferrable from the
full Cartesian Product of a set of columns (if we knew it was
the full Cartesian Product), but are not inferrable from any
subset of the full Cartesian Product.
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o