SUO: Re: Relations And Their Divisitudes
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
RATD. Note 20
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
TJ = Tom Johnston
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.
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.
Tom,
Taking these comments together, I believe that it all comes down to keeping
in mind the distinctions between several different "worlds", the real world,
the world of data that we actually gather in accord with prescribed formats,
and the world of formal or mathematical models, the sole ilk of things that
ever "obey" the axioms and the rules that we prescribe for them. There is
nothing terribly novel about this "3-basket" system of worlds, but I think
that it holds more water than the "2-basket" kind.
With a "flat rectangular" or a "plain vanilla" dataset,
the codebook prescribes what values are valid for each
column, or variable, and the cartesian product is just
the totality of all possible rows, or datapoints, when
there is no consideration of other restrictions on the
data as yet. The rows of a relational table L need do
nothing more than represent the points of a particular
subset of a cartesian product, L c X = X_1 x ... x X_k.
The relational table L might be sufficient unto itself,
as when we make up finite examples of formal relations,
or it may be serving as proxy for a realworld relation,
representing an empirical approximation, falling short
of the real relation M c X that the sample L c X gives
a sign of, then again, on account of coding errors and
other fallibilities, here and there exceeding its writ.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o