ONT Re: Differential Logic -- Series B
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
DLOG. Note B7
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
One more piece of notation will save us a few bytes
in the length of many of our schematic formulations.
Let !X! = {x_1, ..., x_k} be a finite class of variables --
whose names I list, according to the usual custom, without
what seems to my semiotic consciousness like the necessary
quotation marks around their particular characters, though
not without not a little trepidation, or without a worried
cognizance that I may be obligated to reinsert them all to
their rightful places at a subsequent stage of development --
with regard to which we may now define the following items:
1. The "(first order) differential alphabet",
d!X! = {dx_1, ..., dx_k}.
2. The "(first order) extended alphabet",
E!X! = !X! |_| d!X!,
E!X! = {x_1, ..., x_k, dx_1, ..., dx_k}.
Before we continue with the differential analysis
of the source proposition q, we need to pause and
take another look at just how it shapes up in the
light of the extended universe EX, in other words,
to examine in utter detail its tacit extension eq.
The models of eq in EX can be comprehended as follows:
1. Working in the "summary coefficient" form of representation,
if the coordinate list x is a model of q in X, then one can
construct a coordinate list ex as a model for eq in EX just
by appending any combination of values for the differential
variables in d!X!.
For example, to focus once again on the center cell c,
which happens to be a model of the proposition q in X,
one can extend c in eight different ways into EX, and
thus get eight models of the tacit extension eq in EX.
Though it may seem an utter triviality to write these
out, I will do it for the sake of seeing the patterns.
The models of eq in EX that are tacit extensions of c:
<u, v, w, du, dv, dw> =
<1, 1, 1, 0, 0, 0>,
<1, 1, 1, 0, 0, 1>,
<1, 1, 1, 0, 1, 0>,
<1, 1, 1, 0, 1, 1>,
<1, 1, 1, 1, 0, 0>,
<1, 1, 1, 1, 0, 1>,
<1, 1, 1, 1, 1, 0>,
<1, 1, 1, 1, 1, 1>.
2. Working in the "conjunctive product" form of representation,
if the conjunct symbol x is a model of q in X, then one can
construct a conjunct symbol ex as a model for eq in EX just
by appending any combination of values for the differential
variables in d!X!.
The models of eq in EX that are tacit extensions of c:
u v w (du)(dv)(dw),
u v w (du)(dv) dw ,
u v w (du) dv (dw),
u v w (du) dv dw ,
u v w du (dv)(dw),
u v w du (dv) dw ,
u v w du dv (dw),
u v w du dv dw .
In short, eq.c just enumerates all of the possible changes in EX
that "derive from", "issue from", or "stem from" the cell c in X.
Okay, that was pretty tedious, and I know that it all appears
to be totally trivial, which is precisely why we usually just
leave it "tacit" in the first place, but hard experience, and
a real acquaintance with the confusion that can beset us when
we do not render these implicit grounds explicit, have taught
me that it will ultimately be necessary to get clear about it,
and by this "clear" to say "marked", not merely "transparent".
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
http://www.cs.bsu.edu/homepages/mighty/history.html
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o